shock and vibration 1972

Bulletin 42

(Part 2 of 5 Parts)

THEi r _ SHOCK AND VIBRATIONVo BULLETIN

Part 2Ground Motion,

Dynamic Analysis

JANUARY 1972

A Publication ofTHE SHOCK AND VIBRATION

INFORMATION CENTERNaval Research Laboratory, Washington, D.C.

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SYMPOSIUM MANAGEMENT

THE SHOCK AND VIBRATION INFORMATION CENTER

William W. Mutch, DirectorHenry C. Pusey, Coordinator

Rudolph H. Volin, CoordinatorEdward H. Schell, Coordinator

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IFBOultin 42

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THE.SHOCK AND VIBRATION

BULLETIN

JANUARY 1972

A Publication ofTHE SHOCK AND VIBRATION

INFORMATION CENTERNaval Research Laboratory, Washington, D.C.

The 42nd Symposium on Shock and Vibration washeld at the U.S. Naval Station, Key West, Florida,on 2-4 November 4971. 'he U.S. Navy was host.

Office of- The Director of Defense

Research and Engineering

-------------------------------------.--

CONTENTS,

PAPERS APPEARING IN PART 2

Ground Motion

SINE BEAT VIBRATION TESTING RELATED TO EARTHQUAKE.RESPONSE:SPECTRA ..- . 1---E. G. Fischer, Westinghouse Research-Laboratories; ittjsburh, Pennsylvania

SEI1MIC EVALUATION OF ELECTRICAL EQUIPMENT FOR NUCLEAR POWER STATIONS ... 113 H. Prause znd D. R. Ahlbeck, BATTELLE, Columbus Laboratories, Columbus, Ohio

SHOCK INPUT FOR EARTHQUAKE STUDIES USING GROUND MOTION FROM UNDERGROUNDNUCLEAR EXPLOS IONS .......... ........ ...................... ......... 21

D. L. Bernreuter, D. M. Norris, Jr., axd F. J. Tokaz, Lawrence Livermore Laboratory,University of California, Livermore, California

ROCKING OF A RIGID, UNDERWATER BOTTOM-FOUNDED STRUCTURE SUBJECTED TOSEISMIC SEAFLOOIR EXCITATIN ...................................... 33

J. G. Hamm2r and H. S. Zwlbel, Naval Civil Engineering Laboratory, Port Hueneme,California

DEVELOPMENT OF A WAVEFORM SYNTHESIS TECHNIQUE-A SUPPLEMENT TO RESPONSESPECTRUM AS A DEFINITION OF SHOCK ENVIRONMENT ...................... 45

R. C. Yang and H. R. Saffell, The Ralph M. Parsons Company, Los Angeles, California

THE RESPONSE OF AN ISOLATED FLOOR SLAB-RESULTS OF AN EXPERIMENT INEVENT DIAL PACK ...... ................. ..................... #........ 55

J. M. Ferritto, Naval Civil Engineering Laboratory, Port Hueneme, California

A SHOCK-ISOLATION SYSTEM FOR 22 FEET OF VERTICAL GROUND MOTION ............ 67E. C. Jackson, A. B. Miller and D. L. Bernreuter, Lawrence Livermore Laborato'y,University of California, Livermore, California

THE COMPARISON OF THE RESPONSE OF A HIGHWAY BRIDGE TO UNIFORM GROUNDSHOCK AND MOVING GROUND EXCITATION ..................................... 75

N. E. Johnson and R. D. Galletly, Mechanics Research, Inc., Los Angeles, California

DEFORMATION AND FRACTURE OF TANK BOTTOM HULL PLATES SUBJECTEDTO MINE BLAST ..... . . . . . . . .. . . . . . . . ...... .. .. 87/

D. F. Haskell, Vulnerability Laboratory, U.S. Army Ballistic ResearchLaboratories, Aberdeen Proving Ground, Md.

THE IMPULSE IMPARTED TO TARGETS BY THE DETONATION OFLAND MINES . . . . . . . . . . . . . . . . . . . . . . . ...... 97

P. S. Westine, Southwest Research Institute, San Antonio, Texas

CIRCULAR CANTILEVER BEAM ELASTIC RESPONSE TO AN EXPLOSION ............. 109Y. S. Kim and P. R. Ukrainetz, Department of Mechanical Engineering, Universityof Saskatchewan, Saskatoon, Canada

MEASUREMENT OF IMPULSE FROM SCALED BURIED EXPLOSIVES .............. 123B. L. Morris, U.S. Army Mobility Equipment Research and Development Center,Fort Belvoir, Virginia

iii

Dynamic Analysis

THE EFFECTS OF MOMENTUM WHEELS ON THE FREQUENCY RESPONSECHARACTERISTICS OF LARGE-FLEXIBLE STRUCTURES........ ...... . ........ '29

F. -D. Day I and S. R. Tomer, Martin Marietta C-rporationDenver, Colorado

INTEGRATED DYNAMIC ANALYSIS OF A SPACE STATION WITH CONTROLLABLEr SOLAR ARRYS. ............. .. . 137J. A. Heinrichs and A. L. Weinberger, Fairchild Industries, Inc., Germantown, Maryland,and Me D. Rhodes, NASA Langley Research Center, Hampton, Virginla

.. .PARAMETRICALLY-EXCITEDCOLUMN WITH HYSTERETIC MATERIAL

D. T. Mozer, IBM Corporation, East Fishkill, New York, and R. M. Evan-Iwanowski,Professor, Syracuse University, Syracuse, Now York

DYNAMIC-INTERACTIOI BETWEEN VIBRATING CONIVEYORS ANDSUPPORTING STRUCTURE ................. ........................... 163

M. P", Professor, Civil Engineering-Departmen, University. of.Louisville,Louisville, Kentucky; and O.'Mathis, Design Engineer, Rex Chainbelt Inc.,Louisville, Kentucky

RESPONSE OF A SIMPLY SUPPORTED CIRCULAR PLATE EXPOSED TO THERMALAND PRESSURE LOADING ............. 1........ ...................... 171

J. E. Koch, North Eastern Research Associates, Upper Montclair, N.J., and M. L. Cohen,North Eastern Research Associates, Upper Montclair, N.J., and Stevens Institute ofTechnology, Hoboken, N.J.

WHIRL FLUTTER ANALYSIS OF PROPELLER-NACELLE-PYLON SYSTEM ON LARGESURFACE EFFECT VEHICLES ..................... *.................... 181

Yuan-Ning Liu, Naval Ship Research and Development Center, Washington, D.C.STHE DYNAMIC RESPONSE OF STRUCTURES SUBJECTED TO TIME-DEPENDENTBOUNDARY CONDITIONS USING THE FINITE ELEMENT METHOD ................. 195

G. H. Workman, Battelle, Columbus Laboratories, Columbus, Ohio

VIBRATION ANALYSIS AND TEST OF THE EARTH RESOURCESTECHNOLOGY SATELLITE ............................................ 203

T. J. Cokoni.s and G. Sardella, General Electric Company, Space Division,Philadelphia, Pennsylvania

FINITE AMPLITUDE SHOCK WAVES IN INTERVERTEBRAL DISCS ................. 213W. F. Hartman, The Johns Hopkins University, Baltimore, Maryland

ACCELERATION RESPONSE OF A BLAST-LOADED PLATE .................. 221L. W. Fagel, Bell Telephone Laboratories, Inc., Whippany, New Jersey

EFFECT OF CORRELATION IN HIGH-INTENSITY NOISE TESTING AS INDICATEDBY THE RESPONSE OF AN INFINITE STRIP .................................. o235

C. T. Morrow, Advanced Technology Center, Inc., Dallas, Texas


Invited Papers

SMALL SHIPS-HIGH PERFORMANCERear Admiral H. C. Mason, Commander, Naval Ship Engineering Center, Washington, D.C.

v2

<~.-rA~W.CJ rr,~-.'V~rt .'.<fl~ . . . . .- ,--. . . . .

Specifications

SURVEY, OF VIBRATION TEST PROCEDURES IN USE BY THE AIR FORCEW. B. Yarchb, Air Force Flight- Dynamics Laboratory, Wright-Pitterson Air ForceBase, Ohio

SPECIFICATIONS - A PANEL SESSION

SOME ADMINISTRATIVE FACTORS WHICH INFLUENCE TECHNICAL APPROACHESTO SHIP SHOCK'HARDENING

D. M. Lund, Naval Ship Engineering Center, Hyattsville, Maryland

Measurement and Application of Mechanicil Impedance

FORCE TRANSDUCER CALIBRATIONS RELATED TO MECHANICAL IMPEDANCEMEASUREMENTS

E. 'F. Ludwig, Assistant Project Engineer, and N.D. Taylor, Senior Engineer, Pratt & WhitneyAircraft, Florida Research & Development Center, West Palni Beach, Florida

THE MEASUREMENT OF MECHANICAL IMPEDANCE AND ITS USE INVIBRATION TESTING

N. F. Hunter, Jr., and J. V. ,Otts, Sandia Corporation, Albuquerque, New Mexico

A I TRANSIENT TEST TECHNIQUES FOR MECHANICAL IMPEDANCEAND MODA.SURVEY TESTING

J.D. Favour, M. C. Mitchell, N. L. Olson, The Boeing Company, Seattle, Washington

PREDICTION OF FORCE SPECTRA BY MECHANICAL IMPEDANCE AND ACOUSTICMOBILITY MEASUREMENT TECHNIQUES

R. W. Schock, NASA/Marshall Space Flight Center, Huntsville, Alabama and G. C. Kao,Wyle Laboratories, Huntsville, Alabama

DYNAMIC DESIGN ANALYSIS VIA THE BUILDING BLOCK APPROACHA. L. Klosterman, Ph.D. and J. R. Lemon, Ph.D., Structural Dynamics ResearchCorporation Cincinnati, Ohio

MOBILITY MEASUREMENTS FOR THE VIBRATION ANALYSIS OF CONNECTEDSTRUCTURES

D. J._Ewins and M. G. Sainsbury, Imperial College of Science and Technology,London, England

LIQUID-STRUCTURE COUPLING IN CURI.ED PIPES - IIL. C. Davidson and D. R. Samsury, Machineey Dynamica Division, Naval Ship Research andDevelopment Center, Annapolis, Maryland

Transportation and Packaging

A SURVEY OF THE TRANSPORTATION SHOCK AND VIBRATION INPUT TO CARGOF. E. Ostrem, General American Research Division, General American TransportationCorporatic Niles, illinois

THE DYNAMIC ENVIRONMENT OF SELECTED MILITARY HELICOPTERSM. B. Gens, Sandia Laboratories, Albuquerque, New Mexico

HIGHWAY SHOCK INDEXR, Kennedy, U. S. Army Transportation Engineering Agency, Military Traffic Managementand Terminal Service, Newport News, Virginia

v

I¢

DEVELOPMENT OF'A ROUGH'ROAD SIMULATOR AND SPECIFICATION FOR TESTINGOF EQUIPMENT TRANSPORTED IN WHEELED-VEHICLES

H. M. Forkiois and E. W. Clements, Naval Research Laboratory, Washington, D.C.

LABORATORY CONTROL OF DYNAMIC VEHICLE TESTINGJ. W, Grant, U. S. Army Taik-Au"omftive Command, Warren, Michigan

IMPACT VULNERABILITY OF TANK CAR HEADSJ. C. Shang and J. E. Everett, General American Research Division,General American Transportation Corporation, Niles, Illinois

A STUDY OF IMPACT TEST EFFECTS UPON FOAMED PLASTIC CONTAINERSD. McDaniel, Ground Equipment and Materials Directorate, Directorate for Research,Development, Eginegeriv an. Missile Systems Laborator, U. S. Army Missile CommandRedstone Arsenal, Aima, and R. M. Wyskida, Industrial and Systems EngineeringDepartment, The-Universfty of Alabama In Huntsville, Huntsville, Alabama

DEVELOPMENT OF A PRODUCT PROTECTION SYSTEMD. E.:Yound, IBM General Systems Division, Rochester, Minnesota, andS. R. Pierce, Michigan State University, Fast Lansing, Michigan

MOTION OF FREELY SUSPENDED LOADS DUE TO HORIZONTAL .1lP MOTION INRANDOM HEAD SEAS

H. S. Zwibel, Naval Civil Engineering Laboratory, Port Huenem:, California


Test Control

ON THE PERFORMANCE OF TDM AVERAGERS IN RANDOM VIBRATION TESTSA. J. Curtis, Hughes Aircraft Company, Culver City, California

A MULTIPLE DRIVER ADMITTANCE TECHNIQUE FOR VIBRATION TESTING OF

- COMPLEX STRUCTURESS. Smith, Lockheed Missiles & Space Company, Palo Alto Research Laboratory,Palo Alto, California, and A. A. Woods, Jr., Lockheed Missiles & Space Company,Sunnyvale, California

EQUIPMENT CONSIDERATIONS FOR ULTRA LOW FREQUENCY MODAL TESTSR. G. Shoulberg and R. H. Tuft, General Electric Company, Valley Forge,Pennsylvania

COMBINED-AXIS VIBRATION TESTING OF THE SRAM MISSILEW. D. Trotter and D. V. Muth, The Boeing Company, Aerospace Group,Seattle, Washington

SHOCK TESTING UTILIZING A TIME SHARING DIGITAL COMPUTERR. W. Canon, Naval Missile Center, Point Mugu, California

A TECHNIQUE FOR CLOSED-LOOP COMPUTER-CONTROLLED REVERSED-BENDING FATIGUE TESTS OF ACOUSTIC TREATMENT MATERIAL

C. E. Rucker and R. E. Grandle, NASA Langley Research Center,Hampton, Virginia

[F

PROGRAMMING AND CONTROL OF LARGE VIBRATION TABLES IN UNIAXIALAND BIAXIAL MOTIONS

R. L. Larson, MTS Systems Corporation, Minneapoll3, Minnesota

vi

1'

'1p

A DATA AMPLIFIER GAIN-CODE RECORDING SYSTEM

J. R. Olbert and T. H. Hammond, Huhes Aircraft Company, CulverCity, Califorila

STABILITY OF AN AUTOMATIC NOTCH CONTROL SYSTEM IN SPACECRAFTTESTING

B. N. Agrawal, COMSAT Laboratories, Clarksburg, Maryland

Test Facilities and Technlques;

SINUSODAL VIBRATION OF POSEIDON SOLID PROPELLANT MOTORSL. R. Pendleton, Research Specialist, Lockheed Missiles & Space Company,Sunnyvale, California

CONFIDENCE IN PRODUCTION UNITS BASED ON QUALIFICATION VIBRATIONR. E. Deitrick, Hughea Aircraft Company, Space and Communications Group,El Segundo, California

SIMULATION TECHNIQUES IN DEVELOPMENT TESTINGA. Hammer, Weapons Laboratory, U. S. Army Weapons Comna.-, RockIsland, Illinois

A ROTATIONAL SHOCK AND VIBRATION FACILITYRM T. Fandrich, Jr., Radiation Incorporated, Melbourne, Florida

THE EFFECTS OF VARIOUS PARAMETERS ON SPACECRAFT SEPARATION SHOCKW. B. Keegan and W. F., Bangs, NASA, Goddard Space Flight Center, Greenbelt,Maryland

NON-DESTRUCTIVE TESTING OF WEAPONS EFFECTS On COMBAT ANDLOGISTICAL VEHICLES .......................

R. L. Jo_son, J. H. Leete, and J. D. O'Keefe, TRW Systems Group, RedondoBeach, California, and A. N. Tedesco, Advanced Research Projects Agency,Department of Defense, Washington, D.C.

THE EFFECT OF THE FIN-OPENING SHOCK ENVIRONMENT ON GUIDED MODULARDISPENSER WEAPONS

K. D. Denton and K. A. Herzing, Honeywell Inc., Government and AeronauticalProducts Division Hopkins, Minnesota

DEVELOPMENT OF A FLUIDIC HIGH-INTENSITY SOUND GENERATORH. F. Wolfe, Air Force Flight Dynamics Laboratory, Wright-PattersonAir Force Base, Ohio

DEVELOPMENT OF A LIGHTWEIGHT, LINEAR MECHANICAL SPRING ELEMENTR. E. Keeffe, Kaman Sciences Corporation, Colorado Springs, Colorado

TECHNIQUES FOR IMPULSE AND SHOCK TUBE TESTING OF SIMULATEDREENTRY VEHICLES

N. K. Jamison, McDonnell Douglas Astronautics Company, HuntingtonBeach, California

VIBRATION FIXTURING - NEW CELLULAR DESIGN, SATURN AND ORBITALWORKSHOP PROGRAMS

R. L. Stafford, McDonnell Douglas Astronautics Company, Huntington Beach,California

WALL FLOW NOISE IN A SUBSONIC DIFFUSERE. F. Timpke, California State College, Long Beach, Calfornia, and R. C. BinderUniversity of Southern California, Los Angeles, California

vii

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Isolation and Dmping

TRANSIENTRERONSE OF REAL DISEPATIVE STRUCTURES"1. Plunkett, UnWfisity of Minnesota, Minneapolis, Minnesota

DYNAMIC RESPONSE OF A RING SPRINGR. L. Eshleman, lIT Research Institute, Chicago, Illinois

SHOCK MOUNTING SYSTEM FOR ELECTRONIC CABINETSW. D. Delany, Admiralty Surface Weapons Establishment, Portsmouth, U.K.

METHODS OF ATTENUATING PYROTECHNIC SHOCKS. Barrett and W. J. Kacena, Martin Marietta Corporation, Denver, Colo.'ado

P =kGY ABSORPTION CAPACITY OF A SANDWICH PLATE WITHCR!Tft ABLE CORE

D. iXxjcinovic, Argonne National Laboratory, Argonne, nlinois

ON THE DAMPING'OF TRANSVERSE MOTION OF.FREE-FREE BEAMS INDENSE, STAGNANT FLUIDS

W. K. Blake, Naval Ship Research and Development Center, Bethesda, Maryland

OPTIMUM DAMPING DISTRIBUTION FOR STRUCTURAL VIBRATIONR. Plunkett, University of Minnesota, Minneapolis, Minnesota

A LAYERED VISCOELASTIC EPOXY RIGID FOAM MATERIAL FORVIBRATION CONTROL

C. V. Stahle and Dr. A. T. Tweedie, General Electric Company, SpaceDivision, Valley Forge, Pa.

OPTIMIZATION OF A COMBINED RUZICKA AND SNOWDON VIBRATIONISOLATION SYSTEM

D. E. Zeidler, Medtronic, Inc., Minneapolis, Minnesota, and D. A. Frohrib,University of Minnesota, Minneapolis, Minnesota

TRANSIENT RESPONSE OF PASSIVE PNEUMATIC ISOLATORSG. L. Fox, and E. Steiner, Barry Division of Barry Wright Corporation,Burbank, California

EXPERIMENTAL DETERMINATION OF STRUCTURAL AND STILL WATER DAMPINGAND VIRTUAL MASS OF CONTROL SURFACES

R. C. Leibowitz and A. Kilcullen, Naval Ship Research and Development Center,Washington, D.C.

DAMPING OF A CIRCULAR RING SEGMENT BY A CONSTRAINEDVISCOELASTIC LAYER

Cpt. C. R. Almy, U.S. Army Electronics Command, Ft. Monmouth, New Jersey,and F. C. Nelson, Department of Mechanical Engineering, Tufts University,Medford, Mass.

DYNAMIC ANALYSIS OF THE RUNAWAY ESCAPEMENT MECHANISMG. W. Hemp, Department of Engineering, Science and Mechanics, Universityof Florida, Gainesville, Florida

Prediction and Experimental Technlqubs

A METHOD FOR PREDICTING BLAST LOADS DURING THE DIFFRACTION PHASEW. J. Taylor, Ballistic Research Laborator.les, Aberdeen Proving Ground, Maryland

viii

DRAG MEASUREMENTS ON CYLINDERS IN EVENT DIAL PACKS. B. Millis, bitence Research Establishment Sdffield, Ralston, Alberta, Canada

DIAL PACK BLAST DIRECTING EXPERIMENTL. E. Fgelo, S. F. FTields, and W. J. Byrne, General American ResearchDivision, Niles; Illinois

BLAST FIELDS ABOUT ROCKETS AND RECOILLESS RIFLESW. E. Bake]r, P. S. Westine, and R. L. Bessey, Southwest Research Institute,San Antonio, Texas

TRANSONIC ROCKET-SLED STUDY OF FLUCTUATING SURFACE-PRESSURESAND PANEL RESPONSES

E. E. Ungar, Bolt Beranek and Newman Inc., Cambridge, Massachusetts, and H. J.Bandgren, Jr. and R. Erwin, National Aeronautics and Space Administration, Geo'geC. -Marshall Space Flight Center Huntsville, Alabama

SUPPRESSION OF FLOW-INDUCED VIBRATIONS BY MEANS OF BODYSFIRFACE MODIFICATIONS

D. W. Sallet and J. Berezow, Naval Ordnance Laboratory, Silver Spring, Maryland

AN EXPERIMENTAL TECHNIQUE FOR DETERMINING VIBRATION MODES OFSTRUCTURES WITH A QUASI-STATIONARY RANDOM FORCING FUNCTION

R. G. Christiansen and W. W. Parmenter, Naval Weapons Center, China Lake,California

RESPONSE OF AIR FILTERS TO BLASTE. F. Witt, C. J. Arroyo, and W. N. Butler, Bell Laboratories, Whippany, N.J.


Shock and Vibration Analysis

BANDWITH-TIME CONSIDERATIONS IN AUTOMATIC EQUALIZATIONC. T. Morrow, Advanced Technology Center, Inc., Dallas, Texas

A REGRESSION STUDY OF THE VIBRATION RESPONSE OF ANEXTERNAL STORE

C. A,. Golueke, Air Force Flight Dynamics Laboratory, Wright-PattersonAir Force Base, Ohio

FACTOR ANALYSIS QF VIBRATION SPECTRAL DATA FROM MULTI-LOCATIONMEASUREMENT

R. G. Merkle, Air Force Flight Dynamics Laboratory, Wright-Patterson AirForce Base, Ohio

RESPONSES OF A MULTI-LAYER PLATE TO RANDOM EXCITATIONH. Saunders, General Electric Company, Aircraft Engine Group, Clncinnati, Ohio

RESPONSE OF HELICOPTER ROTOR BLADES TO RANDOM LOADSNEAR HOVER

C. Lakshmikantham and C. V. Joga Rao, Army Materials and Mechanics ResearchCenter, Watertown, Massachusetts

INSTRUMENTATION TECHNIQUES AND THE APPLICATION OF SPECTRALANALYSIS AND LABORATORY SIMULATION TO GUN SHOCK PROBLEMS

D. W. Culbertson. Naval Weapons Laboratory, Dahlgren, Virginia, andV. F. DeVost, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland

ix

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THE EFFECT OF "Q' VARIATIONS IN SHOCK SPECTRUM ANALYSIS* M. B. McGrath, Martin Marietta Corporation, Denver, Colorado, and W. F. Bangs,

. National Aeronautics and Space Administration, Goddard Space -'Uight Center, Maryland

* RAPID FREQUENCY AND CORRELATION ANALYSIS USING AN ANALOG COMPUTERJ. G. Parks, Research, Development and Engineering Directorate, U.S. Army. Tank-Automotive Command, Warren, Michigan

U4VESTIGATION OF LAUNCH TOWER MOTION DURING AEROBEE 350 LAUNCHR. L. Kinsley and W. R. Case, NASA, Goddard Space Flight Center, Greenbelt, Maryland

ON THE US E OF FOURIER TRANSFORMS OF MECHANICAL MHOCK DATAH. A. Gaberson and D. Pal, Naval Civil Engineering Laboratory, Port Hueneme,California

WAVE ANALYSIS OF SHOCK EFFECTS IN COMPOSITE ARMORG. L. Filbey, Jr., USAARDC Ballistic R13sear.h Laboratories, Aberdeen ProvingGround, Maryland

STATISTICAL LOADS ANALYSIS TECHNIQUE FOR SHOCK AND HIGH-FREQUENCYEXCITED ELASTODYNAMIC CONFIGURATIONSK. J. Saczalski and K. C. Park, Clarkson College of Technology, Potsdam, New York

Structural Analysis

NASTRAN OVERVIEW: DEVELOPMENT, DYNAMICS APPLICATION, MAINTENANCE,ACCEPTANCE

J. P. Raney, Head, NASTRAN Systems Management Office and D. J. Weidman, Aerospace

Engineer, NASA Langley Research Center, Hampton, Virginia

EXPERIENCE WITH NASTRAN AT THE NAVAL SHIP R&D CENTER AND OTHERNAVY LABORATORIES

P. Matula, Naval Ship Research & Development Center, Bethesda, Maryland

RESULTS OF COMPARATIVE STUDIES ON REDUCTION OF SIZE PROBLEMR. M. Mains, Department of Civil and Environmental Engineering, WashingtonUniversity, St. Louis, Missouri

STRUCTURAL DYNAMICS OF FLEXIBLE RIB DEPLOYABLE SPACECRAFT ANTENNASB. G. Wrenn, W. B. Halle, Jr. and J. F. Hedges, Lockheed Missiles and SpaceCompany, Sunnyvale, California

INFLUENCE OF ASCENT HEATING ON THE SEPARATION DYNAMICS OF ASPACECRAFT FAIRING

C. W. Coale, T. J. Kertesz, Lockheed Missiles & Space Company, Inc.,Sunnyvale, California

DYNAMIC WAVE PROPAGATION IN TRANSVERSE LAYERED COMPOSITESC. A. Ross, J. E. Cunningham, and R. L. Sierakowski, Aerospace Engineering DepartmentUniversity of Florida, Gainesville, Florida

R-W PLANE ANALYSIS FOR VULNERABILITY OF TARGETS TO AIR BLASTP. S. Westine, Southwest Research Institute, San Antonio, Texas

PERFORM: A COMPUTER PROGRAM TO DETERMINE THE LIMITING PERFORMANCEOF PHYSICAL SYSTEMS SUBJECT TO TRANSIENT INPUTS

W. D. Pllkey and 1o Ping Wang, Department of Aerospace Engineering and EngineeringPhysics, University of Virginia, Charlottesville, Virginia

x

STRUCTURAL DYNAMIC ANALYSIS AND TESTING OF A SPACECRAL T DUAL TRACKINGANTENNA

D. D. Walters, R. F. Heidenreich, A. A. Woods and B. G. Wrenn, Lockheed Missilesand Space'Company, Sunnyvale, California

j Ship's Problems

DETERMINATION OF FIXED-BASE NATURAL FREQUENCIES OF A COMPOSITESTRUCTURE OR SUBSTRUCTURES

C. Ni, R. Sccp, and J. P. Layher, Naval Research Laboratory, Washington, D.C.

EQUirVALENT SPRING-MASS SYSTEM: A PHYSICAL INTERPRETATIONB. K. Wada, . Bamford, and J. A. Garba, Jet Propulsion Laboratory,Pasadena, California

LONGITUDINAL VIBRATION OF COMPOSITE BODIES OF VARYING AREAD. J. Guzy, J.C.S. Yang, and W. H. Waiston, Jr., Mechanical EngineeringDepartment, University of Maryland, College Park, Maryland

SIMPLIFIED METHOD FOR THE EVALUATION OF STRUCTUREBORNE VIBRATIONTRANSMISSION THROUGH COMPLEX SHIP STRUCTURES

M. Chernjawski and C. Arcidlacona, Gibbs & Cox, Inc., New York, New York

ix

GROUND MOTION

SINE BEAT VIBRATION TESTING-RELATED TOEARTHQUAKE -RESPONSE SPECTRA

E. G. FISCHERWESTINGHOUSE RESEARCH LABORATORIES

PITTSBURGH, PENNSYLVANIA

Vibration test criteria are developed for evaluating the earthquake re-sistance and reliability of electrical switchgear, including sensitivecontrol devices. A series of sine beat vibrations applied at experimentallydetermined, natural frequencies of the equipment is potentially moredamaging than the original seismic motion. The test table input can berelated to the floor response spectra as calculated for a particular powerplant structure and location in an active earthquake zone.

INTRODUCTION damping for typical nonlinear systems, and suchinformation can be used to authenticate parallel

It is customary to develo;, ,nathematical computer-aided dynamic analyses.models of building structures l] and then makecomputer-aided analyses of their dynamic re- RANDOM MOTION THEORYsponse to seismic disturbances. Similarly,mechanical equipments located in such Random mudon is non-periodic and,structures may require dynamic seismic analy- therefore, unpredictable with time. However,ses, although they become more complicated it can be described in statistical terms basedand difficult to model (2] . In the special case upon "the analysis f noise in communicationof sensitive electrical control devices subject circuits" as pioneered by Rice[3].to possible malfunction during earthquakes (forexample, the safeguards equipment required for Figure la illustrates a random vibrationnuclear power plants), it becomes more practi- excitation which is characterized as broadbandcal to rely upon a conservative method of en- in the sense that it appears to'include all freq-vironmental testing. Experience indicates that uencies in addition to its large fluctuations inthe inadvertent loss of principle function under amplitude. (Earthquake accelerographs of free-shock can pose more of a design problem than field ground motions usually exhibit such char-an obvious strength failure. acteristics.)

To obviate either difficult or question- By way of contrast, Fig. lb illustratesable analyses, earthquake vibration test cri- the corresponding narrow-band vibration re-teria have been developed for evaluating and sponse of a simple (mass-spring-damper)demonstrating the reliability of electrical con- oscillator wherein the frequency is restrictedtrol devices for Electric Utility Systems. In to the system natural frequency, "although theparticular, a series of sine beat v 'rations are amplitude fluctuations persist. As judged byapplied at experimentally determined natural the envelope of the peak magnitudes, this more-frequencies of the equipment. This means that or-less harmonic motion is referred to as athe equipment is conservatively tested by a random sine wave. (Building floor seismic re-procedure which makes it most vulnerable. sponse records exhibit similar sine beat char-

acteristics.)In addition, sine beat vibration inputs

during test can be related to foundation or In general, the original seismic dis-building floor shock response spectra for turbance can be generated In terms of a non-computer-simulated earthquakes. The test stationary random process involving dampedresults provide data on natural frequencies and sinusoids with random frequencies as well as

Envelope Curve of ResponsePeak Magnitudes

: /-Excitation

Time --- ,

(b),Narrow-band Simple System Response In Form of a /Random Sine Wave

TFrequency. u

(a) Random Vibration Excitation Fig. 2 - Frequency spectra showing the outputresponse of a simple oscillator for arandom vibration excitation

Fig. 1- Acceleration-time history records ofbroadband excitation and narrow-bandresponse SINE BEAT VIBRATION

In general, when a random excitation israndom phase relationships [4]. However, the put through a lightly-damped, narrow-band-present discussioncan be simplified if limited pass filterp the outputcresponse usually appearsto a "stationiry random process" consisting of to be a harmonic signal with a slowly varyingan ensemble of motion-time histories, the amplitude. The energy is transmitted primarilylatter considered to be stationary if the sta- in the neighborhood of the natural frequency oftistical properties are not affected by a trans- the system filter. The vibration response wavelation of the origin of time. Consequently, by (see Fig. 1b) appears to be a sine wave at ameans of the ergodic hypothesisp the required single frequency but with pulsating amplitude;assembly-averaging of random data can be hence the so-called sine beat vibration. TheI replaced by the more simple task of time- probability density of the instantaneous valuesaveraging over a single record of long duration, of the filtered response tends to be normal, or

Gaussian (synmetrical)(5].Because resonant vibration buildup is an

important engineering phenomenon, it is On the other hand, the density distri-essential to evaluate the frequency content of bution function of the amplitude variation of theseismic vibration excitation. For this purpose envelope of the random sine wave (see Fig. 1b)the quantity "power spectral density" (PSD) is tends to be skewed. It. can be expressedused as a measure of frequency content of explicitly by means of the well-known Rayleighrandom-type functions. (Important frequency distribution curve, which is employd in studieseffects in earthquake free-field accelerographs of cumulative vibration fatigue. It is alsoappear to be limited to a range from 1 to 25 Hz.) called the two-dimensional error distribution

with reference to the "random walk" problem,Figure 2 shows the frequency spectra which was first solved by Lord Rayleigh(6].

chart of PSD for a random broadband excitation Strength failures caused by random ceismicand a narrow-band response, the latter demon- disturbances involve a relatively ifw loadstrating the filter action of a simple oscillator, cycles, but they can work th, iiat. rial in theHence, the acceleration-time histories shown elasto-plastic range of cumulative fatigue.in Figs. la, b have now been characterized bymeans of a statistical analysis as plotted in A special case of-tho fluctuating sineFig. 2. (Earthquake grcind motion response wave is) of course, the true sine beat vibrationspectra usually appear as broadband excitation, which can be expressed in terms of two rotatingwhereas building floor response spectra corres- (acceleration) vectors as follows:pond to narrow-band quasi-resonance buildup atone or more natural frequencies of the building.)

2

LIP

table motions which, in turn, develop specifiedhh::~ Z ZZ; o eisi test

[ArI Fl, 0'.1 hock response spectra in the mounted equip-l: t;- -- ,.3,-- - -. ,,,g-.j•merit[8, 93.

F+ F.F F Also, steady-state vibration testing atequipment natural frequencies is well-known interms of the Navy MIL-S-167 for shipboardenvironmient. However, for seismic test

criteria the sine beat moion input is muchwhere F - natural (test) freq. preferred because it pro 'uces only a limited

of equipment, Hz quasi-resonance magnification and less cumu-lative fatigue.

Sw freq. of envelope, HzA novel feature introduced by this pro-(2 beat pulses) posed sine beat vibration testing is that the

test results and parallel design calculations fortest frequency, Hz the equipment can be related to the peculi-

2 (no. of cyc/beat) arities of the foundation or building seismicresponse spectra as supplied by the ElectricUtility System fur their installations.

= p c 3 ) TRANSIENT ANALYSIS OF BUILDINGS ANDEQUIPMENT

In actual testing, the frequency of the The well -known normal-mode methodsine beat vibration of the test machine imunting can be applied specifically to mathematicalplate corresponds to each natural frequency in models of buildings and equipment in order tothe equipment as determined by a continuous determine a dynamically equivalent series ofsweep frequency search from I to 25 Hz. dtrieadnmclyeuvln eiso

simple oscillators in terms of the uncoupledHence, the equipment's capabilities are evalu- normal-mode shapes and fixed-base natural

frequencies. In turn, the motion-time historyof the earthquake disturbance can be applied to

In general, environmental testing tech- each oscillator and a summation obtained ofniques are well-established and -several the total transient response of the building ormachines are available for reproducing actual etoipm entearthquake motion-time histories [7 , or test equipment.

Normal Mode Frequencies~2Hz

6 Hz 22 H

(a) Dynamic Model of Building (b) Equivalent Simple Oscillators

Fig. 3 - Computer-aided dynamic analysis of building subjectto seismic disturbance

3

However, a distinction~must be made FILTER ACTION OF THE BUILDINGbetween the broadband seismic iexcitation at STRIICTUREground level and the narrow-ltand floor motionat the various building elevations-the latter Figure 4 illustrates how (a).the broad-resultk.ng from the filtering action of'the mech- -band seismic excitation at the base of theanical-soil-structure system. Alsol i 'the buil dtigcan be' (b) flteredW'andgniied byfollowing discussion an essential distinction the building structure, anl(c) amplified by themust be made between the customary shock equipment response at the coincident buildingresponse spectra used in design-- and the natural frequency of 6 Hz.original motion-time history-used as the basisfor the equivalenitsine beat vibration testing. The resulting floor motion shcwn in

Fig. 4b conslsts of various harmonic oscil-Figure 3 shows (a) a typical mathe- lations depending upon the different paths along

matical model of a nuclear power plant building, which the ground disturbance-has been propa-and (b) thedynamically equivalent seriis of gated. Essentially, this motion under thesimple oscillators. Only the three lowest equipment -has been filtered at 6'Hz and magni-modes of vibration with natuial frequencies of fied 2. 0 times by the building structure. In2, 6 and 22Hz need-be considered, 'since'they other words, it now approximatesfa randomare the only ones calctated within the most sine wave as shown in Fig. 1b, and in-the formimp6tant earthquake hazard range ibr vibration of an, equivalent sine beat vibratlon c iabebuildup from about 1 to-25 Hz. (Similar freq- applied to the test machine mounting~plate inuencies found'in the equipment;model are not order to'evaluate tfie seismic capabilities ofnecessarily coincident with those in -the the equipment.building.)

JIn turn, the test machine will duplicateIt Is appreciated that in complex struc- (or exceed) the resulting equipment response

tures there may be important cross-coupling motion shown'in Fig 4c, which occuriswhen aeffects and fluctuating response motions when natural frequency of the equipment coincidesnatural frequencies are close together. with a natural frequency of the buildingHowever, this situation is usually covered by structure. At 6 Hz with.5 p 'rcenT damping,specifying an envelope-type of building floor the magnification of the' floor motion in-theresponse spectra. (Similarly, when there is equipment amounts to 5. 5 times. (An equiva-interaction of two or more equipment modes, lent sine beat vibratio, itest using 5 cycles/beatthen a more severe response condition might at 6 Hz will produce this same magnification inhave to be recognized.) equipment having 5 percent damping.)

It should be recognized that when a

S.5pek Respon natural frequency of the equipment does notcoincide with a natural frequency of the building

1structure then the quasi-resonance motionbuildup in the equipment will be much lessevident than as slzwn by Fig. 4c. (In turn,

-1 9' " although a sine beat vibration test would be

-(c Manifiecd qulpinenio of 5 a 69 with applied at the equipment natural frequency, the0 5 , Sine Beat 416 Ii , 2 9 Peak input amplitude should be reduced to produce0.} 1 SneBealal ,, ,only the floor response specified at the test

o Mf frequency, which no longer coincides with aTime building structure natural frequency. The net

-0.5 Uresult Is a somewhat conservative sine beat2x byBuIlding Strulure test compared to using actual floor motion

inputs.)0.29 O, 139 Peak Acceleration

0 . BUILDING EQUIPMENT "DESIGN"-0~g I ' lime - PROCEDURE

(a) Rjndm'type Ground Motion Input to Computer Modl ofStcureEqutpent Figure 5 shows a semi-log plot of typi-

cal floor response spectra as supplied by theArchitect-Engineer of the building. It is infor-

Fig. 4 - Typical accelerogranis of OBE hori- mation derived from a transient analysis of azontal motion as transmitted by pertinent building structure model with a basedymanic model acceleration input corresponding to a

4

horizontal motion-time history for anOperl-tting2:0 Basis Earthquake (OBE). Presumably the

_ _ Fmodel includes a soil-structure interactionDamping factor, and the OBE has been established on

1 .i25% the basis of the seismic history of a specific1.56 L9- power plant site.

1-1 The filtering action of the building, as5 previously illustrated by Fig.. 4, was based

W 1. 2 upon the data to be found and interpreted fromH Fig. 5. In other words, the ground motion is__ _ 'random with a peak value of 0. 13 g. At an

. 0.8- upper floor level where the equipment mightI I be located, the input motion has been magnffied

11 -V2 times to give apeak value of 0.27 g corre-. f sponding to the horizontal asymptote for freq-

* 0.iI uencies above 30 Hz. (Compare Fig, 4b withFig. 5). Finally, for 5 percent damping in the

equipment at 6 Hz, which is the coincident

00 1 13 building-equipment natural frequency; the0.30.5 1 3 to 30 maximum equipment response acceleration

Equlpmenthf 'erequency, Hz becomes 1.5 g. (Compare Fig. 4c with Fig. 5&)

5- OBE horizontal floor response spectra Figure 5 also gives floor motionFig. 5response spectra (actually the peak acceleration

at equipment location in building measured on the equipment at each of its natu-

ral frequencies) for several values of percentcritical damping, which can be determined forthe equipment from the motion buildup duringthe continuous sweep frequency test. The

S-6.0 Hz, 2.7 g Peak Respon se 2.7 g

2g =5%, Free Vibration

-29-

(b) Test Equipment Acceleration

-Ig - ,0 Hz, 5 Cycles/Beat 2 Second Pause0.49g Peak Input, Q =5.5 x

(a) Test Table Motion Under Equipment

Fig. 6 - Sine beat motion test for DBE 1.8 x OBE

5

maximum overall magnification of the ground-motion into the equipment at 5 percent damping 25is 1.5 g/0. 13 g = 11.5 times, which shows thatthe selective filtering action of the building Ilstructure can be quite. influential. (For the Steady Stale,

L El Centro earthquake at 5 percent damping-theshock response spectra magnification is only •

The floor response spectra shown in 10 Cycles/Bea

Fig. 5 can be used for equipment design 1-

purposes as follows. The peak horizontal re-sponse acceleration at the equipment cg will be C I I ,CNles/Beat1.5 g at 6 Hz and 5 percent damping. This g 7C\ esI- !means that 1.5 times the weight of the equip- Y 10 ----------..ment should be applied as an equivalent static / . Random,horizontal load at the cg in order to verify that E I _"stressed members are below the yield point. L \ :/j >Also, if the equipment natural frequency is 5 -- - -

30 Hz or greater, there is no further magni- &fication ofrthe basic floor motion which is'onlyt -I-.. I0.27g. 5 7 'BUILDING EQUIPMENT "TEST" PROCEDURE 2 4 6 8 10

E, percent of critical damping

For seismic testing of equipment, thevibration machine input is intended to simulate Fig. 7 - Vibration magnification at quasi-the worst aspects of- the motion of the building resonancefloor at the attachment of the equipment. Aconservative approximation of the earthquake-induced floor motion is applied by means of the equipment natural frequency. In view of thesine beat vibration motion, cumulative fatigue effects associated with test-

ing the equipment in three directions (East-In addition, since the probability of an West, North-South and vertical), it becomes

earthquake worse that the OBE is difficult to apparent that sine beat vibration testing can beagree upon, the Intensity multiplier is used to made very conservative with regard toachieve a more safe Design Basis Earthquake, probable local seismic activity.whereby DBE = 1.8 (OBE). (In practice, theElectric Utility Company and the Architect- In summary, it is Important to recog-Engineer supply the foundation and floor re- nize that for design purposes the floorsponse spectra with a statement regarding response spectra of Fig. 5 can be appliedODE and DBE.) directiy. Howeverj for test purposes it ,.

necessary to apply the actual floor motionFinally, Fig. 6a s huws that a sine beat shown by Fig. 4b, or else apply the conserva-

vibration input consisting of 5 cycles/beat at tive substitute shown by Fig. 6a.6 Hz and a peak amplitude of 0. 49 p is theappropriate test machine input to represent the SINE BEAT VIBRATION TEST METHODdamaging effects of the floor motion under theequipment as shown in Fig. 4b, since Figure 7 provides the theoretical data1. 8 (0. 27g) = 0.49 g. Fig. 6b shows that the needed for the correlation of floor responseequipment response amplitude reaches a peak spectra (Fig. 5) and test table input (Fig. 6a).of 2. 7 g and then gradually decays as a free Based upon the transient analysis of a linearvibration at C Hz before the next sine beat mass-spring-damper model with a sine beatvibration is applied, base motion excitation, the vibration magni-

fication curves (so-called Q-factor at resonance)The total number of successive beat are calculated for both 5 cycles/beat and 10

effects applied to the equipment depends upon cycles/beat over a range of damping valuesthe earthquake magnitude, hence the time (i. percent of critical). This elementaryduration of the corresponding strong motion computer-aided analysis is equivalent to theaccelerograph. As many as 5 successive sine normal-mode evaluation of the equipment re-beats have been used in actual testing at each sponse (see Fig. 4c and 6b).

6

Figure 7 also shows the Q-factors forsteady-state vibration resonance (Q = 100/2 g )and for random (white noise) excitation equal to 50the square root of Q. The quasi-resonance .

buildups produced by the sine beat vibrations S Cri/ 15fall'in between the latter two extremes. By 40 - 5 Damping Damplin / -

comparison, seismic.motion response magni- 6 OIl f-ResConace Slefication factors fall soniewhat below the random Tests /excitation values. The most damaging sections 2

of earthquake oscillographs usually correspond Nto a sine beat vibration excitation of about 3 2-_cycles/beat at various preferred frequencies 1 15from about 1 to 10 Hz.

The previous example for selecting a 1 -sine beat vibration test corresponding to build- 5ing floor response spectra can be summarized 4as follows. At each experimentally determined 3: 0 3 onatural frequency of the equipment, the-ordi- 0 5 10 1 2 25nates for the OBE response spectra curve Resomant Frequency, Hzshould be increased by the DBE factor (1.8times) and then reduced by the Q-factor (5. 5times at 5 percent damping) in order to get the Fig. 8 - Chart for determining number of off-peak value of the sine beat input acceleration resonance tests at 1/2 peak amplitude(0. 49 g at 6 Hz), in this example for 5 cycles/beat. Notice that at equipment natural freq-uencies of approximately 4. 2 and 8.4 Hz, which usually evaluated by testing at successivelyare non-coincident with the building structure larger g-levelsp the danger of fatigue failuresnatural frequency at 6 Hz, the response at 5 is reduced because of lower average stressing.percent equipment damping is only 0. 75 ginstead of 1.5 g maximum at 6 Hz. Hence, the OFF-RESONANCE SINE BEAT TESTSsine beat input acceleration peak should be only0. 245 g at 4. 2 and 8. 4Hz instead of the 0. 49 g The computer-aided analysis used toat 6 Hz. This example shows the need for obtain the Q-factor curves of Fig. 7 has alsoadjusting both building and equipment designs to supplied data for plotting resonance curves ofavoid coincident natural frequencies, magnification versus frequency from 1 to 25 Hz.

Somewhat arbitrarily at the 1/2 peak amplitudeIt should be emphasized that in the ordinate, the corresponding (2) sideband freq-

example of Fig. 4a, the peak acceleration of uencies can be determined along the abscissa0. 13 g corresponds to an earthquake character- for resonance curves at various values of damp-ized as VII on the Modified Mercalli Intensity ing in the simple oscillator system.Scale (6. 5 Richter Scale), meaning damage"slight to moderate in well -built, ordinary struc- Figure 8 shows the above informationtures." However, as shown by Fig. 6b the plotted as a family of fan-shaped curves for apeak acceleration of 2.7 g in the power plant sine beat excitation at 5 cycles/beat. Theequipment under test amounts to an overall upper and lower sideband frequency ordinatesmagnification of 21 times, indicating that sine correspond to the intersections of a verticalbeat vibration testing can be made quite severe. 'line at a resonant (test) frequency abscissa with

the two radial lines drawn for some value ofThis proposed test method is conserva- percent critical damping. (Similar curves can

tive in that should either the damping of the be plotted at 10 cycles/beat and steady-state, 4equipment be evaluated on the high side, or the wherein the sideband widths becomes increas-input "cycles/beat" selected on the low sidej ingly smaller.)the net result in both cases would be a smallerQ-factor and a larger test machine acceleration. The stair-step construction shown byAlso, compared to steady-state vibration test- the dashed-lines gives the maximum spacinging, the Q-factor near zero damping means only between "off-resonance" test frequencies, to-a limited resonance buildup, In other words, gether with the assurance that at least 1/2 peakwhen testing unknown systems there will be no amplitude will be excited.destructive buildup produced inadvertently. Inaddition, since equipment fragility levels are 7I

Of course, when there are many equip- REFERENCESment natural frequencies distributed over therange from 1 to 25 Hz, the stair-step construc- 1. John A. Blume, et al, Design of Multistorytion may indicate that additional off-resonance Reinforced Concrete Buildings for Earth-

-tests are unwarranted. On the otherhand, when quake Motions, Portland Cement Assoc.,there are only one or two obvious equipment 1961.natural frequencies, or only building structurenatural frequenciei to consider, then they can 2. E. G. Fischer, et al, "Mathematical Modelbe used to start the stair-step construction in Analysis for the Dynamic Design ofboth directions to establish additional off- Machinery", SESA, Experimental Mech-resonatice test points. anics, October 1967.

The primary purpose of the previous 3. S. 0. Rice, Bell Sys. Tech. J., 23, 282development is to avoid unwarranted cumulative (1944) and 24, 46 (1945).fatigue and wear of equipment being tested forearthquake resistance. (As a practical matter, 4. J. W. Miles and W. T. Thompson:typical complex electrical switchgear systems "Statistical Concepts in Vibration," Chapterusually introduce nonlinear effects in terms of 11, Shock & Vibration Handbook, McGraw-snubber springs, clearances, friction, cross- Hill, New York, 1961.coupling, etc. Equipment resonances appeaiheavily damped, but persist over a relatively 5. J. L. Bogdanoff, et al, "Response of awide frequency range and they can be excited in Simple Structure to a Random Earthquake-411 three directions of testing.) Type Disturbance," Bull. SSA, 51, 2 April

1961.CONCLUSIONS

6. G. S. Mustin, "Theory and Practice ofIn general, there are insufficient Cushion Design," Shock & Vibration Info.

strong-motion earthquake accelerograms avail- Center, U.S. Dept. of Defense, 1968.able for power plant structures, let alonespecific types of equipment, to establish "an 7. J. Penzien, "Design and Researchacceptable seismic risk". Also, it is generally Potential of Two Earthquake Simulatoragreed that an isolated peak acceleration Facilities," Richmond Fie!d Station, Univ-response is not a reliable indication of damage. ersity of California in Berkeley.In switchgear equipment, where possible lossof principle function is a more important con- 8. G. Shipway, OA New Technique for Seismicsideraton, there does not appear to be any Shock Simulation,)) Wyle Labs; Norco, Cal.obvious correlation with field service reports. and Huntsville, Ala.

The use of computer-aided analyses of 9. E. G. Fischer, "Design of Equipment tobuildings and equipment, and the specification Withstand Underground (Nuclear Weapon)of earthquake response spectra, all emphasize Shock Environment," 28th Shock a.dthe importance of quasi-resonance phenomena Vibration Symposium, U. S. Dept. ofas the source of damage or malfunction. Henc% Defense, July 1960.sine beat vibration testing appears appropriatefor evaluating complex equipment where it ismost vulnerable-at its natural frequencies asdetermined by a vibration sweep at a reducedacceleration input from I to 25 Hz.

Sine beat vibration testing producesonly a limited quasi-resonance buildup and

*avoids excessive fatigue in the equipment, bothof which conditions appev' typical of seismicdisturbances. Also, sine beat test inputs andsubsequent equipment resp-,ses can be directlyrelated to the foundation or building floorseismic response spectra as supplied by theElectric Utility Company. In general, theseverity of the test can be simply adjusted andthe test results can be made conservative withrespect to any specified seismic environment.

8

DISCUSSION

Mr. Gaynes (Gaynes Testing Laboratories): How quencies, and determine the resonance modes by-did you-monitor the relays and-swltches to-determine moitoiligl the dtaplaceiment of the specimen. We al-wheter they were functioning or not functioning? so have gone quite a bit further. We have gone into

the shock specters"., approach by usigrandom trn-Mr. Fischer: Usually we had an electrical hook- sients In determining the ability of the product toup to an oscillograph element. The definition of real- withstand vibration. We will be presenting a paper in

function is quite a touchy point. In some applications, the near future on that particular comparison, and Isuch as computers for the Safeguard system, any kind think you will find that your feelings are correct. Us-of relay flutter is not allowed at all, so one practi- bg the sine beat seems tobe a more severe test thancally has to use solid state circuitry. Again, depend- using random transients, but If the shock spectruming on the application, somebody gives you a defini- can be adequately defned, I think that random tran-tion of what constitutes a malfunction. There are ac- sients would be v.aperkir i;y to go.celerometers located on the structure etc, but as faras malfunctioning, it seems to be the electrical oper- Mr. Fischer: Well it could be more authentic. Iation that is the most significant. Of course ifthe cir- agree, but I started out by saying: "In the interestscult breaker pops open it is pretty obvious that you of simplicity we pretty much stuck to the sine beat.,,are Introuble. We Justify the simplicity by-saying it is a conserva-

tive test. Frankly, I will take testing with simultan-. Heous input anytime. With actual earthquake records,

Mr. Haag,(MTh Systems Corporation): We have as I believe Dr. Piunkett pointed out tids morning,been performing several tests in the seismic shock you just think you are pushing something In acertainarea including the sine beat, and we pretty much con- direction. It can always escape at right angles. Whenfirm your approach. We feel it is a severe test. To you use three dimensional testing you never testfind the resonances, we do pretty much the same as nearly as severely as we are suggesting with sinerequired in MIL-STD-167, where you sweep the ire- beat testing.

9

* -,-- -

'-. t t W . V ' ° ' ii , ' 4, H *. .o=' A - o f. , - ,1$--- . . $,

SEISMIC EVALUATION OF ELECTRICAL EQUIPIENT

FOR NUCLEAR POWER STATIONS

Robert H. Prause and Donald R. Ahlbeck

BATTLLE, Columbus LaboratoriesColumbus, Ohio

A review of current seismic specifications for electrxcal equipment andthe different types of vibration excitation that might be used to satisfythese specifications led to the design and construction of a seismicvibration facility with the capability of reproducing typical buildingfloor earthquake acceleration time histories. The design of the vibra-tion table and control system and the development of the accelerationtime-history command signal are discussed. A comparison of the acceler-ation response spectra for the table motion and a typical command signaldemonstrates the feasibility of using this type of realistic simulationof earthquake motions for evaluating electrical equipment performance.Significant results from this work are included to provide equipment de-signers with some typical experience.

INTRODUT1ION analysis whether the electrical components willperform reliably without damage or temporary

The design of nuclear power gener- interruption of operation during a seismicating stations requires the consideration of disturbance. As a result of this conclusion,many types of possible accident situations in a laboratory facility for the seismic evalua-order to insure that the public is protected tion of Class I electrical equiptrent was de-from potential exposure to nuclear radiation, signed and installed early in 1971 at theThe considaration oi earthquake effects is Battelle-Columbus Laboratories. The design ofparticulLrly important because the forces can this facility was based on the desire tobe extremely large and all pacts of a nuclear achieve a realistic simulation of the earth-station could be affected simultaneously, quake vibration environment.Contrary to popular opinion, nearly all loca-tions in the United States have a history ofearthquakes, although the frequency of occur- SEISMIC RDUIRENTS FOR EECTRICAL EQUIPM Trence and severity vary considerably. As aresult, ail nuclear power generating stations The components and structures of abeing built in this country are designed to nuclear plant that are critical to the -hut-withstand earthquake motions even though they down and maintenance of the reactor in a safemay be located in regions where earthquakes are condition are designated as "Class I". Theseignored for other types of structures. must be designed to remain functional when

subjected to the Design Basis Earthquake (DBE),Modern dynamic analysis techniques sometimes referred to as the Maximum Credible

have proved to be valuable tools for predicting Earthquake, that has been selected for theLhe re ,ponse to earthquake excitation of build- particular plant location. Structures anding structures, piping systems, and many types components whose failure would not interfereof mechanical equipment [1]. Electrical equip- with a controlled shutdown of the reactor ormerit, however, often consists of a light frame contribute to an excessive release of radiationstrcture supporting flexible racks and panels ar generally designated as "Class II", andto which a variety of components such as trans- these are subject to less severe seismic de-

formers, relays, switches and meters are sign requirements.mounted. Analytical techniques can be used todetermine the adequacy of the basic structure Manufacturers of Class I electricaliof an electrical equipment cabinet to survive equipment are now being required to verifyan earthquake; however, it is considered that their equipment is capable of acceptableimpractical, if not impossible, to determine by performance during a seismic disturbance,

Preceding page blank 11

_ _ J

A basic problem faced by these manufacturers is the acceleration time histories of the floorthat only rarely are those persons responsible response could be included directly in equip-for the design of this type of electrical ment specifications, it has been customary inequipment also knowledgeable in earthquake earthquake eniineering to describe the equip-engineering technology. Adding to this prob- ment vibration environment by a response spec-le is the fact that there are no standard trum. The use of a response spectrum-hasrequirements for seismic evaluation, so a particular signLfcance:as & methbdof char-manufacturer receivesa-different set of ieis- acteriziing a complex ti.ansient vibration in amic specifications for each new nuclear power way that is useful lor determiniag h.jenerating station. While-it is expected that simple structure will respond to that vibra-the severity of the DBE wiii vary for differ- tion. In particular, the earthquike-responseent locations in the country, there is also spectrum for a building floor ie the maximumconsiderable difference in the type of speci- response of a series of 3ingle-degree-of-'fications being used to assure compliance with freedom oscillators that are excited by theseismic requirements. The greatest source of floor motion. Each oscillator has a fixedconfusion from nearly all types of seismic percent of critical damping, but a differentspecifications is5 the use of response spectra natural frequency, so the series of maximumas the principal means of describing the earth- responses gives a good representation of thequake vibration environment. Therefore, a frequency content of the floor motion. Thebrief review of the development and implica- floor response spectrum does not describe thetions of earthquake response spectra for acceleration versus frequency characteristicearthquake engineering seems justified, since of the floor motion directly,, and this is theexperience has shown that this-type of descrip- principal sourie-of misun4derstanding-by thosetion often confuses even specialists in other who are familiar with the-use of "the.Fourierareas of structural dynamics and vibrations. spectrum or Fourier series to describe cotp 1,ex

signals.

EARTHQUAKE'RESPONSE SPECTRA

The design of each Class I buildingfor a nuclear power generating station requiresa detailed analysis of the response of that 0building (deflections, accelerations, stresses,eta,.) to a typical earthquake ground mctionselected for that location. The earthquakeground motion will have acceleration componentsin two orthogonal horizontal directions ofabout equal intensity and a vertical component 2ithat usually has lower peak accelerations, _although the energy is concentrated at higher .0frequencies. Fig. 1 shows the component of pground acceleration recorded at El Centro, 0 --California, during the earthquake of May 18, Z1940. Although the ground motion lasted for 0-.-about 45 seconds, the most severe part of the '$' - 4earthquake occurred during an interval of only -'10-15 seconds. The maximum acceleration of 0 4 8 1 16 20 320.33 g measured at El Centro is the strongest T*X,SECONDSground motion that has been recorded (theauthors have no information on very recent Fig. I - Ground acceleration record of

earthquakes such as the one in Los Angeles in North-South component of El

1971). Data of this type are often used to Centro, California Earthquake,

establish the DBE ground motion at nuclear May 18, 1940

plant sites by retaining the time history andby linearly scaling the amplitudes to achievea desired maximum acceleration.

The earthquake analysis of a Class 1 0411 0711,+.o3'o . .building requires a detailed mathematical 0model of the building's stiffness and mass ydistributions. The predictions of the build- I oing floor motions in response to the DBE ground .o. .. . . .motion are then used to establish the seismic orequirements for equipment that will be lo- T... 4s ,

cated in that building. Fig. 2, for example,shows a typical 5-second record from the most Fig. 2 - Typical horizontal building

severe part of the acceleration time history floor acceleration for DBE

preulcted for the floor of tie auxiliarybuilding of a nuclear power station. While

12

The diagram shown in Fig. 3 of a Fig. 4 shows an acceleration responseSseries of mechanical oscillators resting on a spectrum for the horizontal floor accelerationbuilding floor ha proved useful for explain- time history shown in Fig. 2. Most of theing the response spectirum. The floor ancel- horizontal vibratory eitergy fron earthquakeseration time history, '(t), would be predicted occurs at frequencies below 10 to 15 Hz, andfrom a structural dynamics analysis of the the largest response of the building occurs atb-ilding response to the DBE. The equation of its natural frequencies. Therefore, the mostmotion for any one of the michanical oscilla- severe vibrations will be experienced by floor-torn is wounted equipment with natural frequencies

close to those of the building. Equipment2 which is quite rigid (natural frequencies

n 2nzn(t) nn above about 30 Hz) will follow the floor motion

exactly. Therefore, the acceleration ampli-

where tude which is approached asymptotically at highnatural frequencies on the response spec.crum

zn(t) - X(t) - y(c) (2) is identical to the maximum floor accelerationn+ that would be observed from the floor" accel-

is the relative displacement between the mass eration time history (see Fig. 2).and the floor, is the undamped natural fre--quency of the nth oscillator, and C is aselected value of critical damping ratio.The'ire are many techniques available for cal-

culating the displacement response time his-tory, z(t), and the maximum absolute value 60

-of -the response, Iz Imax , can be obtained for 4.eachoscillator. Tie process can be repeated 30

to determine a family of curves of 1z I ma 20

versus natural frequency, wn' ,or diferel .%values of damping, and this is the displace- c _%ment response spectrum for the floor acceler- 1%ation, 1(t). It is a matter of choice whether Jc oGOthe maxicul response parameter of relativedisplacement tz,,I x., relative velocity 04

[in max' or absolute acceleration 1"n + max 0 luis plotted, because they are relAted for zero o3damping by the following equations:

/|I I 'n 0 op I 1 1.1 1 1 I 1 1 111 1 1 I

max ax 0 03 0 O 0 1 20 30 03 40 O0 20 30 40 0

NOalwaO r, rvency. "erttz

2 .1 (4) Fig. 4 - Floor horizontal accelerationI n + I - wunIZnI response spectrum for DBE

mAx max

A detailed discussion of the approximations SEISMIC VIBRATION STUDIESinvolved for small damping can be found'in _Reference (2). The use of the response spectrum has

some important advantages for designing struc-tures to withstand earthquakes. However, itsuse as a specification of the earthquakevibration environment for equipment evaluationhas some significant limitations resultingfrom the absence of all phase angle informa-tion. This makes it possible to devise anynumber of vibratory motions that will satisfy

X1 0, X (1) X3 (t) ftWthe response spectrum for a selected value of• " damping, many of whilch will have little resem-

blance to a typical earthquake floor acceler-ation time history.

Le_ Some of the different types of

Floor vibration currently being used to evaluate

electrical equipment are decaying sinusoidal

Fig. 3 - Series of sechanical oscillators excitation, sine beat excitation, and constantused to determine floor moLion amplitude sinusoidal excitation. All of theseresponse spectrum employ single frequency excitation so that a

number of evaluations at different frequenciesare required to cover the response spectrum.

13

Also, the response spectrum amplitudes for the spectrum of horizontal acceleration, withthese motions vary more than the earthquake minor peaks at 1.5, 2.2, and 6.5-Hz. Negli-response spectrum for changes in damping. gible response was found above 10 Hz. TheFor example, the peak response spectrum ampli- vertical acceleration power spectrum showed atude of 1.1 g at 8 Hz for 5 percent damping predominant peak at 1.0 Hz, with lower ampli-shoin in Fig. 4 can be satisfied by using a tude peaks from 7 to 11 Hz and from 17 to 210.11 g, constant-amplitude (Q - 1/2C - 10), Hz. Thecs amplitude and frequency reuirementssinusoidal vibration at the equipment base. led to the selection of a servovalve-controlledHowever, if the equipment being evaluated has hydraulic actuator to power the vibrationits lowest resonance at 8 Hz with only 2 per- table.cent damping (Q = 25)9 the sinusoidal excita-tion would produce a peak acceleration of 2.75 Since the type of equipment to beg compared with about 1.6 g'Sa for the floor evaluated by this seismic vibration simulatorearthquake motion--an "overtest" of 70 p'rcent. was not expected to respond to large displace-The use of sine beat excitation is preferable ment components at low frequencies (below Ito constant-amplitude sinusoidal excitation Hz), one of the first design decisions was tobecause the variation in the response spectrum limit the total actuator stroke to 6 inchesamplitude with damping is less, but it is still (peak-to-peak). This was done in order todifferent from that of an earthquake motion. minimize the entrapped oil in the hydraulicThis requires making conservative estimates actuator, thereby keeping the oil-columnof the expected equipment damping or measuring stiffness high and maximizing the high-fre-equipment damping at resonances-- a procedure quency response capability. To keep thewhich is both expensive and possibly inaccurate commanded displacement within this 6-inchbecause of nonlinearities. limit, a high-pass filter was used In series

with the acceleration comand signal toThe alternative of using the maximum attenuate frequency components below 1.5 Hz.

flooracceleration amplitude to test at all Analog acceleration signals were recorded onfrequencies may also severely overtest equip- FM tape by digital-to-analog conversion of thement. Furthermore, equipment with several original digital computer data for use asresonant frequencies below 30 Hz will not simulator command signals. Command acceler-respond to single frequency excitation in the ations and the resulting filtcred displacimentssame way it would respond to an earthquake used as a basis for the facility design aremotion, and the large number of evaluations shown in Fig. 5. Peak accelerations of 0.37 grequired might produce fatigue failures that in the horizorital and 0.25 g in the verticalwould never occur during an earthquake, direction were recorded during five seconds

of the floor vibration response data for theFor these reasons, it seemed desir- DBE.

able to develop a more realistic simulation ofthe earthquake vibration environment. The The design of the seismic vibrationprincipal goal was to be able to evaluate facility included consideration of severalequipment efficiently while reducing the tradeoffs necessary to achieve a "cost-effec-possibities of subjecting it to vibration tive" facility. Simultaneous vibration inwhich might be considerably less severe or more vertical and horizontal directions, whilesevere than necessary. It would be quite dan- desirable for achieving the most realisticgerous to "undertest" the equipment, while simulation, was not incorporated because ofovertesting might result in an unnecessary its considerably greater cost and complexity,increase in the equipment cost. The coupling effects of simultaneous vertical

and iorizontal floor motions during an actualearthquake ure reduced in importance by their

SEISMIC VIBRATION FACILITY DESIGN considerable difference in both amplitude andfrequency content. It is quite unlikely that

The first task in designing the the maximum accelerations will occur simmlta-Battelle-Columbus seismic vibration facility neously in time for the two directions.was to deffne the amplitude and frequencyresponse require-ents needed to reproduce the Physical Nize of the simulator wastypical floor acceleration time history shown based on an estimate of equipment sizes to bein Fig. 2. Preliminary procesaing of the evaluated in the foreseeable future. T Isacceleration data consisted of numerical limit was chosen to beobout 3000 pour .s inintegration to obtain velocity and displacement weight with maximum base dimensions it aboutsignals. The horizontal displacement reached 80 by 48 inches.

* a peak amplitude of 12 inches, with a predom-inant 0.3-11z frequency plus some additional To provide a table wit's minimumlower frequency component (;he displacement weight and maximum stiffness, o equilateraldid not return to the initial position within triangle of steel I-bears with three angle-5 seconds). Acceleration power spectral den- bisccting beams In the center was designed.sity curves for the 5-second signals were The single hydraulic actuat-.r is attached belowgenerated by Fast Fourier Transform techniques the table centroid, and the actuator can beto provide some idea of frequency content, rotated 90 degrees to provide motion in eitherLarge peaks were found at 0.3 and 0.9 Ilz in vertical or horizontal directions. Normal

!4

[ Rqproduced 'fn: modes of vibration, of the table when actuateit O lvailable COPY-

at the centroid vere calculated by means of a

finite-element digital computer program, and.ll natural frequencies were predictad toexceed 50 Hz. Support and guidance of thetable in the horizontal mode are providcd byThomson "Roundways", while in the verticalmode a combination of Roundwtys and linear ball -

bushings is used for guidance at the three -

corners of the table. Three Firestone "Airlde'springs connected through a pressure regulatevto shop air are used to support the staticweight of the table and equipment for excit,-tion in the vertical direction. Fig. 6 shristhe seismic vibration table pos.tioned forvertical motion.

,4 ,qo . _. F!g. 6 - Battelle Seismic Vibrationa. Hofqlato Floor Acceletolson Table In vertical mozde

Conflicting requirements of stiffnes(oil-column resonance) and maximu oil flow

t rate at peak vclocity dictated the choice oactuator size. A -1/2-inch bore, I-nhrd

93 on. I , double rod-end cylinder was chosen to rate withV r an available 3000-psi, 25-gpm hydraulic supply

and loog 76-104 (15-gp) flow control servo-b Horizontol FWoo Oocement valve. For this size of actuator, a miim.z

oil-colu=n resonant frequency of 22 11z wasestimated based on an assumed effective oil

4 . bulk modulus of 100,000 psi (in tests this

' I &" " ' m . , .*

IO •Since the hydraulic power supply isI 9. • 25Gmnomum 414" . . located some distance from the simulator, a

1-gallon bladder accumulator, 10-micron fi!ter,c Verhicol Floor Acceleroton and solenoid shut-off valve are located at the

. ,base of the vibration table, with short lengthsj of hose supplying fluid to an actuator-mounted

47,n . manifold. The servovalve and crossover reliefvalve are attached directly to this manifold

toi prvd minu enrpe i on thej j "active" side of the servovalve. Safety fine-

-:. H tions (over-travel limit switches and servo

d veW,cot Floor msvoement over-coand voltebe level) deenergize theshut-off valve in the event of a system fail-

Fig. 5 - Acceleration and filtered ure and stop the table with a maximum qccel-displacement time histories oration of about 0.75 g.of bilding floor response

Design of the servo controller wasAn I-beam frae is used to provide based on the salient fact of life of a

a base for accurate alignment of the guideways hydraulically actuated inertial load: theand actuator. This base is fastened to a oil-column resonance (3). This resonanceheavy "strongback" formed of 36-inch-deep I- results from the mass of the table and equip- +beans located beneath the floor level of the ment oscillating on the effective stiffnesslaboratory. The strongback, fastened to a of the entrapped oil, piston red, and actuatorheavy concrete slab, provides the necessary support structure. Without compensation, theseismic mass against which the simulator can oil-column resonance adds a pair of lightlyreact, damped cnaplex poles to the root locus plot.

Several methods can be erployed to control the

15

L *

effects of oil-column resonance [4]: accelera- Fig. 8 shows the control system intion feedback, pressure feedback (in conjunc- block-diagram-form with linearized-transfertion vith a pressure control servovalve), or functions for the major cimponents. Primarycontrolled bypass flow, for example. servo-system feedback is derived from a post-

tion transducer (DCDT) mounted on the actuator.,Adjustable bypass flow using a nee- The positin feedback signal is subtracted from

die valve and small tube across-the actuator the filtered, double-integrated accelerationis an effective, yet inexpensive, method of command signal to provide the error-signal toc~ntrolling the resonance. The effects of the servovalve. A &odifiedlacceleration feed-introducing bypass flow (based on a linear back signal from a strain-gage accelerometerestimate of system response) are sketched in mounted on the table is also summed with thethe root loci of Fig. 7. For comparable error signal to improve system response in-thevalues of forward-loop gain (the parameter K a 10 to 20-Hz range.represents servo amplifier gain in amps pervolt), the system with bypass flow is far morestable--this is shown by the reduced angle of avector from the origin to the gain-dependent .. ,'.

root. One disadvantage of bypass flow is thelower frequency of the resonant peak for desir-able levels of damping.

Rod 20/Sec 1200 OO

X×z .0 Fig. 8 - Seismic vibration facilityJos control system

1100SEISMIC VIBRATION FACILITY PERFORANCE

at 5 05 0o Closed-loop frequency respons:

=Rod 'oo .1"' evaluations were conducted with a '1500-poundsec dummy equipment load on the table at normal

acceleration amplitudes. A constant-amplitudeiloo voltage command to the double integration

0 No bypol flow circuit wgas varied from 1 to 40 Hz, and theresulting table acceleration was monitoredusing a Quan-Tech Model 304TD-wave analyzer to

0 track the fun "uental component of accelerationwith a 1-lz bandwidth filter. The results ofthe sinusoidal closed-loop frequency responseevaluation are shown in Fig. 9. A second-order

•Rod resonance with a + 3-dB peak ic evident at 10

See Ht, and the effective system bandwidth (-3 dBZo 2oo point) is about 14 Hz for the command signal1amplitude of 0.2 g.

0While the closed-loop frequencyresponse of the control system determines the

o) Icapability of accurately reproducing a complex

,0 0 o2 acceleration time history, the maximum accel-"a X $o, o o0oo eration determined by the actuator stroke,e oo flow rate, and force limitation is the most

significant parameter for any type of sinusoi-,'al testing. Fig. 10 shows the maximum table

,,o accelerations that have been measured for thebctuntor byaw frequency range of 1 to ' 5 lci . AccelerationsX are limited by the maxiatum flow rate of the

S lservovalve at frequencie from abo t 1 to 10hydruic sly and by the taximum actuator force of aboutld7500 pounds at higher frequencies. Since ther , maximn, floor accelerations seldom exceed 0.5

i Fi. 7 Th effct f acuatr byassg for a DBE, the acceleration capability isFig.7 Te efec ofactato byassadequate for frequencies above 2 liz. Lower

flow on root loci of the lower- frequencies are of little practical interestfrequency tormst hydraulic eye- because the lowest equipm~ent resonant fre-

tew With Inertial load quencies are seldom below 5 Itz,

16 *

. iU

ance by comparing the time histories of thecomand and output signals. However, a com-parison of the response spectra, Fig. 12, for

0the floor-acceleration command signal and theo* resulting table acceleration indicates accept-04. able agreement. The amplitude of the tabl e.

1 0 2 0 comand ,al acceleration can be increiiid above the 0.37-gmaximum accelerution for the DBE so that. theresponse spectrum of the table motion exceeds

0 o-the required spectrum at all frequencies whereany type of equipment resonance might occur

t (i.e., above 4 to 5 Hz). The increase in amp-3 litude required to compensate for filter char-

01 oacteristics can be reduced by using an in-0001 creased rate of filter attenuation (sharper

filter) in the control system or by-preprocess-00 Equgment Weight- 1500:b ing the original floor-acceleration time his-05. Mtories with digital computer filter algorithms04 o prior to obtaining an analog signal. High-

frequency noise can be reduced by using ani ~03 actuatori ! ' I

S3 actuator with low-friction seals and extendingrequency, orti, the control system bandwidth by improved com-

Fig. 9 -Table acceleration response pensation for the oil-column resonance.

to sinusoidal, constant-

amplitude acceleratiorn comoandsignal

; ~~~p [ ri, cc or -- -

6.0- 0. Horizontol Floor Accsleration

2.0-

(Al -. .. -- .. . . -- . . . - - ---

3 .0-

06 b. Horizontal Toble Accelerotion

0.6 Fig. 11 - Comparison of horizontal table

Equipment acceleration with DBE floorO Curve Weight, lb acceleration comnand signalo A 1500o 3000 SEISMIC EVALUATION RESULTS

The seismic vibration facility hap

I 0I I 111 I I I I I been in operation at Battelle-Columbus since1. 2 3 4 6 a 10 20 30 40 60 February, 1971, end seismic evaluation pro-

Frequency, Hertz grams have been completed for a variety ofClass I electrical equipment. Results from

Fig. 10 - Maximum acceleration this work are cuumrrized briefly to providecapability of seismic equipment designers with some typical datavibration table for on the natural frequencies, damping factors,sinusoidal excitation and electrical performance of this type of

equipment. Fig. 13 shows a typical electrical

Figures 11 and 12 show the capability equipment cabinet mounted to the seismic

of the vibration table to reproduce a floor- vibration table and being subjected to earth-

acceleration time history. The results In quake excitation in the front-to-back hori-Fig. 11 show that the use of a high-pass filter zontal direction. The equipment is energizedto attenuate low-frequency components and the electrically during the seismic evaluationpresence of some high-frequency noise makes it and all critical electrical performance para-difficult to evaluate control system perform- meters are monitored continuously.

17

LI

VIBRATION CHARACTERISTICS

It is a general practice to conducta brief vibration survey of each equipment

40: ~unit in order to identify major natural fre-20% rCo.. d-,i quencies pr'or to evaluating its electrical

.10 1100 o eqpmtft e.qhl purformance inder specified seismic conditions.• oTable 1 lists the natural frequencies which

have been measured for a variety of electricalrequipment. These natural frequencies-were

1° To*I- ,eonse determi.ed by comparing the acceleration amp-

jO 0540esonso046 litudes from accelerometers located at thei -base and near the top of the equipment cab-- 0376 inet. While this procedure is satisfactory03o for determining the major structural natural02 frequencies of the cel,'4-?s, the identifica-

02 tion of local panel resonances and resonancesof components mounted inside the cabinet

0 1 , ,, 1 I ,,requires more extensive measurements than are02O 0 b SI 2 3 4 6 6 -0 20 30 40 600at,r3l F040i0tnc, Hes practical. Therefore, an evaluation procedure

which is based on exciting-the equipment cab-Fig. 12 - Comparison of vibration table inet with sinusoidal vibration only at natural

horizontal acceleration response frequencies has the risk of missing a reson-spectrum with floor response ance of some small component that could causespectrum for DBE an electrical failure.

TABLE ITypical Natural Frequencies ofElectrical Equipment Cabinets

Equip. NaturalItem Equipment Wt., Frequencies, HzNo. Size, in. lb Horiz. Vert.

- ,1 36 24 90 2350 4.5, >25i : . , • 5.0

2 36 24 qO 700 9.5, >2510

3 60 20 90 3500 7.5,7.5

4 36 24 56 1200 11, 2012.5

5 30 30 75 800 10.5, >25.. ... 11.5

6 60 30 75 2000 9.5, >2510.5

The data in Table 1 indicate that itis unrealistic to expect equipment of this5 _ type to be designed to have horizontal naturalfrequencies above the maximum excitation fre-quency for earthquakes (about 20 l1z). Naturalfrequencies shoikld be increased as much aspossible during design by locating heavy com-por.ents near the cabinet base and sizing thestructural members at the cabinet base totransmit the inertia: loads to the mountingbolts without exceesive deflections. However,experience indicates that it is even more

Fig. 13 - Time exposure of typical equip- important for designers to consider the de-ment during horizontal vibrations tails of the mountings used for the components

inside the cabinets. A brv..en mounting bracketfor a transformer weighing about 100 pounds isthe only structural failure of any type that

18

4 I, . ... .. , . , , ,,.. .... . ::< _., , . , :.., . . . .

has been observed. However, even this failure CONCLUSIONSoccurred only. after extensive exposure tosinusoidal excitation at resonance caused a The resultw'which.have been achievedfatigue condition that would not be expected to date indicate the'feasibility of using thefrom an earthquake, actual floor-acceleration time-historiespre-

dicted for a DBE to evaluate electrical.equip-The data in :able 2 show the appar- ment performance. The use-of tlhs-type of

ent magnification factors and corresponding realistic excitation substantially reduces thecritical damping ratios measured while a typ- possibility of overtesting or undertesting theical equipment cabinet was excited at reson- equipment. Furthermore, the vibration evalu-ance at different acceleration amplitudes. ation portion of a -sismic evaluation programThe selection of an appropriate damping factor can be reduced substantially, since a prelim-is oue of the most difficult and important inary survey for natural frequencies is un-judgments that must be made for any type of necessary and the total exposure to the earth-dynamics study. This type of equipment usually quake environment can be limited to about 60has damping factors no lower than about 3 per- seconds in each direction.cent, and in most cases the damping is in therange of 5 to 10 percent of critical. However, It is expected that the appropriatethe increase in damping with vibration ampli- acceleration time histories can be suppliedtude shown in Table 2 can be expected for this by the nuclear utilities once the advantagestype of equipment, indicating that nonlinear of using a more realistic earthquake simulationeffects are quite significant, are recognized. However, there are-some cases

in which the use of an artificial earthquakesignal has some important advantages. This

TABLE 2 type of signal can be generated using digital

Effect of Acceleration Amplitude On or analog techniques to modify wide-band noiseDamping or ATpcraloElctritce E - with appropriate filters to produce a time-Damping For A ypical Electrical Equip- history signal having any desired responsement Performance spectrum characteristic. Battelle's digitalEquip. Equip. computer facility has been used to generate

an artificial signal having a relatively flatBase Top Magnifi- Damping, (constant amplitude) acceleration response

Accel., Accel., cation PercentAcce.* cc8 Factor Critical spectrum over the frequency range of 2 to 20F C Hx. This type of excitation is desirable for

evaluating equipment that will be installed0.11 1.2 11 4.5 in several different nuclear power generating028 6stations where the seismic requirements are0.25 2.1 8.4 6.0 similar but where the particular time his-

tories and response spectra depend on the0.39 2.6 6.7 7.5 different building natural frequencies and

location of the equipment in the building.0.67 2.9 4.3 11.6

ACKNOWLEDGMNT* - Constant-amplitude horizontal

sinusoidal vibration at 8 1z The authors are grateful to thenatural frequency. assistance of Mr. Charles Rodman and the other

members of the Mechanical Dynamics Division atBattelle who have contributed to the design

EQUIPMNT PERFORMANCE and operation of the seismic vibration facili-ty. They also wish to express their apprecia-

It is encouraging to report that most tion for the cooperation and encouragement ofof the electrical equipment that has been sub- Mr. Julius Tangel of the Public Service Elec-jected to vibration simulating the DBE for tric and Gas Company.nuclear power generating stations has performed

(satisfactorily without any significant designchanges. No examples of cabinet structural REFERENCES

1'damage to the main structural members or hold-down bolts have been observed. The few elec- 1. John A. Blume & Associates, "Summary oftrical failures that have occurred were usual- Current Seismic Design Practice forly caused by high-voltage arcing or relays Nuclear Reactor Facilitios", United Stateswhich malfunction. Failures have been ob- Atomic Energy Report TID-25021, Sept. 1967.served in some meters which are often mountedon a flexible door panel, but a failure in 2. R.D. Kelly and C. Richman, "Principles andthis type of component seldom effects the Techniques of Shock Data Analysis", SVM-5,primary functional performance of the equip- The Shock and Vibration Information Center,ment Naval Research Laboratory, Washington, D.C.,

1969.

19

3. V.H. Larson, "The Control of Accelerationby Electrohydraulic Shaker Systems", 145Systems Corp., Technical Bulletin 840.00-1.

4. L.H. Geyer, "Controlled Dampin ThroughDynamic Pressure Feedback",, Moog, Inc.Technical Bulletin 101.

20

S..' V t J%<AC ~I A.m 4 ~ ~ 4,~ "~ Y .--

SHOCK INPUT FOR EARTHQUAKE STUDIES USING GROUND MOTION

FROM UNDERGROUND NUCLEAR EXPLOSIONS*

D. L. Bernreuter, D. M. Norris, Jr., and F. J. TokarzLawrence Livermore Laboratory, University of California

Livermore, California

Comparisons are made between the ground motion from earthquakes(EQs) and underground nuclear explosions (UNEs). It is shown thatpeak g-levels and response spectra form a reasonable basis for com-parison. Several approaches which attempt to characterize the timehistory are also discussed. It is shown that (1) the peak g-levels-from UNEs have magnitudes comparable to those estimated for thestrongest EQs, (2) the spectra from UNEs are similar to those fromEQs, and (3) a time history comparison shows that both the durationof strong motion and the number of near-peak g-level cycles forUNEs fall within the'range established for strong EQs. Based onthese results it is concluded that ground motion from UNEs can pro-vide an EQ-like environment for testing full-scale structures.

INTRODUCTION rent methods of formulating mathematicalmodels of real structures are not yet developed

The recent San Fernando earthquake to the point where they can readily produce ac-(M = 6.3) provided the first real test of curate models for most structures. This isearthquake-resistant design as practiced in true even for very low level seismic input, asCalifornia. Many structures failed the test. shown by Tokarz and Bernreuter [1). It hasThis earthquake caused 64 deaths and an esti- been very difficult to correlate results obtainedmated $553 million damage. The major cause from analysis with actual dynamic behavior ofof death was catastrophic collapse of multi- full-scale structures, principally because notstory buildings. much detailed data exists on actual dynamic

behavior of structures during earthquakes.Earthquake-resistant design criteria

used in California building codes are based on Dynamic tests have been conducted onstatic approximation of the dynamic loads pro- some large-scale structures in order to de-duced by earthquakes. These criteria require velop more accurate mathematical models [2J.that the structure be capable of carrying a set However, because of the power and mass lim-of static lateral loads whose magnitudes and itations of existing vibration generators, thedistribution are chosen so as to approximate amplitudes of vibration in these tests have notthe effects of the dynamic loads an earthquake been large enough to cause substantial inelasticmight be expected to produce. In view of the deformation. The dynamic response informa-damage caused by the San Fernando earthquake, tion is therefore valid only for small ampli-it is clear that this static approximation tudes of vibration. Since substantial inelasticmethod is inadequate, deformation usually occurs prior to cata-

strophic failure, it is most important to treatThe design of structures to withstand EQ the vibration problem at the relevant ampli-

(earthquake) forces requires a more detailed tudes. This highly nonlinear behavior makestheoretical approach-one involving thorough it extremely difficult to apply the data that candynamic analysis-to guarantee basic surviv- be obtained from model tests [21. The cost ofabTlity. Significant capability for such anal- building a shaker facility large enough to testysis now exists with the computers and soft- full-scale structures has been estimated to beware presently available. Computer methods approximately $20 million [31. Even then theproduce a dynamic analysis by idealizing the size and type of structure would be severelyreal structure into a mathematical model and limited.then determining the response of the model tosome prescribed ground motion. However, due An alternative approach would be to useto the lack of sufficient experimental data, cur- the ground motion from an underground nuclear

Work performed under the auspices of the U. S. Atomic Energy Commission.

21

?4

explosion (UNE) to excite properly located In addition, no clear relation between EQstructures to EQ-like ground motion. Several magnitude and peak g-level has been established.major advantages of using the ground motion Ibis is shown in Figs. 1 and 2. Figure I givesfrom UNEs are: (1) the ground motion is avail- a correlation-of g-level vs distance from theable free as a by-product from nuclear-tests, closest point of observed faulting. The EQ(2),there is no limit on the size or type of magnitudes range from 5.5 to 8.3. It-should bestructure that could be tested, and (3) true noted that the peak g-level from the Sansoil-structure interaction would be achieved. Fernando EQ-was around 1-lg. Figure 2 gives

a correlation by Houcner (111 relating EQ m ag-The ability to test any type of structure nitude to peak g-level and range. Also shown

and the achievement of true soil-structure . in this figure are the Parkfield EQ (M = 5.7)interaction are important. For example, under- and the San Fernando EQ (M = 6.3). As can beground nuclear reactors and storage containers seen, these clearly do not fit Housner's pro-can be tested. In fact, for marr types or large jected correlation.structures this technique of subjecting the full-.scale structure to ground motion from a UNE The response spectrum has been pre-seems to be the only practical way to investi- ferred for structural engineering studies ofgate the structure's response to an EQ. strong-motion earthquakes, because it com-

bines both the representation of the excitingCHARACTERIZATION OF EQ GROUND force and the response calculations. It thusMOTION lumps together under one representation the

major parameters of interest to the structuralThe feasibility of simulating EQ ground engineer. The major disadvantage of the re-

motion with UNE- induced ground motion de- sponse spectrum is that only peak response ispends on the similarity of the two phenomena. determined. The numbei of near-peak responseThe mechanism of energy release is much dif- cycles, which are important for studying post-ferent in a UNE than in an EQ. There are con- yield behavior of structures, is lost.siderable similarities and also considerabledifferences between the induced ground motions The Fourier spectrum of an input functiondepending on the criteria of comparison. The shows directly the significant frequency charac-choice of criteria will depend upon the purpose teristics of the function, and -from it-the time-for making the comparison. For example, cri- histcry response of the system can be computed.teria used by seismologists to compare UNE However, in the study by Jenschke et al. [51 itand EQ ground motion from the viewpoint of was found that results obtained using theseismic detection of underground testing [4] are Fourier spectral method were not satisfactory:quite different from criteria used by structural the sine and cosine Fourier transforms showedengineers whose basic concern is the response high irregularities of the same order as theof structures. ground acceleration function, and no correlated

characteristics could be found. These high ir-In order to show that UNE-induced ground regularities make it difficult to use the sine

motion is similar to that from EQs it is neces- and cosine Fourier spectra for comparisonsary to establish a means of properly charac- purposes. Hudson [6] showed that the Fourierterizing EQs from the point of view of struc- amplitude spectrum is much more regular andtural response. Many studies have been made is very closely related to the relative velocityto develop the best means of characterizing EQ spectrum. It has no advantage over the rela-ground motion [5-101. However, this has tive velocity spectrum and in fact is inferior inproved difficult to accomplish, one reason being that the peak levels may be lost; time-historythat only a few records of strong-motion EQs aspects are also lost.exist (here strong motion is arbitrarily taken tobe a peak g-level greater than 0.1 g). These The power spectral density apprcach isrecords show a large variation in pulse shape, attractive in that it allows a probabilistic ap-time of due'ation, peak g-level, and number of proach to be used. However,. Jenschkecycles. Some of the complex causes of these et al. [5] found that this approach was inade-variations are discussed in a paper by quate since ground motions produced by EQsTrifunac 181. and UNEs are essentially nonstationary phe-

nomena, even for dynamic systems having nat-The most generally used criteria to ural periods considerably shorter than the du-

characterize EQs for structural studies are ration of the ground motion. This is a serious(1) peak g-level, (2) response spectra, problem when dealing with the ground motion(3) Fourier spectra, (4) power spectral density, from UNEs because the duration of shaking isand (5) time history comparisons. None of quite short-abt.ut 5 to 15 sec as compared tothese criteria are completely satisfactory. 30 sec for the El Centro EQ.

Peak g-level has the advantage of being Several investigators have made attemptsextremely simple to compare. It is also phys- to generate artificial accelerograms usingicclly meaningful. Nevertheless, peak g-level various nonstationary processes. But asalone is not adequate to characterize EQs (7]. pointed out by Trifunac 181 the models generally

22

L kp.. Q 'AAA<C ,t.- - .~.. -

5000i I I a 1 ! 1 1 1 1 1-lol

£ Pacoima Dam (San Fernando epicenter distance)

a Boxcar

Mississippi (a) Lo (a/g) 3.5-2 log (D + 80)_

00Lg (a/g) =3.0-2og (D +43) \ 1 .

1I

Pld. o,,ld Calif. 27J.one 19(6-M.). iI. Hegnloire, Moto 17A 09 . 195(M-. 7). \ 6 6.\.C 2. El Centro, Calif. I8 Maky 1940 (M -7.1). 22. Son Jbt., Costa Rica 5sOct. 1950 (M 7.7).\

8 10

10 3. Puget Sound, Wash. 13 Apr. 1949 (M "7.1). 23. Me,.ico City, Mjsxco I I May 1960 (M - 7.2). \, •6

• 5. Long koch, Calif. 1 IMar. 1933 (M 6.2). IS. SonJose, Casio Rica lB8Nov. 1945 (M -7.0)6. Kern Sounty, Calif. 21 July 1952 (M -7.6). 16. Sishop, Calif. lOApr'. 947 (M. 6.4).7. El Contre, Calif. 30 D¢c. 1939 (Me 6.5). 27. Limo, Peru 17 Ocr. 1966 (M *7.5).\ i8. Logan, Utah 30 As.g. 1962 (M * 5.7). I8. Ltuyo Say, Aloslr 10 July 1958 (M *7.8).9. Portland, Oregon 6 Nov. 196? (M -* 4.8). 19. Prince Wliam Soun, Alaka 27 Mar. 1964 (M c 8.3).

20. sontlag, oi . , et 1945 (\ ,7,)

0.01 0.1 1 00

Distance -miles

Fig. 1 - Plots of g-level vs distance from closest point of observed faulting. From Cloud andPerez 010]

used are too simple. This results from a the time histories relative to post-yield be-lack of knowledge of the actual character of havior of structures.strong motion accelerograms. At the present

i time these investigations have not progressed Trifunac (81 suggests using the responseSfr enough to make use of here, envelope spectrum which is a three-dimensional

plot of magnitude of response of one-degree-of-Direct time-history comparisons are usu- freedom oscillators vs time and frequency.ally not made except to obtain qualitative insight Only some accelerograms from the 1040 El

into the nature of the ground motion. Some gen- Centro EQ have been studied in this way; henceeralized time-history characterization of EQ at the present time this approach-while inter-ground motion is needed to study the post-yield esting-does not offer a meaningful character-behavior of structures,. No such characteriza- ization of EQs.tion exi5ts. Cloud and Perez 1101 suggestedcomparing total time the acceleration was above y1he above discussion summarizes the ap-a certain g-level Figure 3 shows such a cm- proaches generally suggested to characterizeparison for the Parkfield and El Centro EQs. It EQ ground motion for structural responseis not at all evident from this figure that the studies. It is clear that no adequate singleEl Centro EQ was by far the more damaging. characterization exists. Based on this discus-

sion we plan to establish a similarity betweenOther investigators suggest counting the EQ and UNE ground motion by showing thatnumber of cycles of a given g-level. Both this () the peak g-levels from UNEs are within the

suggestion and the previous one (total time range estimated for the strongest EQs. (2) theabove a certain g-level) give some useful infor- response spectra from typical UNEs aresimilarmation, but they do not properly characterize to those of strong EQs. and (3) the duration of

gd

behvio ofstrctues Nosuc chracerza- Izaionof23s

tion~~~~ exst. lod ndPeez11] ugese

copain toa ieteaclrto a bv h bv icsinsmaie h p

1.2 * Figure 4- is a plot of peak g-level versusrange for "typical" nuclear explosive yields ofI0, 100, and1000kt(kilotons). For rangesless than I DOB the valke plotted is the initial

1.0- NI Pokfield (M=5.7) acceleration. In this regime, the motion issuch that the top several hundred feet of earth

a San Fernando (M=6.3) spalls away from.the lower layers, often caus-Ing a larger peak'g-level when the spall gapcloses. It would be undesirable to locate test

0.8- _ structures within this spall regime.

It is clear from this figure that the peakg-level of the strongest EQ can be easily du-

*plicated, or exceeded (if desired), by a UNE.-- 0.6 See Fig. I for estimated EQ peak g-levels.

Response Spectra romparisons

E Various types of respobse spectra can be"r 0.4 - generated. These izclude relative displace-

=8 ment, relative velocity, and absolute accelera-0 tion spectra. For low assumed values of vis-

% 7 cous damping (less than 10% critical) all of the0 X7 Nv " above spectra are related 16]. Since all of the

. .- *% spectra to be considered for comparison havegoo low values of damping, only the velocity spec-

0 trum will be used.

L.- ~ ~No standard EQ spectrum has been pn-0 20 40 60 so erally accepted for comparison purposes [71

Distance to fault - miles

Fig. 2 - Plots cf earthquake magnitude and 0.5peak g-levels vs range. FromHous ] 0.4 N (IlV (a) Parkfield, California "I N 6500. Station 2

g-levels fall within the range established by •0.3

major EQs. Vt.2 0.2

COMPARISONS OF UNE AND EQ GROUNDMOTION " 0.1u

Peak g-Level Comparison 0

Most of the published correlations of UNE 0.01 0.1 1 10ground motion data deal with peak g-levels be- Duration of acceleratlion - seclow those of interest [13,14]. To extrapolatethis data to close-in distances leads to grosserrors. The most comprehensive dircusslon of * 0.4- () El Centro, Californiasurfacc ground motion from UNEs In the strong N -Smotion regime is that by Bernreuter et al. [12]. N

The magnitude of the peak g-level at the " -ground surface from a UNE depends on several r 0.2factors [121. These include (1) yield, (2) thedepth of burial (DOB), (3) distance from surface aground zero (range), (4) geology around the ex- 0.1plosive, and (5) geology through which the waveIs transmitted. For a given yield the peak sur- 0 0 . ....... I . ... ,,,,I , , ,,_,_ ,face ground motion can vary by a factor of 20. 0.01 0. 1 10But since devices of a given yield are usuallyburied in similar geologies (and at similar Duration of acceleration secdepths) it becomes more meaningful to talkabout general curves of peak g-level versus Fig. 3 - Comparison of Parkfield and I Centroyield and range. earthquakes

24

10 . t (although several have been proposed). Com-parison herein will be made relative to the ElCentro (1950, NS), Olympia (1949, N8OE), andTaft (1952, N21E) spectra; these are the spec-tra most used for design studies.

There is substantlal variation in spectra% - between different EQs and between the samem i EL recorded at different stations. Figure 5

% % t shows a comparison of the above-mentionedEQs. Figure 6 shows two stations, from the

-Parkfield EQ, both having a peak acceleration10 level of 0.5 g. A comparison of Figs. 5 andi 6

shows even greater differences between the'same EQ at different stations than between dif-

o ferent EQs.

•! 0.1 - 0kt Spectra of the UNE ground motion re-corded off NTS have been extensivelystudied [15,16]. However, extrapolation ofthese studies to locations much closer toground zero Is not valid.

Spectra of the close-in ground motionfrom UNEs have not been extensively com-

0.n puted or studied. This is because most per-10,000 100,000 manent NTS structures are qimple, one-story,

strongly built field-type structures located outRange from surface ground zero - ft of the very strong motion regime. Also, for

close-in survivability, the peak g-level andFig. 4 - Peak radial acceleration for typical change in kinetic energy are the important

UNEs as a function of range for sev- parameters. This limits (at this time) oureral yields choice of UNE ground motion for comparison.

6.28 3.14 2.09

El Centro EQ (48 km)

70- Olympia EQ (72 km)-,"•,

60 e

*6 40

0

0 -20 2 310 /

" "-Taft EQ (64 kin)

0 I 2

Natural period - sec

Fig. 5 Comparison of E1 Centro, Olympia, and Taft EQs

25

- ~. - ~ ,. "L - 4. *7 777- 4- *.,. * - --.

60 _ 1 1 Table I lists four events for comparison

Parkflstd EQ, Cholame Shandon purposes. These events were chosen sinceArray,' Station 1o. 5 N85E they cover a wide range in yield and geological

50 A Sconditions. Figure 7 compares Mississippi (a)3.3 miles from fault and El Centro. Figure 8 compares Missis-Peak g-level, 0.46 sippi (b), Aardvark, and Olympia. Figure 9

.40 compares Blanca and Taft, and Fig. 10 com-pares liallbeak and El Centro.

" 30- -A Study of Figs. 7-40-shows that the re-sponse spectra from typical UNEs are similarto those from strong EQs. Furthermore, the: 20variations between the UNEs and EQs are nogreater than the variations befween the EQsthemselves, as shown in Figs. 5 and 6. Carderand Cloud [17] also noted this similarity be-tween EQ spectra and UNE spectra.

0 1 2 3 4 Time-History ComparisonsFigure 11 shows accelerograms from

k00 E m three.UNEs, namely Aardvark, Mississippi,Parkfield EQ, Cholame and Boxcar (1.2 Mt). (No accelerogram was

- Shandon Array, Station available for Halfbeak; the spectrum was com-[ 80 - No. 2 N65E - puted from a velocity transducer.) The dura-

C 270 ft from fault tion of strong phase motion (peak g-level60 Peak g-lvel, 0.5 greater than 0.1 g) is approximately 2 sec for

60- - Aardvark and over 7 'sec for Boxcar. This fig-ure shows in addition that the duration of strong

"* motion from a UNE increases with larger yieldand also with increased distance from ground

40- zero.

Figure 12 shows the accelerograms from J20-, the El Centro, San Fernando, and Parkfield

EQs. - The duration of strong shaking for theseEQs was Z0, 10, and 1.5 sec, respectively.The number of near-peak g-level cycles ofground motion ranged from 3 for the Parkfield

0 1 2 4 EQ to over 10 for El Centro.Period - ec The accelerograms of the EQs and UNEs

shown are quite different. Nevertheless, theFig. 6 - Response spectrum from Parkfield EQ duration of strong motion and the number of

(June 1966) at two stations cycles of near-peak g-level for the UNEs fall

TABLE 1Sources of UNJ, Spectra

Yield Recording Rango PeakEvent (kt) Geology (it) g-Level Ref.

Mississippi (a) 20-200 'ruff overlaid with 3280 0.98 [5]deep alluvium

Mississippi (b) - 6560 0.45 [51

lialfbeak 300 Saturated tuff over- 7000 - (19]laid with dry tuff

.Tlanca 19 Tuff 5310 0.5 (171

Aardvark 38 Tuff overlaid with 3600 .74 (5]deep alluvium

213

I'-Mississippi UNE (I kin)

.ii

2 ~ 'iv El Centro

, . ~ EQ (48 km)

0 20 40 60 so 100 12

Undamped natural frequency --rad/sec

Fig. 7 Mississippi UNE v El Centro EQ. Relative velocities normalized by dividing originalvelocities by 0.1 g

1.5

R Olympia EQ (72 km)

"10 , Aardvark UNE (I. I km).

0.5 , \...u,,10 A

0 2. 40 6 80 t20

• I-j..'

0

0 20 40 60 80 100 120

Undamped natural frequency - rod/sec

r'ig. 8 - Mississippi and Aardvark UNEs vs Olympia EQ. Relative velocities normalized by dividingoriginal velocities by 0.1 g

27

I~~~~ _ _ _ _ _ _ _ _ _I_

0 80- 1.2 __________

Aardvark UNE

.60 -.

* . -Taft EQ (N21 E)00

2Og4-erlanca UNE -0.

Natural period - sec0.6-

Fig. 9 - Blanca UNE vs Taft EQ O . issip N

*J0.2~90

- ~80 -Halfieck UNE0

* c7 0 ~ ~ ...- El Centro EQ .<-2

k-5 V- -0.4

840- I ' -Cs.6 1 I 1 I> 0 Af 0 1 2 3 4 5 6

1~0 0.4-

0 1 2 3 0.2-24,0 f

7Natural period -sec 0

-0.2Fig. 10 -lialfbeak UNE vs El Centro EQ

-0. 4

2 3 4 5 6 7 8 9 10within the range establisHed by the EQs. It istherefore noncluded that from the standpoint of The- seea time history comparison, the ground motionfrom any of these UNEs could be considered as Fig. I I - Accelerograms from Aardvark,being representative of a future major EQ. Mississippi, and Boxcar UNEs.

SUMMARYThe oreoingcomarisns how hatnear-peak g-level cycles for UNEs fall within7be oreoingcomarisns how hatthe range established by strong EQs.

from the structural engineer's viewpoint,close-in ground motions from UNEs are sim- From these results it is concluded thatliar to those produced by major EQs. More typical UNEs generate ground motion equiva-specifically: lent to that of a possible future major EQ.

Therefore, by predetermining the yield and(1) T7he peak g-leveis from UNF~s are location of UNEs (relative to the location of acompar., -? to tho~se estimated for the strongest test structure), a test structure can be excitedHQs. by grouand motion comparable to that of a futuremajor EQ.

(2) The upectra !rom typical UNEs areslnr~ir to tliis,2~r EQs (i.e., the envelope However, to study post-yield behavior ofUCVelOpcir .. ypical UNE spectra would structures it will be necessary to generate UNEmatch an envelope from strong EQs). ground motion with longer pulse durations and

also more cycles of near-peak g-level. This(3) A time history comparison of ground can be accomplished by sequentially firing

motion from typical IJNEs shows that both the IJINEs or by subjecting test structures to groundduration of strong motion and the number of motions from UNEs fired over a large time

28

0.4 Parkfield EQ, 6-27-66, 2026 PST-

0.2 Cholme Shandon No. 2

, - _V PA, % A % &-- - I

0 2 6 8 10 12 14 16 18 20 22 24 26 2830El Centro EQ, 5-18-40, 2037 PST

0.2 -. 1 Strong motion

0 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I 1.2 -San Fernando EQ

Pacoima Dam S160E*~1.0-

0.8-

0.6-

4 J0.4

0.2 -

-0.2 -

-0.4 --

-0.8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time - sec

Fig. 12 - Accelerograms from Parkfield, El Centro, and San Fernando EQs.

span. Both of these possibilities need much One important fact to note is that lowerfurther study. yield explosives can be used. The only reason

for using very large yield explosives is to in-crease the duration of shaking. This is of

In order to properly locate structures a some importance since a large number of lcwerstudy must be made to correlate the ground yield explosives are detonated as compared tonuction spectra from UNEs with yield, geology, the very high-yield explosives. Springer a--'and depth of burial of the explosive. Once this Kinnamannj 181 give a list of U. S, UNEs. Thisis accomplished it would be possible to subject list indicates that more than a sufficient num-test structures to increasingly strong motion ber of UNEs are exploded every year to supplyranging from elastic response to fi.allysevere the required ground motion for any type of testdamage, program.

29

REFERENCES 4th World Conf. Earthquake Eng., Vol. I,pp. A2-119-A2-132, 1969

I. F. Tokarz and D. L. Bernreuter, "Com-parison of Calculated and Measured Re- 11. G. W. Housner, "Intensity of Earthquake3sponse of a High-Rise Building to Ground Ground Shaking Near the Causative Fault,"Motions Produced by Underground Nuclear Proc. 3rd World Conf. Earthquake Eng.,Detonations," Lawrence Livermore Labo- Vol. I, pp. 111-94-111-115, 1965ratory Rept. UCRL-50977, Dec. 23, 1970

12. D. L. Bernreuter, E. C. Jackson, and2. D. E. Hudson, "Dynamic Tests of Full A. B. Miller, "Control of the Dynamic

Scale Structures," pp. 127-149 in Earth- Environment Produced by Undergroundquake Engineering (R. Wiegel, editor). Nuclear Explosives," in Proc. Symp. onPrentice-Hall, Englewood Cliffs, N.J., Eng. With Nuclear Explosives (Las Vegas,1970 Nev., 1970), U. S. Atomic Energy Com-

mission Rept. CONF-700101, Vol. 2,3. J. Penzien, J. G. Bouwkamp, R. W. pp. 979-993, May 1970

Clough, and D. Rea, "Feasibility Study ofLarge-Scale Earthquake Simulator Facil- 13. J. R. Murphy and J. A. Lahoud, "Analysisity,' Earthquake Engineering Research of Seismic Peak Amplitudes from Under-Center, University of California, Berkeley, ground Nuciear Explosions," Bull. Selsm.Rept. EERC-67-1, Sept. 1967 Soc. Am., Vol. 59, pp. 2325-2341, 1969

4. J. R. Evernden, "Identification of Earth- 14. W. V. Mickey, "Seismic Wave Propaga-quakes and Exlosions by Use of Tele- tion," in Proc. 3rd Plowshare Symposiumseismic Data,' J. Geophys. Reg.. Vol. 74, (Davis, California, 1964), U.S. AtomicNo. 15, pp. 3828-3856, 1969 Energy Commission Rept. TID-7695,

pp. 181-194, 19645. V. A. Jenschke, R. W. Clough, and

J. Penzien, "Characteristics of Strong 15. R. D. Lynch, "Response Spectrum forGround Motions," Proc. 3rd World Conf. Pahute Mesa Nuclear Events," Bull. Seism.Earthquake Eng., Vol. 1, pp. 111-125- Soc. Am., Vol. 59, pp. 2295-2309, 1969111-142, 1965

.udson, "Some Problems in the 16. R. A. Mueller and J. R. Murphy, "Seismic6. D. E. Hudon o m robems to Spectrum Scaling of Underground Detona-

Application of Spectrum Techniques to tos"EvrnetlRsac opStrong-Motion Earthquake Analysis," Bull. tions," Environmental Research Corp.

Seism. Soc. Am., Vol. 52, pp. 417-430,1962

17. W. K. Cloud and D. S. Carder, "Ground7. D. E. fludson, "Design Spectrum," Motions Generated by Underground Nu-

pp. 93-106 in Earthquake Engineering clear Explosions," in Proc. 2nd World(R. Wiegel, editor). Prentice-flall, Conf. Earthquake Eng., Vol. III,Englewood Cliffs, N.J., 1970 pp. 1609-1632, 1960

8. M. D. Trifunac, "Response Envelope 18. D. Springer and R. Kinnamann, "SeismicSpectrum and Interpretation of Strong Source Summary for U. S. UndergroundEarthquake Ground Motion," Bull. Seism. Nuclear Explosions 1961-1970," to beSoc. Am., Vol. 6, pp. 343-356, 1971 published in Bull. Seism. Soc. Am. See

also Lawrence Livermore Laboratory9. S. C. Liu and D. P. Jhaveri, "Spectral Rept. UCRL-73036, February 1971

and Correlation Analysis of Groured-MotionAccelerograms," Bull. Seism. Soc. Am., 19. If. F. Stevenson, "Structural ResponseVol. 59, pp. 1517-1534, 1969 to Close- in Horizontal Ground Motion

from Underground Nuclear Tests at Pahute10. W. K. Cloud and V. Perez, "Strong- Mesa," Holmes and Narver, Inc., Las

Motion Records and Acceleration," Proc. Vegas, Nev., Rept. IIN-20-1039, 1969

30

DISCUSSION

Voice In terms of the duration at certain am- soil conditions in the surrounding areas. For in-plitudes in those plots that you showed, what was the stance, the acceleration levels in a particular earth-origin of that.data? quake may vary from place to place depending on the

type of soil or rock encountered. Are you aware ofMr. Bernreuter: That was from a paper by Cloud any studies that have been done?

and a coauthor in the fourth world earthquake confer-ence. He was looking at how long the level remained Mr Berreuter: Yes, there have been quite a fewabove a certain value; for example,how many seconds studies. Harry Seed, University of Southern Calif-it remained above 1/2 g or 1/4 g for different earth- ornia has made studies tryiny to relate ampliflca-quakes. tions of base rock motions to soil structure. There

are also several other papers on the subject.*yjjThen this was not the actual time hedwelled at any particular level?

Mr. lUernreuter: No. Mr. Haag: Have they been conclusive inany way?

Voice Was it the time after the initiation of the Mr. Bernreuter: I do not really feel that theyevent? have been conclusive. One main reason being simply

the lack of recorded motion. Once the San FernandoMr. Bernreuter: Yes. records are studied in detail, one might be able to

understand this better. This is the first time wbhaveMr. Haag (MTS Systems Corporation): I am in- had so many recording stations for a given earth-

tcrcsted in knowing if any one has done any work In qtake over varied geologies. So there is some hoperelating the earthquake accelerations to the type of that this may be clarified.

31

ROCKING OF A RIGID, UNDSRWATER BOTTOM-FOUNDED

STRUCTURE SUBJECTED TO SEISMIC SEAFLOOR EXCITATION

J. G. Hammer and H. S. ZwibelNaval Civil Engineering Laboratory

Port Hueneme, California

This paper describes in analytical study of an early-generation seafloor struc-ture subjected to severe seismic excitation. The structure Is assumed to

consist of a spherical pressure hull supported on legs, which in turn rest onpads on the sea floor. Friction between the pads and the underlying soil and

the shear resistance of the soil prevent sliding of the structure, but thestructure tends to rock rigidly in a manner that compresses the soil beneathit and may cause lifting off.

The problem is formulated by second order differential equations containingnon-linear terms. Effects of drag, added mass, and foundation reaction areconsidered. A numerical methud of solution is followed that generates solu-tions at successive time intervals that are automatically adjusted to conformwith preestablished accuracy criteria. The behavior of the foundation dependson the vertical forces and the load history. The ground excitation is a simu-l.ated earthquake motion generated from a random process of prescribed powerspectral density, multiplied by appropriate envelope factors.

An example structure is analyzed and its response computed for three different

support conditions. The results are shown in terms of displacement and accel-

erations of the structure and shearing force and vertical reactions of thesupporting medium.

INTRODUCTION

A structure placed on the floor of theocean must be adequate for the loading condi-tions expected during its useful lifetime. Oneof the most severe conditions, and one that isnot too well understood, would occur when therewere heavy seismic disturbances in the oceanfloor. This paper describes a study and analy-sis of the behavior of a particular kind ofstructure under such loading.

The structure considered is an early- hgeneration type that would be constructed outof the water and lowered into position on th(ocean floor. It might be a self-contained,unmanned scientific or observation station; crit could be a manned station complete with lifesupport systems and means for ingress-egress.As envisioned, it would consist of a negatively.buoyant spherical pressure hull supported byframed legs, which in turn rest on footings.The footings would rest on or in the bottommaterial, possibly with additional fasteners toprevent sliding. The structure could look like Fig. 1 - Simplified structorethe simplified model shown in Fig. 1.


The structure is assumed to be in moderate described by the holding forces to prevent

to deep water. The spherical pressure hull is sliding relative to the ocean floor and by thedesigned for the hydrostatic pressure at that vertical forces acting on the footings.depth. The footings are proportioned so as tosupport the weight of the submerged structurewithout exceeding the bearing strength of the PROPERTIES OF THE OCEAN FLOORbottom material. The mass of the structure isassumed to be concentrated at the center of the Probably the largest unknown factor inspherical hull. The main drag effects of the the prol.lem being considered is the nature ofwater are assumed to be those acting on the the material on which the structure is likelyspherical hull. The position of the structure to be placed. Submarine soils apparently varyat any time is given by two coordinates as as widely as do terrestrial soils. One of theshown in Fig. 2. most coinplete surveys of available data is Ref.

[1], which considered about 250 reports andabstracted data from the most relevant of them.The actual samples taken from the deep oceanfloor are very few. Ref.[l] reported that only300 or so cores have been obtained from theNorth Atlantic, for example, which averagesonly one sample for each 30,000 square miles.

/ The quantity of data from Pacific Ocean sampl--/ T ing.is even less representative.

/ What is believed, however, is that thefloor of the Pacific Ocean is covered chieflywith elastic, red clay deposits consisting ofvolcanic and terrigenous colloidal matter.The predominant sediments of the Atlantic and

-.___ -__ I Indian Oceans are calcareous oozes, with anmy °ooze being defined as a material containing

.greater than 30% organic material. Closer to

- the land masses, terrigenic deposits that havecome from rivers and currents cover the ocean

- ' floor on the continental shelves. The portionof the ocean floor that is solid rock isbelieved to be a very small percentage.

Fig. 2 - Coordinate system

Ref. [1] gives the following estimates asto the composition of the ocean floor. About

The rotation clockwise from its initial posi- 82 of the area of the sea floor is terrigeniction is given by 0, and the vertical downward shelf sediment in an average water depth ofdisplacement of the center of mass is given by 100 meters. About 46% is an ooze and 28% is ay. The horizoraal position of the center of red clay. These latter two compositions occurmass relative to the base of the structure is in the deep ocean and are pelagic sediments.proportional to 0 for small rotations since The remaining part of the ocean floor isthe structure moves as a rigid body. Its total believed to be hemipelagic muds, plus the small

displacements are the sum of the relative move- amount of rock outcropping previously mentioned.ments and the movements of the supportingmedium. These ocean bottom sediments have accumu-

lated over long periods of time. The rates ofThe loading situation is as follows. The deposit are of the order of a centimeter or

ocean floor undergoes a violent horizontal less per thousand years. Based on this kindshaking, which in turn shakes the footings and of rate, the total depth of sediment would bethe rigid structure that they support. The expected to be a few kilometers. However,system moves in a combination of rocking, hori- seismic surveys indicate a sediment thicknesszontal translation, and vertical settlement of of only 0.1 to 0.5 kilometers. It is believedthe footings. Forces acting on the structure that below this depth gradual hardening of theare the inertia of its mass and the added mass sediment has taken place so that the layers areof the water, the drag forces caused by the intermediate in composition between the uppermotion of the water relative to the structure, sediment and the basaltic bedrock underlyinggravity forces due to the submerged weight of it.the structure, and the reactive forces of theocean floor material on the footings. All of The core samples that hav been ,aken ofthe motion is assumed to occur in the single the uppermost few feet of the ocean bottomvertical plane that contains the horizontal sediment indicate very low shear strengths.axis of movement of the base excitation. The These range from 0.5 psi to 2.5 psi. Theresponse of the structure is described by time water content varies from 80% to 150% in therecords of accelerations, velocities, and dis- majority of samples. The porosity was in theplacements. The foundation requirements are range 60% to 80%.

34

- Ti,, -- -2 ., . .e. .. '- >..,, , = . .... . .. . . i

The shear strength seems to increase almost classical way, and solutions usually used arelinearly with depth below the water-sediment those due to Rankine and to Coulomb (Ref.[7J).interface. The void ratio apparently does not It would be possible, therefore, to make adecrease with depth. This might indicate that rough estimate of the horizontal restraint on athe older, lower deposits have become stronger sunken footing, given the properties of thethrough age rather than by consolidation. At soil material and the structural forces andthe soil-water interface, however, one can con- geometry. Lacking specific information aboutelude that the material supporting the struc- the sediment properties, it is convenient toture will be very soft with a low shearing and assume that sliding is prevented, and to com-bearing. In fact 89% of the samples reported pute the horizontal restraint required duringin Ref.1] had bearing strengths in the range the response. In a specific location the capa-0.5 to 2.5 psi. bilities of the bottom material could be com-

pared with what is needed to prevent sliding.When a negatively buoyant structure is

'placed on the ocean floor, there will be some The thick layer of soft material over-imnediate settlement due to an almost elastic laying the bedrock will tend to alter thedistortion of the bottom material. Following nature of seismic disturbances occurring inthis, there will be a much slower, and greater, the bedrock as they are transferred up to thesettlement duc to consolidation of the material sediment-water interface. Studies have beenunder the additional weight of the structure, made of layered soil masses to show the effectAn equilibrium is eventually reached, and this on a seismic disturbance applied at the lowestcondition is assumed at the time the seismic layer and transmitted by shear from horizontaldisturbance occurs. layer to layer until it reaches the uppermost

layer (Ref.[8]). These studies seem to showIf a footing has settled into the sediment that an input earthquake such as that recorded

it will resist being pulled out because of at El Centro in 1940 will emerge with similarsuction formed beneath it. These break-cut low frequency components but with attenuatedforces are commonly experienced in anchor and high frequency components. Of interest is thesalvage work. It is believed that pullout fact that the peak accelerations of the move-finally results when a failure mechanism ment do not change appreciably since they seemoccurs in the soil material surrounding the to be associated with the lower frequencies.footings. It is sometimes considered that the The particular structural size ad type that is

average foundation stress resisting pullout of of interest here responds primarily to thea footing is comparable to the ultimate bearing lower frequencies, and it is therefore assumedcapacity of the soil material. This was that ignoring the moderating effect of the

assumed in Ref.[2], and was substantiated by layers of sediment would not alter the struc-actual experience reported in that study. tural response significantly if the bedrock in-

put resembled a large land earthquake. TheIt is believed that the rate of loading problem, of course, is that one cannot predict

has an effect on the static resistance of the what the disturbance will be, even in thesoil mass. There are several studies (Ref.[3], bedrock.[41) that have looked at the dynamic propertiesof soil under footings of structures on land. What can be done is to take a record of aRef.[51 assumes a homogeneous, isotropic, elas- strong typical earthquake and use it for studytic medium and derives expressions for a soil purposes as though it were applied to the foot-resistance function involving a linear combi- ings of the structure. The analytical proe-nation of the deformation and the velocity of dure established for this can be refined asdeformation. The deformation is multiplied by information concerning sediment properties anda coefficient K, and the velocity by a coef- depths and bedrock motions becomes available.ficient C. K and C are functions of the shear-ing modulus of the soil, its mass density, thesize of the footing, Poisson's ratio of the EQUATIONSsoil, and the frequency of the periodic forcethat is loading the soil. Ref.[6] uses a simi- Fig. 3 shows the system of forces actinglar soil resistance function to study dynamic on the structure when it is displaced so thatfield tests on a soil-pile system, with the 0, y and xb are all positive and the motion isadditional assumption that C is also propor- such that the three accelerations 0, y and xbtional to the depth of the hole. are also positive.

Another possible motion of the footing The center of mass is accelerated horizon-(assumed to rest on or in a horizontal ocean tally by the combination of two motions: afloor) is horizontal sliding. Sliding motion motion relative to the base as the rigid struc-would be resisted by friction and by the shear- ture rotates, and the motion of the base itself.ing resistance of the sediment material. It is The total horizontal acceleration of the massbelieved that the shearing resistance manifests is thus h0 + xb, and the inertia force isitself in a way that offers passive resistanceto slipping within the soil along some interiorsurface outward from the footing. This kind of F = 0N(hO + xb)passive resistance has been studied in a

35

'_'

*y

Fig. 3 - Force system (OyxbOyi b positive)

where H is the mass of the structure and 8 is a with H the mass, a the radius, and Y the propermultip)ier that incorporates the added mass of coefficient so that YMa 2 is the mass moment ofthe water, inertia of the spherical structure about its

own diameter.The structure is considered to move through

the water with a velocity resulting from the RL and RR describe the static and dynamicsame two motions: A horizontal component of resistance of the soil beneath the footings.the rotation about the base, plus the base They are assumed to have the following formmotion. The total velocity uf the structurethrough twe stationary water is then h6 + ;b,and the drag force acting on the sphere is RL = RS(L) + t LAL)

F D - 1/2 CD Tra2p ~h6 + )(h5 + 1)RR " RS (R )+R i("R'4R)

where A. and A are the vertical movement ofwhere CD is a drag coefficient, ra2 is the. the fooings and AL and AR are the accompanyingcross section presented to the flow, and p is velocities. From the geometry of Fig. 3the mass density of the water. The quantity inthe second parentheses is shown as an absolute bvalue so that the velocity squared will have AL - y - 0the same algebraic sign as the velocity; andthe drag force will have a direction consistent - y + b0with the direction of the flow. R 2

These horizontal drag and inertia forcesacting thrcugh the center of mass are opposed The static resistance of the soil material isby a force FS acting through the base in a given by RS(A), in which the resistive forcehorizontal direction opposite to the algebraic depends on the vertical compression; andsum of the inertia and drag forces. RD(Ali) is the additional effect caused by the

rate of loading A and the depth of the hole A.There is an inertial resistance to rota-

tion of the sphere about its own diameter. The other vertical forces act through theThis resistive tcrque is center of mass. 'Tey are the inertia force

10 yHF - OMY

36

? a ~ aaA~a' c ~ ~ a

and the submerged weight Eq. (3) is a set of simultaneous nonlinearfirst order differential equations. The ini-tial conditions at time zero are

Xl(to) =X2(to) "X 3(to) -X 4 (t o ) " 0 (4)

where Mg is the weight in air and a is a multi-plier to Cive the weight in water.

While Eqs.(3) and (4) cannot be solvedFrom the force system shown in Fig. 3 directly, they can be handled readily by numer-

equilibrium equations can be written. Summing ical methods. The procedure used here is tothe moments about the center of mass begin a step-by-step solution using the Runge-

Kutta method to predict values of each of theXn coordinates at the end of successive time

-M2 -M~b + _ ) R, + h)- -O intervala. The values for any given time are2 hsubstituted into Eq. (3) to find the time

derivatives.

-1/2 CDra20h(hi + hb)(h6 + :t) (1) After the Runge-Kutta pronedure has givenpoints for four times and the correspondingfirst derivatives have been obtained, Haming'smodified predictor-corrector method is used toadvance through succeeding intervals of time.This method is a stable fourth order integra-

(2) tion procedure that requires approximately halfBH4y - c~g - RL - RR the calculations per step that other methods of

comparable accuracy require. It has the addi-tional advantage that at each step the calcula-

where RL and RR are funecions of B,y,i and ; as tion procedure gives an estimate of the localpreviously explained, truncation error. The procedure is therefore

able to choose and change the size of the timeEqs.(i) and (2) can be transformed into interval based upon a pre-established accuracy

four first order equations by making the fol- criterion. The particular subroutine used islowing substitutions given in Ref.[91.

X Y/h AN EXA1PLE STRUCTURE

In order to demonstrate the application

X 2 = of the procedure, a number of assumptions weremade as to values of the parameters, and aparticular structure was analyzed. As far aspossible the assumptions relating to the prop-erties of the soil media are consistent withthe reported data of Refs.[J1, [2] and 17]. No

X4 h attempt was made to make a detailed design of a4 g structure, nor was an optimum configuration

sought.

These equations can then be arranged in the It is assumed that a steel structure Is toform be placed on the level ocean floor at a depth

of 6,000 feet. The structure resemb as that

f shown in Fig. 1, with a radius a of 10 feet, aI 1 (X3) base dimension b of 30 fect, and angle 0 of 60*.

If the spherical hull is designed for the hydro-2 f2 (X 4 ) static pressure at 6,000 feet and an estimate

made for the weight of the legs, footings andinterior loads, the whole structure is assumed

= f3 kXI X2'xy x4 ) to weigh 418,000 pounds in air and 148,000pounds In the water. For symmetry, it isassumed that the structure has four legs and

4 4four footings, with each footing about 12 feetsquare and having a bearing pressure of about1.8 psi when resting in the static position.

where b and ;b are the acneleration and valo-whee x an x ar th acelraton nd .Il- The added mass of the water Is assumed to

city of the support medium, and g is the accel- b e ate mass of the voe of aterration of gravity. be one-half the mass of the volume of waterdisplaced by the spherical hull. The drag

37

coefficient is assumed to be unity. The mass numerically equal to one-tenth the slope k ofmoment of inertia of the structure about a the static resistance. This static slope washorizontal diameter of the spherical hull was assumed to be 6,000 lb/in, for both types (b)assumed to be 2/3 Ma2 with no added mass effect and (c); and the coefficient c was thereforewhen the sphere rotates. taken as 600 lb-sec/in.2 .

Three types of support media were con- The above assumptions imply that the struc-sidered. The first was an almost rigid rack- ture placed on type (a) material will not settle.like surface that has the static resistance When the seismic loading occurs, the footingsfunction shown in Fig. 4(a). The second was a will tend to rock or slide, but will not depress

the supporting medium.

In type (c) material an initial settlement

of

(Ws4)(l/K) - """'oo b" .n Z 6 in., \ 4 A 6,000 Wi/n.

(a) Rigid bottom material was assumed under each footing. In type (c)

material an initial settlemint of 6 inches wasassumed due to elastic distortion, and an addi-tional settlement of 18 inc'ies was assumed due

r barltin to consolidation.uliabearinh

CaacitY Type (a) material was assumed to have un-{ limited bearing strength. Types (b) and (c)were assumed to have an ultimate bearingstrength three times the static bearing pres-

(b) Soft eiato-plastic bottom sure. The pullout strength of type (c) mater-ial then becomes the force necessary to liftthe footin a distance equal to the consolida-tion settlement against a resistance of 6,000lb/in, plus the addition velocity dependentviscous force.

ultimatebearin i rle structure-soil-water system is thencapacity assumed to be excited by a horizontal motion

of the base resembling a strong earthquake onpullout land. This obviously will not be the samefor, motion as occurs in the bedrock below the sup-

porting material, except perhaps for materialtype (a); but indications are that the layersof sediment will have a small effect on the

(c) Soft llterir exhibiting frequency component that affect this structurethe most. Using the base motion as though it

Fig. 4 - Static resistance functions were applied to the base of the structure isassumed for seafloor material therefore not considered unconservative; and

of course one is not sure anyway of the natureof the movement in the bedrock. It is pos-

soft material that deformed elasticially under sible to model the effects of the sedimentthe dead load of the structure but did not con- layers on the motion passing through thcm, butsolidate. When loaded further it followed the this could be done if more specific informationpattern shown in Fig. 4(b). The third type were available about a particular structuraldeformed initially under the dead load both by site.distortion and consolidation. It follows thepattern of Fig. 4(c). An additional property The input used in this example is an arti-of type (c) material is that it offers a ten- ficial accelerogram, taken from Ref.[10], thatsile type resistance to having the footing lift is believed to contain all the characteristicsout. of a very strong earthquake, stronger than that

for which recorded data exist. The earthquakeThe type (a) bottom material does not Is of 120 seconds total duration and is designed

change with rate of loading so its resistance to represent an upper bound for the grounddoes not depend upon A. Types (b) and (c) are motions to be expected in the vicinity of theassumed to be velocity dependent and to have a causative fault during an earthquake of magni-resistance function of the form R - kA + cLA. tude 8 or greater.The coefficient c was asaumed to have a value

38

Ref.[i0] followed the following procedure. having the same support materlal, did not liftAn approximation to white noise was passed off at all. It instead moved horizontally withthrough a filter to give the process the desired the ground.frequency content as determined by the powerspectral density. The resulting stationary In Fig. 6(b) a departure from 6 - 0 doesGaussian process was then given the desired non- not necessarily mean that lift-off has occurredstationary properties by multiplying by a suit- because the rotation may be due to verticalably chosen envelope. The record was then movement of the footings. Lift-off did occur,processed and corrected to filter the spurious however, for this case on both sides. This canlong period componets. Response spectra were be seen from Figs. 7(a)and 7(b), which are plotsthen calculated and compared to those of re- of the force exerted by the left and right foot-corded motions. Undesirable frequency compo- ings as a function of time. It is interestingnents were removed by filtering again. Finally to note that the reaction force can momentarilythe accelerograms were scaled to the appropriate exceed the static bearing capacity becLuse ofintensivities of shaking. Fig. 5 shows the the dynamic effect. Comparing the final por-resulting simulated records, which are used as tions of the curves in Fig. 6(b) and Fig. 7the input for this structure. shows that the structure is settling into a

i" h1k& .6 Ig h"Iip

10

IN''t-

."

.M 25 A 5M 4 L. . . .0.j W

,, tx, V

Fig. 5 - Assumed base excitation

RESULTS

Fig. 6 shows the displacement xr of the canted position even though the foundationcenter of mass of the structure relative to ith reactions are fairly balanced.base and the angle of rotation in degrees, bothas a function of time for the first 80 or 100 In Fig. 7(c) tie response is less becauseseconds of the earthquake. Fig. b(a) As the of the resistance to uplift of the footings.response if tie supporting material is type (a), In this particular case, none of the footingsFig. 6(b) is for type (b), and Fig. 6(c) is for broke loose to lift off.type (c).

In computing the response the assumptionIn Fig. 6(a) any departure from 0 - 0 means was made that the structure was restrained from

a footing has lifted off. The maximum lift at horizontal sliding. The- results in Figs. 6 and0 - 7* is about 3-1/2 feet. The same structure 7 do not include any sliding relative to theresponding in air to the same excitation, and support material. The total horizontal

39

2.0

2.0

I II Iq I I I I I I I

0 0 20 to w0 of02 2 W ~ I

6(b) tYr- 9b)

1.0

2.0

-2.0L

0 201 0 20 40 50 40 0 70

0 typo Wp

Fig. 6 - Response of sample structure

40

0 20 3 0 50 6 0 4

02.

1.

o¢ lo

,AA hmAAAh AA .^. .^.41OV ' Z

I V I a I I I0 1O 20 O0 Q0 50 GO 70 O 00 s0

t (coo)

2.0

0t

Fig. 7 - Vertical reactions on footings

restraining force required to prevent slid'ng (c) would have a very low scatic shearingwas calculated, however, and the values were as strength, possibly not exceeding 1 psi. It

follows: would appear that types (b) and (c) might de-velop these forces under the footings assumedif all the footings were acting. The structure

Table I on the type (a) materiel would have to developits horizontal restraint principally through

friction between footings and support material.Type of Bottom Maximum FS if The force required would be nearly equal to the

Material no Sliding submerged weight of the structure, so it ap-pears doubtful that the required frictional

(a) 130,000 lb force could develop. Some kind of shear fas-

(b) 111,400 teners might be used to hold the footings(c) 104,600 against shear. These fasteners would of course

affect the tendency toward lifting off, so the

problem would require re-analysis.

These horizontal forces must be resisted Finally, the peak values of absolute ac-

by friction between the footings and the bottom celeration of the center of mass of the sphere

material, and by the dynamic shearing resis- were computed during the response time in eacitance of the material. The horizontal force of the three cases. The peak acceleration was

would probably be shared unequally by the four C.250 g for the structure on type (a) material,

footings if none lifted off; and by perhaps two 0.202 g for type (b), and 0.189 g for type (c).

of them if lift-off occurred. Whether or not The Input acceleration of the base had a peakhorizontal1 resistances of the m.ignitude shown value of about 0.44 g that did not coincidein 7able I could be developed is; ,ot certain, with the peak response-; of the structure.It is certain that materials of type (b) and

41

F I,

CONCLUSIONS [6] K. W. Korb and H. H. Coyle, "Dynamic andStatic Field Tests on a Small Instrumented

The kind of approach and solution des- Pile," Research Report No. 125-2, Texas

cribed here could be extended and applied to Transportation Institute, Texas A&M Univ.,

the analysis of a great number of situations. Feb. 1969.

The method itself is not limited to simplifiedabstractions, and if more accurate information (7] K. Terzaghi and R. H. Peck, Soil Mechanics

were kno,.i about a particular structure and in Engineering Practice, John Wiley & Sons,location, a more realistic analysis could'be 1948.made. The purpose of this paper has been todiscuss some of the problem elements and to [8] I. H. Idriss and H. B. Seed, "Seismic Re-

demonstrate a method of solution. sponse or Horizontal Soil Layers," J. ofSoil Mechanics and Foundation Div., Pro-

Even such a simplified analysis, however, ceedings of the ASCE, Vol. 94, Jul. 1968.

should be useful. It can permit a rapid pre-liminary analysie of any structure being [9] IBM System/360 Scientific Subroutine Pack-planned for the ocean floor. From the analysis age (360A-CM-030 Version III, Programmers

rough design criteria can be formulated for the Ma nual, p. 337, Fourth Ed. 1968.

strengths required in the various structuralcomponents. The possibility of a failure of a [10] P. C. Jennings, G. W. Housner and N. C.footing in bearing or shear might be predicted Tsai, "Simulated Earthquake Motions,"and avoided. In some cases, failure by gross Earthquake Engineering Research Laboratory,movement in overturning or sliding might be California Inst. of Technology, Apr. 1968.indicated if additional restraints were notprovided. The predictions of peak accelera-tions might show the need for shock mountingdelicate equipment. The maximum displacementand final displacement might be useful if pre-cise positioning were a requirement for thestructure.

Continuing the approach described here tostudy structures of varying size and mass oughtto provide considerable additional insight Intothe behavior of ocean floor structures in gen-eral when they are subjected to earthquakeforces.

REFERENCES

[I] W. E. Schmid, "Penetration of Objects Intothe Ocean Bottom," Final Report, ContractN62399-68-C-0044 for Naval Civil Engineer-ing Laboratory, Mar. 1969.

[2] W. D. Liam Finn and P. M. Byrne, "Analysisof Ocean Bottom Sediments," 1971 OffshoreTechnology Conference Paper No. OTC 1471,Apr. 1971.

[3] R. V. Whitman and F. E. Richart, "DesignProcedures for Dynamically Loaded Founda-tions," J. of Soil Mechanics and Founda-tion Div., Proceedings of the ASCE, Vol.93, Nov. 1967.

[4] R. T. Ratay, "Sliding-Rocking Vibration ofa Body on Elastic Medium," J. of SoilMechanics and Foundation Div., Proceedingsof the ASCE, Vol. 97, Jan. 1971.

[5] T. K. Ilsiech, "Foundation Vibrations,"Proceedings of the Institution of CivilEngineers, Vol. 22, May 1962.

42

NOTATION

Symbol Meaning

a radius of spherical hull

b base dimension of structure

c velocity coefficient in soil resistanceg acceleration of gravity (32.2 ft/sec2)

h height of center of mass above base ,f structure

k displacement coefficient In soil resistance

q ultimaate bearing stress of soil

t time

xr motion of center of mass relative to base

xb , xb, xb displacement, velocity, and acceleration of baseexcitation

y, vertical displacement, velocity and accelerationof center mass

CD drag coefficient tFD drag force

F% inertia force

FS composite force acting horizontally through baseof structure'

.q mass of structure

RI,, RR left and right vertical reaction forces on footings

Ws submerged weight of structure

coordinates of transformcd equations

a ratio of weight of structure in water to that in air

0 ratio of effective mass of structure and water to rhemass of structure

y coefficient of MIa2 in expression for mass moment ofinertia

Al, AR vertical displacement of left and right facings

AL, AR vertical velocity of left and right footings

a 0 0 angular displacement, velocity, and acceleration ofrigid structure

p mass denqlty

base angle of structural configuration

43i

I ,% ' --. .

DISCUSSION

Voice: Did you include any kind of damping to Mr. Zudanz (Franklin Institute), The modelaccount for the soil radiation? looked to me like you had a weightless type of found-

ation at each support. 'You did not have any couplingMr.' Hammer: We have run cases where we cou- between different support points. Is that correct?

pled the damping of the soil with cquivalent struc-tural damping to get a new equivalent factor, but inthis study we just assume 5% critical damping for Mr..Hammer: No. There were only soil springseach mode. to ground and not soil springs from support to support.

44

DEVELOPMENT OF A WAVEFORM SYNTHESIS TECHNIQUE -

A SUPPLEMENT TO RESPONSE SPECTRUM AS A

DEFINITION OF SHOCK ENVIRONMENT

Robert C. Yang and Herbert R. SaffellThe Ralph M. Parsons CompanyLos Angeles, California

A procedure is developed for synthesizing a time-historyto describe a transient disturbance such that theresponse spectrum of the disturbance matches a givenspectrum and the amplitude ratios and phase relationshipsof its frequency components remain within assigned ranges.

INTRODUCTION In most analyses and in all tests,therefore, the disturbance must be de-

Analyses and tests to determine the scribed by time-histories. If theresponses of systems exposed to transient strength of the disturbance is boundeddisturbances often employ a response by a spectrum, however, spectra of thespectrum as a means for bounding the synthesized time-histories must closelyseverity of all probable disturbances, approximate the design spectrum.As the response spectrum describes onlypeak responses to the disturbance of Peak responses of any system exceptlinear, undamped oscillators expressed an ,ndamped single degree-of-freedomas a function of their natural frequen- linear system to a motion time-historycles, the spectrum does not define whose spectrum matches the design spec-unique amplitudes and phase angles of trum may differ significantly from thosethe frequency components of the disturb- bounded by the spectrum. The magnitudeance itself. Au infinite number of of these differences will depend both ondisturbances, each with different corrpo- the characteristics of the system and onnent amplitudes and phase angles, can the amplification ratios and phasegenerate peak responses in the linear relationships of the frequency compo-undamped oscillators which are essen- nents of the disturbance. If alltially identical. Indeed, as transient responses boundcd by the spectrum aredisturbances are rarely unique, this the result of disturbances which must begenerality of the response spectrum has considered in design, strict compliancebeen one of the factors which has with the spectrum would require that anprompted its extensive use as a means infinite number of time-histories befor defining shock severity. ccnsidered in analyses and tests, each

with frequency components whose ampli-The response spectrum alone as a tudes and phases were different.

criterion of input motion in analyses ortests of many practical systems, how- Practical considerations, however,ever, may not be sufficient tr. pvoperly limit to a very few the number of time-define system responses. First, as the histories which can be considered inspectrum bounds the peak motions only of analyses or tests, sometimes to only ali:tear, undamned systems, responses of single one. Thus, in selecting a fewsystems with other characteristics may specific time-historiles to represent thebe significantly different. Second, entire family bounded by the spectrum,while in some respects the generality of the characteristics of the system whichthe spectrum may be desirable, in others will be exposed to the motions must beit may be too broad in that other infor- examined and the parameters of the fre-mation concerning the disturbance may be quency components of the time-historiesavailable which might be cmployed to chosen so that, hopefully, criticalreduce the size and perhaps the severity responses of the system will beof the family ci possible disturbances. generated.Third, of course, shock test machinesmust generate motions which are describ- The family of motions bounded by aable in time rither than In frequency. spectrum can be reouued in size if an

45

77 77,~~~ --. 7,;

-;7T 7 X

examination of the basic phenomenon Hwhich resulted in the spectrum indicates W(t) = mA m (t) (2)that limits can be established, even mgrossly, for the ranges of possibleamplif-ication ratios and phase relation- where Am are constant coefficients de-ships. For example, all possible ining the amplitude of the functiondisturbances might be describable as r(t) In synthesizing a time-history

"pulse-like or as "oscillatory" the problem is to select a sequence ofMS implying that the amplification ratios functions, fro(t), (m = l,2,. ..M), with asso-

for the former will be low while for the ciosd fiets m s tht the

latter they will be higher. Or, perhaps, response spectrum of the composite wave-

it may be known that all the energy is fcrm not only matches i. points on a

introduced at essentially the same time, given response spectrum but at inter-

such as might occur ' impact, thus mediate frequencies as well. To match aproviding some insight into the phase spectrum exactly at all frequencies, anrelationships. infinite number of terms would be

required. However, as the responseSynthesis of a time-history of a spectrum to be matched usually involves

realistic disturbance thus involves not some measure of uncertainty, smallonly matching the design response spec- deviations may be accept.ole in fre-trum but also incorporating specified quency regions lying between theamplification ratios and phase relation- selected frequencies wn so long as theyships for each frequency component. do not exceed prescribeC bounds.Response spectra, in nondimensionalform, for pulses, such as a sine, versed An analytical sclution of Equation 1sine, ramp, terminal peak, square, and for the forcing function fo(t) which wouldothers have been treated extensively in reduce the problem to an explicitthe literature, and little problem is solution of a set of M simultaneoususually encountered in synthesizing a algebraic equations for M unknown value08family of such pulses which will satisfy of A. is not possible since an infinitethese three requirements. Where the number of such functions can, be founddisturbance Is more oscillatory, how- whose spectra will closely match theever, the problem of matching closely required spectrum.not only a response spectrum but alsothe amplification ratios and phase Forcing Function. If a uniquerelationships of the original disturbance forcing function is to be defined,is more difficult. therefore, it is necessary to postulate

a function fr(t) such that its amplifi-This paper describes a technique for cation ratio and component frequencies

synthesizing oscillatory waveforms such may be varied to satisfy the equation.that their response spectra closely Further, since the function will be usedmatch a given spectrum and the amplifi- to represent a real transient event, itscation ratio and phase relationship of initial conditions must be zero and Itseach of its frequency components approx- terminal values of both acceleration andimate specified values, velocity must vanish. For some phenom-

ena, displacements must also terminateTHIEORETTCAL FORMULATION at zero.

Princioles. The equation of motion In constructing such a function, con-of a simple linear oscillator subjected sider first the case where all frequencyto base excitation is: components are in phase. EHquation 3

describes one forcing finction whichd2 Y satisfies all specified conditiors when-4t y =

2W(t) (1) ever N, the number of half-cycleb, is

any odd integer except unity.where y is the absolute acceleration ofthe mass, w. Is the natural frequency of fn(t) = sin 2nb t sin PrNmb t 0 < t < Tthe system, t Is time, and W(t) is the

( )

base acceleration. 'rThe response spec- = 0 t > 'I'mtrum of the excitation function W(t) Isa plot of the maximum responses of the The frequency of each component ofsystem defined by Equation 1, as a con- the forcing function isN!mb and the fre-

tinuous function 01 w,. quency of its envelope 1, , where bm isexpressed In liz. Then Tm , the duration

w(t) can be expressed as a linear of the forcing function, is T.1 = I2b m'combination of a;celeratton functions This forcing function where N = 5 isr(t), such that shown in Figure 1.

46

Solution of Equations. The solutionof Equation I (Reference 1), substituting A --

£ Equations 2 and 3, is

M

whereFigure 1. Selected Acceleration

Forcing Function, N - 5

c ;2 -cos 21nt cos 2 n(l

- )b t ]

C(N )22 -O;

co 2Z t cos 2nnt- Con [ n (t - (T)

J+(iN 2b2 - 2 (T M t)2(5)

= 24 , and H is the leaviside Equation 5 is determinate for allfunction. The first term of Equation 5 values of' t, , and bm except whendefines the principal responses of the bm= ;n/(l - Nr)"and b= n/(i + rm). As bmoscillator to the forcing function while approaches either of these values, how-the second term defines its residual ever, the limit of Qm can be obtained byresponses. applying L'Ilospital's rule. Thus,(Equation 5 may be rearranged

N -1)2 os 2nt - cos 2n(--t [[+ -[ao M-1n I~~~ It sin tn

_ N4 .i)2 + 6 - .+.)2(. ( - t)(

+ Nl(U-.1),, [1+' 6(1m-1)] f6t (N -1)2cos 2nr t + 2 sin 2,, it] (+ M 14[2N + 6(Nm+l)2(Nm -)] r 1 m

where 6 = Wnl.N5 - b) and 161<<i, and

(11M+l)2 1cos 2 rnt - cos 2 7, -+ 16( m+a ) 1 nt]

lJ-211+4(11_1) 2 (Ifm+1)

- ,-5 ["(i+i)t cos 27nTt + sin 2r.r,,t! if (' - t) (7)

+ 1 (N +1) n[1+26 (u1 +1)j [6,(1m+1) c~s 2Ct + ? sn 2ntj

+ f ~~1-21; +6 2Ift T

where 6 n and j6 <«1.

47

='

Amplitude Coefficients. From the Thus, at each iteration step, theresponse spectrum to be matched, M variation of Am is a single valuednumbers of Yk maximum responses corre- function of Yk and the process converges.sponding to frequencies 1 (k - 1,2,...M) To generate large diagonal elements incan be selected. If the response spec- [Qkm] , the value of b. for a given Nmtrum is to be matched closely at must be selected so as to produce a nearintermediate frequencies, the choice of resonant response; that is, the maximumthe number and relative 'values of the response at a selected system frequencyfrequencies is not completely arbitrary should be dominated by one frequencynor is the selection of component fre- component of the forcing function andquencies bm. Procedures for establishing none of the other frequency componentsthese parameters are discussed later. should contribute substantially to it.

Assume that values have been assigned Amplification Ratio. The amplifi-to Yk, Zk, and b, and that a trial set of cation ratio of each individual frequencyamplitude coefficients Al, A2,,... based component of the waveform can be deter-on the desired amplification ratio, have mined from its normalized responsebeen selected. Either Equation 5, 6, or spectrum (Figure 2). Maximax responses7 may be searched numerically for the are indicated by the solid lines whilemaximum response occurring during the the residual responses are shown by thetime period o Z t V (Tm + 1/ 2lTk) and for dotted and dashed lines.the time tk at which the maximum responseoccurred.

A set of matrix equations can be 00- I N. -written - L; _TII]I

6.[-MINM SECIRA - V " " 1

= Q X k,m =1,2,3 ... 1 (8 RSIDLAI. SPCIU.--9---.URk Lkm] I'm 4.0

where k is the maximum value of AQ,, 3 z.0 : -*corresponding to Zk , and Qkm is evaluated /at t=tk.. - ... t " -I

Usually, the maximum response 9k at h --frequency k calculated from the first N1trial values of Am will not coincide .6 - .with the maximum responses indicated ongiven spectrum at the corresponding -frequency. A new set of Am values can 11 . II-i ...... 1:0then be calculated as . 4

jAm) = k' j k,m = 1,2,3 ..... M (9)-

Using the modified coefficients, a .01 .0? .0 .6 .0 .1 .Z .4 .6 .8 1.0new set of values for 5k and t k can be ,oIZED F0(%,C RATO Rb./

determined. This process may then berepeated until the calculated maximumresponses are approximately equal to the Figure 2. Normalized Maximax andresponse values indicated on the response Residual Response Spectraspectrum; or lYk- 50 < 9 1 where S is an of the Selected Waveformarbitrarily small number. Component for N = 3, 5,

7, and 9The iteration procedure employed to

obtain coefficients Am is not necessarily It may be noted from Figure 2 thatconvergent for any value of waveform the maximum normalized response amplitudecomponent frequency bm. From Equation 8 ratios for components where N is 3, 5,it may be noted that if the diagonal 7, or 9 are very nearly equal to theterms of the square matrix (Qkm] are value of 11 for that component. Where Iilarge compared to tLe off diagonal trms is equal to 3 or 5, the value of theof the same rcw, the diagonal terms of normalized frequency parameter bm/; isthe inverse matrix [Qkm] - ' will also be equal to 0.28 and 0.19 respectively.relatively large. it each iteration For N = 7 andN = 9, the normalized fre-step the new value of coefficient Am of quency at which maximum response occurseach row is influenced principally by is even closer to 1/lj. Thus, if thethe required maximum response value Yk forcing function frequency Nmbm iswhere In and k have identical index selected to correspond to a proper rationumbers and correspond to one system of the selected system frequency wn thefrecuency k on the given spectrum. amplification ratio at the frequency wn

48

in equal approximately to the selected To demonstrate the method considernumber of half-cycle oscillations the trapezoidal spectrum shown in FigureN. 3 and the three collinear points which

are to be matched. The problem is toselect the three frequencies, ;1, ;2,

As an example, assume that it is and ;3 such that both the solution willdesired to match a given spectrum at a converge and, In the region of the spec-frequency of 10 Hz with a single fre- trum between these frequencies, thequency component with an amplification spectrum of the synthesized wave willratio of 7. The selected component lie close to the line which includes theshould have N n 7 half-cycle oscillations three points.at a frequency of bm a 10/7 - 1.43 Hz. How-ever, if the forcing function is toinclude components at other frequencies,convergence of the iteration procedure 0can be assured only so long as the 0response of any single frequency compo-nent is not influenced substantially by C-3those due to the remaining components. 0Note also, that the amplification ratio 2of a single component can only be an odd tinteger and that the minimum amplifica- 0tion ratio is 3 in order to satisfy the 0requirement that all motions vanish at LOG FREQUENCYt = Tm.

Selection of Frequencies. If the Figure 3. Three Selected Match Pointsiteration procedure converges, the spec- on a Trapezoidal Responsetrum of a waveform synthesized as Spectrumdescribed above will match M points onthe given spectrum and the specified In determining the frequencies, twoamplification ratios will be approximated adjacent match points are considered atat these points. The problem which a time. In the region between two match*remains is that of selecting frequencies points, the spectrum is influenced morewk such that they are sufficiently by the component at the higher frequency.separated to ensure convergence of the Thus, in the procedure, if the componentssolution but not separated so far that for the two match points do not have theat intermediate frequencies a close same value for N, the value correspondingmatch between the two spectra cannot be to the higher frequency component shouldobtained. be assumed to be applicable to both

points.The complexity of the equations

describing the response motions precludes If the match points lie on a line ofthe use of a closed form solution to constant relative displacement, theidentify optimum match points. However, stipulation can be madea semi-empirical approach has beendeveloped which, although perhaps not Yk yielding optimum values, has been found -to give good results and is simple to k-1 -I)apply. While if they lie in a line of

Frequently a design response spec- constant pseudo-velocitytrum, when plotted on four coordinateoaper (such as shown later in Figure II) Yk Wkis trapezoidal with a low-frequency Y-=-'1range of constant relative displacement,a mid-frequency range of constant pseudo-velocity, and a high-frequency region of If they lie in a line of constantconstant absolute acceleration. While acceleration, of course Yk =Yk-i'spectra of few, if any, physical phenom-ena are of this exact shape, spectra of Using these relationships, a largemany occurrences of a phenomenon are number of cases were calculated in whichfrequently bounded in this manner. the separation ratio a, where a n k]Zk.1,While this spectrum shape is not and the parameter 11, were varied.essential to the technique for selecting Optimum separation ratios for differentapnropriate frequencies, it does sim- values of N arc shown In Table 1.pilfy the selection.

49

L

TABLE 1 (Reference 1) Dr (2Nmm) td, (M+1), M

OPTIMUM SYSTEM FREQUENCY RATIOS FORDIFFERENT VALUES OF N Note than when Dr ; N, the time-

.. histories of the two frequency componentsN 3 5 7 9 11 13 occur consecutively. For match point

frequency ratios shown in Table 1, aWi c1.35cose correlation with the desigh spec-

S=1 1.35 1.6 1.3 trum in the region between thesek_1 frequencies cannot be achieved where the

time delay ratio exceeds N.

Thus, in the example, if II = 5, 12 = 7,and N13 = 5, and assuming , = 1.0 Hz, ;2 Ranges of time delay ratios forshould be 1.6 liz and w3 = 1.6 x 1.35 = 2.16 values of N from 3 thru 13 which willHz. minimize deviations from the design

spectrum in the region between matchPhasing of Waveform Comnonents. point frequencies are shown in Table 2.

Phasing of the waveform components can These ranges were calculated for threebe represented by specifying a delay in frequency components whose frequencythe starting time of each waveform com- ratios wer, ai indicated in Table .1,ponent. Mathematically, the waveform and whose values of N were equal. Forwith phased components can be written other frequency ratios and combinationssimilar to Equation 3. Thus of N, of course, other ranges of time

delays might yield equally acceptableN correlations with the design spectrum.

m=1 TABLE 2

fm= 0 0 < t < tdm RANGE OF ACCEPTABLE TIME DELAY RATIOS= sin 2i, bm(t-td) sin 2, Hmb(t-t) (10) FOR DIFFERENT VALUES OF N

RANGES OF TIME DELAY RATIOS, Dr

tdm i t <tdm + Tm

where td5 Is tne time of inititation of '.--.3 '., .the mth waveform component from time ". .zero.

Equations b thru 7 remain valid evenwhen the time delay is included in then. MWAVSYN Computer Program. A computerThus, the time delay term will have program, designated MWAVSYN, was devel-

. little effect on the responses at the oped to solve Equations 5, 6, and 7 forspectrum match points. In the range amplitudes Am and to perform the iter-between the match points, however, the ations necessary to ensure that theeffect of the time delay can be much response spectrum of the composite time-more pronounced and match points history match the design spectrum withinselected to minimize error in the inter- an assigned tolerance 5.mediate ranre are no longer valid forall time delays. Input data for the program consists

of a definition of the required trape-Usln' the searatton rrenuencies zoidal response spectrum, amplification

ratios shown in Table ], responses at ratios and time delays as functions ofthe intermediate frequencies were calcii- frequency, and the accuracy tolerancelated for various time delays of any imposed on the spectrum of the synthe-three adjacent frequency components and sized waveform. Using the optimum systemthe ranves of time delays aere identified ratios given in Table 1, the programin which deviation was found to be a selects the system frequency matchminimum. Tf tdm is the time delay of points on the spectrum. An Interpolationthe mth frequency component measured subroutine then establishes the requiredfrom time zero, the time delay between response amplitude, amplification ratio,the (M +l)th and the Mth components is and time delay for each frequency compo-td, (4+1), 1. The time delay ratio Dr Is nert. Amplification ratios are reduced,leefnqd as the ratio or the time delay to Integers which are nearest to theof two idjacent component: to the period odd-numbered half-cycle oscillations.or a hile-cvcle or the comoonent at Based on the haif-cycle oscillations atfrequency NMbm. ''hujs the selected system frequency points,

50

the time delays are forced to stay with- -

in the rangps shown in Table 2. Aminimum accuracy tolerance of 15 percenthas usually been found to be easily,achievable.

Output data consists of time-motionhistories of the composite waveformexpressed in terms of acceleration, |velocity, and displacement. Time-motionhistories canobe punched on cards and/orplotted by a CALCOMP machine. A detaileddescription of the program is presentedin Reference 1.

EXAMPLE

Dynamic analyses of structures sub-jected to earthquake or nuclear weaponseffects involve the calculation of time-motion histories at many interiorlocations. Owing to the many assumptions Figure 4. Comparison of the Verticalinvolved in these analyses, such as the Spectrum of a Typicaldetail nature of the external loads, the Structure and the Spectrumsoil and site properties, and the Generated by the Synthesizeddynamic characteristics of the structures Waveformthemselves, responses calculated at anygiven location can be regarded as onlyone of a large number of possibleresoonses. To account for possibledifferences in the details of the -

responses due to possible variations in - .the parameters, the shock environmentwithin various zones of structures arefrequently bounded by spectra envelopingthe spectra of the calculated responsesof all noints lying within that zone.

-- l eQUenD AIR SPECTRUM

For example, an envelope of spectra I ( Iof predicted vertical responses of a -- SNTHESITZo WAVCFOM

typical structure is shown in Figure 1; AIR CT UM I I

by the dotted lines. In the design of 01 05 ,o so 100 200

shock isolation s'ystens supported by the ,RIQU1NCV. HZ

third floor, time-histories of thedisturbance were required. As no single Figure 5. Ccmparison of the Requiredtime-history of the predicted responses Amplification Ratio Spectrumhad a snectrum which matched the and the Amplification Ratioenvelope, and as it was not feasible to Spectrum of the Synthesizedinvestigate the behavior of the shock Waveformisolation system to all time-historieswhose spectra were included in theenvelope, it was desirable to synthesizea time-history which would incorporate the low-frequency responses of the"worst" conditions, structure were the first to be energized

by their exposure to an outrunningBased on a consideration of the fre- ground motion, that as the airblast

quencies of the modes of tht shock pressure wave approached closer to theisolation system, it was concluded that structure higher-frequency modes wereIt would be necessary to match the energized, and that the arrival airblastdesign snectrum only within the range of pressure wave at the structure wasfrequencies from 0.1 to 20 Hz. Time- accompanied by the highest-frequencyhistories of the calculated structural responses.resnonses were then filtered within thisfrequency bandwidth to determine their Input requirements based on theseamnlilication ratios (Figure 5). Phase estimates and the calculated amplituder' lationshIrn were estimated f'rom an coefficients of the synthesized waveforn

examination of the nredicted time- are summarized in Table 3. The responsehistories a' the loading rhenc:,ena spectrum, generated from the synthesizedwhich nroduc Pd them. It was noted that waveform is shown by the solid line in

51

Figure 4. As noted, the accuracy iswithin +15 percent at frequencies of . ..-. -lower than 0.3 Hz and better than +10 _percent at higher frequencies. Therequired amplification ratio spectrumand the amolification ratio spectrum of - - _,the synthesized waveform are comparedin Figure 5. Acceleration, velocity, 1- 4t i I -.i

and displacement time-histories areshown, in Figures 6 thru 8. 47

SUMMARY

A technique has been developed which -d dsimplifies the synthesis of an oscil- .1i illatory waveform incorporating specified ramplification ratios and time delays and - Tmatching a given response spectrum. The . . . ... .nature of the basic synthesized waveformlimits the frequency components whichcan be included in the waveform and Figure 7. Velocity-Time Function ofrequires that their amplification ratios the Synthesized Waveformbe equal to odd integers greater thanunity.

Desnite these restrictions, however,the technique is a significant improve-ment over the older cut-and-try methodsand, in addition, has the importantadvantage of oermitting the systematicvariation of amplification ratio andphase, two parameters of vital importanceto the responses of most practicalsystems. ,,

!..,~ ~ ..... ,. .... .. FT .~- ,--- -IrIT - i ....A- _JaJ. I a. _ , T F...

_i _I j-

~A;WII W_.;4 -M4- .~Figure 6. Acceleration-Time Function of Figure 8. Displacement-Time Function of

the Synthesized Waveform the Synthesized Waveform

52

TABLE 3

SYSTEM FREQUENCIES, COMPONENT FREQUENCIES, TIME DELAYS, ANDCALCULATED AMPLITUDE COEFFICIENTS OF THE SYNTHESIZED WAVEFORM

NUMBER OR SELECTED NUMBER OFFREQUENCY SYSTEM COMPONENT HALF-CYCLE TIME AMPLITUDECOMPONENT FREQUENCY FREQUENCY OSCILLATIONS DELAY COEFFICIENT

Sm1z bm Hz Nm tdm, sec Am in./sec2

1 0.100 0.028 3 0 0.372 0.160 0.023 7 0 0.6143 0.232 0.026 9 0 1.023 0.336 0.037 9 0 1.715 0.1188 0.054 9 0 .11436 0.706 0.078 9 0 7.517 1.025 0.1111 9 0 16.428 1.1187 0.165 9 0 31.909 2.152 0.239 9 0.925 76.70

10 3.120 0.3147 9 1.563 153.11111 11.530 0.503 9 2.005 219.11512 6.110 0.555 11 2.889 23'.0513 7.950 0.612 13 3.707 2116.131l 10.310 0.793 13 4.006 312.3715 13.1130 1.033 13 4.006 280.7016 17.48 1.3115 13 4.006 250.35

REFERENCES

1. R. C. Yang, Modification of theWAVSYN Computer Program, DocumentNo. SAF-U2, The Ralph M. ParsonsCompany, 30 April 1971

I

THE RESPONSE OF AN ISOLATED FLOOR SLAB-RESULTSOF AN EXPERIMENT IN EVENT DIAL PACK (U)

J. M. FerrittoNaval Civil Engineering Laboratory

Port Hueneme, California

This paper outlines a test of a horizontal cylinder covered with an earth berm subjected tothe pressure and drag forces in the 300,psi overpressure region from the detonation of the500-ton high-explosive shot of Event DIAL PACK. The objective of this test was to obtaininformation on the response of an isolated floor slab placed on a soil fill inside the concretecylinder. Data from seventeen channels of active instrumentation, composed of a pressurecell, velocity gages and accelerometers, were recorded. Reduction of the data was made.

INTRODUCTION Shock Isolation

Objective An essential element in the design of protectivestructures is the provision of a reliable shock isolation

The prime objective of this project was to obtain system for personnel and equipment. The levels of shockinformation on the response of an isolated floor slab inside to which a system acting at its designed structural capacitya horizontal cylinder covered with soil and subjected to pres- may be exposed can be very high, with ground accelerationssure and drag forces in the 300-psi overpressure region. The over a hundred g's and ground displacements over two feet.specific objectives were: Very few devices arc capable of withstanding shocks of this

1. Determine the absolute motions of the floor magnitude without serious damage. A more serious limita-

slab caused by the blast loading. tion is the peak shock which an unsupported human beingcan tolerate. Yet the unimpaired functioning of both per.

2. Determine the motions of the slab relative to sonnel and equipment immediately following an attack mustthe cylinder, be preserved if the facility is to fulfill its intended mission.

3. Evaluate the survivability of the structure In the design of shock resistant equipment, the required

including the performance of the retaining strength of the equipment is controllkd by its response to the

wall. shock produced by the dynamic loads on the structure.Expensive conventional shock isolators could increase the

Background cost of the structure to an extent that might be unacceptable.An alternate approach is to isolate the floor slab by "floating"

There is a requirement to store sensitive equipment if) it iii a layer of soil! within the cylinder. The cylinder is a

aboveground hardened shelters. The equipment must be closed structure capable of being constructed economically

shock isolated from the shelter. It is important to limit the and supporting high overpressures with minimum disturbance.

blast induced motions transmitted to the floor slab to reduce The floor slab must be strong enough to carry the equipmentload and resist the longitudinal and torsional motions fromthe shock. The motions of the floor slab are related to the

crete slab on a soil fill inside a horizontal concrete cylinder motions of the surrounding concrete cylinde, transmittedis being considered as a possible means for diminishing the through the soil fill. However, as the cylinder is displacedshock motion. This structure, if satisfactory, will provide downwardh the iefi owevisolated floir slab tdnas tosimple, inexpensive shelters which can be rapidly constructed reduce the peak accelerations felt on the floo.and will eliminate the cost of providing expensive shock iso-lation platforms.

Preceding page blank

$ 55

EXPERIMENTAL PROGRAM Instrumentation

Test Structure Seventeen channels of active instrumentation consisting

of eirht structure velocity gages, eight accelerometers, andThe isolated floor slab test was planned as part of Event one pressure cell were used. The layout of the instrumenta-

DIAL PACK [1], a high explosive field test of 500 tons con- t. ca is shown in Figures 6 and 7. The first letter, A or V,ducted at the Dcfence Research Establishment Suffield indicates an accelerometer or a velocity gage; the second(DRES), Ralston, Alberta, Canada. The test structure, letter, V or H, indicates vertical or horizontal orientation.located at an azimuth of 120 degrees 270 feet from ground The data were conditioned, amplified, and recorded on 32-zero, was at an anticipated side-on pressure range of 300 psi. track tape recorders located in a bunker 2,700 feet fromThe structure location and construction detcils are shown in ground zero. An inter-range instrumentation group timingFigures 1 and 2. It consisted of a 6-foot-inner-diameter 7- system was recorded on one track of each tape recorder andinch-thick horizontal right-circular cylinder aligned later used in the data reduction. Timing from an NCEL-perpendicular to the direction of propagation of the blast designed timing generator was also recorded on one trackwave. The invert of the cylinder was 1-1/2 feet below the of all recorders. A detonation zero pulse provided by DRESnatural grade. The cylinder was covered with 2-1/2 feet of was recorded as received directly onto the last track of eachcompacted fill forming a tapered earth berm extending along tape recorder to provide a reference for data located on dif-the sides of the cylinder and one end at a 3:1 slope. A 2- ferent tape recorders.foot-thick retaining wall held the other end of the earth The gages were mounted on steel plates cast in or boltedberm in place; a bolt-on steel closure plate was incorporated to the structure. The pressure ceh was installed in a speciallyin the retaining wall to provide access into the cylinder. One designed concrete mount which was cast in the wall with theand one-half feet of compacted soil was placed inside the heat shield flush with the face of the wall. Passive instrumen.cylinider, and a 6-inch concrete slab was cast over the soil. tation consisted of tree orthogonally oriented reed gagesA spring-mass system, Figure 3, was installed 4 feet from mounted on the slab and a scratch gage (Figure 7) mountedthe rear end of the slab to simulate equipment on the slab. between the retaining wall and the cylinder.Figure 4 shows the structure under construction, and Figure5 shows the completed structure.

:= 53' " - 2'-,-

2' 6" - cylindercberm I_\&wallZ

"B' 420" closure

Nt.Se l renoreenomtte ndaing.1.. Section, B..-B u ,-

Figur 1T

A

I..- ' o" --

1''6,_ ........ _ _T_ _

Noe:Stelrenfrcmetomttendaing. Jectionoam

19' 6i8' urY 1.' 6onstrucionnplan

F3 pie cyinde

270ft

ground zero I C L

270 ft

53 ft x 42 ft (berm dimensions)

120-degree azimuth Figure 2. Layout of Project LN322.

1/4.in.-thlck end plates welded to beamsA.. 7"ItF 1/4.ln. end plate"

~~~ 3/4-In. plywoodbled to flanges

~2 2 x4 angle0 2 x 2 x 1/4 angles, equally spaced,

welded to beam flange

Spring-mass system filled with SecionAdbags to a total weight of 4,OU0 lb. Fgr .Srn~asla ytm

Figure 4. Test structure under construction.

597I

!P7. -

figure 5. Completed test structure prior to, test.

groujnd zeroj

AVi and AH V4ad V

VH57 andVH

AAV1 and AH6

Figur 6.Lgaigefsei ags

AV3 and 58

~AV1 and AH2

AH 4

Figure 7. Test gages.

The Pressure gage Was Set in the outside (aec of the RESULTS AND DISCUSSIONretaining wall to measure thc fre.icld sidc-on overpressure.Horizontal and vertical accclcromcters wcrc set on thle cvI Observed Damagcindcr ncar each cnd to measure thc horizontal and verticalcomponents of tlic cylinder motion.. The -lab instrumnenta- Thc detonation of the 500-ton highi-explosivc chiargetion was placed in tlac same cross-sectional planes as thc occurred on 23 July 1970. At D+lI hour project personnelcylinder instrumentation to evaluate the relative motions returned to 6r.L LN322 site. The fireball Wa blackened theof the slab with respect to the cylindei. A vcrtici'l and two retaining wall and the berm with an layer of carbon dust.horizontal acclerMCM' etes re set at the center of the floor Twelve inches of crater ejccta were deposited in front ofslab to record floor accelerations in three orthogonal direc the ground zero end of the retaining wall. This level wastions. The two vertical velocity gages at eath end were reduced to about 2 inches in) front of the closure plate.installed near opposite edges of the floc. slab to record IThe maximnumn size of the ejrcta was estimated to be 8rotational motions of the floor slab. The three orthogonal inches. Approximately 18 inches of the berm on thereed gages were used to provide records of horizontal and groueMd Meo side were compressed and/or blown tway.vertical shock spectra. The scratchi gage was used to recordrelative motion between the retaining wall and the cylinder.

59

A pattern of major diagonal cracks was noted on the Postshot measurements revealed that the slab rotatedwall. S,,vceal of these cracks were observed on the top sur- clockwise approximately 3 degrees. The permanent relativeface of the v'all and extended completely through the wall. horizontal translation of the slab was 1/2 inch away fromThe wall had about 1.degree permanent rotation into the the ground zero side of the cylinder. The center of the slabberm. Some cracks in the concrete were noted around the remained at the same elevation relative to the cylinder. Theclosure plate. At D+I day the closure plate was removed permancat horizontal translation of the cylinder with respect

revealing additional cracking in the wall. These cracks to the retaining wall was 3-5/16 inches away from groundappeared to go completely through the wall. zero.

The concrete cylinder sections suffered significantdam. 6v. A horizontal line of compressive failure was located Reed Gagesat about 30 dcgrees from the top of the cylinder on the lec-ward side of all the sections. A region of major apalling and The plates from the recd gages were removed, andtensile cracking was noted at about 90 degrees from the com- measurements of the traces were taken and converted topression zone (60 degrees from the top of the cylinder) on displacements. These values, plotted in Figure 9, give thethe windward side of all of the sections (Figure 8). The first shock frequency spectra of the slab.three sections had been unintentiorally oriented so that the The reed gage consists ot'a number of masses on a rigidsplice in the circular reinforcing steel was located in the bs,. The response of equipment mounted on the slab to aregion of the tension failure. Th6 splice appeared to have given shock can be determined fr3m the measured responsefailed, and a section of the pipe was pushed inward approx- of a reed having the same frequency and damping as theimately 4 inches along this line. The floor slab was littered equipment to the shock. Thus, the shock spectra diagramwith spalled concrete and sand. can be used to determine the shock isolation requirements

of the equipment.

1 , -Active Instrumentation

All of the active channels of instrumentation functionedI satisfactorily. The analog data tape was returned to NCEL

where the data were converted to digital form using theS, , NCEL Analog-to-Digital Converter. Accelerations were sam.

(: .... pled at increments of 0.1 msec for 200 resec, and velocities• -. were sampled at increments of I insec for 2,000 msec. No

filtering was used. The data were automatically plotted usinga CDC 6600 computer.

Figure 10 is a plot of the side-on pressure data recordedat the midpoint of the retaining wall above the closure plate.' The peak pressure recorded wai 317 psi. The shock wave

4 had an arrival time of 32 msec and a total duration of thepositive phase of 103 nsec. The actual duration of thepositive phase was longer than the expected value of 60i nmsec.

AIn several radial horizontal velocity plots, the velocityat late time did not retun to 7ero, indicating permanenttilting had occurred. Visual measurement made after theshot indicated a permanent tilting of the slab of about 3degrees clockwise when viewed from the retaining walllooking into the structure. Correction factors were sub-tracted in an attempt to remove the tilting effect. Theaccelerations and velocities were integrated. A summaryof the peak values is presented iu Table I. Positive values

I.s indicate motions down. away from ground zero. and to the

Figure 8. Postshot view of interior of structure. right looking at ground zero.

60

11000Shock Spectra

0~~~ HorzotalRaia

9W

ElHorizontal TransverseSVertical

011 10 100100-4

Frequency (cps)

Figure 9. Shock spectra of floor slab.

350-

C

-70'Time (msec)

Figure 10. Pressure-tirre curve, gage PSI.

61

Table 1. Peak lnstrumer.tation Values

Peak Peak PeakType of Gage Acceleration (g) Velocity (fps) Displacement (in.)

Instrumentation Negative I Posit iti Nive Positive

Cylinder

Vertical accelerometer AVI 19.6 146.8 2.3 20.1 0.0 8.5

Vertical accelerometer AV7 112.8 173.9 7.6 22.4 2.4 5.6Horizontal accelerometer AH2 90.2 52.6 9.9 5.0 2.3 3.2Horizontal acceleromctar AH8 72.4 86.0 6.3 11.2 0.4 13.9

Floor Slab

Vertical accelerometer AV3 23.7 28.5 4.8 10.7 0.0 4.9Vertical accelerometer AV4 15.5 11.3 1.1 10.5 0.0 5.5Vertical velocity VV1 5.4 11.3 0.0 5.3Vertical velocity VV2 2.0 9.2 0.0 5.4Vertical velocity VV3 1.0 9.1 0.0 4.8Vertical velocity VV4 2.0 7.4 0.0 6.6

Horizontal transverse accelerometcr AH6 3.0 2.2 0.0 .5 0.0 .4

Horizontal transverse velocity VH6 0.7 .9 0.4 .6

Horizontal radial accelerometer AH5 13.5 19.4 0.0 3.8 0.0 2.8

Horizontal radial velocity VH5 1.0 3.4 0.6 3.8

Horizontal radial velocity VH7 1.3 3.9 0.3 4.0

Spring-Mass

Horizontal radial velocity VH8 0.7 3.2

Note: 1. Accelerometer data taken for 200 msec, and velocity gage data taken for 2,000 msec.

2. Positive direction is downward, away from ground zero, and to the right facing ground zero.

Cylinder Displacement

The horizontal and vertical displacements obtained fromidouble integrations of cylinder accelerations were used to -31produce a plot of cylinder movemcnt (Figure 1 I). Although -1 ground zerothere may be some error associated with this procedure, it is -

believed to be accurate enough to relate direction of motionand approximate orders of magnitude of movement. Similar -3 downplots, Figures 12 and 13, were made for the slab movementusing the horizontal and vertical displacements ubtainedfrom the integration of the velocities. T =5

10 lmsec

> -7

! '1

-9Horizontal Displacement (in.)

Figure 11. Displacement of cylinder.

62

-

i -1 1

II I I I ' I I l l I-1 1 3 -1 1 3 5

ground zeroundro

down

0-U

>

-5

Horizontal Displacement (in.) Horizontal Displacement (In.)

Figure 12. Displacement of floor slab, windward side. Figure 13. Displacement of floor slib, leeward side.

Figur:- 11, 12, and 13 indicate tie motion of the inches. The amount of rotation shown is about 3 degrees,structure was initially downward and away from ground confirming the ficld measurement. From Figure 14, thezero, chcn downward and toward ground zero. Subsequent rotation of the slab began at about 125 msec and rotatedmotion at about D+100 msec was upward and away from the full amount at about 200 mscc.ground zero. The motion toward ground zero is quite Generally the data obtained in the first few hundredunusual and unexpected, but is believed to be valid. Pos. milliseconds are very reliable. After this period secondarysible sources of error such as gage rotation, reversed effects may adversely affect the data. The peak values andcalibration, or an erroneous gage were considered and wave shapes in early time (several hundred milliseconds) arefound not to have been present. This effect was obse'ved quite reliable and are relatively the most important. Accel-on four independent sets of instrumentation. Ovaling of crometers are usually more reliable than velocity gages whenthe cylinder was also dismissed as a possible explanation rotation of the gage is suspected. Vcloc.ity gages experiencingbecause the direction of ovaling near the gage location rotation as slight as three degrees may erroneously indicatewas inward rather than outward. The initial downward apparent motion of 100 percent of the anticipated value inmovement was caused by the direct compressive wave. A magnitude. Evidence that this has occurred is noted whenpossible explanation of the horizontal movement toward velocities at late times (2 seconds) fail to return to zero.ground zero is the reaction to the magnitude shear wave Accelerometers are relatively insensitive to rotation and arewhich enveloped the berm producing planes of opposing in.;uenced only by the prc luct of the sine of the angle of *shear forces. These forces gave rise to motion as slippage rotation and the component of acceleration iii the perpcn-occurred. The reverse movement of the structure was dicular direction. Thus, for small rotations the influencecaused by a combination of rebound and direct induced is negligible.ground shock arriving about 100 mscc after detonation. Integrations of acceleration data give reasonable

The time to peak positive accelerations of the cylinder indications of wave shape and velocity. l)isplacenentswas about 12 msec after the arrival of the blast wave- how- from integration of accelerations should be capable of givingever, the time to peak rositive acceleration of the floor slab an order of magnitude of niovemnent and direction. The inte-was about 110 insec after the arrival of the blast wave. The gration of data represents a second level of confidence.magnitude of the floor slab accelerations was substantiallyreduced compared to that of tile cylinder. Once set in Shock Isolationmotion, the movement of the floor slab was independentof the cylinder. The slab appears to have remained station- Usiiig all isolated concrete slab "flo.itinmg' on sand withinary during the initial movements of the cylinder, amid then a horiLontal cylinder reduced the peak a.celcration of 174g'smoved downward until coming to rest on the soil. on the cylinder to about 2 8 g's on the floor slab. The 22-fps

Figure 14 shows the relative vertical displacenient of peak vertical vclocit) of the cylinder was reduced to abouttwo velocity gages located on tl,e slab aid separated by 36 iI fps. Table 2 compares tihe peak motions and shows the

63

Ai

shock isolation of the isolated floor slab. Table 2. Comparison of Peak Motions

Velocity AccelerationCONCLUSIONS(p)(g)(fps) WgS)

The objectives of this project were satisfied. The !

motions of the floor slab were determined, and the surviv- Verticalability of the structure evaluated. All of the instrumentation Cylinder 22 174functioned satisfactorily. Data were obtained to compute Floor slab 1I 28the shock isolation requirements for equipment to be storedin the shelter. The significant findings and conclusions arc: Horizontal

1. The isolated floor slab significantly reduced the Cylinder 11 90transmitted motion. Floor slab 4 19

2. Shock spectra data for designing the required shockisolation system has been| determined.

REFERENCES

1. NCEL Technical Report R-726, "Dynamic Response ofan Isolated Floor Slab-Results of an Experimental Test inEvent DIAL PACK." by J. M. Fcrritto, May 1971, PortHuenene, California.

6-

VV2

~VVll

1.000 2.000

-2

Time (msec)

Figure 14. Rotation of floor slab.

64

~I,

DISCUSSION

Voice: Concerning the measurement on the floor about 4,800 degrees of freedom, so it is quite a largeslab, does this refer to the isolated mass sitting on problem. We are attempting first to look at it struc-top of the springs, or ig the slab sitting on top of the turally, and the time step is made as large as possi-sol? ble keeping the economics in mind in order to be able

to run the problem.Mr. Ferritto: All the measurements were made

on the actual concrete floor slab. The spring mass Vcice: Do you solve 4,000 dynamic degrees ofsystem shown in the slides is simply used to provide freedom?an equivalent equipment on the floor slab.

Mr. Ferritto: Yes, We are running approximate-Voice: Is it correct that you were not trying to ly 300 time increments to approximate about 100 mil-

establish the effect of isolation using a layer of soil? liseconds, I believe.

Mr. Ferritto: Basically that is correct. The Voice: What is the highest frequency of the mod-springs of the mass and the weight of the mass were el you are putting together?selected to provide the frequency that we would ex-pect from a typical piece of equipment placed on the Mr. Ferritto: It is a very complex problem be-slab. It actually represented what we tried to approx- cause the loading is quite complicated. We have aimate in the model, traveling wave. We have a very complex soil-struc-

ture interaction. The extent to which the berm partic-Mr. Keen (Bell Telephone Laboratories).: The ipates in the problem still remains an unknown. This

high accelerations ii your last slide uould indicate is one rescn we are using a plane strain, finite ele-the presence of very high frequency data. You also ment analysis with a quadrilateral element rathermentioned that you planned to use nonlinear finite el- than another type of modeling procedure. The periodement analysis techniques. Would this predict any of of the cylinder by itself in compression is about 2the high frequency phenomena which I assume to be milliseconds. The period of the cylinder in flexure ispresent? about 11 milliseconds.

Mr. Ferritto: To this date we have been running Mr. Zudans (Fraklin Institute): I am interesteda structural analysis primarily interested in the op- in your 4,000 dynamic degrees of freedom. Are youtimization of the structure itself without looking at modeling it as a two-dimensional infinitely long typethe isolation characteristics. We have made several of strip?runs. The time step and other information we haveused in sizing the finite element mesh has been scl- Mr. Ferritto: Yes, it is a plane strain model.lected, basically, to satisfy the structural character- The quadrilateral element that we are using has 12istics. The structure, as we are now looking at it, has degrees of freedom and approximately 400 elements.

65

A SHOCK-ISOLATION SYSTEM FOR

22 FEET OF VERTICAL GROUND MOTION*

E. C. Jackson, A. B. Miller and D.'L. BernreuterLawrence Livermore Laboratory, University of California

Livermore, California

Shock isolation of fragile equipment from severe ground motion induced byunderground nuclear detonations requires special techniques for inexpen-sive, reliable performance. Two shock-mitigation systems that have beenused successfully for the past several years are described. These systemshave allowed equipment to be located closer to the explosive source,resulting in considerable savings in diagnostic cable costs. A-newsystem has been designed for even more severe ground motion. The newdesign, the instrumented testing program, and test results are discussedin this paper.

INTRODUCTION REQUIREMENTS AND ENVIRONMENT

Experimental data from underground nuclear For maximum utilization and mobility, mostdetonations are conditioned and recorded by electronic equipment is housed in truck-trailersensitive electronic equipment. Normally, the vans or portable buildings on skids. Mostexperiment requires the equipment to be located trailers are conventional highway type, but arelatively close to the explosive source. In few have been fabricated for higher shockmany cases, ground motion induced by these requirements (7 g vertical). Loaded weightsnuclear explosions requires shock-isolation vary from less than 10,000 lb to 72,000 lbs forsystems capable of supporting heavy instrumen- trailers and up to 140,000 lb for skidtation vans and isolating them from a very buildings. There is a large variety ofsevere three-dimensional dynamic environment, electronic equipment and accessories. Shock

fragility levels vary from 1/2 g to more thanDuring the past several years we have 50 g. When properly mounted, most equipment

designed, developed, and fielded several falls within our medium fragility range - 6 gdifferent shock-mitigation systems. Two basic vertical.systems have been standardized and are discussedin this paper. These standard designs have The ground motion induced by an undergroundallowed us to reduce signal attenuation and to nuclear detonation varies considerably,reduce very large cable costs by locating depending upon yield of device, geology, andequipment close to the source. Shock-isolation location of interest. The optimum location,costs themselves have also been reduced, and with respect to reliability and overall costsoverall reliability has been increased by this for diagnostic and other portable instrumenta-stat.dardization. tion bunkers located on the surface, regardless

of yield, is safely outside the subsidenceTo meet requirements for even more severe crater, but not more than half the depth of

ground motion, we have designed a shock- burial from surface ground zero. In this areamitigation system to withstand vertical ground the surface usually spalls and follows amotion up to 32 ft/sec, or about 22 ft ballistic path. Figure 1 is a time historydisplacement. The new system is a modification curve of surface ground motion for a largeof one of our standard designs, with the usual event, but the characteristics are typical ofcrushable materials replaced by a columnar the spall region. The vertical slapdownenergy absorber. This energy absorber allows a acceleration pulse is usually followed by alonger stroke without an increase in payload horizontal radial pulse, which can be eitherinitial height. away from or toward surface ground zero.

Horizontal tangential pulses are also occasion-In this paper, we discuss the new design, ally significant.the instrumented testing program, and the test

results. Work performed under the auspices of the

U.S. Atomic Energy Commission.Preceding page blank67

The basic requirement of all shock-

mitigation systems is controlled relativedisplacement and force transmission betweenthe shock input and the package or system tobe isolated. The fundamental classificationof shock and vibration systems is the mannerin which it stores, absorbs, or dissipatesenergy. A great many different materials andmethods are used in shock and vibrationisolation. A complete discussion of allA --,., pare-eters involved in shock mounting items for

. V underground detonations would be quite long;therefore, in this paper we will describe only

'dII 0 h....,l a few methods that we have been using." :,4 In some cases, accurate final position or.1 V alignment of the equipment with respect to the

ground is important and must be consideredin the overall design. However, in most casesposition and alignment are iot important. Thispaper concerns only these cases. Thisvariance allows us to design around the three-

-- ------.....------ %,6 dimensional ground-motion environment bydecoupling the horizontal shock components.Decoupling is achieved by placing the systemon surfaces with very low friction. Verticalaccelerations are isolated by constant forcevs displacement energy absorbers. Ideally,these absorbers should have negligible reboundcharacteristics. With this condition thaconventional analysis is relatively simple

o-.(see Ref. 1). Accurate measurement of th6 pay-load weight and center of gravity and the

"-" appropriate sizing of the energy absorbersminimize the amount of differential verticaldisplacements (tilting) of the system. Whenrequired, excessive horizontal displacementsare controlled by nylon tethers. Energy-

Fig. I Typical time history of ground motion absorber deceleration set values are based onwithin the spall region. equipment fragility levels and a structural

amplification factor of 1.75. This factoraccounts for the elasticity of the shock mount

DESIGN METHODS structures, equipment mounting brackets and thetrailer structure. Dynamic effects of impactvelocity on energy absorber forces are

The overall problem blends together: (1) considered separately.definition of input, i.e., ground shock

parameters; (2) fragility level or shocksensitivity of equipment to the input and We have developed several shock mount1..,ose (i.e., does it record or transmit systems varying in complexity and cost. Thethrough shock arrival time?); and (3) design selection of a system for a given event dependsof a fail-safe shock mount system to modify primarily on the estimated maximum verticalthe shock environment when required. ground motion as shown in Table 1. The energy-

absorber system does not necessarily have aThe reliability of a system is strongly factor of safety greater than unity with these

dependent on a foreknowledge of the input maximum estimated input conditions, except forconditions. Prediction of surface ground the incorporation of fail-safe features. Themotion for our purposes relies heavily on respective nominal ground motion values areempirical data and methods (see Ref. 1). The much less.shock-isolation design for a given event isbased on nominal and maximum ground motionestimates in order to obtain a factor of FOAM AND REUSABLE CRIBBING SYSTEMsafety based on energy. The maximum estimatesare based on maximum credible yield and maximum For many events the estimated maximumscatter of applicable empirical ground-otion ground motion is less than 10 ft/sec and we usedata. Whenever possible, the design is based the foam and reusable cribbing (F&RC) systemonly on peak input parameters since detailed for trailers and vans. It includes permanenttime histories of ground motions are very reusable wood cribbing with an angle iron basedifficult to predict.

68

TABLE IShock-Mitigation System Limits

[Maximum Vertical Ground MotionSystem Velocity, ft/sec Displacement, ft

Foam and reusable cribbing 10 nu 2-1/3F Universal guided column 24 " 12

Full-stroke guided column 32 % 22

tied to the trailer (Fig. 2). This forms asolid extension of the trailer structure to thenormal suspension height and is capable of with-standing the horizontal loads. A simple foamand plywood crush pad is bolted under thecribbing. Styrofoam Is used because for the Umrequired heights and surface areas, it isstrong enough to withstand the horizontal shearforces without additional guide structures,The crush strength of the polystyrene foam isconstant for the first 35% of deflection andincreases gradually to about 140% at 65%.: .'.',, ...deflection. Because very-low-friction surfaces . . ..- ',

are not required, the crush pads sit directly - - .on plywood ground pads.

Fig. 3 Typical installation of UniversalUNIVERSAL GUIDED-COLUMN SYSTEM Guided Column (UGC) system.

The universal guided-column (UGC) systemwas designed for ground motions exceeding theF&RC system capabilities and up to a maximum The bottom af the column is attached to a metalof 24 ft/sec or about 12-ft displacement. A disk by means of a pivot joint. The crushabletypical installation is shown in Fig. 3. It is material sits between the metal disk and thean energy-absorption system that can use almost beam as shown in Fig. 4. All horizontal loadsany crushable material. Commercially available are transferred to the metal disk and thereforepolystyrene rigid foam has been the most into the metal guide column. Horizontal loadsinexpensive reliable material. The vertically into the disk are controlled by antifrictioncontrolled crush load is transferred to the surface pads made of properly sized Teflon,trailer via a cross beam. Each end of this grease, and acrylic. The pad design is basedbeam contains a guide in which a column is on many laboratory friction tests and fieldinserted. This column extends through Jhe experience. Measurements of the actual full-crushable material down to the surface pad. size system have consistently indicated a 0.06

coefficient of friction. The horizontal forceis transferred to the beam by means of themetal column, which imposes a twisting momenton the beam. Because the beam is a box-typestructure, it can withstand the torsionalloads. The magnitude of this twisting momentis a function of the coefficient of friction

/for the surface pad dnd crushable material-- height and crush load.

c1 We have found that the maximum horizontal0 displacements occur after slapdown and after

the crushable material has compressed;Fig. 2 FARC shock-mitigation system: (A) therefore, the surface pad Is designed to have

replaceable foam pad bolted to very low friction for a limited horizontalcribbing, (B) turnbuckle tie, (C) displacement while the structure is mountedwood cribbing with plywood scabbing, high and then after slap-down the column(D) angle iron frame, (E) reusable moment arm is smaller and the system iscribbing assenbly. allowed to slide onto higher friction

69

A primary difference in the system is the typeand location of the ene.rgy absorber (see Fig.5). All remaining components function the sameas the UGC system.

BThe constant-force columnar energy

absorber is called a TOR-SHOK and is manufac-Momrent due tohrnl f o tured by A. R. A. Products, Inc. A single

stage of this device consists of two concentrictubes with a coil of ductile wire forcedbetween them. The interference fit between

oC the wire and the tubes is sufficient to preventsliding and to force the wires to rotate.Rotation of the wires is similar to rotating"the ring of a torus inside out. The resultingtensile and compressive strains are in the

F plastic range. Contracting or extending the

I-Veitical force

Fig. 4 UGC characteristics: (A) column, (B)crossbeam, (C) crushable material,(D) pivot joint, (E) metal disc, (F)Teflon, (G) anti-friction surface pad,exploded for clarity.

SECT A-Asurfaces. Nylon rope tie downs are also usedto prevent excessive horizontal displacementand to absorb some of the energy. Colun

This system has been used reliably on morethan twenty occasions with actual ground Rubbermotions exceeding 12 ft,'sec or 36 in. displace- isolatorment. Vertical energy-absorption strokes havereached 40 in. and horizontal displacements upto 8 ft have occurred. Energy

absorber

FULL-STROKE, GUIDED-COLUMN SYSTEM

A new shock-mounting system has recentlybeen designed to meet requirements for verysevere vertical ground motion up to 32 ft/secor about 22-ft displacement. This design is amodification of the UGC system, with the crushmaterials replaced by a full-stroke, constant-force proprietary energy absorber. For crushmaterials (honeycomb, Styrofoam, etc.) theenergy-absorption stroke relative to the Crossbeam

original height is designed to be about 35% fornominal design conditions and/or 65% formaximumi credible conditions, depending upon pua ewhich case prevails. The crush materialsbottom out at 65 to 75% deflection. The full-stroke system allows tl! original height to be Crushablereduced by one third, and therefore the materialstructural requirements for some of the basiccomponents are also reduced by one third.

The structures for the full-stroke, / / -§//'-guided-column (FSGC) system are similar to theUGC system except that they are stronger. The Fig. 5 Characteristics of FSGC system.

70

tubes apart axially forces the wires to rotate ratings and cluster arrays of energy absorbersmany times. The repeated cyclic plastic were tested. At some locations the clustersstraining results in almost constant energy were arranged to impart very high eccentricabsorption per cycle of rotation (or inch of loadings into the columns. The FSGC system setlinear stroke) until eventual fatigue failure. up before drop testing is shown in Fig. 6.

( Before incorporating TOR-SHOKs into a Test instrumentation consisted of acceler-i

shock-mounting system we purchased some for ometers and high-speed movies. Thirteenstatic and dynamic tests of individual units accelerometers were placed on the top of thein both the extension and contraction columns, on the FSGC beams and on the equipmentdirections. During the static tests loads were inside the trailer. All channels were recordeduniform and increased breakaway loads occurred on magnetic tape, with seven selected channelsonly a few times. Impact acceleration pulses on m scietigraph , f i c ev aluatio n(three times the average deceleration values)were measured on the payload in about half ofthe extension and compression drop tests. The The drop tests are summarized in Table 2.axial component of the lateral resonant Tilting occurred during the first three tests,vibration was quite apparent during the but averaged deceleration strokes agreed withcompression drop tests. In one case, this was test parameters. An erroneous center ofa function of two stages and not the overall gravity location, which was discovered beforelength of the column. In all tests drop height, testing began, caused the tilting. After thestroke, and the average deceleration valuesagreed very well. There were no changes in loadduring the tests, even though one unitexperienced five complete strokes.

In order to use the full-stroke capabilityof the guided-column system, we attach theenergy absorber between the top of the columnand the beam, as shown in Fig. 5. We are usingthree-stage energy absorbers with a compressedlength of 58.5 in. and 8-ft stroke capabilitywith capacities up to 12,000 lb. Each columncan accommodate up to four energy absorberswith a total deceleration load capacity of48,000 lb. Rubber isolators are used inmounting to insure that no bending moments aretransferred into the energy absorber.

DROP TESTS

A series of full-scale drop tests using a30,000 lb trailer was conducted. Various load

Fig. 6 Prototype FSGC system before testing.

TABLE 2Drop-Test Results - Average Values

Deceleration

Drop Velocity Desi!n Conditions ActualTest Height at Impact Force Stroke StrokeNo. (in.) (ft/sec) (g) (in.) (in.)

1 21.5 10.7 2.7 7.9 7.62

2 38 t4.3 2.7 14 13.5

3 74 19.9 2.8 26.4 26.94 12 8.0 3.0 4 4

5 180 31.0 3.17 57 55 - left side77 -right side

t Gravity force not included LThree energy absorbers failed and system bottoned oqit on right side.

71

)-

third test the energy absorbers were reset(compressed) and redistributed in accordancewith the correct center of gravity location.Figure 7 shows the trailer after the fourthtest during preparation forthe 15-ft drop ofthe fifth test. Fifteen energy absorbers wereused during the fifth test, and three of themfailed. They represented 26.6% of the total _

deceleration force. Two failed on the side 'qthat bottomed (see Fig. 8). They represented34% of the deceleration force on that side..Peak accelerations inside the trailer, when thesystem bottomed, were 12 g (see Fig. 9). Noneof the equipment inside the trailer was damaged(including the fluorescent lights) and the onlydamage to the trailer was a wrinkled skin inone area requiring replacement of a few rivets.Motion of the trailer during the third, fourth,and fifth tests is shown in the movies.

Typical unfiltered acceleration responseof equipment inside the trailer is shown inFig. 9. Transducers 12 and 13 are at oppositeends of the trailer. An initial peak pulseoccurred only at one end during the second testand at both ends during the fifth test. Theseinitial peaks occurred randomly during thetesting. Impact acceleration at the top of thecolumns was 100 to 300 g.

Failure of the three energy absorbersduring the fifth test occurred when individualstages ,;2 ced the end of their stroke and thespot-welded bands failed to retain the wires(see Fig. 10). Modified energy absorbers werethen successfully tested. Several units werestatically loaded to 25,000 lb (twice the Fig. 8 Right side of FSGC system aftermaximum absorption load), and one unit was 15-ft drop test, showing two

separated energy absorbers.

A AA

Fig. 7 Prototype F5GC system ready for Fig. 9 Response of equipment inside trailer15-ft drop test. during tests 2 and 5. ,

A ff~T7~7T~N

A, 72

impact-tested to failure. This unit finally pads. These are under the protective polyeth-failed at a swaged tubing flange. ylene covers at the base of each column in

Fig. 11.Our comprehensive test program identified

discrepancies in the as-built FSGC system.These were corrected before using the system ACKNOWLEDGMENTunder actual severe ground-motion conditions.Normally we desire fail-safe shock-mitigation The authors would like to acknowledgesystems. In actual field applications of this the cooperation and assistance provided bysystem, we have installed rigid-foam backup Holmes & Narver, Inc., (Nevada Test Site) in

connection with the F&RC and UGC systems andEG&G, Inc., (Las Vegas) in the drop testportion of this program. In particular, theefforts of Mr. E. Fuller and Mr. H. Montalvo

,, Wer Nbof H&N and Mr. P. Hulhall and Mr. R. Nakanishiof EG&G are gratefully acknowledged.

: REFERENCE

1. D. L. Bernreuter, E. C. Jackson, and'o A. B. Miller, "Control of the Dvnamic

Environment Produced by Underground NuclearExplosives," in Proc. Symp. on Eng. WithNuclear Explosives (Las Vegas, Nev., 1970),U.S. Atomic Energy Commission Rept. CONF-700101, Vol. 2, pp. 979-993, May 1970

II

Fig. 10 Cross section of energy absorber atend of extension stroke.

I "

Fig. 11 Field installation of FSGC system.

73

DISCUSSION

Mr. Fox (Barry Wright Corporation): How much Mr. Fox: You were only concerned about thedeflection did you get out of the foam itself? Was inital shock"blast. You did not care what happenedmost of the deflection taken up under the shock? after the ocital blast? Is that correct? h e

Mr. Bernreuter: Yes, the actual extensionof the Mr. Bernreuter: Yes.TOP SHOCKS themselves took up most of the shock.

Mr. Peralta (Bell Telephone laboratories): InMr. Fox: Was that fifteen feet? sizing the shock absorbers, do you do any kind ofcalculation as to how much energy they should beMr. Bernreuter: No, I think their stroke was able to absorb?designed to be twelve feet capacity. In this case theystroked 12 feet because they bottomed out. In fact, a Mr. Bernreuter: Yes we do quite a bit of calcu-couple of the units failed. We picked the trailer up 15 lation based on the weight of the trailer, the expectedfeet off the ground and dropped it, so it had a velocity level, etc. The formulas are worked out, and we haveof about 28 feet per second when It struck the ground, performed a compute*, analysis.

74

THE C014PARISON OF THE RESPONSE

OF A HIGHWAY BRIDGE TO UNIFORM GROUND SHOCK

AND MOVING GROUND EXCITATION (U)

Neil E. Johnson and Robert D. GalletlyMechanics Research, Inc.

Los Angeles, California*1This paper compares the dynamic response of a highway bridge structuresubjected to:

* The Uniform Ground Acceleration Time History for the May 18.1940 El Centro Earthquake

* Moving Ground Acceleration

9 An Average Acceleration Shock Spectrum for Strong Ground Motion

An analytical procedure is !outlined for predicting the response of ahighway bridge structi.rewhose ground motion varies between its supportsand is dependent upon seismic wave propagation characteristics.

A typical six span highway ibridge is presented. A finite element modelA of this bridge is develope4 including representation of soil/foundation*1 stiffness and damping characteristics. The response of this structure

is evaluated for the various ground motion conditions using normal modetechniques. The dynamic analysis of this st'ucture was predicted usingthe tMHI/STARDYNIE Structural, Analysic System.

INTODUCTION seismic wave train must be connidered as ittraverses the structure the assumption of uni-

Considerable interest is currently being ex- form ground motion for the structurallarsemblyprorced so to the behavior of civil structures can nn longer be Juatified. If it car. be antic-uzidr thf. effects of seluinic excitation. Tra- ipated that the ground motion may differ In,iior,.l analyris techniques applied to such phase and magnitude among several structural:'tr et.ure include the use of quani-statie load supports then the resulting structural responserar'tors, n:hock spectra and, in some eases, must be reviewed carefully. This paper Rd-tr .e,1rration tlm,- histories. This paper ad- drersen n hypothetical case where a strong: -el-'Ir,~"':,'.a ,;pecial casve of time history analysis mic disturbnce propagates along the prounl-'aer,'n th" propalration velocity of a seismic supportino, a typiclu two lane, six span, sinrh,watvr 'movinfg !on(itdinally along the structure ped^ntal r:upport highway bridge structur, nt .A

1con::idered. liRults from this moving groun, relatively slow, uniform velocity.motion nnalycsi technique are compared with more The method of:inalysin utilired for attacKin-tra,,itionil time history and shock spectra tech- The pro othe lya ire ne or' a trc-niuen through the application of all three the problem of the 4lynamic response of a jtrue-

tenique toh the iplicatinh ofa three tural system to moving ground motion Ir eutlin,4techniques to an idealizrd hirhway bridge, in detail. Particular emphaiiis IN placel upon

For .tructural nnalv-' the ,tsnumed ground utilizing standard atructural coessoitia nrd

motion is often given in the f'onri of an accel- analytical techniques. The recults ofr thi.eration timn history recorded durini a neinmic -ffort were then compare-. to re.,ultn eonidrit,

,rint FI or an artifJ'Inlly i.enerated time uniform ground motion for tl,- -amo ieismic di.;-hi,'. 'y with certain nimrlni qn,1 statictical turbance and to reculto conil srldeng a typlenlqurntitJ,:; rettined [,I. For purpo.es of truc- average sheock speetrn. for wtrone, ,round mot.n.

t;ural 'voluqtion thin g:rond ttotion if, generall:Y A discuarion of ths fridpo mn-I I r jv"%,, This" rpplre,-to the foundation and 'rfonol suppoktr inelmior the geotrmry qn .e.rn ie.irrlptleo of

Jn n unif'orm manner, that is. with thr' diftlr- the bridge ansembly. A 1trler ummary ir %r'bnlriv(! of the prond exactly the c am, tit each aseumetd aoil properti Is lndleate'. Ptercura. porint. 11,,r n,.h time. .tmutrel was %Aste.ll 1n I urrtel - fvn',i-

When tho ,ho ar't,,rirtit- iengt,, of a Qro with ltuc ur e '. * h,.,e tri lo' r .. , ' a1 ., m

I'; Inri, ,,noii, '1 that the popoWf'tflrn t, imn ,1 r) frxs'1 tte' pre' - l

75

Best Available COPY

Nodes with Lumped 2

2 23

Coordinate System

Pig. 1 " Blridge Physical Characteristics and Idealized)'thematica. M4odel

A finite element model of the bridge struc- to minimize the number of dynamic degreos oftural system was developed. The natural fre- freedom for the problem. Note that a much morequencies, mode shapes and related properties detailed model might be desired if an actualarc identified. Normal mode techniques were highway bridge were to be analyzed using the

utilized in the problem formulation. The techniques described herein. Such a modelSTARDYNE Structural Analysis System and its would probably include a more extensive three-auxiliary subroutines were used exclusively for dimensional model considering both beam andthe modal vibration analysis and the dynamic plate finite elements and a more detailed de-response computations. scription of soil/structure interaction includ-BRIDGE MODEL ing pile foundations, if used. The assumed mass

distribution is summarized in Ta.,le 2.A mathematical model of a typical 5ix-span, The roadway superstructure and support columrns

singe pdesal sppot hghwa brdgewaswere modeled as a series of beam mrembers. Si.c-developed to illustrate the analysis techniques tion properties evaluated for these beams are -described herein. The bridge, wnich is shown

i~ Fg. m~ Fi. 2 is n iealzaton f agiven in Table 3. These section propertiesn Fg. a:d Fi. 9 isan dealzaton f aassume concrete as the basic material withtypical structure and is patterned after typical aprpit adfctosfrrifrigsel~~~~two-lane highway bridge designs. prpit oilain o enocn te.:

Using the finite element model describedThew lnidealized bridgeconsiststh of six 120 foot above, the natural frequencies and normal modes -

long toln spn wih heroadway superstruc- of' the bridge were determined through the use ofture placed approximately 50 feet above the teSAON tutrlAayi ytmcmueground urface. Reinforced concrete construe- thegram.eTNE STA l Anas Semomputer

tion ~ ~ ~ ~ ~ Fg wa asue. SpotclmsaBrid ge scaphrct risisan The azdYN rgadvlpdb

ens ae assumed. toe supporte bridge sv pa- and in use by a large number of engineering firmtpe fodane a s, worldwide through CoNtrol Data Corporation Data

typefoudatins.Centers, is a large (up to 6000 DOF) itatic andFoundation/soEf interaction was idealized as dy oamic analysis system based on the finite ele-

a series of linear support springs. These ment, normal mode bethod of analysis.springs have been designed to give reasonable The lowest sixty (60) normal modes were

translational and rotational stiffness proper-dermnd Ths oswihcnb gnral

utiled i the prolme fyeo raion. The seldtehniued. Teodes her i.Sch a meogeel l

sri te aSsrued nye ob Ta e soi classed as either horizontal bending, vertical

spxilaring r srutne s e i d T ex bending or longitudinal are summarized in TableThe example bridge was modeled as a lumped . The generalized weight and modal particpa-

mass, finite element screi ap shown in Fig. 1. tion factors correspnding to each o c theseThis modtl consists of 45 unconstrained nodes mldes are also given in Table I m. A brief de-each having 6 degrtes of freedom. Only 2 n es scripton o f the norml r ode ethod an the terswere selct.ed as m, ss poi dt locations in order in Tble T is given in Appendix A. e

76

tyia srcue n s atredatrtyia asm onrt s h ascmtrilwt

Due to the symmetry properties of the ideal- Table4 were used in the various dynamic responseized structural system, it should be noted that analyses described below. A modal damping fac-only a portion of the normal modes listed in tor of .05 was used for all response analyses.

17"-0" TA= I

SUMIARY OF SOIL SPRING CONSTANTS

8 4'-6" Direction Soil Spring Rate

pr[ 121"X) 1.2P'5 x 106 lb/in

61J X2 1.215 x 106 lb/In

Assume 2 sq. in.steel per foot X3 2.1450 x 106 lb/inof length

,4 9.1475 x 109 in lb

10"

S6 9.)47,5 x 109 in ib

SUMARY OF MASS PROPERTIES

t Element Mass Property

Roadway Superstructure 84;76.32 lbs/ft

Fig. 2 - Typical Cross-Section Support Columns 23414.8 !bs/ft

Pad Foundations 201;,282 b.; Each

R3,AM PROPERTI ES

Beam Area J 12 I3 Shear Shape FactorsType In in in in 2

Roadway 8111,, V2,71 ),60 h, )71,662 -0,o5",240 0.35) .

Supp__rtColu-n 23114 ,522, , 1i3 1,21l .44 ].2. .,, ,l'h.

77

TABLE 4

RESULTS OF THE MODAL ANALYSIS

Mode Frequency Genralized Weight Modal Participation Factors otbde De-cription

No. (cps) (kips) x x3

1 0.577 3360 0 1.247" 0 Horizontal Bending2 1.41 9 3h27 0 0 0 flor zontil Bending3 2.005 6557 i.0414 0 0 Longi tul nal4 2.717 3570 0 0.50,)5 0 llcrizontal Bending5 3.021 17211 -0.1081 0 0 Vertical Banding6 3.037 1396 0 -).6009 Vertical Bending7 3.185 165) 0.0588 0 0 Vertical Bending8 3.319 2669 0 0 0.);782 Vertical Bending9 3.1431 1958 0. 0240 0 Vertical Bendingt

10 3.502 '°84 0 0 1.57)1 Vertical Bending11 3.997 2557 0 0 0 Horizontal Bendin

4.15) 43 , ) 0. 3618 0 nori zontal13 1.454 3204 0 0 0 iforizonzal Rotation11h 5.058 15 A4 1) 0.5317 0 Hori zontal Pending15 ).4O6 1811O) 0 0.181,3 0 Horizontal Bendingi( 5.731 2188 0 0 O. 111 Vertiva! Bendin.117 u.058 3412 . 170 0 0 Vertical Ber.ding38 6.449 1773 0 0 0 Horizontal Bending]( '.186 ,827 0 0 0 Horizontal Bending20 6.512 3207 0 0 -0.0062 Vertical Bending21 6.981 58O 0 -0.P120 0 Vertical Bending22 7.013 35P2 0.033, 0 0 bongitudi hal23 7.1)0 753 0 0 -0.03b2 Vertical?)l 7.20) 585 0 .0,413 o ';orlzontal25 7.2141 1227 0.8))5 0 0 Longiltudinal26 7.256 620 0.0160 0 0 Longit utdinal27 7.259 5)2 0 0 0 Horizontal28 7.259 701 0 0 -0.0183 Longitudinal29 7.278 761 -1.0444 0 1 lon1i tudinal30 7.434 15)2 0 0 0.218P Vertical Bendin,-31 7.492 5 0 1. 00.56 0 Horzontal32 7,572 1250 .Y17. 0 0 Vertical Pending3 7.676 3178 f 0 .(P3% Vertical Bending34 7.709 27 0 0 0 ' ors i on35 8.4 0 38. ) -. (7)4 0 dor zontali 8.511 1 3. 2 0 O. ,, * Ion,,it-dinal

R,.7 i -0. )! ,2 0 'iorni.on'al BLndin18 10"' 0 0 0 iorl zo:It-]3' j i.7? 2 -o. ,10, 0 10 orn 4(".t'tho 11.73 ]80 0 0 1; *¢:2 on tai ending, I 12.3 32 1 82 0 ! 0 aoOh

42 13.176 1188 0 O. "17-3 0 8or! zontal4 13.88 1053 0 0 0 "orizontal14 1;. 315 10, 0 -0. ) 00 I 8ori zoistal Bendi n,45 I4.4hil 15 ,(l')5c 0 ,ori.zontail iending40, 14.77 * 15 Y) 0 0 0 orizontal47 14 .e1 5 1, 0 -Q.j' -7 0 orl zont i]

48 15. 0' 0,'43 oQ.t )~f 10. on1 :t'l i'llI 17. V1' 0 ) ori,'ontal Bendin.:

51 18.718 i -,. , Ion, "i .'dinal'13 1 .774 1 .' "'.0. 'id, i rwtl

54 1.13 1,1 -'. )' 0 e I"0I . 2') "'4 - ,. )' or :or i,'c ,l Be,,W r,

f 5'0 .IA> -' " - ).,) I121 P!I',tP

78

PRESENTATION OF THE METHOD OF ANALYSIS Rearranging Eq. (2), the equations of motion

The earthquake response of structures is de- become

scribed in many sources (5, 4]. A variety of 4I I N cl' 1xl - Wflyl + (5)computational tools are available to the struc-tural analyst. Both transient response tech- Just as the terms on the left hand side of theniques (5] and shock spectra methods (6] have equations can be considered as the forces actingachieved wide acceptance in predicting the be- upon the system in the absence of ground motionhavior of structural systems subjected to ground (lyl =jW =(Y' -0), the terms of the right handmotion. This paper addresses the problem of the side can be viewed as those forces Induced uponresponse of structures with large characteristic the system by the ground motions lyf when thelength and considers the case where unattenuated mass points are constrained (lxi = .il = x] =O).strong ground motion traverses the supports of a This provides a straightforward way for construc-structure at a uniform velocity. The method is ting [c') and [').developed In a straightforward manner and is out- For seismic analysis the ground motion islined below. often described in terms of ground acceleration.

Consider a system with n degrees of freedom Eq. (3) may be rewritten in the following mannerand r resilient supports to ground. Fig. I shows to accommodate these terms.a typical system. Assume that the equations of Introducing a new setof auxiliary coordinatesmotion are to be written in center-of-mass Jz(, the forces induced by ground motion on thecoordinates and the mass matrix of the system constrained system (right hand side of Eq. (3))Nt] is diagonal. For simplicity, assume thedamping is small and proportional to the stiff- may be writtenness. This is done for ease of the present K]IZI+ [CilC I =']lyl + ['Jlfl ()analysis; other restricted types of dampingcould be considered. Modal techniques will be Substituting the transformationintroduced, modal damping will be utilized andmodal coupling due to damping will be neglected. Izi = [T)JYj (5)

In the absence of ground motion the equationsof motion are: where

rrmIYJI +CI1 -, [K] XI=0 0) niT] is a transformation of ?onstant

wcoefficients;where n1

n nIz I is an auxiliary set of coordinatesnNq is the diagonal mass matrix of the into Eq. (4), it is seen that

system;nn [K][Tflyl + [C][Tfl I . ,Jjyl + ,]iJi (61

n[C) is the damping matrix of the system; [

n n Comparing like coefficients in Eq. (6) it may[K) is the stiffness matrix of the system; be written that

n[x] describes the absolute motion of the [KI(TI =I]:mass elements of the system.

For ground motion of the supports the forces in- and (7)A duced upon the system are described in terms of'

relative motion between the mass elements and [C][TJ= !c')the support elements. Considering this effect,the equations of motion can be written Eqs. (7) may be rewritten to give

~,~( 'I[C]l -l c'IJ] + [K]ix!- PkI+'I'h o (P) (T) -- f 1 K f'C [c'1 (8)

where The two right hand portions of Eq. (0S) arer identical expressions since it was previouslyn[kI] is a rectangular array of stiffness assumed that

coefficients relating ground attachnodes to adjacent internal nodes of [C) =o{KI and 'c'] =a .'J (9)the system;

r Furthermore, the existenc- of iK1 is assuredn , a rectangular array of dampini: since !;,] repre-ents the stiffness %atrix of acoefficients relating ground attach constrained zyster.nodes to adiacent internal noles of Next, u.'in F1. (',, F'. (1) h..ay be writkentree system;

1 -.j'zx C , i *JK ;W 'KJlr7 + ( 1l )

r ly! d'scribes thp absOlute ivround1 motions(displacements) nt each of the r hoarrn{ in,'rrr, Fh. (10, .eco.essupports.

79

N4I'i+[C(ikl-fli)+[K(txl-Izi),o (11) Then

Defining y,(t) (Y(t)IeI =xl-f, 0 )]lY'(t){ Y (t- t2) I

and subtracting 1JYiJ from both sides, Fq. (11) ii= Y~) = Ytt)(8may be written 1

I'v~~~~~~~~~lY Yi' cII KII: I' i ,(t -tr)

Noting thatFig. h shows the time histories 6ir a ground dis-

"I = [T] I = f[K Ill (1i) turbance for the special case of several evenlyspaced colinear ground points.

Eq. (13) may be written

I! +{c]I +[K]I f -Nt [KM"I [k'1 J (15)4.,

or 4 C

P4N14l +[C]II +IKJIB -[D][k') l (16) 0

where (D] =-N[K I1 is the dynamical matrix forthe constrained system. H

For the case of moving ground motion, it canbe assumed that the basic ground disturbance hasa plane wave front and a time history given by5'(t). Further, let the velocity of propagationfor the disturbance be v and the time the distur-bance encounters the first ground point be t D-. , 4-1

Then, the disturbance will arrive at successive 4 .

ground points at a time a

t dcosoiI VV Vdie i 0

t v (see Fig. 3) (17)

where 04)

t is the time of arrival of the disturbanceat the ith ground point;

di is the distance from the first groundpoint to tie i t h ground point; 43

O is the angle between a line adjoining thefirst ground point and the ith ground LAAA AApoint and the direction of wave propaga-tion; 0s..

v iz the velocity of propagation. 0

$s4Plane Wave Front ' -

Time

tr Fig. 4 - Time History for MovingGround Motion

13- ; t OrcuindPoint

ith Ground Consider a solution to Eq. (16) -f the form

VPoint 'f' 'Ii (19) r

where

(j is an arraj of mode shapes o& the un-di damped constrained system;

IIin a set of Penerallzerl (modil) co,)rdi-

Fig. 3 - Wave Train Propagation nates.

80

-

Substituting and premultiplyine, by [O]T, Eq. (16) that for the May 18, 1940 El Centro earth-becomes quake. The vertical component of ground

acceleration equal to 0.6 times the hori-S- V zontal component (Case 2);

0) . The horizontal component of the groundacceleration in a direction 45

° to theOn defining, in the usual manner, the generalized longitudinal direction of the bridgc withparameters a magnitude equal to that for the May 18,

1940 El Centro earthquake. The vertical[¢]T N([I = r'-aj component of ground acceleration equal to

0.6 times the horizontal component (Case3).I JT(CJ(4], =

RESULTSJ I¢J=X Using the loading conditions described above,

,JI I ' ( ' = a series of nine independent analyses were per-formed. Each of these analyses resulted in the

The equations of motion become computation of displacements, velocities andaccelerat.,ns at the various nodes on the I leal-

~JI'i + + (21 ) ized bridge structure. These results can befurther extended to include equivalent nodal

After solving Eq. (21) for I1, the absolute forces and internal member loads and stresses.

accelerations of the system become To briefly illustrate some the results ob-tained and to provide a basis for comparing the

' ] = li! - IZ - (TJYJ three methods used, the following representativedata are presented.

and 72) Acceleration responses of a point on the road-

way at the midspan of the bridge are presentedI~x! 4,1 J*.fTHY Iin Fig. 7 through 12. Figs. 7, 9 and 11 show

INPUT DATA representative acceleration time histories usingthe Uniform Ground Motion (UGM) :ethod for each

For this study three methods for solving the of the three load chses. Figs. 8, 10 u,!. 12 showdynamic response of a structural system to seis- similar acceleration time histories using theinic inputs were used. For each method, three Moving Ground Motion (MG4) method.loading conditions were considered. The methodswereFigs. 1, 14 and 15 show the distribution of

maximum values of acceleration for each load* Transient response of the structural sys- condition as a function of bridge span (see Fig.

tem subjected to the uniform ground accel- 1 for span locations). Shown on each figure areeration time history for tL. 'ay 18, 1940 results from the Uniform Ground Motion, MovingEl Centro earthquake (North-South compo- Ground Motion and Shock Spectra (SS) analysisnent); methods. It should be noted that the accelera-

* Transient response of the structural sys- tion values shown on these figures for various

tem subjected to moving ground motion with span locations do not necessarily occur at the

a constant propagation velocity of 400 ft/ same point in time. Consequently, some caresee and an acceleration time history given should be used when interpreting these analyisby the May 18, 19110 El Centro earthquake results.(North-South component); CONCLUSION

9 Hespon~e of the structural system subjectedto the average acceleration shock spectrum A number of conclusions may be drawn from a

for strong ground motion 153. review of the analysis results presented above.Some of the more significant of these are de-

The time history of the May 18, 191.0 El Centro scribed below.earthquake (North-South component) is shown inFig. 5. The average acceleration shock spectrum It is seen from Figs. 7 through I that thefor. strong grage mo eeratios n shon sp. time range of peak response of the bridge isfor strong ground motion is shown in Fig. .generally of longer duration for the Moving

The loading conditions considered were: Ground Motion cases than for the Uniform Ground

" The Lorizontal componcnt of the ground Motion cases. This phenomena would increase the

acceleration in the lateral direction of probability ofachleving n extreme peak response

the bridge with a nagr.it We~ equal to tat due toa fortuitous combinationof modal responses.for the May 18, 19O F.l Centro earthquake. An examination of Fig. Is show. that for Uni-The vertical component of ground sccelera- form Ground Motion, the longitudinal response istLion equal to 0.( times the horizontal Aignificintly higher than for Moving Ground Mo-component (Case I); tion. This is due to very largerol.alparticipa-

* The horizont-! component of the ground tion of" the first longitudinal mo'le of the ;yrt-n

acceleration in the longtudinal direction when both end supports have the identical ground

of the bridge with a magnitude equal to excitation. For the Movinv Grounli c-ase, a reIur-

81

.50

0215

0.5 -

0 5 10 1 5 20 25 50Time, shc

Fig. - North-South Coirponent of Horizontal GroundAcceleration of El Centro Earthquake May 18, 19140.

0The response spectrum selected for this con-parison was a composite average of acceleration

: .8 shock spectra for several strong ground motioni events [3. Since this spectrum does not bound

- Percent of Critical Damping that derived from the ground motion time history

~utilized for this problem the resulting responsedoes not necessarily encompass the results ob-

.6 tained from time history studies. For a shockO-spectra analysis the phase relationships are lost

= .oo nd a pessimistic summation of modal response isI assumed. For a given ground motion time history

o4 / 2a respo~nse derived from shock spectra analysis

.. ,0

'-.4 will bound the response developed from a time-- '- .... history analy'sis for either the Uniform or Moving

e Ground Motion method..In summary, it can generally be concluded that"

.c =0. an evaluation of Moving Ground Motion effectsSresult in different, structural resp-nse than

.would be computed from a Uniform g round Motiono analysis. The probblity of higher responsestends to increase as the number of supports in-

_ _creases. Consequently, it can be recommendedthat Moving Ground Motion should be considered in

0 0.8 1.6 2.4 the seismic design of multiple support brde .

Undamped Natural Period, Sec It can be further concluded that time historyresponse analysis of either the Moving or UniforGround Motion types will give moretmehnondful

to .. 6response predictions than those obtained from>Shock Spectra analysis when many modes contributeSpectrum Curves to the response of the uystem. For thi reason,

realistic and economieal highway bridge design01 practice Ghould consider tine history effects.

,i.n in th2. ptrticipation of the first longitu- REFdRet .

cinal mode w observed when the ground excita- Efh.tinn r 'ch nndupportwa distintly different. 1. Robert L. Wiegel, Earthquake Fgineerin,ue to 'he' bridpe symmetry, the resiponses to Prentice-Hall, New eiersey, 1970

1.h,. Uniform G;round Motion exhibit :.;iraitry prop- P.M. Amin tnd A. Ang "Nonstationary Stoehn Ic

,rties ,ihout theei'pan of' .he briIe. ThiS Models of Earthquake Motion", o rn l of tibprnpnr'y n tte. reSponse rersult doot occur Applied Mechsnics Division, ASCE, Vol. 94f'or '-i Mwvlni' g;round M'tion method. This ii due No. M, Procurement hpr a0y . April si

S,* v'rltpian in ceround .oon it time h orides eects.u[ I ,,,"- ,t+h,' , 1. mif dusturhane,, travets's V'. rl M. Harrnd, Voals , .'raw-le, h~ 'n* '. ,r p enerally, of the rosp on gothe n:

*wn m'd wiffer b n t ' hethev produein hcit

tiron ofc opnde l,pen upon both the loafflnt. 1.. Robert. Hurtyand ieg, heF. F.tinst in, :)ynai, .. , -I1.Vr*an 'in- lre tlon consid-red. )~,trtures, Prentlce-ePt Hll, New .Jesrsy, 970

82 s

-V

__..1". 0

-. 23. -Ae AWAA

Z .25

Time. moo Time. see

flt. 7 - Hid Span'Horitontal Acceleratioo History Fig. 8 K id Span Horizontol ActeleratlonHistorythiforn Oroua Mtion ov4'rg Grond lotlon

-|

Case 2 UGH Came 3M

0-

,oNc v vvv0 2 3 4 50 1 3 4 3

Tie*, see Time see

Fig. i -d1 Span Longitudinal Acceleration History Fig. 10 M kid Sjpn Longitudical Aceleraton HistoryUniform Ground Mtion Moving GrOund W.tio

Came 3 3CM

.3 8.2

.22

7v 0

v. -. 2-

-.2

0 1 2 3 4 5 0 1 2 3 4Tio'. see Ti- , e

Figure It K id Spun Vertical. Acceleration History ?Ig. I 1 4 Mid~f lpnertical " lraln.Wlifors Crownd Flotion ivll ud tin

-.1

1- UGM1.0 A- lMM

o a- SS

. ..o.. .. o

.- \--,.,-.0*<. o\--'." -.-It ",I,, ".-1-'d ",- , "0 , 10%" %

< --, .. . --0-

ot m I I , I I I , i ,I ..

1 3 5 7 9 11 13 15 17 19 21 P3 25Span Location

Fig. 13 - lMhimun Horizontal Acceleration Distribution(Case I)

195

---O... <)-- -O ... O --<'- -O ... - 0 - - -0 - - - - 0O- --O0

~' 1.0 A- MGM

0

.94

,4 [3- SS

.5

0

.. ..---. -- A --G . --.- A--.-o-A-. - -

01 I I I ,, a I ; I I 2; 21 3 5 7 9 ! I1 1 7 9 >

Span iocationFig. 1, - 1aximt1 . Lonitudinal Acceleration Dintributon

(Case ")1.0 0- UG'

A- IJM

0 V

43/" / /1\.1/ r%, // a-l0.5 / .

o - _

1 3 5 9 11 15 1 1) 1 19 21 25 25

Span Location

Fi . 1) - Maximum Vertical Acceleration Distribution(Case ")

84

5. Richard Rosen, et al, STARDYNE: User's Manual, Note that modal participation factors are com-Mechanics Research, Inc. and Control Data puted for each of the threetranslational degreesCorporation, 1971 of freedom.

6. Richard T. Haelsig, DYNRE IV User's Guide,Mechanics Research, Inc., 1968

APPENDIX A: DEFINITION OF ANALYSIS TERMS

The STARDYNE System is formulated around the"Stiffness" or "Displacement" method of struc-tural analysis. In using this method, the struc-tural system can be represented by the followingequations of motion (for free vibration):

r?4Ji 'j + [KF]Ix =1.0 (1)

where

M4 is the diagonal mass matrix;

[Ki is the stiffness matrix;

jxj is the system coordinate.

The cigenvalues (natural frequencies) andeigenvectors (normal modes) of the system can beextrar'ted from the following reformulated versionof Eq. ()

K =j 1,( 0~ (2)

where

o 2 is the 1.th elgenvalue;

(r) is the rth eigenvector.The generalized mass of the structure is given

by

The generalized weight is given by

(4)

where

g s tho Gravitational constant.

The rodal participation factorr :j~r) are "a-

fined to be

,mil'ir

Mrwhere

Pi(r) is the modal participation fact-r for'he rth mode in jth translatIona direc-tion;

m] is the oecntL of rars .- soeiati withIth d'oreo 'f freedon;

'1ir ie tho component of rth 7o1al vectora srocinte'l with i t !' dereop -f frre'ioq

r11r is the r'enerali7e'] mafrr for rth mo,1.

'C ~85 -

p.

i1'

DE:OI1MATION AND FRACTURE OF TANK BOTTOM

HULL PLATES SUBJECTED TO MINE BLAST

Donald F. HlaskellVulnerability Laboratory

U.S. Army Ballistic Research LaboratoriesAberdeen Proving Ground, Md.

An analytical study has been performed to develop a method for pre-dicting the deformation and fracture characteristics of flatrectangular tank bottom hull plates subjected to blast attack fromshallow buried mines located under the plate's center. The analyticalmethod developed does not require electronic digital computers in itsapplication, and is an easy to use, accurate, and directly applicabletool for vulnerability assessments and engineering design. The methodof analysis is based on large structural plastic deformation, a semi-inverse energy method and a reasonable description of material behaviorincluding a static stress-strain curve and strength failure criterion.Blast loading is characterized by the energy associated with the blastwave. Plate deformation is calculated by equating blast energyimparted to the plate to the strain energy absorbed by the plate inreaching its final deformed shape. These formulations developed correlatewithin an average error of 7% with available aluminum and steel platemine blast test results.

The results of the effort are presented in simple graph and nomogramformat for rapid armor areal weight determinations and mine blast-tankbottom plate evaluations. This method of analysis will facilitate vuln-erability assessments and engineering design of armor subjected to mineblast attack.

LIST OF SYMBOIS U plate strain energy, in-lb• 1 mine explosive charge weight, 11)

A maximum transverse plate deformatinn i.e.amplitude of displacement w (sec Fig l),in a,b width an'J length respectively of rectan-

VE reflected mine blast energy flux density, gular plate [see Fig I), inM in-lb/in 2 e base of the natural system of logarithms

Fy plate tensile yield strength, psi el: plate tensile no,'mal fracture strain.

K plate fracture criterion constant with the h plate thickness, infollowing values: 1.315 - mean fracture; h no-fracture thic,ness i.e., plate thick-

0.835 - sure fracture (95% probability of ness with a 951 probability that fracturefracture); 1.791 - no fracture (951e witl h a oc .upoa mit hat frtactrprobability that fracture %ill not occur) will not occur upon mine blast attack, ip

M typi-al measurement I sure-fracture thickness i.e., thicknesswith 95: probability of fracture, in

mean value of tyi ical measurement.Ih mean thickness at hlich fracture occurs,

I standoff (plate-bottom to mile-top idistance i.e., see Fig 1), ii H number of mcastirement-;

S r 1. (1 * Y ) F eF: plate strain energy p normally reflected pressure from the blastabsorbed to fracture wave, 1'si

per ui;it plate ,,rea, in-lb/in'

11 plate strain energy of deformation, -n-b/ standard deviation of thentc tI average relative error.

Preceding page blank 88"7

_ 2:

,7 ".,-

INTRODUCTION under the test plate in a predug hole leaving adistance of 17 inches between the top surface

The vulnerabi!ity analyst, munitions de- of the mine and bottom surface of the testsigner, and armored vehicle designer need a plate. The mine was then covered with approxi-methodology for determining the effects of mine matel:' 3 inches of loosely packed soil. Afterblast attack on the integrity of armored vehicle detonation of the mine, deformation and thehull bottom plates. To be efficient and effec- extent ef cracking, if any, were recorded.tive, this methodology should be reasonablyaccurate and account for the various factors The A-my BDARP mine category data collected

that influenL, tank bottom plate integrity in described damage to the equipment including suchactual mine blast encounters. In addition, it factors as damage location, systems affected andshould be simple i. form so practical answers threat involved.can be easily obtaied in a short time. Such amethodology has been unavailable up to the pre- Tables I and II list the deformation andsent time. In the ast the information required fracture test results of rolled homogeneousto assess mine-tank bottom encounters had to be armor steel plates and 5083 aluminum plates.obtained by rather expensive and time consuming Charge weight ranged from 2.5 to 24 lbs with atest programs - analytical methods to develop constant stan'off and burial depth of 17 inchesthis information were not available. This and 3 inches, respectively. Since mechanicalsituation was caused by the very complicated properties test results for cach of the platesnature of the problem, tested were not available, estimates of the

necessary parameters were obtained from theTo avoid these complicationa, an approxi- material specifications. Yield strength and

mate method of aiialysis was selected to treat failure strain for the 5083 aluminum alloythe subject problem. The result of this treat- plates here obtained from MIL 111)K 23 specifica-ment is a simple method for rapidly predicting, tions. In regard to the steel armor plate,with reasonable accuracy and without recourse measured or specified hardness values were con-to a high speed electronic digital computer, the verted to yield strength and failure strain bydeformation and fracture cl-aracteristics of neans of well known relations (Ref. i).armored vehicle hull bottom plates attacked byblast type mines. The particular equations The steel plates ranged in thickness fromdeveloped are applicable to flat rectangular 3/4 in. to 1 1/2 in. while thickness of theplates of uniform thickness with fully clamped, aluminum plates ranged from 3/4 in. to 3 inches.or built-in, edges that are subjected to blast All plates here essentially the same size -from shallow buried mines centrally located either .10 in. by 65 in. or 44 in. by 75 in. Alltunder the plate (see Figure 1). These equations the aluminum plates vere apparently of the sameshould be particularly useful for quick vulner- strength and ductility whereas the teel platesability estimates and in the initial design ranged in strength and ductility from 90,000 psiphases of land mines and most armored vehicle to 172,000 psi and 0.15 in/in, to 0.23 in/in.design projects. respectively. Correlation betheen these test

results a:d predictions of the analysis i!"ILST I)hSCRltIP'ION discussed in the next section.

All the data presented in this report has CORIU:I.A'IION 01' ANALYSIS A.I) TF.ST RE.SIULTSobtained from available test results and battledamage records. The data consist of restuIts leformation and Fracture Relations. Thefrom armor plate - mine blast tests performed deformation and fracture analysis developedat Aberdeen Proving Groutd as well as combat yields the foliloing relations for plate defor-damage data collected by the BDAI6P (Battle nation and fracture thickness:D amaqe Assessment and Reporting Program) teams I laml, I2 F 1/2operating in South Vi.tnam. A - F ()

In the APG tests 'he mines includedpancake-type service and experimental casedmines as well as experimental bare pancake K I.mcharges. The test procedure was as follous. A h1 - m (2)test plate a i show n ijure 2 wa, placed on (l4) I' .top of two steel beams measuring apt ro~imate lytwelve inches square by sixty inches in length where 0.23 -0A4I " 1

wi tir the beams resting on the ground. A rec- -- 1.35 ; e (3)tangular Otccl frame heighing three and a half Iitong and an 8 ton plate for a total hold-dohn- 1.31.5 for the meedian fractur(to-test plate height ratio of 28 were then (placed on top of the test plate. (Tis hold- I or sught may ie.S 8, for sure fracture, i.e., 95'.down-to-test plate weight ratio may be compared k probaih, lit ot fracurtto the typical tank-to-botton plate heightratio of 24.) 1is fixture provided eseit ally 1.791 for no fractur(, that is, 9.'clamped (fixcd) boundary conditions all around Iprobahii : that fracture hill not

the plate. A mine %as then positioned centril) ,,Ikor

88

2 55

Deformation Correlation. Table I lists theoret- action. Iowever, until this analysis isical results for plate deformation calculated by improved to bring the theory into closer agree-use of Eq. 1 as well as the deviation of these ment with test, adequate fracturc predictionspredictions from the test results. As indicated, may be made by using the least squares data fitthe average positive error is 14% and the nega- equation as the fracture criterion. The aveiagetive error is 18.1% with the overall average absolute error between this least squares fiterror -6.4%. The average absolute error is equation and the plate test data is 13.2%.16.6%. These test and analytical results are Besides, as indicated by Figure 4, the actualplotted in Figure 3. In this figure the test tank mine damage data points which were notdata has beei. plotted as reduced deformation used in arriving at the least squares equationamplitude A defined by: and its 2a limits fall well within the 2a prob-

= 2A a2 b2 F h I 11/2 ability limits of the least squares fit equation.

Ia2b2 Y DISCUSSIONversus mine blast energy flux density E . Them General. The deformation relation, Eq. 1, wastest data is as follows: circles represent 5083 derived from first principles without making useal-iminum and the squares class 2 rolled hon.o- of test data to develop the equation. Firstgeneous steel armor. Lach data point represents principles were also used in developing the forma single test plate or the average of either two of the fracture relation, Eq. 2, although theor three tests. A least squares fit of the data :xact value of the fracture criterion constant Kis shown as the solid (- ) line with the 2o, was obtained from a least squares fit of testor 95% probability of occurrence, limits as the data.dotted (----) lines. The dashed line is thetheoretical curve. As indicated, the theoreti- A relatively simple and straightforwardcal curve is slightly below the least squares means for calculating the thickness of a givencurve. Consequently, plate deformation pre- plate that will fracture when attacked by minedicted by the present theory is, on the average, blast is presented by the nomograms of Figures6.4% lower than would be obtained in practice. S through 7. These nlomograms have been preparedIf a conservati'.e method for predicting deforma- from the fracture thickness equatio)n (Eq. 2).tion is desire(* tile equation corresponding to Figure 5 i. a nomogram for mean fracture thick-either of the 2o limits could be employed. For ness bosed on Eq. 2 with K r 1.315 where thisexample, use of the upper 2o curve to predict value of K has been determined by a least squaresdeformation would yield results that 95% of the fit of the plate fracture data. As indicatedtime are higher than would be obtained in actual previously, the average error of this relationpractice. is -0.6% and the average abiolute error is 13.2%.

Figure 6 presents the conditions for plateFracture Correlation. Plate fracture data thickness with a 95% (or + 2u) probability ofli.sted in Table 11 is plotted in Figure 4. In no fracture, hlr. That is, plates of thisthis figure plate strain energy absorbed to the thickness have a 95% probability of retainingpoint of fracture per unit plate area S is thines a e a m n atty Thisplotted versus mine blast energy flux cisity their integrity after a mine attack. This.m according to Lq. 2. Along with the test data nomogram is based on Equation 2 with K 1.791.

points this figure shows the least squares fit Plate thickness that will be breeched by 95%curve of the test data (equation 2 with k of the mine attacks may be obtained from Figure1.315), 2o probability limits (equation 2 with 7. This figure is a nomogram for sure fractureK equal to 0.835 and 1.791) and the theoicti-al thickness, |1., i.e., it represents 9;% proba-fracture Iine. As in Figure 3 the circles and bility of fracture. It is based on Eq. 2 withsquares represent 5083 aluminum and class 2 K = 0.835, the equation for the -2o limit curverolled homogeneous armor steel plates respec- of Figure .1.tively. Tile triangles correspond to actual tankdamage from nine blast in South Vietnam. Through five scales, these nomograms in

a

The theoretical criterion for fracture is ) tough

elsentially ness II, fracture thickness hi., scaled standoff1 '3

* , and mine charge weight h. In adit ioithere are two unlabeled scales used in the

whereas the l-ast squaies fit equation for the alc tonl"est dta iscalculation%.:est data is

S - 0.533 I. TLe procedure for using the fracture notno-m

That is, the -lope of the least squares data f:t grams is as follows. The sure-fracture nomogram,is s g~lht ly hl~'ler than half that of the theorct- Figure 7, will be used as an example. To cal-

is~~~~clt thihtl thii% titan hal tht ohi th teoiiL i1 phte fracture critorio. here may be two culate the thioknes, h" at %illch a plite ofa

posible reasons for this diserep.iny between width-to-length a:pect ratio with train energ)theory and test results, namely, inadequacy ofthe t r.icture cri terion and neglect of bending R

89

11

with 95% certainty under attack from a mine ofcharge weight W at a standoff R:

1. Locate the aspect ratio on the 9 scale,

point 1, the plate toughness on the U, scale,

point 2, and draw a straight line through thesepoints intersecting line a at point 3.

2. Similarly, locate the charge weight onthe W scale, point 4, and the scaled standoff on

the R/W scale, point S. Draw a straight linethrough points 4 and 5 intersecting line b atpoint 6.

3. Draw a straight line through points 3and 6 interesecting the sure fracture thicknessscale hsF at point 7. This va!ue is the thick-nes which fracture occurs.

Two equations have been developed, one fordeformation and one for fracture of flat,rectangular, clamped plates subjected to blast Pattack from shallow buried mines. The averageerror between predictions of these equations andtest results, including both plate and tankdata, is less than 7% with the average absoluteerror less than 17%. From the fracture criteriona set of plate fracture nomograms have been pre-pared that yield plate thickness for 95% prob-ability of fracture under mine blast attack, 95$probability that the plate will not fracture,and mean fracture thickness. Fracture thicknessand areal weight can be determined from thesenomographs in little more than the time it takesto draw three straight lines on the nomograph.

REFERENCES

I. T. Lyman, Ed., Metals Handbook, pp. 108-110, 8th Ed. Vol. 1, Properties and Selection ofMetals. American Society for Metals, 1961.

y

IV)

Figuue t - Mnme-R~ctongulor Plate confhqu¢,tlon

90

PLAN VIEWOF

TEST PLATE

HOLD-DOWNt PLATE

FRM

TEST PLATE

MINE

SIDE VIEW OF TEST ARRANGEMENT

Figure 2 -Typical Plate Test Arrangement.

300 f2o

200- 0 0 01

'11/ 'LEAST SOS. FIT -

0L6bI9.oil 0 '4

100 00

010 20 30E m~ los IN-LB/IN!

Figure 3 -Plate Deformation; 0: 5083 Aluminum Test Data, 0:- Rolled Homogeneous ArmorSteel Test Data,-: Least Squares Fit of Data,--: Theory

91

.. 4) .* .~.a* *q 00O4

4Jq t .4 -1% N'. SNO) O 4 Nq (1 .4C ;-1( 4

+ + +I I. 1 4 +4 +I + II

0 m wo"l '., m N'ae ~N m

4J 11 In .11 4.+1.44 .

0N. ~ e -N r ) r tfN 0 -.4 1"NL) -too l

*m -q r, ~O N a) N ID t, 0 0) r, It in Go N In -TQO

F-.0

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cl-

CC 9

00 W

4l o '4o

CL

41

i-in

CSA4r- rt,(1 NrtCD( I I 00 CS a0 92

LIN

Ic

Table II. Fracture Data

Stand-Off 17 In. (This Includes 3 In. Mine Burial Depth)

W h a a Fy ef Deviation

Test Charge Plate Plate Plate Yield Failure FromCase Weight Thickness Width Length Strength Strain Least SqsNo Material lb in in in KSI in/in

1 5083 Al. 4.0 0.75 44 65 33 .16 +25.72 6.5 1.25 44 65 33 .16 -0.23 12.0 2.0 44 65 33 .16 +15.64 11.9 2.0 44 65 33 .16 -14.65 11.0 2.0 44 65 33 .16 - 6.26 15.9 2.5 44 65 33 .16 -21.77 19.9 3.0 44 65 33 .16 -25.28 14.5 2.5 44 65 33 .16 -10.69 17.S 2.5 44 65 33 .16 -27.010 Steel 20 0.75 44 60 135 .17 -16.611 20.7 1.SO 44 60 112 .20 +55.812 20.2 1.0 40 65 137 .175 + 7.613 20.2 1.0 40 65 116 .22 +14.414 21.4 1.0 40 65 161 .155 + 5.515 20.1 1.0 40 65 163 .1S - 0.116 21.9 1.0 40 65 137 .17 - 4.617 20.8 1.0 40 65 160 .16 +11.S18 17.6 0.75 40 65 172 .13 -14.019 24 1.25 40 65 1FO .16 +13.620 19.9 1.0 44 65 90 .225 - 4.021 19.9 1.0 44 65 90 .23 - 2.422 19.9 1.0 44 65 114 .20 + 7.823 19.7 1.0 44 65 137 .17 +10.224 19.9 1.0 44 65 110 .21 - 2.8

Average * -0.6%Error

AverageAbsolute = 13.2%Error

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0 ~ ~ 0 ¢394

Fqgure 4- Plate Froture. 5083 Aluminum Test Dot 0. Po'led homoi$ ecs Armor Steel T¢il

00!0: 0, Ton Test Ooto6 LeOst .u' s rI Ot "ti - -T

t :yt -

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962

,I

THE IMPULSE IMPARTED TO TARGETS BY THE

DETONATION OF LAND MINES

Peter S. WestineSouthwest Research Institute

San Antonio, Texas

Numerous expensive testing programs have been conducted to deter-mine how land mines damage vehicles ox' armor plates; however,very few analytical studies'have been conducted because the load im-parted to a structure from a land mine explosion has never beendetermined. The few analytical calculations which have been madeassumed that the target was loaded by an air blast. In this paper,the writer shows that the impulse imparted to the target is not en-tirely an air blast phenomenon. The impulse is caused principallyby the momentum in the soil particles surrounding the buried explo-sive. A technique is developed for predicting the impulse impartedto targets such as wheels and armor plates. Several Illustrativecalculations are compared with experimental test results to establishthe validity of this procedure for e.stimating the impulse imparted toa target from a land mine explosion.

INTRODUCTION study; however, the authors did recognize thelimitations of using air blast data directly and

Although the effects of land mines on com- attempted to account for the effects of shockbat vehicles have constituted a serious problem wave reflections from the ground plane byfor a long time, very few theoretical or care- doubling the explosive charge weight. A lim-ful experimental studies have been made of this ited set of experiments measuring blast loadsproblem in the past. Almost all past efforts from buried 5-lb TNT explosive charges washave been purely empirical; that is, one simply conducted by Aerojet General Corporation 14]blew up vehicles or armor plates with land and compared with air blast data. The con-mines, and determined what thicknesses of clusions made from these experiments were:various armor materials would be necessary to"defeat" given mines. Numerous testing pro- I) The impulse from an explosive charge ingrams of this nature are listed in a recent the ground is a function of charge weight,bibliography [1]. stand-off distance, and depth of burial.

All of the analytical studies which have 2) Buried charges demonstrate a greaterbeen conducted assumed that the loading is an efficiency in transmitting impulse to aair blast phenomena. Personnel at Cornell target than does a blast wave from aAeronautical Laboratories [2] used 13RL data spherical charge in air.on reflected impulses and pressures from ex-plosive charges in air to estimate loads on In this paper, the author will demonstratetank hull bottoms from mine detcnation. These that the impulse imparted to a target by deto-air blast data were applied without considering nating a land mine is not an air blast phenome-ground reflection factors or accounting for the non. The basic premise behind these earliereffects of burial. In another study conducted studies, chiefly that the loading wab caused byat the Cleveland Army Tank Automotive Plant, a shock wave propagated through air, was in-AllIson Division of General Motors 131, the correct. Aerojet General's conclusion thatsame BRL data for impulsive loads from air buried charges are more efficient was a cor-blast were not properly used as in the Cornell rect observation; however, they were

97

incorrect in comparing the loads to air blast spheric density, or velocity of sound in air,ones. The impulse imparted to a target from can be found in Eq. (1), The specific impulsea land mine explosion is caused by the momen- imparted to the body being loaded is causedturn in the explosive products from the charge primarily by the momentum of the explosiveand primarily from the soil encasing the products rather than being caused by a shoc.kcharge. This paper considers only the i- wave propagated through air. Although thispulse, i.e. the area under the positive rres- equation is Zor a spherical explosive charge insure history of the load imparted to a target air, a land mine represents a very similarfrom a land mine explosion. Peak pressures circumstance, One can think of the groundand transient loading histories are not inwhich surrounds the mine as a charge sur-eluded, and cannot be discussed until direct rounded by a weak case of soil. The mass

measurements are made of these effects, term, M, is thus the mass of the mine and anThese limitations will not affect structural equivalent mass of soil. The philosophy rep-studies of hull plate response or vehicle over- resented by Eq. (1) is applicable to a land mineturning provided the response of the system explosion because the mass of the engulfed *i-being stiadied falls within the impulsive loading at standoffs typical of mine attacks is veryrealm. T41.4 paper will indicate how specific small relative to the mass of the explosiveimpulse is di-tr* ,ted over the surface nf products. If we assume that a 20-lb land mineseveral targets, particularly wheels and flat is covered by 3 in. of earth and that the hull ofplates. a structure is 17 in. above the center of the

mine, thin we can demonstrate the validity ofLOADING FROM A MINE BLAST this claim. The density of air under ambient

sea-level atmospheric conditions is 0.0765The nature of the loading applied to P lb/ft3 and the volume of this air enclosed within

structure auch as the hull of a combat vehicle a sphere 17 in. in diaimeter is 11. 9 ft3 . Thus,from a land mine explosion involves very com- the weight of an equivalent amount of engulfedplex phenomena. A shock wave propagated air is 0.0765 times 11.9 or 0. 91 lb, which isthrough air is only a minor cause of loading very small relative to 20 lb of high explosive.and not the principal one. Rudimentary calcu- Actually this ratio of air mass relative to masslations indicate that the mass of earth and ex- of explosive products is smaller than 0.91 toplosive products impinging upon the floor plate 20. 0, because the effective weight of soil (ap-contribute considerably to the loading. There- proximately 100 Ib) is 5 times greater than thefore, the loading is a very complex wave form weight of the charge in this illustrative example.that differs considerably from the classical, Later discussions will show howto estimate theexponentially-decaying wave that is usually effective maos term, M, in Eq. (1).associated with blasts.

Jack and Armendt [6] have measured tran-Baker [51 has shown that very close to the sient pressures at the surface of a plate from

charge the normally-reflected specific im- spherical pentolite charges detonated in apulse from a spherical charge detonated in air vacuum. This loading is believed to be some-equals: what analogous to loads caused by land mines

because the pressures are primarily caused by(2 ME)I/2 the explosive products. A typical pressure4iR S2 (1 ) trace as recorded by Jack and Armendt may be

seen in Fig. I. Observe that this normally-where M = total mass of explosive and reflected pressure history differs markedly

engulfed air from conventional air blast waves. Two shockfronts may be seen at A and B in the pressurehistory in Fig. I. The rest of the wave has a

S = stand-off distance from center gradual rise time and a rounded shape. Theof charge gradual rise time and rounded shape are

caused by the mass of the explosive productsSreflected specific impulse, impinging upon the plate and the pressure

This equation is applicable whenever the mass transducer in it. Theoretical studies such asof the engulfed air is less than approximately those by Lutzky [7] indicate that explosions inone-tenth the mass of the explosive products. an absolute vacuum would have these charac-If the explosive charge represented by Eq. (1) teristics. Jack and Armendt feel that the ini-is encased, the mass, M, represents the mass tial, sharp-rising shock front at point A is anof the case and explosive. Observe that no Initial reflected air blast wave that would dis-parameters defining ambient atmospheric con- appear entirely were they to have had a corn-ditions, such as ratio of specific heats, atmo- plete rather than partial vacuum. They offer

98

* Pressure3

Time

Fig. 1. Normally- Reflected Pressure History TakenIn Vacuum Near a Spherical Pentolite Charge

no opinion as to the cause of the second shock so that the spLCific normally reflected pulse,front at location B ini Fig. 1. The second shock ipR, can be computed, we will consider howfront may be a secondary shock (,sometimes Eq. (1) can be applied to real targets whichcalled a "pete" wave by other investigators). possess complex configurations. Eq. (1)Ordinarily, these shocks-are less intense than allows one to compute the normally reflectedthe first shock; however, under these circumn- impulse per unit area at various standoff dis-stances the wave has an atmosphere of gases tances. This equation does not tell one howfrom the explosive products through which it to calculate the total impulse, 1, imparted tocan expand; whereas, the Incident wave has a wheel or to a plate where the impulses im-almost no medium through which it can be con- parted to differential areas on these targetsducted. A land mine blast will probably give are not normally reflected. Total impulsea loading somewhat similar to that seen in can be deterrilned by knowing the projectedFig. 1; however, the relative magnitude of area of a target and the peak normally reflect-first shock, explosive products, and second ed impulse imparted to this area :)y usingshock wave pressures will vary considerably. Eq. (2).One can observe in Fig. I that the impulse(area under the pressure history) is princi- IR =iR Af (2)pally caused by the explosive products andthat air shocks produce very little impulse. The term I in Eq. (2) is a shape factor whichProbably in a land mine explosion, the initial is a function of target shape and standoff con-peak would be larger because of the pre- -ditions. This shape factor will be calculatedsence of an atmos.-here, but the secondary for wheels and rectangular plates, both ofshock would be greatly diminished, because of which are common targets.reflections being transmitted far into theearth and because the cylindrical shape of a SHAPE FACTOR FOR A WHEELland mine would disperse reflections ratherthan focus them as in a spherical charge, A To calculate the shape factor for a wheel,strong possibility exists that a peak pressure consider encounter conditions as presentedwill occur which is a shock rather than being in Fig. 2a. The wheel being considered is

*caused by momentum from soil flung upward located directly over the center of the explo-by a land mine explosion: however, the vast sive charge. We determine the total verticalmajority of the impulse is caused by momen- impulse imparted to the wheel by consideringtum in soil products. the specific imnpulse imparted to a differential

area located on the rim of the wheel. TheLOADING IMPARTED TO COMPLEX total impulse is then computed by integrating

TARGETS the differential areas around that portion ofBefore ,iscsslng wat num,, ,ecal valuethe rim being loaded. Fig. 2b shows the 4

Befoe dsct~sin wha nuericl vluespecific impulse being applied to a differentialshould be assigned to the mass, M, in Eq. (1) area, dA.

99

I

r cote 'rfl-cosO)N rsinO~

L._

(a) (b)

Fig. 2. Wheel Traversing a Land Mine

If the wheel is a thin strip of thickness, h, This impulse may be divided into its verticalthen the differential area, dA, equals: and horizontal components. We are only in-

terested in the vertical component as all hori-dA = hrd0 (3) zontal components cancel because of sym-

metry. The vertical component, dl, of theThe specific impulse, i, as shown in Fig. 2b impulse equals:at the location of the differential area equalsafter substituting for S in Eq. (1): dl = i cos (* + e) cos e dA (8)

(2 ME) 1/2 The total impulse imparted to the wheel is ob-= 4 nr(l-cos e) + d 2 (4) tained if one integrates Eq. (8) over the loaded

L cos segment of the wheel. Because this loading iscaused by soil particles impacting the wheel,

Geometric considerations which are seen in all segments of the wheel are loaded until theFig. 2a indicate: impulse vector, i, becomes tangent to the

wheel. The impulse vector which is parallelr sin5 to a line from the center of the charge to the

r(I - cos 0) d differential area becomes parallel to the wheelwhen tan€ is a maximum. This occurs at:

The cos ?/ can be calculated from the tan t andequals after substituting Eq. (5) for tan t: d

(tn 0r(l-cosO) +d

Cos*-r n (.oer (6) .rcose (d+ r-r cos 0)+ r 2 Vin? =0

or atWe will assume that only that component of thespecific impulse which is perpendicular to theCo(9 cos 8 r - (9)differential area imparts momentum to the r -, dtarget. This assumption would be invalid werethis to be an air shock; however, it is valid Let us define a nondimensional quantity, 6,

because we are considering momentum from suoh thatmoving particles of soil. The impulse whichis applied to the differential area equals: 6 1 +A (10)

diA i cos I dA Substituting for d in Eq. (9) and rearrangingrterms indicates that the limits of integrationare:

100

e arc coo (-) () The differentl:n area, dA, equals:dA:. dy dx (17)

Because of symmetry, the total impulse may

be obtained by multiplying Eq. (8) by 2.0 and Eq. (1) indiciaes that the specific impulseintegrating the resulting expression from 6 directed at the differential area is:equal to zero to the limit expressed by Eq. (11).This integration gives: i (2ME)I/ (18)

e 41R 2

I =fzi coso( e) cosO dA (12)

0 Once again we assume that the component ofthis impulse which is tangential to the differ-

Substituting Eqs. (3) and (4) into Eq. (12) and ential area does not load the differential area,recognizing that 2 rh equals the projected and that all of the impulse imparted to thisarea, A, of the wheel yields: differential area is caused by the normal com-

0 ponent of the impt,.se, i, expressed in Eq. (18).

J= (2ME) 1 2 A _d2 f cos 2 $cos(*+0)cos0 dO The impulse imparted to the differential area4TS 0 (d+ r-r cose)2 equals:

S'A = /X 2 7 2 $ ( 1 9 )

But Eq. (13) is Eq. (2) provided the shapefactor, 0, for a wheel equals:

0 One calculates the total impulse imparted toSird)2 f cos2# cosO cos(*+9) do the entire plate by substituting into Eq. (20).

(d+ r-r cos 0)2 X Y

If 4f f aAdA (20

Eq. (14) for the shape factor may be simplified 0 0further if the appropriate substitutions from

Eqs. (5) and (6) are made into Eq. (14). Making Substituting Eq. (16) into Eq. (18), Eq. (18) intothese substitutions and gathering terms yields: Eq. (19), and Eq. (19) and (17) into Eq. (20)

e yields:2 (6 Cos -1) cos 0 d90= 62 (15) x Y

f6+1-26os3= f f (2ME) _/4Sdxdy (21)

0 0 4r(x2+y2 52 )3/ 2

Although this integral is a very difficult one to

compute in closed form, a numerical solution If one multiplies and divides Eq. (21) by S2 XYis easily obtained on a computer. The results and recognizes that 4 XY equals the projectedfrom a numerical investigation are presented area, A. then:in Fig. 3 where shape factor 0 for a wheel ispresented as a function of 6. 2 s

i (2ME) 1/ 2 A $- r Sdxdv

SHAPE FACTOR FOR RECTANGULAR 4n S2 00 (x2-+y2 S2)3/2

PLATESBut Eq. (22) is Eq. (2) provided the shape fac-

To calculate the ahape factor for a rect- tor. 0. for a rectangular plate equals:angulhr plate, consider a plate with a rect-angular x-y coordinate system having an origin rS3 X dyat the center of the plate. Because of the sym- E r f d3metry which exists when the charge is buried J 0 x y2 + $2)3/2

directly under the center of the plate, we willconsider the vertical impulse imparted to a After performing this integration, one obtainsdifferential area located in only one quadrant of for the shape factor of a rectangular plate:the plate. The plate will have half spans ofX and Y. The slant range, R. from the center ! a (Y/S) (XIS) 1 2of the charge to the differential area equals: (X/S)(Y/S) a iY/S)(IS) 2 i 4 1]

R F/(x24y2+S2) (16)

101

Ye,

4-

2

~7/4- -- '/

1 2 4 6 8 10 20 40 60 80 100(I + d IR 3061

Fig. 3. Shape Factor For a Wheel

MODEL ANALYSIS TO DETERMINE crater produced by detonating a buried explo-EFFECTIVE MASS sive, indicates what soil properties should be

selected. In order to determine the impulseIf one knows that value of effective mass is from the detonation of a buried explosive, we

appropriate for a given size of explosive charge are interested in the momentum imparted toand depth of burial, he can compute either spe- soil particles. The explosive cratering analy-cific impulse or total impulse by selecting the sis must concern itself with both the momen-appropriate shape factor and substituting in tum imparted to soil particles and the influ-Eqs.(1) and (2). Effective mass, M, will be ence of gravity on the trajectory of these par-determined by using the analytical results ex- ticles. Reference [87 uses the bulk density ofpressed by Eqs. (1) and (2), conducting a simil- the soil, p. the seismic velocity of the soil, c,itude analysis to interrelate charge weight and and the acceleration of gravity, g, to charac-depth of burial, and applying some experimen- terize soil properties. In this study, we willtal results to devise a numerical relationship use two of these three parameters, the density,between the various parameters. p, and the seismic velocity, c; however, we

will delete the acceleration of gravity, g, fromTo conduct a similitude analysis, one be- our analysis. Deleting this term from the

gins by listing the relevant parameters. We analysis is equivalent to assuming that gravi-wish to measure specific impulse, i; this para- tational effects are insignificant. Such anmeter is our response parameter. The im- assumption is appropriate as the gravitationalpulse is created by detonating an explosive field of the earth is not strong enough to appre-charge of weight, W, which was buried at some ciably reduce the momentum in soil particlesdepth, d, from the surface of the soil to the in the relatively small distance between mostcenter of the charge. Impulse is to be mea- land mines and targets. The six paramete'ssured at some standoff distance, S, from the which have just been presented as defining thiscenter of the charge. The most difficult as- problem of the impulse from the detonation of apect in conducting this similitude analysis is land mine are summarized in Table A togetherthe selection of appropriate parametc rs for with their fundamental dimensions in the engi-characterizing the soil. Fortunately, a recent neering system of force, F; length, L; andanalysis [8] of a similar problem, the size of time, T.

102

F -'r

TABLE A I fI '6Parameters For Determining Impulse d 73) ,1

From A Land Mine ExplosionEq. (26) defines a three-dimensional space.

Fundamental This space can be reducedi in a two-dimen-Symbol Parameter Dimensions sional space by squaring the second pi term

and multiplying the result by the first pi termi specific impulse FT/L 2 to form a new dependent parameter. AfterW charge weight FL performing this operation, Eq. (26) may be re-d depth.of burial L written as Eq. (27).S standoff distance Lp density of soil FT 2 /L 4 iS2 = f"(, (27)c seismic velocity L/T d3 d3

Eq. (27) is Eq. (1) without an explicit expres-Many different textbooks present several sion for the appropriate functional format.

different procedures for developing the three Eq. (1) states that impulse varies inversely asnondimensional ratios called pi terms from the square of the standoff, S. Inasmuch as thethe six parameters listed in Table A. In this effective mass, M, in Eq. (1) annot possiblypaper, we will omit the algebraic procedures be a function of S, Eq. (1) ir.diated that theand present the results. Table B lists three three-dimensional space expi essed by Eq. (25)pi terms which can be obtained, should be rewritten as the two-dim-nensional

space, Eq. (27).TABLE B

Pi Terms - Impulse From Eq. (1) does not tell us how effective mass,Land Mine Explesion M, relates to charge weight, W. and depth of

burial, d; thus, it does not furnish the de-i sired functional format for predicting impulse

'r d Scaled impulse from a land mine explosion. Eq. (27) which is

Eq. (1) expressed in a slightly different formr Geometric similarity does not provide the desired functional formd either. The functional format relating scaled

W charge weight, W/d 3 . to scaled impulse,Tn3 .Scaled charge weight

pc 2 d3 (i S2 )/(d 3 ), must be obtained from experi-. . . . .. ... ........ .. mental test data. Experiments must be con-

ducted so the measured scaled impulse,The first pi term is normalized or scaled (iS 2 )/(d 3 ) can be plotted as a function of

impulse. This pi term io a unique function of scaled charge weight, W/d 3 . Before makingthe other two pi terms, a statement of geo- this plot, Eq. (27) will be modified by substi-metric similarity and scaled charge weight. tuting Eq. (2) to form Eq. (28) which is an ex-We write this functional relationship as Eq.(25). pression for total impulse rather than for

specific impulse. Analyzing test data in terms(f3 W (2) of total impulse rather than in terms of spe-

pcd d pc2 d 3 / cific impulse is more convenient.

2The soil parameters P and c can be deleted IS f.(2'from Eq. (25), because they are essentially Ad 3 (d 3 )constants. Soil d,.ity varies very little overa wide range oi soil conditions. Seismic velo- SCALED IMPULSE AS FUNCTION OFcity for soil in a -olid state does vary over a SCALED CHARGE WEIGIITlimited range- however, the soil which pro-vides most of the impulse from a land mine Reference 141 describes a limited seriesexplosion acts more like a fluid than a solid, of tests in which 5.0 lb charges were buried atWe will assume that the seismic velocity of all various depths and a rigid mass with an ex-soil "fluids" is corstant. A careful perusal of posed surface area of 50 in. 2 suspended at

A Reference [8] would show that relatively accu- various heights directly over these charges.

rate predictions of crater size can be made The masses were not constrained or preventedwhen P and c are treated as constants. De- from moving vertically; however, they couldleting P and c from Eq. (25) because they are not rotate or translate horizontally. By mea-constants yields Eq. (26). suring the displacement of the mass for dif-

ferent depths of burial and standoff distances,

103

'~' g

normally reflected total impulse could be The wheels were held by a yoke whose othereasily calculated from Eq. (29). end was attached by means of a torsio.a spring

to a rigid wall. This test configuration isV 2gh(9 effectively a spring-loa~ed pendulum with a

2 Vwheel as the ball of the pendulum. By mea-surIng the maximum rotation of the yoke and

Table C contains the test number, depth of wheel system as a result of the detonation ofburial, air gap b~tween ground and bottom of various size explosive charges, the impulsethe mass, and the calculated total impulse for imparted to a wheel could be calculated. Thethese tests. The depth of burial was mea- explosive charges used in this test variedsured from the ground to the top of the charge from 0. 106 lb to 0. 468 lb of pentolite. Therather than to the center of the chrge. *By as- charges were rectangular parallelepipedessuming that the 5.0 lb charges were cylinders with a constant surface area of 2" by 21" and awith diameters of twice their thicknesses, thickness that depended upon the size of thedepths of burial and standoff distances fromthe explosive charge. 1/2" of soil was placedcenter of the charge were estimated. Because over the charges. Steel wheels which were 7"the masses being loaded by the explosive in diameter and 1" thick were in direct con-charge and soil products are very small, the tact with the ground. On a few occasions theshape factor § is essentially equal to 1.0 in yoke held only one wheel which was centeredthese tests. The final two columns in Table C directly over the mine; however, most ex-present scaled impulse and scaled charge periments were conducted on a 3-wheel arrayweight. In subsequent discussion, these two with a 1. 75" center-to-center spacing betweenquantities will be plotted to determine the successive wheels. The middle wheel in anyfunctional format for Eq. (28). The quantities array was always over the center of the land(IS 2 )/(A~d2) and W/d 3 are dimensional be- mine. In a 3-wheel array, the impulse wascause the soil constants have been deleted applied to all 3 wheels. To estimate the im-from this analysis. Throughout the rest of this pulse imparted to the center wheel only, thepaper, the units for I will be (lb-sec), A will following equation was applied and solved forbe (in 2), S will be (ft), d will be (ft), and i the impulse impartedwill be (psi-sec).

TABLE CI S2 W- Versus - Using Rigid Mass Test Data

A§d3 d

Soil Cover IS2 J WTo Top Total A§ d3 d3

TestNo. Of Charge Air Gap Impulse d S /Pss tS.\ I lb(in) (in) (lb- sec) (ft) (ft) it /

101 12 4 123.5 1.125 1.46 3.68 3.51102 12 0 166.0 1.125 1.125 2.95 3.51105 4 12 80.3 0,458 1.46 35.5 51.8106 8 8 118.3 0.791 1.46 10.2 10.1107 12 4 154.2 1,125 1.46 4.60 3.51108 12 8 90.9 1.125 1.79 4.07 3.51109 12 0 198.9 1.125 1.125 3.54 3.51111 4 16 39.5 0.458 1.79 26.3 51.8113 8 12 75.5 0.791 t.79 9.68 10.1114 4 8 177.5 0.458 j. 125 46.6 51.8115 8 4 202.9 0.791 1.125 10.3 10.1117 4 8 170.7 0.458 1.125 44.6 51.8

A second group of experiments which mea. ic + 2i c sin 0 13 (30)sured impulse imparted to targets from landmine explosions are from a series of unreport- where tan (L d-ed model tests conducted by Mr. Bruce Morris 1.75)at MERDC, Fort Belvoir. In these experi-ments, wheels were placed over buried charges. to the center wheel, ic . The experimental

104

data indicate that Eq. (30) proportions the im- A straight line fits the experimental datapulso appropriately. In developing Eq. (30), we presented in Fig. 4 very accurately. Theassune that the impulse imparted to the out- equation of this line as obtained by a least-aide wheels is applied parallel to.a line from squares fit to the data in Fig. 4 gives Eq. (31).the center of the charge to the bottom of thewheel, and that the upward momentum of out- I W0 . 72 d0 . 84

side wheels is caused by the vertical compo- #A = 1. 725 s2 (31)

nent of this impulse. Table D contains thetotal impulse measured by this test arrange- Eq. (31) is the explicit expression for eitherment, the number of wheels in the test array, Eq. (27) or Eq. (1). If one substitutesthe equivalent impulse imparted to the center 1.4 x 10+6 W for E in Eq. (1) and equates thewheel only, the charge weight, depth of burial; left hand side of Eqs. (1) and (31) after makingand standoff distance. Before computing the the expressions dimensionally consistent, hetotal impulse imparted to a center wheel, one obtains the effective mass term, M. M whenmust look up the shape factor for the wheel in expressed in slugs is given by Eq. (32).Fig. 3. Both 6 and # are listed in Table Dfor each test configuration. The dependent M(slugs) = 3.49 W0 4 4 d1 . 6 5 (32)quantity, (IS 2 )/(At d3 ), and independent quan-tity, W/d 3 , are computed from this informa- Usually the reader prrers to think of mass intion and listed in the last two columns of units of pounds. Mass in pounds is given byTable D. Eq. (33).

TABLE DI S2 W

A Versus - Using Wheel Data

IS 2 WTotal Impulse on AI d3 d3

Test Impulse No. Center d S 6 f (psi-secNo. (lb-sec) Wheels Wheel (ft) (fit) \ft/ ft3

2 23.5 3 12.1 .0832 .3749 1.285 4.3 98.3 400.3 32.6 3 16.1 .0930 .3847 1.319 4.0 105.5 353.4 18.6 3 10.7 .0607 .3524 1.208 5.45 155. 475.5 27.7 3 14.6 .0778 .3695 1.266 4.55 131.5 425.6 28.5 3 14.1 .0930 .3847 1.319 4.0 92.6 353.8 53.3 3 24.2 .117 .4084 1.402 '.25 110.5 261.9 28.1 3 13.9 .0930 .3847 1.319 4.0 91,2 353.10 43.2 3 20.1 .110 .4017 1.378 3.4 102. 286.11 47.1 3 21.4 .117 .4087 1.402 3.25 97.5 261.13 71.4 3 31.7 .125 .4167 1.430 3.1 130. 240.14 13.7 1 13.7 .0930 .3847 1.275 4.0 90. 353.15 15.9 1 15.9 .103 .3947 1.354 3.6 90. 311.16 23.9 1 23.9 .113 .4047 1.387 3.4 113.5 274.

Tables C and D contain all of the ea- 1surements of impulse imparted to objects M (lbs) = 112.5 W0 . 44 d I 68 (33)from land mine explosions that are known tothis writer. In spite of spending many years The author had substituted into Eq. (33) beforeand dollars on explosive plate bulge tests, no making his earlier statement that the mostone has measured the impulse imparted to the significant mass in the explosive productsplates in such experiments. In Fig. 4, one from a land mine explosion was the mass ofsees a plot of scaled impulse, (IS 2 )/(A§ d 3 ), the soil products.as a functlon of W/d 3 for the data contained inTables C and D. These experimental data ap- Fig. 4 also shows that experimental re-pear to collapse into a unique function as pre- suits agree well with Eq. (31). The scatter isdicted by Eq. (28). small considering the inherent scatter in ex-

perimental test results. Although data from

105

600 ' ' ' ' '' 1 ' '

400 x IMPULSE ON THREE WHEELS+ IMPULSE ON ONE WHEEL

200 - 0 IMPULSE ON BLOCKS

100 X _

60-

is' 40 r

10 m7f \ . . , WO-7d o. 8+

1.7254-"4 " --"s" -+

44

2I I h II I I ii,1l I1

2 4 6 10 20 40 60 100 200 500W Il-b"'

Fig. 4. Scaled Impulse Versus Scaled Charge Weight

only two types of experiments are used in the ground. On the other hand, if the chargedeveloping Fig. 4, a significant number of is located at the surface with no cover overdata points (25 points) from a variety of test it, the impulse begins to be an air blast phe-conditions is used to develop this curve. The nornenon. Three data points were not includ-charge weights in the wheel experiments were edt in Table C, because in those rigid massvaried by a factor of 4. 0. Depths of burial on tests the top of the charge was flush with thethe rigid mass tests ranged from 4" to IZ" of stil'face of the ground, and no soil covered thesoil cover, and the air gap varied from zero charge. Because the center of the charge wasto 16". By combining these facts with the ob- below the surface of the ground, W/d 3 equaledservation that theae data involve two different a finite value of 2570. Extending the least-types of targets, one can see that 4, W, d. and squares fit in Fig. 4 would predict that scaledS have all been varied in Eq. (31). inipulse, (I S)/(d3 ), should equal 490. Using

Eq. (t) as applied to air blast with a groundSUMMARY reflection factor of 2. 0 predicts that scaled

impulse, (I SZ)/(d 3), would equal 4745. TheFig. 4 and Eq. (31) do have limits of W/d 3 average of the three tests on the rigid masses

for which they apply. Obviously, if the depth was a scaled impulse, (I S2 )/(d 3), of 3227.of burial is too great, the detonation of a Obviously this test was in a transition rangeburied explosive will not disturb the surface of where the impulse was changing from being

106

caused by momentum in soil particles to being The author is indebted to Mr. Brucecaused by momentum in the explosive products Morris, the Army's technical monitor on thisfrom the charge. Some finite soil cover is re- project and Mr. Alexander Wenzel, SwRIquired for Eq. (31) to be valid. Provided project leader, for being given the opportunityscaled charge weight, W/d 3, falls between of probing into this problem. In addition,1 and 1000, the analysis procedure recom- Dr. Wilfred E. Baker of SwRI is herebymended in this report should be valid, thanked for reviewing this text and making

several helpful suggestions.The other major restrictions to this analy-

sis procedure is that the standoff distance to REFERENCESthe target must be sufficiently small so that theweight of an equivalent sphere of air must be 1. J. Sova, "Summary of Armor Materials andless than 0. 1 times the weight of the explosive Configuration Tests at Aberdeen Provingproducts given by Eq. (33), For a 20 lb charge Ground, "Combat Vehicle Mine Protectionburied 3 in. deep, this observation means that Conference (U), U.S. Army Weapons Con-the standoff distance must be less than 38 in. mand, 28 June 1967 (Confi( -itial Report).for this approach to be strictly valid, Pro-bably this standoff distance can be doubled 2. J. K. Cockrell, R. Anderson, et al., "Phasewithout causing serious error. Most targets III Parametric Design/Cost Effectivenessare much closer to the ground than several Study for a Mechanical Infantry Combatfeet. This final restriction does not appear to Vehicle (MICV), "Cornell Aeronauticalbe very restrictive. Labs., Report 6M-2144-H-4, 20 February

1968 (Confidential Report).

In this paper a procedure has been devel-oped for predicting either the specific impulse 3. A. B. Wenzel, R, C. Young, and C. R.or the total impulse imparted to any target ex- Russell, "Structural Response and Humanposed to a land mine explosion. We have seen Protection From Land Mines (U), "Allisonthat the impulse is not primarily an air blast Division of General Motors Corp., Cleve-phenomenon. The loading is caused by mo- land Army Tank-Automotive Plant, TR 3481,mentum in the enplosive soil products. To June 1968 (Secret Report).calculate the impulse imparted to compleatargets, one must determine a shape factor 4. W. L. Kincheloe, "Reduction of Blast Ef-which Is a function of the geometry associated fects, " Final Quarterly Report, 0477-01with encounter conditions. The shape factor (04)FP, Contract DA-44-009-ENG-4780,for a wheel may be obtained from Fig. 3 and May 1962.the shape factor for rectangular objects suchas plates may be calculated using Eq. (24). 5. W. E. Baker, "Prediction and Scaling ofThese shape factors are substituted into Eq. Reflected Impulse From Strong Blast(31) to compute the total impulse imparted to Waves, " Int. Jour. Mech. Sci. , 9, pp. 45-51,any target. The specific impulse at any loca- (1967).tion on a target may be estimated by substi-tuting Eq. (33) into Eq. (1) and taking the appro- 6. W. H. Jack, Jr. . and B. F. Armendt. Jr.,priate component of the resulting specific "Measurements of Normally Reflected

impulse. Shock Parameters From Explosive ChargesUnder Simulated High Altitude Conditions,"

ACKNOWLEDGMENTS BRL Report No. 1280, Aberdeen ProvingGround, Maryland, April, 1965.

This paper is a direct outgrowth of con-tract DAAK 02-70-C-0579 between the U.S. 7. M. Lutzky, "Explosions in Vacuum,"Army Mobility Equipment Research and NOLTR 62-19, White Oaks, Maryland,Development Center, Fort Belvoir, Virginia, November 1962.and Southwest Research Institute, San Antonio,Texas. Under the terms of this contract. 8. P. S. Westine. "Explosive Cratering, ",Southwest Research Institute is to design, Journal of Terramechanics, Vol. 7, No. 2,develop, and test a new mine clearing roller 1970, pp. 9-19.system. A rational design of a new rollersystem required that we determine the natureand magnitude of the loads from a land mineexplosion.

107

[I

CIRCULAR CANTILEVER BEAM

ELASTIC RESPO1JF4 TO AN EXPLOSION

Y.S. Kim and P.R. UkrainetzDepartment of Mechanical Engineering

University of SaskatchewanSaskatoon, Canada

The response of a circular centilever beam subjected to a planetransverse air blast was obtained. From this response, the drag co-efficients of the circular cylinder under the unsteady flow conditions ofan air blast wave were determined using the domain conversion method.Using this method, a conversion of the response was made from the timedomain into the frequency domain and then from the frequency domain backto the time domain.

Two circular aluminum cantilever beams of lengths 5 ft. and 2 ft.,and diameters 3 in. and 2 in., respectively, were tested at the 11 psinominal overpressure location in a 500 ton TNT field explosion (Dial Pack)From the measured response, the drag force was obtained, and using theempirical equation for dynamic pressure, the drag coefficients wereobtained in the region of Reynolds number 7.81 x 105 to 3.7 x lO5 and Machnumber 0.41 to 0.23.

INTRODUCTION efficients under unsteady-state conditions suchas those from the air blast wave have only

Baker [1] obtained maximum responses of recently begun to be investigated (3][4].rectangular cantilever beams subjected to an air Bishop [3] obtained drag coefficients of ablast wave. The loading was separated into a circular cylinder and pressure coefficientsdiffraction phase and a drag phase, each having around the cylinder when subjected to an airpressure decreasing linearly with time to zero. blast wave of nominal pressure 20.7 psi. TheIn considering the drag phase, a mean drag co- obtained drag coefficients were in the range ofefficient was used for its duration. This Reynolds number (Mach number) from 3.75 x 106duration was determined from the linearly decay- (0.6) to I x 106 (0.25). The results agreeding time function which maintained the same drag roughly with steady state values. Mellsen [4]impulse as that obtained from the Brode [2] measured drag coefficients of circular cylindersempirical relation for dynamic pressure. For a by the free flight method at 12 psi and 8.5 psistiff beam, the predicted and the experimental nominal pressure locations. The blast waveresults showed good agreement. However, for source was a 500 ton TNT field explosion. Theslender beams, poor agreement was obtained, obtained drag coefficients were in the range of

Reynolds number (Mach number) fiom 1 x 106The elastic response of a circular canti- (0.433) to 0.44 x 106 (0.205). The drag co-

lever beam subjected to a transverse plane air efficients oscillated about the mean values ofblast wave was studied in this investigation. 0.67 and 0.48 for the 12 psi and 8.5 psi locat-The loading was separated into two phases as was ions, respectively. It was mentioned thatdone by Baker. However, the drag phase analysis Reynolds number and Mach number appeared to varyconsidered the actual drag loading which was ex- jointly in such a way that the drag coefficientpressed in polynomial form. Drag loading is did not change with the time varying flow of thedrag pressure times projected area and drag blast wave.pressure is drag coefficient times dynamicpressure. In this work, the drag loading (drag co-

efficients) from an air blast wave was obtainedThe drag coefficient will be a function of using the elastic response record of a circular

time due to a continuously changing Reynolds cantilever beam subjected to the air blast wave.number and Mach number during the air blast To obtain the actual drag loading, the elasticloading. The drag coefficients of a circular response of the beam was separated into a quasi-cylinder under steady-state conditions are static response and a sinusoidal response. byalready well known. However, the drag co- expressing the drag loading (forcing function)


in a polynomial form, the relation between the in the wake flow.forcing 'fnction and the quasi-static r6sponse Jwasobtained. Thus the forcing function was Mellsen and Naylor [6] arrived at a non-determined from the "known" quasi-static res- dimensionalized time for double vortices toponse. The quasi-static response was obtained fully form and start shodding. It was photo-by converting the time domain response into the graphically obtained from the interaction of the

freouency domain and then from the frequency air blast waves (oC peak ovcrpressures 10 to 17domain back to the time domain. iii)-with circular cylinders. This time is

given 'byBLAST LOADING

The overpressure-time record at some dis- - 2.S (4)tance from an open air explosion source can be- D

represented by Friedlander type decay [5]'whichis given by where U is shock front velocity, D is diameter

of the cylinder, and T, is the time for doublep(t) a P (I e- Kt/T. vortices to fully form and start shedding. This

is then the start of steadily decaying flow.

where P is peak overpressure, T+ is the posi-- As already mentioned, the net transversetive duration of the overpressure and K isa loading on a circular cylinder as a result ofdecaying constant. For low values of P -(about the air blast wave is separated into the dif-10 pAi), the overpressure can be represented fraction phase and the drag phase. Fig. 1 showswith reasonable accuracy by setting KR: 1. this type of loading. In the approximate pro-

- cedure for determining the net transverse load-The dynamic pressure in the air blast wave ing during the diffraction phase, the net trans-

is given by the emprical relation [2] verse overpressure is considered as a pressuret -t/T d decaying linearly from the normal reflected peak

q(t) - q0 T1 d (2) overpressure at time t = 0 to the drag pressured at time t =T I. The time T1 is cbtained from

(4). The drag phase then follows the diffrac-where q is peak dynamic pressure which can be tion phase.obtaine3 from the Rankine-Hugoniot relation

~17.S.qo X P, (2.S -- (3)

where r is the ratio of the peak overpressure -

to the ambient pressure. In (2), T is the 2duration of the dynamic pressure an Is usuallyslightly larger than T. due to the inertia ofmoving air. For low values of qo, the dynamicpressure can be expressed-with reasonable ac- .curacy using 8 a 2.

When the shock front hits the leading edge(stagnation point) of a circular cylinder, thepressure rises instantaneously to the noirmalreflected pressure. As the shock front passes Fig. I~over the cylinder, regular reflections occuruntil the angle between the shock front and theinteracting surface reaches a critical valuewhich depends on the shocX strength and the RESPONSE IN TIM DOMAIN

radius of curvature. At this stage, Mach stems Small deflection, linear elastic theoryarising from the incident and reflected waves (the Euler beam equation) will be used for pro-begin to forn at each side of the cylinder.

dicting the response of a circular cantilever~These Mlach stems cont-Inue to envelope the sur-

face until they cross over at the trailing edge beam subjected to a transverse plane air blastand start to go back around the cylinder. For wave.small shock strengths tip to about S psi peak The solution of the Euler beam equationoverpressure, the waves continue right back to with respect to deflection is represented bythe front of the .ylinder and croQ4s over onceagain at the leading edge. After this, thewaves become very weak and are rapidly dissipat- y(xt) - nl Xn(X) gn(t)ed. In the case of stronger thocks (for exampleabout 20 psi), the waves, on passing back across where X (x) are normal mode functions and g (t)the cylinder, form their otn comfplex Mach stems. are time functions which can be determined ?romThes2 never really reach the front of the cy- Lagrange's equation. When the beam is subjectedlinder because they break up and are dissipated to a uniformly distributed forcing function

110

p(t) a P f f(t), the time function equation CM = H - m if (H-m) is evenbecomes

PO = M - m - if (H-m) is odd.in (t) + Co )2gnCt) nft 6

n (12) can also be expressed as

where w are natural frequencies, . a%6 modal i 113participation factors, and Mn are generalized n ml i-0,2,4,..c-1) ) (13)masses. Dots in the expression represent time "(n)differentiation. m * 0,1,2.

The solution of (6) with nonvanishing Rewriting Gn(t) of (11) givesinitial conditions becomes

g (~)~ = ( nGn(t) gn*(t) g*(O) cos wnt - sin wnt

(t _ Po (0Gnt*ld n n n si wngn n

(wn)'Mn(14)An (o)

+ gn(O)cos W t + An sin wnt (7) g*(t) An sin({nt n) (IS)nnn'

wherewhere T is a dummy variable. get) -rO +M + C-tt + 6Mt

The strain (for zero initial conditions) at

the stagnation point at distance x from the root a f(t) 1 2) 1f()t. .of the beam becomes t n)n f M+) (t) -. +

n ~ )

M2f(M)(t)e(x,t) - P J, En(x).G (t) (8)(14/ H

4D2 tn dn(0) 2 h

where En(X) = 4 2 (V n 2 X (9) A_ ([g(O)J2 + [---- n

n n K) X (L) 2 Tx1 nOW g*(O)

In (8), Gn(t) is the nth mode response time On = tan k ;n (17)function; in (9), D is diameter, E is modulus of nelasticity, I is moment of inertia, L is lengthof the beam, Pn are constants of the normal In (15), G (t) was separated into two parts whichmodes X (x), and Kn are related to the roots are g*(t) and a sinusoidal response. gnCt) willAfKO of the frequency equation, be called the nth mode quasi-static responsen time function. This quasi-static response is

The normalized (in magnitude at t = 0) diffe-ent from the static response which can beforcing function f(t) is a polynomial of the Mth obtained from the Euler beam equation by settingorder in time. the inertia term equal to zero.

t2 + The strain corresponding to g,(t) becomesf(t) - a0 + ait + a2 to becme

for 0tsT (10) e*(x,t) P p En(X) g (t) (18)= 0 for t>T

g(t) can be expressed In matrix form asFor this forcing function, the nth mode responsetime function Gn(t) for 05t!T becomes g*(t) = (t) T (C (19)

(Cn) (19)f )si n~- dt n

n Ct) = onwheesint0n (t-Tdsiwhere {tT = [, t, t'M . .

= CO + c~t + C t2 .. + CrVn n n n n ,

CC-"Cn cos wnC - - sin wnt el1) C

n

where the coefficients are given by n

m ('1)1/2 (mmi)l,* a +i (1

n i=.0,2,4,. * ml le

m 0,1,2,..,M eIn (12)

The constant colunn vector (Cn) can be expressed

111

Z l4

by substituting fjTxof (10) and the infinite serifeexpansion of en nI into (23). The coefficients

[Cn) [Sn . (A) (20) are thus

here (A) is a constant column vector of the n (,2,4. (.)i / 2 ( u i ) l (29)l'ocin fuctin ft) nd S ]is 'Nx H upper n &,,..__

triangular square matrix which can be obtainedfrom (12). Then substituting (19) and (20) into In (29), A are the constants of f(T)e n w(18) gives expanded irin infinite series. These are given

e*(x,t) - P (t)r nl -n(x)[Sn]'(A) (21) by 0 (Cn'n)(m*i'j)

In (21), P (t)T(A) is the forcing function P(t). m~i * Jo ( j (30)

Thus if e*(x,t) is known, the forcing functionP(t) can be obtained from (21) by expressinge*(x,t) in polynomial form. where aj 0 for j > M.

So far dmping has not been considered. With the consideration of damping, the nthHowever, in the actual case there is a slight mode quasi-static response time function,damping in the response due mainly to internal g -4(t) e'¢nunt, can be expressed in an infinitedamping and aerial damping. When damping is 9sIe asconsidered, the strain becomes

1 2 2--~t*(t) en nt c0 + t+ tc + ~ 2 + c' t

ed(xt) P n l End(x) Gnd(t) (22) gn +C ne Cne ne

+... (31)where End(x) ' En(X)/[-(Cn)21

In (22), G (t) is the daired nth mode response wheretime funct gn which corresponds to Gn(t) in the m (-rnwn)(case of no damping. n cm E 132n

ne j=o (Mj) CndJ 32)

Gnd(t) • 0nd 6t f(,)e-4nns(t-r)in wnd(t.i)dBy taking terms up to the M'th order (!'Q M) in

(23) (31), gnd(t) e-Cnwnt can be expressed in matrix*d( 0 ) form as

[g d(t) - g d(O) cos w dt - - - n tgd(t) e = (t)T(Cne (33)

sin w ndt]oennt (24)

(t)eCWt - In (33), (C ne =(B n](C nd (34)

where [B I is a M' x M' lower triangular square

e-cnwn t matrix which can be obtained frol (32) and (Cnd)is a constant column vector of C d (j = 0,1,2,..

Se t -Ac t + * '). (Cnd) can be expressed as• gd(t) e'nn . nd'oS(wndt *d ) *

e'%nJnt (25) (Cnd) = ISnd]D](A) (35)

where i n " n (26) where RSd] is an upper triangular square matrixwhich ca be obtained from (29), and ID ] is a

+ Ct + C2dt2+ (27) lower triangular square matrix which ca bend nd nd obtained from (30).

2 (0) } .non~hen the strain corresponding to g*d(t).And '([gnd2(O)J

2 + cd 2 becomes

-I Wnd g9nd(0) C -1(x,t) • NOT}T F Dn)(A) (36)nd = tan ,d ( n) tan (l/tane nd) Cn(x t) = P't) ' xn

(28) Thus if the quasi-static response es(x,t) andCoefficients CO Cd . the damping ratios Cn are known, the forcing

nd' Cd' nd. are obtained by function P(t) can be obtained from (36) by

112

expressing e*(xt) in polynomial form. For this where 4case, an error will result due to truncation ofthe infinite series when taking terms up to the n(w) • ) Rf(w) (45)

N'th order only. However, if N' is taken suf- n)2 -?

ficiently large, the error can be minimized.

RESPONSE IN FREQUENCY DONAIN(46)

Taking Fourier Transform (FT) of (7) withzero initial conditions gives Taking IFT of a (w) using the real part only as

given in (44) yfelds

Pn POan W)Mwnfiw)Jf(w) (37) M - n [g~)# 2 (t)] (47)

n n gn (tan)2% wnn n~

Substituting (37) into FT of (5) yields a(t) and g2(t) are time functions obtained from

Px n((w) and d2CW), respectively.n i)( P(2) (38) n

2

where F1 (w) is FT of impulse response functionsin w t of the beam. This impulse responsefunctdon is causal, i.e. sin w t is zero for g(t) J 0[Xf(w)6(w-tn) - Xf(w)6(w+wn)].t cO. Thus

W cos Wt dwn

fn (w) = F{sin wnt u(t) " j

a ~ (0 = ~ 2 - f(wn) cos nt (9

[6(w + ton) 6 (w- n)] (39) If the response in the frequency domain

n(w) is known, the response in the time domainwhere 6(w) is the delta function, can be obtained from (48) and (49). Also, if

the response in the frequency domain is known,Forcing function f(t) is also zero for M(w) can be readily obtained because _2(w) has

t < 0. Let FT of f(t) be delta functions only. Knowing (hw), Af(w) canbe obtained from (45). Then by taking IFT of

No= f + j f (40) Rf(w), the forcing function f(t) can be obtaine.If the imaginary part of %n(w) rather than thereal part is used, a parallel analysis will give

Substituting (39) and (40) into (37) will give the same result.an(w) which has real and imaginary parts.

The domain conversion is made possible

FT of a real causal time function h(t)(71 using the Fast Fourier Transform (FFT). loweverIs FIT does not have the characteristic property of

FT of a causal time function. Thus the respons,() M(w) + jX( ) (41) likewise the forcing function, in the frequency

domain obtained from the above analysis and thatwhere obtained using FFT will be different. Actually,

R( NO cos w obtaining the forcing function directly from(45) is possible In theory only. In practice,

R) h(t) sin wt dt (42)' the quasi-static response can be obtained usingthe characteristics of FFT. The forcing functon

can then be obtained from the derived relationInverse Fourier Transform (IFT) of 11(w) is between the quasi-static response and the forc-

2 o. ing function. FFT Is simply an efficient methodh(t) ; R(w)cos wt dw of computing the Discrete Fourier Transform

(DFT) [8).2 RMX sin wt dw (43)

As shown in the Appendix, a sinusoidalAs indicated by (43), the causal tine function function (sine and/or cosine) can be representedcan be obtained using only the real or thefrequency domain. This Iscainary t d u g oy taccomplished by taking the sampling duration asimaginary part of FT. an integer number of periods corresponding to

The real part of Gn(w).is the frequency of the function. Impulses on thep n imaginaiy axis correspond to a sine function and

R a (w)) n (c.)J (44) Impulses on the real axis correspond to a cosine

n {wn )) } n n function.

113

Cutting (or smoothing) thase impulses in The FT of this response was obtained usingthe frequency domain corresponds to removing the 1024 sampling points (N) for the duration ofsinusoidal function in the time domain. This eight times the fundmental period (the same"cutting impulse technique" can thus be used to sampling duration which was used for the compil-obtain the quasi-static response of the beam. ation of experimental results for theS ft.- 3 inThe sinusoidal response is removed by making use beam). Since no damping was considered, sampi-of this domain conversion method. Ing was started at t = 0 (with sample intervzl

being AT'. The result is shown in Fig. 3. (FTA slightly damped sine function can still is shown for positive w only in all Figures).

be represented by impulses at the corresponding In Fig. 3, an impulse at wl on the imaginaryfrequency on the imaginary axis, except that axis corresponds to the sine function and anthere will now be a small side lobe across the impulse at wI on the re-al axis corresponds toimpulses. The sampling duration must be the the cosine function. Since the sampling dura-same as for the case of no damping. Thus, using tion is not an integer multiple of the periodthe imaginary part of FT only (refer to the corresponding to w2, the response in the frc-Appendix), the cutting impulse technique can quency domain at w2 is not of impulse form.still be used to remove the damped sine function.In this case an error due to the existing smallside lobe will result. However, the error isquite small as shown In the Appendix. 1500 ..........- REAL MGNRPPRr

A slightly damped cosine function will have ... IMAGINARY PART Aimpulses and small side lobes on the real axis M.1024

in the frequency domain. The cutting impulse 21 &

technique can also be used to remove this func- 00a Nt

tion, this time by using the real part of FT I. /NATonly.

Thus, when slight damping exists in the re-sponse, the sinusoidal function must be made a s0sine or a cosine function only with a zero phase -angle in order to use the cutting impulse tech- 2

nique. This is possible by starting the sampl-ing at time z fI

C .o . .. ..

nd nd aT n_ or T. in (25) -S 'nd : 'nd /o /

- Soo0

In Fig. 2 the response (strain) of an o0 4 0 600

aluminum cantilever beam of length S ft. and FRoUECY I a fa

diameter 3 in. (which was also tested in a fieldexplosion) subjected to a uniformly distributedexponential forcing function is shown. Here -,CC¢modes up to the 2nd were taken into account and 499 4

damping was neglected. The units of the respore Fig. 3 Response in Frequency Domain of Circularare psi, which is the result of dividing the Aluminum Cantilever Beam Subjected to Uniformlystrain by n W.l Dn(x)" istributed Exponential Forcing Function e-kt

(W 1 14S.5 rad.sec., k=20, L=5 ft, D:3 in.)

" Cutting the impulses at w1 by linear inter-polation using adjacent points implies removing

" , , the sinusoidal response corresponding to w! inthe time domain and also implies that the quasi-static response in the frequency domain has a

/ value at w1 which is the average of the two ad-, .-. jacent point values. After cutting the impulse.%

I FT was taken using both real and imaginaryparts. The result is shown in Fig. 2 and Iscalled the modified response. As shown In Fig.2, the fundamental mode is absent from this re-

Pig. 2 Responses of Circular Aluminum Cantilever sponse. From this modified response the quasi-Beam Subjected to Uniformly Distributed Exponen- static respnnse can be readily obtained and Istial Forcing Function e-kt (k * 20, 1, 5 ft,, also shown in Fig. 2. If the sampling is taken1) 3 in.) fine enough such that an integer number of

periods corresponding to w2 can be obtained foranother sampling duration from this IFT, the

114

sinusoidal response corresponding to w2 can also mately l.S%.be removed if needed.

Instead of using both real and imaginaryparts for the IFF, the real part or imaginary REAL PARTpart only can be used. After cutting the im- - AGINARY PARTpulses, IFT was taken using only the imaginary N , 1024part of FT. The modified response is shown in OT. t.-Fig. 4. This modified response is different 1000 W, Nfrom the modified response shown in Fig. 2 due f . I/N.Tto the folding effect refer to the Appendix).The forcing function e - Kt is around 0.001 at theend of the sampling duration. Thus at t - 0, -the error in the quasi-static response due to Soothe folding effect is around 0.1%.

4 0.0

.! ..... ... .. .........,.....10 1 40*',.OU

- -- I"t011.0PO o

+ co.+,,.,,<,, o ..,,,'++I o. o 1, o€* '¢u i 200 400 60

Fig. 5 Damped Response in Frequency Domain of-- ................ Circular Aluminum Cantilever Beam Subjected toc. 0Uniformly Distributed Exponential Forcing

D, l ,t,, Function e-kt (w, ' 145.5 rad./sec., k = 20,

Fig. 4 Responses of Circular Aluminum Canti- = 0.0074, L - 5 ft., D = 3 in.)lever Beam Subjected to Uniformly DistributedExponential Forcing Function e'kt (I. - S ft., To avoid ambiguity, the exponential forcinp

D 3 3 in., k = 20) function and the quasi-static response obtainedfrom the modified response considering damping

For the S ft.- 3 in. beam, a damping ratio were not shown in Fig. 4.of 0.0074 for the fundamental mode was observedfrom initial laboratory tests. The beam res- EXPERIMENT AND EXPERIMrNTAL RESULTSponse was obtained again, this time taking thedamping into account. To make the sinusoidal Two aluminum cantilever beams, eiic offunctioa a sine function (with zero phase angle) length 5 ft. and diameter 3 in. and the other ofonly, sampling was started at time Ts - 0.96 ms. length 2.5 ft. and dihmeter 2 in., were testedThe FT is shown in Fig. S. After cutting the at the expected 12 psi nominal overpressureimpulses at w1 (of the imaginary part), IFT was location in a field explosion of 500 tons of TNT.taken using the imagi nary part only. The result This test was conducted on JuJy 23, 1970 atis shown in Fig. 4. The differences between the Defence Research Establishment Suffield, Ralstogtwo modified responses of Fig. 4 are due to: Alberta. Circular aluminum rods of smooth sur-

face were fixed to steel bases using a shrink1. Sampling starting time T . fit method. rbe steel bases were then fixed in Z

the field by bolting then to concrete bases.2. Existing damping effect on the modified Four strain gauges (one facing charge one on theresponse. Damping effect on the quasi-static opposite side, and the other t4o similarlyresponse is very small and the difference located, but higher on the beam) were Installeddue to damping was estimated to be only about on each beam. Two strain gauges were used for0.3% each strain output. The straip gauges were

connected through bridges to r magnetic tape re-3. The remaining sinusoidal difference cor- corder which was located in . bunker.

responding to 1. This is duo. to the exist-ing side lobe in the frequency domain when An overpressure rec,,d obtained from adamping Is considered. The maximum differ- piezo-electric gauge at lhe expected 12 psience is about 2% (refer to the Appendix). nominal overpressure lo~ation Is shown in Fig.6.

From this record it coa be seen that the actualThe similarity between the exponertial peak overpressure is ipproximately 11 psi and

forcing function used and the quasi-static re- the duration is ahoyo. 0.23 sec.sponse was examined. The quasi-static responsewas smaller than the forcing function by approxi-

115

All A

Predicted(rad/sec) Actual(rad/secl

S ft.- 3 in. 145.5 144.0

a 2.5 ft.- 2 in 388.1 3955

!....? .TABLE 1 Fundamental Frequencies

, , To make the fundamental mode of the sinu-

soidal response a sine function only, samplingFig. 6 Overpressure obtained from 500 Ton TNT was started for the 2.5 ft.- 2 in. beam at Ts =Field Explo)sion at the Location of Expected 16 AT and was carried on for the duration of

12 psi Nominal Overpressurc, twenty-two times its actual fundamental period.For the 5 ft.- 3 in. beam, sampling was started

Strain outputs from the 5 ft.- 3 in. beam at Ts s 33 AT and continued for the duration ofat the location 1.0 in. from the root and from eight times its actual funJamental period. Thenthe 2.5 ft.- 2 in. beam at the location 0.75 in. a conversion into the frequency domain was madefrom the root are shown In Fig. 7. The responses using FFT. After cutting the impulses (on theshow that the ground shock arrives at the beam imaginary axis) corresponding to the fundamentalbefore the air blast wave. Due to the excita- frequency wl, IFT was taken using only the imag-

inary part to get the modified response. The

small free oscillation of the beam, mainly of result for the 5 ft.- 3 in. beam is shown infundamental mode type, is initiated. When the Fig. 8. The modified response is made up of ashock front of the air blast wave arrives at sinusoidal response (which har *odes higher than

the beim, the initial conditions of the beam 2nd) superimposed on the quasi-stat.c responserespOhseare no longer zero because of these (dotted line). The actual quasi-statlc responsesmall'free oscillations. (up to 65 as) was obtained by eye from the quasi-

static responses (which were obtained by takingtwo more and two less sample points at the

_____________-I Ibeginning)- using an "averaging" method (refer toithe Appendix). This response is shown in Fig.

by a dashed line. The actual quasi-static re-sponse does not have any sinusoidal responseremaining In it.

SMA Of LtI6TM * V nD SD4WTII *I

---- ACTUAL OVASI.STATIC IaCSPOS

WI~t~MACi1YIIMA CD - - -o..,,,..

6 8IA Of LIOTP $FT AND 1AMIT 3 4% -

Fig. 7 Response (Strain Output) of Circular IAluminm Cantilever Beams Subjected to Air

Blast Wave of Peak Overpressure 11 p.,iThe beam response from the air blast wave A 0 o t: &a G . ©-4 - .o

consisted of a slightly damped sinusoidal re- TIE lSt€oftsi

sponse (mainly of fundamental and 2nd mode type)superi mposed on the quasi-static response. The Fig. 8 Responses of Circular Aluminum Canti-domain conversion method and the cutting impulse lever BELan of Length 5 ft. and Diameter 3 in.technique can be used to remove the sinusoidal at II psi Location (AT = 0.125 ms., N = 2790)response and so yield the quasi-static response.

The modificd response and the actual quasi-The response was sampled by digitizing static response for the 2.5 ft.- 2 in. beam were

using a sample interval (AT) of 0.125 ms. The obtained using the same 1.-ocedure and are showntime t I 0 was set to be the time when the shock in Fig. 9.front arrived at the beam. Fundamental frequen-cies (w ) of both beams were obtained by taking Several attempts were made to remove theFT of the responses after 0.23 sec. (slightly sinusoidal response corresponding to the 2nddamped free vibrations now exist). The actual mode (..2) from the modified response using thefundamental frequencies obtained, as well as technique described previously. However, due topredicted values, are shown In Table 1. the difficulty of f'l=ing an integer number of

116

either of the two parts of (50) and (51) in eachcase. However, differences will be small be-cause the slope differences at the intersection

----- -ACtuAL os stATIC RsPt points are small.h ~ -QASI-STATIC R[SIPONS[ NO1 If-V94~

DAAG-- o PRIsIIIR Actual drag pressures were obtained from(50) and (51) using (21). In (21), modes up to

A the 6th were taken. Since slight dampingexists in the actual response, (36) should have

S been used. However, because damping ratios of10 modes higher than the second were very hard to

1 [ i iobtain and the damping effect was in fact verysmall,(21) was used instead of (36). The drag

os /UIIIJ'I1IM pressures obtained are shown in Figs. 8 and 9ind can be represented by the following:

ocL ,-----.-- -; -- -- 5 ft.- 3 in.beam0 01 00 003 0I ~ 00 1 coo0co

• I M, IsEco*os 0 - 35 ms Pd(t) = 1.396-24.986t-S87.10t2+26127.7!"3 (52)

Fig. 9 Responses of Circular Aluminum Canti- 2(+ lever Beam of Length 2.5 ft. and Diameter 2 in. - 5 ms Pd(t) -5.625-404.82t-7990.08t

235 55ms Pdt) -5625-074.8t379.t

at 11 psi Location (AT = 0.125 ms., N a 2796) +50734'8t3

2.5 ft.- 2 in. beamcycles of the 2nd mode from the modified response,it was not possible to construct an impulse at 0 - 15 ms P 1.630-42.55t(4774523t2+

w2 in the frequency domain. This difficultyarose as a result of the relatively large sample 15 - 35 ms Pd(t) a -11.265+S72.18t-22080.6t2

interval for U2 and the accumulation of error in .267967.9t 3

obtaining FT and IFT using FFT.As mentioned when dealint with response, the

An empirical relationship was fitted to the existence of the intersection region introducesactual quasi-static response time data (data an error. The error will be small, however,were obtained using a time increment of 1.25 ms). because the intersection region is very short.The type of curve fitted was a polynomial obtain- Bec'use this error will accumulate as more partsed using polynomial regression. Due to the are taken into account, the drag pressures (and"bumps" (a fairly rapid rise followed by a also the actual quasi-static responses) were notgradual fall) in the responses, the relationship obtained after 65 ms for the 5 ft.- 3 in. beamfor each beam was obtained in parts as follows nor after 35 ms for the 2.5 ft.- 2 in. beam.(actual quasi-static response Ps in psi and timet in sec): Drag coefficients can then be obtained from

Pd(t)5 ft.- 3 in. beam

Cd(t) = (t

0 - 35 ms: Ps(t) 1.446-31.643t-S87.1Ot2+

26127.7t 3 (50) For approximately the first 0.5 ms the flow3S 6S ms: P s(t) =-.369392.87t-7990.8t

2 is conditioned by reflection and diffraction of+50734.8t3 the incident shock and hence there is no true

drag coefficient until a steadily decaying flow2.5 ft.- 2 in. beam is established around the cylinder. However,

0 - 15 ms: P (t) 1.687-53.006t-4774.23t2+ the "equivalent" drag coefficient, as it may be299549.3 3 (51) termed, is useful for the purpose of construct-

15 - 35 as: P (t) *3.571+563.53t22080.6t2 ing the loading configuration on the beam as

s- 76.90 shown in Fig. 1.-267967.9t3

Drag coefficients obtained from the responsesExpressing the actual quasi-static response of the two beams as a function of Reynolds

in parts using polynomials gives rise to the number and Mach number are shown in Figs. 10 anddifficulty in matching the magnitude and the II.slope (rate of change) of the responses at thepoints where two parts meet. With regard to (50) DISCOSSIONand (51), there is a slight difference in theslope at the point where the magnitude of the Drag Coefficientsresponses Is made to coincide.

Drag coefficients obtained from the responseA smooth curve was drawn through the inter- of the S ft.- 3 in. cantilever beam for the

section point to connect the two parts for each duration of the initial 65 ms are for Reynoldscase. The regions of inaccuracy resulting (intc' number 7.81 x 105 and Mach number 0.41 (at timesection regions) are actually very short. The t 0 0) to Reynolds number 4.1 Y 105 and Machintersection regions cannot be expressed by tumber 0.23 (at time t 65 ms). Mach numbers t

117

,- I-+* r' = .- . -+ . .. -1 <' - -

differences were observed:

1. The upper critical Reynolds number is higherin the case of air blast drag coefficient values

2. The drag coefficients above the criticalReynolds number range are a little higher com-pared to the steady state values even consider-Ing the Mach number effect.

, a-3. The levelling off of air blast drag co-efficients at Reynolds number around S x 105 wasnot expected. Of course this region is in thecritical region where flow conditions vary con-tinuously. Thus, any small shift in the flow

0conditions can cause considerable changes. Thismay be the explanation for the "levelling off"

011- of the drag coefficient at Reynolds number aroud5 X 105. Variations in the drag coefficient

04 (the same kind of phenomenon as noted above)were also observed for the 2.5 ft.- 2 in. beam.

0, 045 J5 oI 0J 06 07 0,5 08 Drag coefficients obtained from the 2.5 ft.-Ol4l 10|45 0o 4 10101 01 I111 f# IIIIo1|r 0

40*1.I 103011 45,I 1614S.61 Wo 4838 2 in. beam response for the duration of theREYNOLOS No..,- initial 35 ms. are for Reynolds number 5.2 x 105

IAcH and Mach number 0.41 (at time t = 0) to Reynolds

Fig. 10 Drag Coefficient vs Reynolds Number and number 3.7 x lO5and Mach number 0.3 (at time t

Mach Number obtained from Elastic Response of 35 ms). Drag coefficients as a function ofCantilever Beam of Length 5 ft. and Diameter Reynolds number and Mach number varied between a3 in. at 11 psi Location maximum value of around 0.61 and a minimum value

of around 0.41 for the range of Reynolds number

and Mach number observed. It appears that thedrag coefficient increases considerably before

os - levelling off as Reynolds number decreases below3.7 x 10

07 -Elastic Response and Maximum Strain

0The response of a cantilever beam subjectedto an air blast wave will exhibit a side oscill-

2 ation perpendicular to the direction of blast

wave travel due to vortex shedding. This side05-

0oscillation is very small compared to the re-0sponse in the direction of the blast wave. Since

IX04 the shear Centre and centroid coincide, the bend-

ing vibrations about the two perpendicular axesare independent of each other. Thus, the ob-

03 tained strain outputs from the strain gauges(placed in the direction of the blast wave) didnot record any strain from the side oscillation.

03 035 04 045 05 055

,. , . , ,It" o, ,o40 The maximum displacements at the mid-sectior,of the two beams were:REYNOLDS No (. 10-

6)

(MACH No 5 ft.- 3 In. beam 0.01 ft.

Fig. 11 Drag Coefficient vs Reynolds Number and 2.5 ft.- 2 in. beam 0.004 ft.Mach Number Obtained from the Elastic Response As shown above, the displacements of the.of Cantilever Beam of Length 2.5 ft. and Dia- beams were very small and hence use of the Euler

meter 2 in. at 11 psi l.ocation beam equation is justified.

are lower than the critical Mach number (which Since the response was mainly of fundamentalIs approximately 0.41). Thus with regard to the mode type, the maximum velocity at midsectionsteady state values of the drag coefficients, t was obtained using the maximum displacement atReynolds number effect on drag coefficients is midsection and the fundamental frequency. Thus,expected to be dominant in this Mach number r.ngc the maximum velocities were:

5 ft.- 3 in. beam 1.55 ft/secDrag coefficients of a circular cylinder

subjected to an air blast wave generally follow 2.5 ft.- 2 in. boam 1.45 ft/secsteady state values. However, the following The velocities of the beams are very small com-

118

pared to the air particle velocity during the The forner is symmetric about the folding samp-blast loading (at t a 0, the air particle vel- ling du.ation tf(k z N)and the latter is anti-ocity was 504.S ft/sec). Therefore, the effect symmetric about tf.of the velocity of the beams on Reynolds numberof air flow around the beams can be neglected. Expanding R(x(k)) for a point k* (k I N

For both beams, maximum strains were ob- givestained near the quarter point of the first cycle R(x(k*)) * [R{(i)cos WI-I{ (i))sin 2 i]of each fundamental frequency. Analytically, i-0 _Nmaximum strains were computed using the loadingconfiguration of Fig. I together with initial 1 N-i N-i 2-k* 21

conditions. For the drag loading, mean drag iN kEOcoefficients of 0.48 and 0.562 for the S ft.- 3in. beam and the 2.5 ft.-"2 in. beam, respect- 1EI I(x(k)csN-i sk* 2kively, were used. Mean drag coefficients are N k-0 icOthe mean values of the drag coefficient for theduration of a quarter of the first fundamental I N-I N-1 2k* 2kperiod of each beam. Computed and measured N kE0 R(x(k)1ii0sin -- 'sin -W- imaximum strains are shown in Table 2.

I N-I N-I 2vk* 2%kComputed(uin/in) k4p N kWO- I

S ft.-3 in. 'beam 6 53(A.3)

2.5 ft.- 2 in.beaml 490 510 N-i 21 k* 2lk•In (A.3), lie cos -W--i-cos N izO Ni

TABLE 22r Nl 27Tk*. .21Nki NiExperimental and Computed Maximum Strains sin 2k- , N sin -ix*sn - and t=0sin

The computed maximum strains show close

agreement with the experimental values. Instead 2rk* 2o 1of using mean drag coefficients with dynamic N are analogous to the integralpressure of Friedlander decay type as was done formsabove, maximum strains aere also obtained usingthe actual drag pressures of (52) and (53). The c smaximum strains so found were very close to the 0o 2"k*y'cos 2wky dy, Icos 2wk*sin 2wky distrains determined using mean drag coefficients. s 2ak*ysin 2wky dy and 1sin 2aky.cos 2xkydy

The good agreement between computed maxi-mum strains and experimentally obtained maximum respectively. Thus R(x(k*)) becomesstrains implies that the obtained drag coeffic-ients up to at least 4 of the period of each * N-Ifundamental frequency are correct. Also, the 2-N k=O (k))}kk**good agreement suggests that the loading con-figuration used is justified. I R{x(k)N- 6

+N k=O k** kk**where 6kk,, is the Kronecker delta

The Fourier Transform pair for continuous k*i

signals can be written in the form k** Is k* and N - k*

X(W) z .ft x(t) e'jwtdt rk** ' I when k** is k*2 (A.1l) k

x(t) z f' A(l) )Jwtdw = -1 when k** is N -

The analogous Discrete FoL.,ier Transform pair In (A.4), the first term resulted fromto (A.1) is N-1 2nk*N- ~ j 2lI.)0 R{X(i)}cos -r - and the second term from

X(i) NI x(k) i- k 2A kFN~i) 1 x~k ) e1 N N1- ,{Rqi)j 2ak F the sameN=- (A.2) 1s2 N 2

x(k) = i X(I) -3 N result will be obtained. Therefore, it can be

where N is number of sample points, seen that T0 R((i))cos -- i is a symmetric

When X(i) and x(k) of (A.2) are separatcrd expression of R(x(k) ) folded about the foldinginto real and imaginary parts, it can be shownthat R(x(k)) is composed of two parts, sample duration tf and - i=O l(X(1))sin i isN-i 2Nk N-I 2Nk

1 R(X(i))cos L i and iO l(X(i))sin L i. an antisymmetric expression of WR(x(k)) folded

119

t! 2,

about tf. Thus R(x(k)) :an be obtained usingonly the real part or the imaginary part of X(i)If Rfx(k)) is finite up to tf. The same is true NNfor I(x(k)). 40

Thus, when x(k) is real, R{R(i))is symmet-ric and IT{(i)) is antisymmetric about the fi ding 400 jfrequency ff (i N).

o0 - REAL PART

Bt taking IFT using R{((i)) only (setting IMAGINARY PART

I((i)) equal to zero), a modified h x(k) which , .oo074is symmetrically folded about tf in the real 300 ,.1455pazt will be obtained. Likewise, by taking IFT N • 1024

using I({(i)) only, the modified h x(k) which is - - T . -antisymmetrically folded about t in the real W, WI

part will be obtained. Since R(f) is inter- - , II8T

preted as being periodic from the sampling 200 S

theorem, the Fourier coefficients between N/2and N-i can be viewed as the "negative frequency"harmonics between -N/2 and -1.

Likewise, the

last half of the time function can be interpretedas negative time. ,oo

The DFT of a sinusoidal function depends o ,upon the sampling duration taken. If the samp-ling duration is taken as an integer number of 0periods of the corresponding frequency, there ,10)will be impulses at the corresponding plus and ,t

minus frequency points iii the frequency domain. .soImpulses will be obtained on the imaginary axisfor a sine function and on the real axis for a Fig. A.I Slightly Damped Sine Function incosine function. The magnitude of an impulse is FrAquecy Dmin

.f the sampling duration is not an integerimber of periods, taking DFT will not produceimpulses. The fact that the occurrence of im- UNI

pulses depends on the sampling duration of theDFT can be explained by the sampling theorem 171.

Whlen damping exists, the sinusoidal function 40 j , .00o74cannot be represented with impulses only in te 05 ,.,4sfrequency domain. Sin )1t e 1

4 lt with w1 = N .1o24145.5 rad/sec and 41 = 0.0074 is shown in Fig. [ ,L2 *A.I. There are impulses at ± wI and small side N, N

lobes around ! wI in the imaginary part while the 2 NOA

real part is far different from that when there /ME. , N-is no damping (the real part is zero when there W M" 0

is no dimping). Cos w t e' l1lt will have the "2"same FT as sin Wlt e "C -1lt except that the real s #

part and imaginary part will be Interchanged.

In Fig. A.2, the imaginary parts of sin wlt. ,

e"41W lt are shown for different sampling durat- "ions. The sampling duration of I is eight times 0 (-Si Lthe period of w1 , the sampling duration of I1 Isshorter than that of I by AT,1 , and the samplingduration of Ill is longer than that of I by TI " ,o 0As shown in Fig. A.2, taking more or less samppoints than the exact number makes the side lobechange sign across the corresponding frequency. Fig. A.2 Imaginary parts of Slightly DampedAfter cutting the impulses, IFT was taken using Sine Function Obtained with Different Samplingonly the imaginary part for each case. Fer I, a Durationfunction very close to a slightly damped sinewave (with zero phase .ngle) of initial ampli- cases I1 and III, if the middle of the two re-tude 0.02 was recovered. For II and III, func- covered functions is taken (an "averaging" used),tions very close to slightly damped plus and the error dtic t, the existing damping can beminus cosine waves (with zero phase angle) of minimizedinitial amplitude 0.02 were recovered, respect-ively. Thus, for these cases, the error due tothe existing side lobe was 2% at most. For

120

From the above discussion, it is evidentthat when slight damping exists, the sinusoidalfunction must be a sine function only or

a co-

sine function only to make use of the cuttingimpulse technique; while in the case of no dampAing, the sinusoidal function does not necessar-ily have to be a sine or a cosine function only.

REFERENCES

1. W.E. Baker, W.O. Ewing, Jr., J.W. Hanna,and G.W. Burnewitch, "The Elastic andPlastic Response of Cantilevers to AirBlast Loading", Proceedings of the secondU.S. National Congress of Applied Mechanics,1962, pp. 853-866.

2. N.L. Brode, "Numerical Solutions of Spheri-cal Blast Wave", Journal of Applied Physics,June 1955.

3. V.J. Bishop and R.D. Rowe, "The Interactionof a Long Duration Friedlander Shaped BlastWave with an Infinitely Long Right CircularCylinder. Incident Blast Wave 20.7 psi,Positive Duration 50 ms, and a 16 cm Dia-meter Cylinder", AWRE Report No. 0 - 38/67.April, 1967.

4. S.B. Mellsen, "Drag Measurements on Cylind-ers by the Free Flight Method-OperationPrairie Flat", Suffield Technical Note No.249, Jan. 1969.

5. S. Glasstone, "The Effects of NuclearWeapons", Published by the United StatesAtomic Energy Commission, 1957, Revised in1962.

6. S.B. Mellsen and R. Naylor, "AerodynamicDrag Measurements and Flow Studies on aCircular Cylinder in a Shock Tube", SuffieldMemotandum No. 7169, May 1969.

7. A. Papoulis, The Fourier Integral and ItsApplication, McGraw-Hill Book Co., Inc.,New York, 1962.

8. G.D. Bergland, "A Guided Tour of the FastFourier Transform", IEEE Spectrum, July 1969

ACKNOWLEDGMENTS

The work reported herein was supported bythe Defence Research Board (Grant No. 1678-09)and the National Research Council (Grant No.A-3384). The authors wish to thank DefenceResearch Establishment Suffield personnel fortheir helpful suggestions and for the oppor-tunity to participate in event Dial Pack (500ton TNT field explosion).

121

j MEASUREMENT OF IMPULSE FROM SCALED BURIED EXPLOSIVES

Bruce L. MorrisU.S. Amy Mobility Equipment Research and Development CenterFort Belvoir, Virginia 22060

I-dimensional anaiysiswais peorfoR-d to determine the physical scaItngparameters governing the response of wheels to blast loadihg. Hopkinsonscaling was used to determine the proper charge size and location for one-iquarter scale blast tests. The total energy imported to the test wheelsby the detonation was determined, and the scaled specific impulse was1 clwculated. The test and calculation procedures are descrbed.J

INTRODUCTION IT tLl/2p 1/2 Time scaling

.* The utilization of mine neutralizationhardware require that this equipment operateunder the intense pressure of near-field ex- T2" .L3..plosive detonations. Since accurate theoretical "0Pknowledge of this explosive-target interaction Soil conditionsis limited, designers have had in the past to 113. cM1/2

resort to full-scale explosive tests to evaluate 3-02P/and prove their designs. This process is ex-pensive and time-consuming, so this Centerelected to use scale models to evaluate materi- 7r4- rtals and configurations for mine clearing rollerwheels capable of withstanding the blast effects Irs5 9 . Initial conditionsof 30 lbs of explosive. Z p and restraints

DERIVATION OF SCALING LAWS Z f

In order to correctly interpret the experi-mental results, it was necessary to determine z7= Ethe dimensionless products, or Pi terms, POgoverning the interaction between explosive and .L-target. Since the target wheels are in contact /f8

= aMwith the ground, parameters describing ambient 2 Response scalingair conditions were omitted from the analysis.It is believed that a major portion of the im- Tig= IL112

pulse imparted to the target is caused by the p1/2 M112soil being thrown out o? the, crater and im- Ipinging on the target. Westine [1] concludes /i0= Q

that density and seismic velocity, rather than Pa stress parameter, best describe the soil con- Lm=,ditions. These parameters, along with others For replica models, E /7 where m andgoverning this phenomena, are listed in Table I p denote model and prototype respectively. Ifalong with their dimensions in a force-length- the same material is used in the model andtime (FLT) system. MM 3."prototype wheels, I? = X Ie assume equality

Ten dimensionless products, or Pi terms, of blast pressure, ie, Pm Pp. These con-

can be formed from these 13 parameters. There straints are then applied to the above Pi termsare many techniques for creating this list of to establish the scaling law below.terms, and no matter which method is used, the Ianalysis is not modified as only the algebra isdeleted. Listed below is one set of dimension-less products or Pi terms.


• ; .

A7_ _ _ _ _ _ _ _ ---- - - - '; - " - . i

Table I. Physical Parameters Governing Explosive-Target Interaction

Symbol Description Units

P Blast pressure FL- 2

t Time T

p Mass density of soil FL-4T 2

c Seismic velocity of soil LT-I

L Characteristic length L

ri Shape of System -------

M Mass of target FL']T 2 -

g Acceleration of gravity LT-?

f Total load on wheel F

E Energy absorbed in wheel system FL

a Acceleration of wheel under blast load LT-2

t Impulse applied to wheel FT

<- Stress in target - FL"2

(T1 tm=t P Time scales as length TEST EQUIPMENT AND PROCEDURESratio

The test wheels were placed in a 3.5byV2 p m/Op Sam soil for model 2.5 by 2.5'foot dirt-filled test rig as seen in

and prototype Figure 1. The wheels were connected with a773 cm= cp common axle and secured to the test box by

rubber torsion spring with a total 'calibratedrr4 (ri)m=(ri)p Geometric similarity rotational stiffness of 1180 inch-pounds per de.,

gree of rotation.fl5 gm I g Since model and

5 prototype tests are The explosive charges (2 inch square blocksconducted in same of C-4 explosive cut to provide desired weight)gravitational field, were placed under th- center wheel with one-this term is dis- half inch of soil cover over the Lharge. Thetorted as an engin- charges were-detonated by M,.6 blasting caps,eering judgement. and the spring rotation wai measured by-,the

scribes in Figure 1.16 fm=Xfp Loads scale as square

of length ratio toprovide equalstresses.

17 7 Em- AE p scribe1.

f8 m=l a p springsykes

kmYl or Veel

fr- 0 Tmp 0 cagHopkinson has shown that blast pressure is afunction of stand-off distance R and chargeweight W as P=f(R/WI/3 ). Thus, if Rm=ARp,ur= A.JWp to produce equalblast pressures.

Figure 1. Test box 5howing rubber torsionsprings, scribes, and test yokes.

124

5l

IMPULSE CALCULATIONS -(Mo0 + -- 1For computational purposes, the wheel

torsjon-s rtfig system is as shown in Figure 2. 0 + ( K )Q = "( - ) (5)-

Boundary conditions on eq. (5) at t-O are O-Oand 0=orig. Thus,

I borig t

%% i5orig"

1.i For small angles (sin 0 0), the solution ofeq. (5) is given by

moFigure 2. Q= A cos w t + B sin w t - Mo "(6)

o=-A w sin w t + B w cos w t0 prelcad angle where

90max maximum rotation

K spring constant Substituting the boundary conditions into eqs.(6) and (7) yields

I total impulseA= Mo and B= Iitotal energy K

Mo initial spring moment .Omax occurs at a time tmax when 0 = 0, or when

A length of yoke tan w tmax = B/A

mass moment of inertia Sin w tmax and cos w tmax are calculated from

T torque tan w tmax , and the results are substituted into

Ignoring gravitational effects, the total energy eq. (6) to determine Qmx,absorbed in the test system is given by

E= ForceX Distance= [k(Op ) X Omax ] x A X 1 +O Y V (B/A)2 K

E- K(Qp+ max). (1) Substituting for A and B and solving for i2- yields

From conservation of energy, K.E.= P.E. andV= 2E/H (2) i= 2K1 max I + 2 Mo

From the impulse-momentum relationship,But Mo=K Op and the above equation reduces to

I a mv VIEm. (3) eq. (4).

VT Equation (4) gives the impulse imparted to5ubstituting eq. (1) into eq. (3) and realizing the three-wheel test rig, but the impulse on thethat Izintl yields center wheel alone is desired. This is approx-

1 meimated oyi_ -= VYK-1 Qma, A1+ 2W (4)

.ma x ic + 2 ic sin $ = i3 wheel

Equation (4) can also be derived using a where ic = impulse on center wheelconservation of momentum approach. Here,

T= 1 0 where 0 and 0 denote derivatives withrespect to time. d = depth to center of charge in feet

125

~!

Westine [2) has developed a method of REFERENCEScalculating the impulse imparted to a targetfrom a land mine detonation given by [1]. P.S.,Westine, "Explosive Cratering,"

J. Terramichanics, Vol. 7, 11o. 2, pp. 9I = A 0 to 19, 1970.

where I a total impulse [2). P.S. Westine, "Impulse Imparted to TargetsheeI - specifi impulse [] by Detonation of Land Mines," in SympesiumI - specific impulseA - projected area of target 7 in2 for on tn Detection and Neutralization,

these tests March 24-25, 1971, Vol 1 of 2, Fort0 = shape factor which is a function of Mc Nair, Washington, D.C.

target shape and standoff conditions [3]. W. D. Kennedy, "Explosions and Explosives

Shape factors are determined for the various in Air, " in Effects of Impact and Ex-configurations tested under this program. These plosions Volume I, Summary Technicalshape factors, together with the total impulse Report, NDRC, Washington, 1946.as calculated from eq. (4), are used to cal-culate the specific impulse generated by thedetonations. This impulse is transformed intoscaled specific impulse by dividing by the cuberoot of the charge weight. These scaledspecific impulses are presented in Table IIalong with the scaled distances and other dataitems and are compared to previous extrapolateddata [3] for TNT in Figure 3. The TNT data hasbeen adjuated to C-4 explosive. Data generatedfrom these small charges is thus seen to fallwithin the limits of data generated using ex-plosive charges of up to 550 lbs.

Table II. Scaled Specific Impulse

Shot Charge wt. Scaled Dist. Impulse on 1 Wheel 9 Specific Imp Scaled SpecificNumber (lbs) (ft/lb"3) (lb-sec) (psi-sec) Impulse

pst-mseclbl/3

2 .229 .068 12.1 4.30 .402 656

3 .285 .062 16.1 4.00 .575 875

4 .106 .088 10.7 5.45 .281 594

5 .200 .072 14.6 4.55 .457 782

6 .285 .062 14.1 4.00 .504 766

8 .420 .056 24.2 3.25 1.063 1420

9 .285 .062 13.9 4.00 i .496 755

10 .381 .058 20.1 3.40 .845 1165

11 .420 .056 21.4 3.25 .942 1260

13 .470 .054 31.7 3.10 1.460 1880

14 .285 .062 13.7 4.00 .489 744

15 .342 .060 15.9 3.60 .631 900

16 .395 .057 23.9 3.40 1.005 1370

126

2000 - iii

1000 *800 4+600 +

1 -40

03 20B %"L .""-

1SYBL SOURCE TYPE OF CHARGE 42 4o

V) 8 A BRL Bombs, all sizes -

6B8 RRL-ARD 8-550 LB bare chargeD DATA SHEET Bombs, all sizesP PUS 0.5 LB rect blocksU UERL 10 LB bare charges A+ MERDC Current tests P

2 \

1 11111II I lI I I II I !

.06 .1 .2 .4 .6 1 2 4 6 10 20 40

SCALED DISTANCE (FT/LB1/ 3)

Figure 3. Scaled Specific Impulse vs Scaiea bistance.

127

DYNAMIC ANALYSIS

THE EFFECTS OF MOMENTUM WHEELS ON THE FREQUENCY RESPONSE

CHARACTERISTICS OF LARGE FLEXIBLE STRUCTURES

F. D. Day III and S. R. Tomer

Martin Marietta CorporationDenver, Colorado

A computer program using existing mathematical techniques has beendeveloped for use In performing linear frequency response analysesof large elastic systems that contain axisymmetric momentum wheels(gyros). The program incorporates the state vector form of thedynamic equations of notion, and uses complex eigenvalue/eigenvectortransformations to yield a set of uncoupled equations that are easilysolved for the steady-state responses of the system. The structuralcharacteristics are input In the form of normal modes of the systemwith nonrotating momentum wheels (i.e., all translational Inertiasand the rotacional inertias about the axes normal to the spin axis

are included). These system modes may be derived directly from afinite-element model or through the modal coupling of a number ofsubsystems. The program was used to calculate transfer functions onthe Skylab Apollo telescope mount both with and without the controlmoment gyros. A comparison of these transfer functions is presented.

INTRODUCTION

The generally accepted method for deter- equations of notion only when modal damping orraining transfer functions on large systems is damping proportional to either the mass or stiff-based on calculating the normal modes of the ness matrix is assumed. The introduction of thesystem and using them to uncouple the equations momentum wheel terms, however, recouples theof motion. This method is based on the assump- modal damping matrix, and a second eigenvaluetion that the modal damping matrix is diagonal. s3lution must be determined. This is done usingThere are times, however, when the assumntion of the state vector form of the equations of motion.a diagonal damping matrix is not valid; for ex-ample, when a structure has gyros or momentum This meLhod was used to calculate transferwheels. The introduction of the momentum term functions for the Skylab Apollo telcscope mountproduces a skew-symmetric, rather than a diago- (ATM) in order to determine the effects of thenal, damping matrix. In the past, analyses of control moment gyros (CMGs) on the structure'sthis type have neglected the effects of the mo- behavior. The importance of this example andmentum wheels. The purpose of this study is to the reason for choosing it lie in the fact thatinclude the momentum term and develop a method these same transfer functions are used In de-to facilitate the calculation of the transfer signing the experiment poitting control systemfunctions so that their effects on the structure (EPCS) for the ATM. Because of the proximity ofcan be evaluated. the COGs (which are used for overall cluster con-

trol) to the ATM, their effect on the EPCS son-The method makes use of existing mathemati- sors could be significant. Transfer functions

cal techniques to develop the state vector form with and without the C1Gs were computed. Theof the dynamic equations -i motion. From these results are compared in this paper.equations, complex eigenvalue/eigenvector trans-formations are then generated and used to un-couple the equations of motion. Once the equa- THEORETICAL DEVELO. -NTtions are uncoupled, they are easily solved forthe harmonic steady-state response. The derivation of techniques used to gener-

ate acceleration responses or transfer functionsUndamped normal modes are used to describe for large flexible structures with momentum

the characteristics of the flexible structure, wheels will be developed in four parts:These modes are sufficient to uncouple the

Preceding page blank129

9~~- -- 17-777.W,~ ~ 7~

1. A brief review of the equations of and for steady-state responses,siotion for the elastic structure with-out momentum wheels; {4(t)) = jqo} e ift, (4)

2. Equations of motion for a spinningrotor as a free body; where the modal coordinate vector 1qol consists

3. Equations of motion for a flexible of complex numbers yielding magnitude and phase

structure with spinning momentum wheels information.

or gyros; Substituting Eq. 3 and 4 into Eq. 1 yields,

4. Procedure used In the digital computer when the normal modes are ortho.-ormalized on the

program for uncoupling and solving the mass matrix,

response equations.(j~ - ~] i 2CUwoRJ) io}

1. Equations of Motion for Large Flexible . 1T o (5)Structures (0 (5)o

sinceResponse analyses of large flexible struc-

tures generally employ component modal substi- - ol0tution or modal coupling [Ref 1] techniques to [meqJreduce the number of equations-required for the when the modes are orthoncrmalized on the wasssolution. These techniques rely on the ortho- matrix. Solving for 4qojgonality of the undamped normal modes of the

structure or substructure to produce diagonalequivalent mass and stiffness matrices. In ad- lqol (I2r21 2 +I r2~c-fl1)dition, modal damping is usually assumed ao thatthe final equations are uncoupled and readily 10]T f (6)solved. T{o")

In terms of the generalized or modal coor- Substituting Eq. 6 into Eq. 2 yields a ma-dinates, the uncoupled equations are of the form trix equation relating responses to forcing

functions in discrete coordinates:ml ()+ [rw1 { .2a ](q) -1

q P 0eq eq- iF(t)) (1) {Xo}= [41 - I + i 2;. o

where 01 Tfol" (7)

{q} - vector of modal coordinates; Fcr iirgle-unit input forces, Eq. 7 describes

m [41 T a 4,j the steady-state transfer function.[(in]

- generalized mass matrix; The uncoupled set of equations formed by

Eq. 6 and 7 can be solved quite efficiently by

2 1 ,T r1digital computers. The solution involves onlySmeq t [k matrix multiplication and the inversion of a

- generalized stiffness matrix; diagonal complex matrix. As will be shown inthe following sections, including rotating mo-

4W m 0Tc 0 mentum wheels fully couples the modal equationso eq] of motion, greatly increasing the computer cost

involved in determining responses from this for-generalized damping mulation of the problem.

(F(t)) - {T (f(t))

- generalized force, 2. Equations of Motion for a Spinning Rotor asa Free Body

and the discrete coordinates, (x), are relatedto (q) by the modal coordinate substitution Consider a rigid body spinning about one of

its principal axes of inertia. If the coordinate(x) - [] {q). (2) system is fixed In the body at the center of

mass and aligned with the principal axes of in-Now, for a harmonic forcing function, ertia, Euler's moment equations can be expressed

as follows:

(f(t)) - fo e t; (3) Io + 0 I 0 0 O;

x x y y

130

-mI + In the previous section, the equations of

y V X X notion in modal coordinatei,'Eq. 6, were un-z I + 6 1y - y 1X (8) coupled and could be-readily solved for re-

sponses. Equation 12, however, is fully coupledin the velocity coefficients. If one pvocaeds

For an axisymetric rotor (Ix -y = Is) with as in section I ari lets

constant angular velocity X about the z axis, intEq. 8 becomes (f(t)) = If t e ;

M. -X I x Ox + 6 y ( Iz " 1 s) X; (F(t)) - IFo1 e ift ;

y - 60 -Ox - Ao ; (9) (q~t)) -ilol

z 0. for steady-atate response to a harmonic input,the solution to Eq. 12 would be in the form

Since the inertias about any axes in the x-y Tplane are the same, the coordinates can be fixed - - 02 + i + []at the center of mass with the z axis coincident [r10 j onjwith the spin axis and the x and y axes nonrota- (13)tin&. The angular velocities and accelerations, Cr1 [ ] / fFo}.

6 and Oi' respectively, are then descriptive of The solution then involves invertiog a fully

the coordinate system's notion, while the spin coupled matrix that is also dependent on therate, A, relates the body motion to the coor- frequency of the forcing function. Since thedinate system. inversion must be performed for each forcingfrequency, the determination of responses from

In matrix form, letting y = I z - Io A, Eq. 13 is very time-consuming and costly. How-~z ever, these equations can be uncoupled by re-

Eq. 9 becomes writing them in state vector form and generatingcomplex eigenvalues and eigenvectors. The pro-

1 0n . 0 0 ( 1 0) c t b f e ul

00 F0 00 0 4. Procedure Used to Uncouple the CombinedL System Equations of Motion

The acceleration coafficient matrix is in-

dependent of the spin rate and can be included Rewriting Eq. 12 asin the system inertia matrix when the momentumwheels are coupled with the elastic structure. {q) + [B) {) + [E] (q) ( (F) (14)The velocity coefficient matrix, however, .isdependent on the spin rate and will couple the wheresystem modal equations. _

B [ + ]T [] [w;

3. Equations of Motion for a Flexible Structure [E]-with Spinnip Momentdm Wheels t @Wriig the equations of motion for the equations of motion can be expressed in termsirtng teeutosomoonorthe of a state vector by making the olon sub-

combined system in discrete coordinates, stt [ef. 2 a: following

m] (x) + (c] + [r]) fx) + (k] (x) - (f(t)) q+

where [m) now contains the rotor inertias about -.'-s nn.-mal to the spin axis, and [F] contains 1the skew-'ymmetric rotational-dependent terms, -y, transformed to the system coordinates. JR] - -- ;1 .0, 0 j(s

A rvird. nate transformation, (x) 10)€ (q), E __ !uz:6 "1v, undamped normal modes yields (15)

30 -1I

(q) + +2W ] T [,F] ) ;) [,I I

131

The modal equations of notion can then be ex- -pressed as hC -3 ItoI 17 t ill

JR) {b + [HI Z} = ID). (16) or

Preaultiplying by -[R] yields -o = a -170 -iR 3 ] Jol. (24)

-(i) -[R]-i [H] {z} -JR]- ! (D) This formulation of the equations of motioninvolves the inversion of a forcing frequency-

or dependent matrix that Is diagonal. If responses tor transfer functions are to be generated for

JU] (Z) - {7) - (J), (17) more than just a few forcing frequencies, it be-comes much more efficient to generate the com-

where plex eigenvalues and eigenvectors once and to[ 0 ,.0 ] invert the diagonal matrix r'o - i0. each time

[Rh L-.- "'-.-i, than to invert the coupled matrix In Eq. 13 for

11.o.0 -B each forcing frequency.

which must exist, and Substituting back to discrete degrees offreedom from Eq. 12 and 17 into Eq. 24 yields

[UB -B -0](H-0 -------

-JR] - -B -10 (f(t))$

The homogeneous form of Eq. 17 is the standard and, sinceformulation for an eigenvalue problem. Now,letting q }(Z) = €

{z) =( 0 eat, (18) q

the homogeneous form of Eq. 17 then becomes and(Jul - arl-o.]) {(o - (o), (19) {)-[ q H{)

the discrete coordinate accelerations can be ex-

where the set of values nj are complex eisen- pressed as functions of the discrete forces by

values with corresponding eigenvectors . -I

Assembling the eigenvectors into a matrix xi - i a (j 1f21 -a - n.]

[Y], the coordinate substitution (

(z - [V] (0} (20) [

into Eq. 17 produces T

Ju] I) w 1 - .j. (21) where 172] is the lower half of the matrix

Since, for the elgenvector matrix [7, [ -- .

Jl IT] - (v) t ,.J,

premultiplying Eq. 21 by 1[]- uncouples the SAMPLE PROBLEXsystem of equations In the form The following example shows how the method

-1 Was applied to include moentum wheels when de-1mJ (F,] - (.) - [] - (J). (22) termining the frequency response of the Skylab

ATH and associated substructures pictured inFor r harmonic input and steady-state responses Fig. 1. The ATM is the experlment pertion of

Skylab that houses the photographic and tele-

(J) - J Odi t; scopic equipment.

{(J-} 4 e ¢o e (23)

ao that Eq. 22 becomes

132

I

FIGURE 1. ATh RACK, DEPLOYMENT ASSEMBLY, SO0AR ARRAYS

Figure 1 shows the ATM rack, deployment The design of the EPCS depends on transferassembly, and ATM solar arrays. The rack is the functions calculated for various points on themain support, and houses the structure contain- ATM system. The goal of this task was to de-ing the experiments; in addition, it houses the termine the effect of the spinning CHGs on theCMGs, which are used to stabilize the entire EPCS transfer functions. Although the clusterSkylab cluster in orbit. These three MGs, control system, of which the CHGs are a part,which have a spin rate of 9300 rpm, are the mo- and the EPCS are independent control systems,mentum wheels considered in this analysis. the elastic motion of the CMG support structureFigure 2 shows the orientation of each of the will cause the wheel (gyro) to generate a torquethree CMGs to the Skylab cluster, that may be picked up by the EPCS sensors and

cause the system to respond.Figure 3 shows the ATM spar, canister, and

gimbal ring assembly (GRA). The spar is the Using existing models of the various ATMstructure on which the various experiment pack- substructures for the vibration analysis led toages are mounted. The GRA is used to aim the a total structural model with 1321 degrees ofvarious photoeraphic experimenLs. The GRA is freedom. Two factors associated with this studycontrolled by the EPCS, which has motion sen- dictated that the size of this model be reduced:sors on the spar. first, the computer cost to obtain an eigenvalue

I; - -... ............. ............ ......

TY. ,. ,-/

eTZ TX+Z

y

FIGURE 2. CM, SPIN AXES

133

solution to this large a model was excessive, To perform the transfer function analysisand second, the current transfer function pro- on the ATKA force input points and accelerationgram could only accommodate 50 modes due to output points were selected on the basis of theircomputer storage limitations. In light of these effect on the EPCS. The input points selectedtwo factors, we decided to use the constrained were the GRA flex actuator torque motors and thecomponent mode substitution method [Ref. 3], or rotor imbalance moments on the CMGs. The outputinertial modal coupling method, as it is more points selected were the fine sun sensor and thecommonly called, to determine the modal proper- EPCS rate gyros (both of which are on the spar),ties. To use this method, the ATH system was the flex actuators, and the CMGs.broken into three substructures, consisting ofthe spar and GRA, the ATM rack, and the canis-ter. For this analysis, the deployment assembly RESULTS AND CONCLUSIONSand ATh solar arrays are considered part cf therack. The results shown in Fig, 4 thru 7 indicate

that the CMGs on the ATh have a distinct effectBy using this method, the eigenvalue solu- on the ATM transfer functions. In general, the

tion for the final modes and frequencies was effect is random: that is, the amplitude of theperformed on a 288-degree-of-freedom system, transfer function for a given input/output com-rather than on a 1321-degree-of-freedom system. bination may remain unchanged. increase, or de-These 288 degrees of freedom represent component crease, depending on the frequency range inmodes of the various substructures selected with question, Prcdictably, the greatest effecta frequency cutoff criterion. The frequency of occurs when either the input or output point isthe modes obtained ranged from 0.009 to 68 Hz. near the CMGs, as can be seun in the figures.

* E--5

Despun CMGs

- I- I Spun-up CMS.: .I II ; ; II I I I I I ii I I

-u -_-_=

FREQUENCY (HZI i FREQUENCY IHZFIG. 4.- FINE SUN SENSOR VS. FLEX ACTUATOR FIG. 6- SPAR CENTER V1. CMG

Z-6

~~E-6E - - E- - 1 4

FREQULNCY IZ FREQUENCY (1111FI./-FIN. 5-U SCMSO VS. FLEX ACTUATOR FIG. - CMNTG VS. CG

13

-aR

-i -- .. .. . .....----- ---

- UiI/I . -4

-Rvv ULjJY HlZ) 6lR RQUENCY (ttzI

FIG, 5- CMG VS, FLEX ACTIU 0 FIG. 7- CMIG VS. CMG

134

Although the results of this study indicate r j1.0 Ithat including momentum wheels or gyros in the [Ri - ..--I- [ = state-vector velocity co-calculation of transfer functions for structures [1.0J 0 efficient matrixcontaining them can influence the magnitude of E Ithe transfer functions, the governing factors state-vector displacementon the magnitude of their effect are not clearly [ = -----|0 cotaefictoi paeent mti

understood. Limitations on the scope of this 0 -1.0] coefficient matrixstudy prevented investigation of some key fac- -tors, such as the proximity effect exhibited in -[R] [H] - state-vector characteristicthe example and the effect of higher-mode trun- matrix

cation on the validity of the transfer functions [I] - matrix of complex eigen-near the truncation frequency. vectors

[a J = diagonal matrix of complex

- matrx of udanpednormaleigenvalues~f[~NOMENCLATUREegnaus

1711 - upper half of [Y)

s[2] - lower half of [I]

[) =matrix of undamped normal Vectorsmodes, either generated di-rectly from a discrete coor-

(q} - vector of normal mode coordinatesdinate model or indirectly

through component mode ayn- (F(t)) - vector of generalized forces in normalthesis mode coordinates

[m) n mass matrix in discrete co- Tordinates - Ii MW

(ki = stiffness matrix in discrete (f(t)) - vector of discrete forces

[k) - damings matrix in discrete vctrodireefceapiuscoordinates (x = vector of discrete coordinates[c) - damping matrix in discrete. .fol - vector of discrete force amplitudes i

coordinatesd aeqo- vector of modal coordinate amplitude/

m diagonal generalized mass phase coefficientsN Jq - atrix

- [1.0J if the normal modes { - vector of discre.e coordinate ampli-

are orthonormalized on the tude/phase coefficientsmass matrix {O] - vector of moments in rotor coordinates

2m - diagonal generalized stiff- (O - vector of rotational degrees of freedom_ j ness matrix in rotor coordinates

v2oW o - diagonal generalized damping [ol - vector of normal mode coordinate gen-0.q matrix eralized force coefficients

[r] - matrix of assembled rota- Z} - state vectortion-dependent terms

No. of Rotors

[ (D) = generalized forces in state-vector for-J -l mulation of the equations of motion

Y 0 0 [T1 tJ) l -[RJ (D) {2}-- 0 0 ('*T - complex eigenvector

-] - vector of complex mode generalizedcoordinates

[T] = coordinate transformation rtmatrix relating the rotor 1 ol - coefficient matrix in (JI o elocal coordinates (0) to thesystem coordinates xl Scalars

[B] - coupled velocity coefficient f = circular frequency of forcing functionmatrix for inclusion of mo-mentum wheels W - undamped natural circular frequency of

_ T th mode

[E] 0 - generalized stiffness matrix

135

I -mass inertia about principal axis

e - rotational degree of freedom about axisj- constant angular velocity about rotor

spin axis

y = angular momentum

a complex eigenvalue

Symol diagonal matrix

[ I transpose of matrix-1

- inverse of matrix

[--] - partitioned matrix

- partitioned vector

REFERENCES

1. W. C. Hurty, "Dynamic Analysis of StructuralSystems Using Component Modes," AIAA J.,Vol. 3, No. 4, pp. 678-685, April 1965.

2. P. W. Likens, "Dynamics and Control of SpaceVehicles," NASA TR 32-1329, Rev. 1, Jet Pro-pulsion Laboratory, Pasadena, California,January 15, 1970.

3. W. A. Benfield and R. F. liruda, "VibrationAnalysis of Structures by Component ModeSubstitution," AIAA/ASME lth Structures,Structural Dynamics, and Materials Confer-ence, Denver, Colorado, April 22-24, 1970.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the effortsof Mr. Jack Nichols and Mr. Wayne Ivcy, ShE -

ASTN-ADS, Marshall Space Flight Center, fortheir aid in obtaining permission to use theSkylab ATH modal data obtained under ContractNAS8-24000.

136

T

i'a

INTEGRATED DYNAMIC ANALYSIS OF A SPACE

STATION WITH CONTROLLABLE SOLAR ARRAYS

Joseph A. Heinrichs and Alan L. WeinbergerFairchild Industries, Inc.Germantown, Maryland

Marvin D. RhodesNASA Langley Research Center

Hampton, Virginia

An integrated dynamic analysis and corresponding digital computer simulationfor application to a space station with controllable solar arrays are presented.The analysis and simulation have been developed for the primary purpose ofevaluating dynamic load Interactions between the solar arrays and the spacestation which can result from orbital perturbations of the combined system.Integrated into the analytical formulation are the dynamics associated with thespace station, the solar array flexibilities and their respective control systems.Application of the simulation is made utilizing present concepts of a spacestation with large area arrays and of typical control systems. A structuralanalysis of Ahe flexible solar arrays Is initially required to providemodal data for the simulation; and analytical results for an array concept aregiven.

A verification of the structural dynamic methods used Inthe simulation ispresented. This verification is accomplished by the application of the simu-lation to a problem of known solution, a uniform beam subjected to a unit stepload applied at mid-span.

INTRODUCTION design results, and the primary array frequenciesmay fall within the control system bandwidth.

The solar cell and battery system has been A digital computer simulation for evaluating thesuccessfully used on many small spacecraft; dynamic interactions of large solar cell arrayshowever, space stations of the future will have power and orbiting space stations has been formulatedrequirements which are much larger thin those and considers the dynamic characteristics ofwithin the present design experience of solar cell the array structure and the required systems forsystems. Therefore, the solar cell arrays used attitude and orientation control. The objectiveson future spqce stations must be relatively large of this simulation were to (1) provide an auto-and be capable of tracking the sun in a manner mated methodology of Interaction loads analysisthat does not restrict the desired space station for use as a design tool, (2) analyze present arrayorientation. This Is usually accomplished through structural concepts which are to provide 100 KWthe use of an orientation control system (OCS) for of electrical power to future space stations andthe arrays. A potential problem exists duo to (3) obtain an indication of space station stabilityundesirable Interactions between the solar arrays by its real time motions.and space station caused by required control andstabilization forces combined with external portur- The equations of motion for an orbiting spacebations. station with attached controllable arrays have been

generated and were digitally programmed for solutionSpacecraft Inmtabilities have been observed by numerical integration techniques. In the develop-

in the past when flexible appendages are par' of the ment of the system's motion equations, a modalsatellite structure. This past experience is sum- synthesization technique was employed whereby themarized by Likins and Bouvier (1) . Because of the elastic characteristics of the arrays were describedrequiremento imposed upon large area solar arrays, by a finite set of orthogcnal cantilever ndes.a weight-efficient design rather than a stiffness Only rigid body motions of the space station were

137

'p ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Lt a-C2JSr~s<~ ~~~"' I 4 4 C~ .k

considered. The rigid body/flexible body Interface synthesis analysis of the array structure waswas described by space station acceleration in terms performed by a stiffness matrix method utilizingof Induced inertial loads forcing the flexible arrays; equivalent discrete element structural models Iin like manner the flexible array root forces and representing 600 Inertial degrees of freedom. The tmoments acted as forcing functions on the space modes utilized in the simulation were chosen onstation. Structural mode descriptiona of the arrays the basis of a significant percentage of loadwere required as Input to the simulation; therefore, participation in Interface force and moment.a structural analysis of the elastic system was Simulations were performed on the orbitingrequired prior to the performance of this simulation, structural system perturbed by Initial attitudeProvision was made In the simulation for closed errors and external forces representing docking.loop attitude control system dynamics of the spacestation and OCS dynamics for the solar arrays. The To demonstrate the adequacy of the method-latter control system provided the desired orientation elegy which has been simulated, a problem ofof the arrays with the sun by controlling the rotation known solution was selected --- the uniform free-about the orbit-adjust and seasonal-adjust ax.. free beam planar response to a unit step loadOutputs of the simulation include interaction forces applied at mid-span. The flexible appendage solarand moments, magnitudes of all motion variables arrays were represented as cantilever uniformand control parameters as functions of time. beams, having the first five bending modes as

flexible degrees of freedom. When coupledThe formulated simulation has been applied Inertially with the rigid body translatien mode, the

to an extendible solar array structural concept cantilever mode solution yielded results for freq-

and space station which are presently undergoing ueny and loads which compared favorably with theseparate engineering evaluations. Also, candidate exact free-free beam solution.

array orientation control and space station attitudecontrol systems have been mathematically described CONSIDERED PARAMIETERS

and digitally programmed for this application. Two The presented analysis and correspondingattitude control systems were provided for the space simulation Is intended to be applicablo to futurestation; they are the reaction jet and control moment space stations with controllable solar arrays suchgyro (CMG) systems. The necessary modal as that shown In Figure 1. Structural concepts

II

Fig. 1. Space Station and Solar Array Structural Concepts

138

2,of arrays and space stations as shown and associated liroviding sun aignment within a specified'timecontrol systems are presently undergoing separate after leaving tie earth's shadow. In addition,engineering developments without regard to a total the OCS mus, meet accuracy requirementssystem dynamics criteria for minimizing inter- despite experienced space station disturbancesaction loads. One objective of the present analysis andf provide minimum dynamic excitation to theis to assist In the development of dynamics arrays. Two generic types of OCS drive systemscriteria for each of the cormponent structures from have been considered in the simuiation and aresanalytical results of total system characteristics. he continuous and non-linear drive system, s.'

The continuous-type drive system employs eitherIn order to account for all of the significant a DC torque motor or a variable-frequeny

dynamic influences upon the space station and synchronous motor as its drive element. Asolar array load interactions, the following block diagram of the cortnuous-drive-OCS modelparameters were considered as basic and are contained In the simulation is shown In Figure 3.accounted for in the simulation

Conhns.aio "4colr

0 Solar array flexible body dynamics in __

terms of generalized modal coordinates.

* Space station rigid body dynamics. nik M

* Space station control system dynamicsincluding guidance and command.

* Solar array OCS dynamics Includingguidance and command.

It was assumed that the space station 71 - m or lmecon.st

structural frequencies are significantly higher than Kn " r co:fic.,.,

the solar array frequencies and are decoupled from IA - sr(, ,ttis biourotation axis

the control system. The analysis considerations K g..?- &W ti.s vm.,stanM for lopcorn;timtIon

can be best described by the block diagram shown K - 8E gear r4oIn Figure 2. This illustration shows the breakdown 0, - sSe nfiu

of the basic parameters together with the considered Fig. ., Contiruous Drive OCS Modelcounling paths.

The non-linear drive OCS Is similar to the contin-a Auous drive system with the exception that the

or.ynarl control logic of the non-linear OCS is operatedin an on-off manner. When the array errorexceeds some preselected threshold value, themotor Ia turned on until the array Is driven to

Control a null position at which point the motor is rwitched,Sp¢ eIatlomi. off. This threshold value has been made a user

option in the simulation so that its affect uponloads can be evaluated by parametric studies,

Two generic types of control laws/orquersJ:,,i .Solar Array have been Incorporated into the simulation forCenerator i i OrlevvIaln " ,Driver attitude cotrol of the space station and are the CM,

and the reaction jet control system (ICS). The

Analysis ConsideratIons CMG control system is used for precision attitudestabilization against eyclic disturbance torqueswithout the need for propellant exr'enditurc. It

The constraint placed upon the representation consists of three two-degree-of-freedom controlof the array dynamics Is that they bp described in moment gyros with parallel outer gimbals andterms of their orthogonal cantilever modes. The with their momentum vectors Initially equallyorientation and space station control systems spaced In the orbit plane (Figure 4.). Thisthat have been Incorporated Into the simulation partict,'ar CNIG configuration permits simple Aare those designed to complement the present steering laws and a planer, rather than threespace station and array configurations. The dimensional, enti-hangup law. The CMG controlprimary requirements of an OCS are maintaining dynamics included in the simulation (2] ha? aa desired accuracy with the sun vector and system frequency of 1.4 IN and a damping ratio

139

Z Axis C Outer Gkim Spbas

X Aded ot lomon KKsu) _° 1+Odter

GimbalAxis

MO Axisl Sc a Commended Attitude Anigle

0,; - Attitude Artle: late

IVASr K*, Kh Attitude Argle Galn. Rate GalaGimbl P v Control Loop DeadbadAxis

T a RCS Torque L lI w Sp-e Station Moment of tnertla

Fig. 6. RCS Model

ANALYTICAL FORMULATIONFig. 4. 3 PM Centrot Moment Gyro Array

Space station and solar array motion equationswere formulated together with the Interactive

of 0. 707. The RCS Is primarily used for dynamics provided by the respective attitude and

reference attitude acquisition maneuvers and the orientation control systems. The method givenmomentum desaturation of the CMG system. It by Likino [4] has been used as a basis for theis also an alternate to the CMG for controlling methodology provided in the simulations. Theattitude of the space station. The RCS is depicted in simulation developed from the employed mathe-the sketch of Figure 5. It is comprised of 4 matical models has been successfully run on thesets of quad thrusters providing redundant control CDC 6600 computer. The simulation modelabout the pitch, roll and yaw body axes. All utilized for the mathematical system is describedmaneuvers using the RCS are performed by firing below.the thrusters in pairs. The RCS model includedin the simulation is depicted in Figure 6 and * The space station and the two arrays arethe corresponding control equations are given In each modeled as interconnected bodiesReference 3. with each of the arrays permitted

controlled rotations about the spacecraftIn order to account for space station disturb- attachment points. The allowed axes

arices such as crew motions and docking forces, of rotation consist of those parallel to theprovision has also been made In the simulation to space station roll axis and the arrayallow for the application of time dependent forcing vane axis.functions.

* The flexibility of the solar array Ismodeled by means of a truncated set of

+ yau Z cantilever modes which Is excited bythe acceleration of the array support.

+ A difference equation technique [5J isutilized to obtain the modal response.

* M~aneuver and attitude control of the spacestation together with the solar array or-ientation control are modeled In terms

%, of the transient forces and torques pro-duced by closed loop guidance equations.Both the space station attitude and array

* l'itch orientation guidance commands arecomputed external to the structuraldynamics section. These provide thespace station with a fixed orientationrelative to orbit coordinates and point

Fig. 5. RCS ,Jet Location/Function the solar arrays at the sun.

140

The array driver gear train for the. axis The following notation appliesparallel to the roll axis Is modeled asan ideal mechanical transformer. The J - lndex.of solar array, J is equalvare driver axis is directly driven, and to 1 or 2either or both motions about these axesmay be rigidly constrained. FAj T A " qare the transient forces and

J torques produced by the flexible9 The simulation orbit generator uses array dynamics

Lyddane's method (6) for near earthorbits. The generator is included In T - torque exerted on space stationthe simulation to provide a reference for by hinged body along constrainedguidarnco commands. A block diagram axesrepresentation of the simulation programis presented in Figure 7 where Impor- mT - total system masstant logical switches and function Inter- Trconnection have been clearly delineated. 0 - Newtonian reference point

RIGID BODY AND FLEXIBLE BODY DYNAMICS 0 - Spacecraft reference pointFORMULATIONS B

Salient equations and techniques used in the CS. S. Spacestationcenterofgravity

simulation are shown below. A complete fornu- CG - Center of gravity for the entirelation c! all equations may be found in Reference 7. systemThe space station and array are modeled as a THsubsystem of Interconnected bodies whose motion CGj - Center of gravity for the JIs described by the Newton-Euler equations of hinged bodymotion. RO Radius vector In ECi

The connected space station and solar array Rbodies In the Earth Centered Inertial (ECI) FR External force applied to spacecoordinate frame are show,, In Figure 8. The statiouifollowing three vector equations are utilized. sai

System Force Li Angular momentum of J array

body abouts Its CG f2 - L Angular momentum of space

mT d 2 (Re+C = + FR (1) station about Its CG

h - Vector from space stationreference point to space station/array interface

Space Station Moment-jvector from the space station/

d ri array interface to the CG of(( ) x 1 1 + 'xI the jTH array body (on space~--( 0 ) jstation)

it - vector from point of external

+ T -r FA -r- (2) force anplication to spaceA i J rej ' I A - station CG

- vector from space station+ T + T reference to system CG

AC CNIGTl

T - Interface torque on .J array

Jth Array Body Moment body produced by OCS

d r' - (fo- h ) force moment armdt (J -T - rj x(30._FI (jL) T -rj xF F - Interface force exerted by space

I station on jTlI array body

141

. , - ~ ~ ~ ~ .t n- ~r~~i&r Va z. 's4 .~ ra~n~z n~z ~ MudS~aas 3I~....Eg ai!

-777"N

d reference frame. This requires the updating of

implies differentiation w. r. t. specific direction cosine matrices during thet an inertial reference frame simulation. The principal coordinate frames and

. -direction cosine identities utilized are as follows:T CMG -control torque exerted by the

control moment gyros (or s Lo X where iS the vector

reaction jets) T In the (ECI) coordinates,IX [ {X5 J Xs Is the vector in space

T 1station coordinates, andInittatisalion IXA I Cj X XA is the vector In solar

2 array coordinates.

The direction cosine matrices are calculated indit Routine the simulation in terms of Euler angles and areEi Rupdated periodically by the following equation:

ICMiajor Cycle Furtiona i

o Appendage Equation Updateo space Station Guidance '113 ito Solar Array GuidanceIntermediate Step 0mi2

411. 12 ir 3 Vwherec l, w12, W13 are the rotational rates about

Dynamica Equationa the ith coordinate frame axis. The rigid bodyo CNIG/teaction Jetso Appendage ynamics scalar equations derived from those presentedo Rigid Bodies Dynamics above reduce to the matrix form shown in Figure

9. The submatrfees (Aij represent the linearBlk 4 i term coefficients, Vol and ol0 represent the rigid

Inisgration Packsage 1 ody translational and rotational accelerationsrespectively and LAI the rigid array rotationalaccelerations relatiA to the space station

Fig. 7. Simulation Flow Chart coordinates for the two unconstrained axes ofrotation. The right hand side of the equationsrepresent the applied forces, torques (control

Space sttion torques Included) and all non-linear terms.Controlling torque profiles are computed in the

- CG simulation at designated time increments by0 S Space the appropriate control equations and are used toSeCO staion force the above matrix equations. The array and

r RF space station attitudes are referenced to theirrespective coordinate systems and are periodically

"- updated during the simulation.

42.4 ' The flexible array dynamics formulations areo adapted from the flexible appendage equations

;0 09 developed by Likins [4. Shown in Figure 10is a sketch of the flexible array geometry utilizedin the analysis. it is assumed that the particlemasses of the array have negligible inertias and

ON deflections are sufficiently small that linear struc-tural analysis it valid. The force on the ith mass

Fig. 8. Rigid Body System Geometry Is given In Equation b.

d2F, = in1(1 -+1

Equations 1-3 were formulated Into a set Fof matrix equations for the facilitation of The terms used tare defined in Figure 10.digital computations. The system force and spacestation moment equations were formulated withrespect to the space station reference frame andthe array body equations were written In th array

142

V V.

LPVZVL t>V-, - -V - ~ t.I'V~t4I

-- -- -- -- M is the mass matrix

[A] [ 5] [,] [A,3] ,.) K is the symmetric stiffness matrix

Do(3 L is the matrix of rigid body inertial loads

(Reference 4)

i i i0 ri (I B and G ace force coefficient matrices which1 2 [6 1 ] 14 ] ! 2 F2 are dependent upon rotation rates (Reference 4)

0o3

- (4) The order of the above equations of motions are3N where N is the number of discrete masses

, comprising the dynamical system. Following the

[A 3 ] [ 7 It 1 1 [I, I ,A1 method of Reference 4, It Is of convenience toLA transform Equation 6 Into orthogonal coordinates

representing cantilever modes of vibration. Thesystem of equations in orthogonal coordinates cfhn

then be truncated on the basis of some chosen

4A2 F4 i - criteria or engineering experience. This procedure1 A4 1 L 1 1A 1 *A21 F4 permits orders of magnitude reduction in the number

4] Z522 of equations describing array flexibility. The

discrete coordinate equations can be transformed tonormal coordinates by orthogonal transformations

Fig. 9. Rigid Body Matrix Equations prodticed by most automated matrix methods ofstructural dynamics analysis. The transformedequation then becomes

where Il, 7, arc N1 X 1 matrices and M is thehlef. AqI. number of cantilevered modes utilized. Note that

Iling' It""% a modal damping term (2ecr) has been arbitrarilyinserted in the classic manner of structural

analysis.I'lexillet Arrav

+ 7 The assumption has been made in going fromPl.~l I 4r Ir Equation 6 to Equation 7 that the motion dependent

If- - e'ctot i,.hn of ITit I , matrices which are functions of rigid body rotationmat particle of flexlii h ass 1'oin rates, are small and have a negligible effect upon

asv,;r pirn. the resulting transformation procedure. Withoutcor sion o

T it this assumption the simulation would be requiredfalp pnrice for urml.,rmed to he performed in discrete coupled coordinatesaippervIlsge

- t ior ileflhtlion of iast with resulting manipulations of large order

porticle matrices.

Fig. 10. Flexible Array Geometry The left hand sides of Equations 4 and 7 areconstructed in the simulation from computeddirection cosines, the rigid body inertia tensor,

Substitution of the appropriate direction cosine center of gravity and appendage attachment locationsmatrices and consilderailon of the appropriate In the space station coordinate frame and modalproperties resulting from elastic deformation properties of flexible appendages. The lattergives the following: includes deflection coefficients, frequencies,

damping coefficients and masses, for a chosen

nunlber of modes.[MNl*q +[K~q -[G~q -[11)4 + 1, (6)

ri 1 1 2 2 2 N Rloth the rigid body andi elastic equationswhere q " 2 3 1 2U U1 are solved sequentially employing a finite

N N] difference method. A change to the IntegrationU2 U procedure in the simulation is presently being

1143

4.4

made and incorporates the simultaneous solution exercised independently and the results correlatedof the equations. Excitation of Equation 7 is with known data. At that point, the completeaccomplished by the internal loading acting on each program was checked for continuity and a problemof the discrete masses resulting from the trans- of known solution was then executed to verifylational ani.rotational accelerations of the rigid the structural dynamics methodology contained inbodies. The rigid body equations are in tirn the simulation. The problem --- a free-free beamforced by the application of the flexible body with zero damping subjected to a concentratedinteraction loads, control system torques and force at mid-span --- was selected because itexternal forces. The simulation computes was considered to be a good test for solutioninteraction loads from the following definitions, convergence. It also provided information

concerning the accuracy of the analytical approach

F = m Transient and programming techniques. Closed form solu-A J 1. 71 force tions for the modal response of free-free beams

subjected to concentrated forces are provided by

T r in O " Transient Leonard (8J. In addition, two other solutionsA [ moment for the response of a free-free beam were obtainedusing numerical integration methods. One

solution was obtained by the method provided indefined by aReference 5 and the other by an independently

derived method using a variable order Adams

F -F + C F integrator. These solutions provided a basisINTj IIJ Aj A for verification of the simulations.

T + JT It was necessary to approximate the free-freeTINT = H + j TA (CjJ JrC beam configuration in the simulation by two canti-

lever beams, attached at their points of constraint.where Figure 11 shows a sketch of the cantilever geometryFw is the total force exerted by the th and associated coordinates. The "space station"

INTflexible array/hinged body combination mass has been set to zero and the "solar arrays"

on the space station.0. Symmetric About

TT total moment exerted by the J Ttflexi-ble array/hinged body combination on x,the space station. '

FIf , T It are the hinge forces and moments A*J Jon the jTiI rigid hinged body. _

and

Cj is the appropriate direction cosine matrix Z(internally coniputed). F)

T(t) - aoThe above completes the outline descriptionof the formulations and the computations used Fig. 11. Cantilever Beam Simulation of ain the simulation. The simulation computes all Free-Free Beamtransient variables at specified tine incrementswith the option of automatically plotting these represented by cantilever-free beams. An analysisvariables with the Calcomp plotter. As prey- was initially made for system elgenvalues andlously mentioned, a basic assumption of rigid elgenvectors of the two-cantilever arrangement inspace station dynamics Is made in the mathematical order that comparisons could be made withdev'elopment. Work I; in progress for modifying corresponding theoretical free-free modalthe present simulation to include spqce station properties. These comparisons were necessaryflexibility in terms of free-free modes of vibration, to determine if cantilever beam modes could he

used to accurately represent free-free beam

SIMULATION VEJITICATION modes. Motion equations for the system of twoconnected cantilevers were derived to facilitatethe frequency analysis: these equations are

Verification of the simulation program was presented below.accomplished in two distinct parts. Eachfunctional subprogram --- Lyddane's orbit Ngenerator, CMG space station control, OCS linear A X*(t)+ i i (2 (iand non-linear solar array drives --- was N A O

144

MEQ1 + M 2 derived and are shown below. The subscript "F"1 (1) EQ (a refers to free-free in these equations.

+ () X (t) = 0 (9) *.Ft)( 2)1 2 X M(14)

where 2MIA

i I, 2,3... N Fi2#1 it) 0 F

and F EQF 2 o (15)

M A = Maass of the cantilevered beam 2F40 LL ~(t) 0 0 12V1 ( Modal shear force coef. in the fTtt Flt) (1-coss t1) (16)

cantilever mode at mid-span hEQ fl2Fi

of the simulated free-free beam Ft

F 0 applied step load at mid-span It should be noted that anti-symmetric modes of thefree-free beam were not excited due to the

X(t) = coordinate for rigid body motion positioning of the disturbance force at mid-span.

hEQ = generalized mass of the ITil Modal data for both the free-free and canti-

I cantilever mode lever uniform beams were obtained from stzndArdreference tables (9] in order to compare the

o = iT1 cantilever mode natural cantilever and free-free beam formulations.frequency (uncoupled) Numerical data assumed for this comparison were:

MA = 5, 0 slugs, Fo = 0. 5 lb. I 1 = 12. 566 rad/sec.(t) = generalized modal coordinate The evaluation of the numerical coefficients of

for the ITI1 cantilever mode Equations 8-16 was based upon a 25 pointdiscretized mass representation of both the canti-

= coupled frequency of O.bratior. lever and free-free beamt. The frequenciesof the system, approxlmates the obtained by an orthogonal coordinate transformationiTil free-free frequency analysis of Equations 8 and 9, for the cantilever

beams, are given in Table 1. These frequencies,The solutions of Equations 8 and 9, using only along with corresponding free-free beam

the fundamental cantilever mode, are given by the frequencies have been normalized with respect tofollowing: the frequency of the fundamental free-free beam

mode. The rate of convergence in this frequencyF2 comparison is demonstrated by the successive

X (t) " - + _ cos g t (10) number of modes used. Similarly, the degree ofA A ( A Q 1 I Vcorrelation and convergence in the modal~amplitude domain can be seer. in Figure 12

which is based upon use of three cantilever modes.F The comparlson, show that the cantilever beam

-2 formulation can be used to accurately descrbr(AMEQ l- ) free-free beam modal properties If a sufficient

numiber of cantilever modes are usedl.

VIF Although motion histories of the flexible

o (1-cos Qt)(12, appendages were of Interest, the primaryI p2 I 2 tijectlive of the slmulation was to obtain Inter-(MA FQ1 actio. loads between the space station and solar

array fo desln evaluation. Therefore, forpurposes of :',rification with known solutions

where I N5, 8] shear foi e histories at the one-quarterA EQ1 cmae h ha ifr bandfo h s

-21 13,span beam locatlotm were obtained. Figure 13ft - ca) I (1) compires the shear hior "tainel Arom the use

To -A ) of one cantilever mode It 'he simulation with theshear calculated using the n. "dal accelerationmethod and one free-free mode. Excitation at

Similarly, the corresponding analytical the base of the cantilever beams b. a zero nasssolutions for the perturbed free-free beanm were space station posed no problem for ti, simulation.

145

T'ABLE 1

F'requency Comparlisan of (Uniform Beam) Cantllor~r + Rigid Body Mode Representationof a Free-Free Uniform Beam

Frequency Ratios: f ,n I Jrei-free

Reference Feeqiency Ratios Calcilated Coupled JM1ede Frequerncy Ratios

I Mode 2 'Mode 3 Mode- 4 Mode 5 ModeUncoupled Canti- Canti- Canti- Cant[- Canti-

Symmet ric Free- Canti- lover lever lever lever lever

NorFe oe em+ Ba em+ Ba en Beam 11~nm Rigid Body Rigid Body Rigid. Body Rigid Body Rigid Body

1 1.000 0.632 1.0101 1. 0006 1. 0003 1.0000 1.0000

2 5.404 3.958 --. 5.548 5.420 5. 401 5. 411

3 13.344 11.0114 1--- -13. 7*t 13.410 12. 367

4 24.814 21.652 --- -- 25.584 24.965

5 3,q.812 36.861 -- - ----- 41'.010

1.03

Note Obse CalculaW. Us.ing Mcdal Dipac fl I.*..

-. 33r 3. 0341 at (i

Sm.0.t abyrn-rI 'L4o ______.__._ _1_If__1

"s .6-ree

4

FilM. 3. .. Timel.V 13Mise313t... 0

ZdSymomerc Mod. Figt. 13. Com parison of Cantilever & Free-Free1.0 Sm. shut C eam Shear tit 114 Sr~aai vs. Time for n Unit Step

Force~ Applied at riMI-Spart.

Figure 13 ase contrastr thr, modal sr-crat Ion Anx]mods~i di.%place-ncnt methiods o)f computing totol

.6 . 4 0 3.31 loadr. The calculation of force hiatcrv co:-rep-

pondi-ig to the Modal diaplaecmrent Method was-. 11 !>a Ptat . mzari'-?Indtl ndcntly of the oyrautatlon. at The

-1.6on-quarter .'p21, the' ditferevcet In loadt frontthe two tnds were not ovignttica.-A. With

Fig. 12. Com~parison of Mode Shapes for a1 respect to computing total (quasi-dtcady plusFrce-Free Uniform Bean) and a 3 Cantilever dynimir! ioads, Bimpiinghoff (1io, Pointe out

Mode Approximation the dert uilitv of the modal accelerr Pion metho'I

146

which Is the method used in the simulation. .3

The comparisons given by Figure 13 show an .2

excellent agreement between the simulation results .1 . 2 .0 .06

and those obtained by an independent method 0 I.rutilizing the varlable order Adams numerical -. 1 . ... I-Integration technique. .3

Figure 14 presents a comparison of simu-

I. ~ ~~lation results using five cantilever modes with 7 ? -- ,os,independently calculated results using free-free ,ith

modes. The low frequency waveform and magni- NI]2 Ro.. ts.uise frst rl,,c,-tlvr wed"tude in both histories is seen to be in good agree- . .

S

ment. The small differences in the high frequency :",.the degree of convergence In approximating the Tie I.laseScontent between the histories are attributable to h.' .l , ,. . .

given number of free-free modes by the same

number of cantilever modes. -,4

Variable Order Adams Numericul Integration Reults VaiI the rFrat fo rn mntc Fr ree- r.. Beem Modes

._.2. -. 1 (I/4, t)- -M (i/4) 9 (1- S. V (IA) Yl (0

' ~ ~ ~ ~ l 0 6f/o, .08 .,o fIt .,, .,6 .1, .10 .11'14-.6-e *Fig. 15. Uniform Beam Comparisons of Shear

... __ . istory @1/4 Span for a Unit Step Force Applied

..4 at Mid-Span

" A mode-by-mode comparison of the maximashear force components derived from the simu-lation and those given in Reference 8 is presented Inin Figure 16. Again, the differences Indicated

simlaltion Ietutt. teg fir r ite Carfiler .ode. are attributable to the degree of convergence in

approximatltug the given number of free-free modesby the same number of cantilever modes.

l08 .o 0 . 12 . 14 . 16 . 18 . 20 . 22.,

The above presented comparisons betweenTime.e cantilever an free-free beam response re-

suits demonstrated that the structural dynamicsmethodology contained in the simulation wassufficient for the accurate evaluation of Inter-action loads.

-. 8 . t -Il I -M 11/4) 9(1) - Z V 41/4, I )

Fig. 14. Uniform Bean Comparisons of Shear Maximum her at Span x. Nldelistories @ 1/4 Span for a Unit Step Force Applied 1 Mro

at Mid-Span icw00 Ilekrence 890 Figure 9

80 Sim latlonFigure 15 presents a comparison of simulation C70 ant lr Aroximaion

reflults using five cantilever modes with resultsobtained using tho Nastran "Direct Transient 40Itesponse Method" [5]. The free-free .beam representation was discretized Into 40 30

masse; for use i this method. Modal truncation 20

was not considered in this method; each of the o Fdiscr2te masses was allowed two Inertial degrees 0 2 3 4 5

of freedom co.'responding to planar bending of l (ilHfd-Dlmly) symmetric hlode Number. Ithe beam. In general, good agreement exists

between the frequency content of the two shear Fig. 16. Comparison of Modal Shear Forcehistories. Also, the maximum shear force Participation for a Uniform Beam Subjectedgiven by the hitorles is in excellent agreement. to a Unit Slop Load Appli.d at Mid-Span

1471 i'

• 4

TYPICAL SIMULATION RESULTS 3 degrees of freedom. The stiffness of the arraymembrane was considered as a function of applied

An extensive analysis of the parameters that tension loading, and the central boom was modeledinfluence the structural dynamics of solar arrays as a beam column. A detailed description of theis planned. This analysis will form the basis of structural modeling and resultant modal data isa generalized dynamic design criterion for solar reported In Reference 3. Selected frequencies arearray structures. Ithough this analysis has not listed in Table 2 in terms of in-plane and out-of-been completed, some preliminary results have plane modes.been obtained.

The selections of modes to be used In theThe solar array/space station conceot being simulation was made on the basis of those contri-

evaluated is shown In Figure 17. The space station biting a large percentage of load participation.is a rather stiff structure (modeled as a rigid Load participation for symmetric modes isbody) which contains 96. 5% of the total mass. evaluated by calculating the shear at the array tThe solar array is a large flexible structure attachment point due to a base translationalcomposed of membrane strips stretched between acceleration. The participation is equal tothe Inner and outer structural support members. (Z mi 0i) 2/Rn Zmi where mI is the discrete mass,An extendible boom in the center of each vane 0i is the modal deflection coefficient and Rnapplies a tension load to the membrane strips. is the modal mass. Load participation for anti-The solar cells and associated interconnects are symmetric modes is evaluated by calculatingcemented to the membrane substrate and generate .the moment at the array attachment point due tothe power required for space station operations, a base rotational acceleration. This participationThis array was designed Iy the Lockheed Company function is equal to (Zmo i ri)2AZn mr12under Contract NAS9-11039 and is reported In where r[ is the distance from the vane axis ofReference 11. the array to the mass point. A typical mode shape

seais shown in Figure 18 for an out-of-plane anti-Outter Strxtursl SRtdst) symmetric mode. The load participation factor

Support for this mode was 67. 7%. The results of the2! . analysis also Indicate that modes with a significant

load participation ;all within the frequency band-width of the space station attitude controlsystem. Therefore, coupling of the attitude

.. control system with array modes can be expected.'y 7

Selected results from one simulation areM ram presented and correspond to the configuration of@triph Figure 17 with small Initial attitude errors. TheInner Structural 1104" s space station was perturbed by a force history

srepresenting the docking of an orbiting module. Thedisturbance force is directed along the X axisof the space station. The continuous OCS drive

Sollup Solar Arrays Fully Deployed system was used In this simulation for array

(Flexible Body) control.'4(10,000 PT

| Array WetSs - 7,890 Ib.Space Station Weight -710,090 Tb. Figure 19 is a graphical series of results

space Station Inrtias from this simulation. The docking load profileand resulting roli axis acceleration, is given in

1, 2.4x 10 UK_-Fj.a Figure 19a. It Is seen from Figure 19a thatI, " 5.4,iX I s lug-Pt.:

I, • s.s x 1o sut.rt. 2 the magnitude of feedback force from the flexiblearrays upon the space station is small because of

Fig. 17. Space Station/Solar Array Concevt the non-oscillatory nature of the acceleration history.for Dynamic Interactions Study (Rollup Flexible Figure 191) shows the Initial array orientation errorsArray) with respect to the sun vector as a function of

time. Figure 19c shows the time variations ofmodal acceleration for two array modes - an

A structural analysis was performed on these out-of-plane mode forced primarilly by OCSsolar arrays prior to simulation to obtain required torque and an in-plane bending mode forcedmodal data. The matrix method of structural primarily by space station acceleration. Spaceanalysis considered in Reference 12 was utilized, station control torques were not initiated duringThe array was modeled using a discrete mass this simulation since the angular dead-bandtechnique with 200 mass points, each mass having contained within the reaction jet control system

148

TABLE 2Frequencies and Modal Participation Factors of Selected Roilup Array Modes

Out of Plane

_______ Symmetric Anti symmetric

Frequency Percent Frequency Percent(Hz) Priiton(Hiz) Participation

6 .0734 50.4 6 .0557 67.736 .203 12.5 31 .162 5.456 .287 3.5 57 .318 7.3

62 .340 7.7

In-Plane

Symmetric Antisymmetric

Mode Frequency Percent Mode Frequency Percent(Hiz) Participation (Hlz) Participation

2 2.26 57.0 1 .291 55.02 1.94 3.17 2.71 6.58 4.12 2.7

2500

2000 Applied Docking Force and Space Station

L 1 1h<' 500 Acceleration istories (Holt1 Axls Direction)

1000Foc

I~~ 1000

Force

- Accelerationx

- 500

0 0.5 .0 1.5 4 6 0 10Timne (Second@)

"NC IFig. 19 (a). Simulation Rlesults for an ExternallyApplied Docking Force

was not exceedled. Figure l~ci presents the timcvariation of interaction force along the spacestati-m X axis; and Interaction moment sbout thespace station V axis. liiitia! high frequency tran-

Freq~vocy - 0.0557 Ha sient loadhi are seen to decay rapidly. This IsX2 Particiption - 67.7% caused by a relatively high value (0. 05) of the

Inpuit modall damping parameter. These limitedoo results which arc presented show the type of design

X3 data that can he obltained from this dligital siniu-lation of the integrated dynamic analyis. A

Fig. 18. Out-of-Plane Antisymnietric Motion complete set of (data, corzrespondling to variations

149

in initial parameters and basic structural data, interaction Force ad Moment Histories

will provide the basis for ihe derivition of a 200otructural design critetion.

.1 Solar Array Anglar Error Hitorlos

Seasoal Adjust Axis

.05 asOrbit Adjust Axle~

0 -

-. 05 - --

0 2 4 6 10 10

Fig. 19. (d)

Time (Seconds)

Fig. 19. (b)

Modal Coordifte Acceleration Histories CONCLUSIONS )le Mio- c e An Integrated dynamic analysis method has

(Acceleration x 10) been developed and implemented in a digitalcomputer program for eimulating the structural

0 dynamic interactions between a space station andcontrollable solar arrays. Simulation verification

lot Antlaymetric In-Plan studies demonstrated that the analytical formulationBending Mode and the modal synthesization technique employed

provide an accurate method for evaluation ofdynamic Interactions. In addition, the verificationstudies showed the programming of the almulation

S-2 to be correct.

-3L

0 2 4 6 a 1O

Time (Secors)

Fig. 19. (c)

150

REFERENCES 7. "Structural Interaction Simulation System",

Technical Report R104, Revision 1, Wolfi. P. W. Likins and H. K. Bouvier, "Attitude Research and Development Corporation, River-Control of Non-rigid Spacecraft," J. Astronautics dale, Maryland.

and Aeronautics, May 1971/Vol. 9 No. 5 Pg.64-71. 8. R. W. Leonard, "On Solutions for the Transient

Response of Beams", NASA Technical Report2. "Preliminary Synthesis and Simulation of the R-21, 1959.

Selected CMG Attitude Control System," GeneralElectric'Report EL-506-D, 5 March 1970, 9. D. Young and R. P. Felwar. Jr., "Tables ofGeneral Electric Company, Binghamton, New Characteristic Functions Representing NormalYork. Modes of-Vibrition of A Beam," University of

Texas Publication No. 4913, 1 July 19)49.3. "Interim Report, The Study of Dynamic Inter-

actions of Solar Arrays with Space Stations and 10. R. L. Bisplinghoff and H. Ashley, Principles ofDevelopment of Array Structural Requirements" Aeroelasticity, pp. 344-350. John Wiley andFairchild Industries Report 8581R-1, February Sons, Inc., 1962.1971, Fairchild Industries, Germantown,Maryland. 11. "Evaluation of Space Station Solar Array

Technology and Recommended Advanced Develop-4. P. W. Likins, "Dynamics and Control of ment Program,", First Topical Report LMSC-Flexible Space Vehicles" Jet Propulsion Labor- A981486, Lockheed Missiles & Space Company,atory Technical Report 32-1329, Revision 1, December 1970.Januay 15, 1970.

12. "Stardyne User's Manual", Mechanics Research5. "The NASTIRAN Theoretical Manual", NASA Inc., Document, Los Angeles, California,

SP-221, Section 11. 3, 1970, Office of Technology January 1971.Utilization, NASA, Washington, D. C. %

6. R. If. Lyddane, "Small Eccentricities orInclinations in the Brouwer Theory of theArtificial Satellite," Astronomical Journal,Volume 68, No. 8, October 1963, p. 555,

DISCUSSION

Mr. Zudans (Franklin Institute): When you indi- Mr. Mains (Washington University): You said atcated on one of these diagrams the docking force inone step that you were using a 600 degree-of-freedomtroduced into the system, how was it introduced rel- system for analysis. Is that correct?ative to the mass-center of the system? You seem tobe getting the moments, but there were no indications Mr. Weinberger: That is correct.on the slide as to how this was done.

Mr. Weinberger: In this particular case the Mr. Mains: Did you use a direct integrationdocking force was applied at the aft end of the space technique to get those response curves that youstation along the axis of the space station. There showed?were some out-of-plane motions. These motionswere due primarily to a slight misalignment of thesolar arrays that were active with a linear controlsystem. This means that even 0.1 of a degree atti- Mr. Weinberger: Yes we did.tude error would cause some motion of the solar ar-rays, and hence some torques Into the system. Butthe docking force was applied along the axis of the Mr. Mains: How do you have any handle on thespace station. There was no eccentric force or load- meaning, the reliability, of a solution of that size forIng. this kind of problem?

151

DISCUSSION

Mr. Weinberger: The 600 degrees of freedom Mr. Weinberger: We are examining this problemsimply referred to the structural analysis model that in connection -with the tension loads that are trans-was developed. From the structural analysis model, nritted through the boom and so on. This is one of thethe generalized mass and modal coordinates were areas in which we have made certain assumptions inused for the modes that were selected. In this case linearizing. The validity of these assumptions has notwe used 12 modes, so the 12 elastic modes were been established at this point, although we do haveused for the solar arrays plus the 6 rigid body de- some preliminary test data from Lockheed on the ar-grees of freedom for the space station. We did not ray of this type. This was a Lockheed array geometryhave a 600 degrees of freedom model for direct into- that we were studing. From what we have been able togration. ascertain, we have chosen the model In such a way

that the results agree fairly well with the test data.XrMjps Did you then do an elgenvalue solu- We get z correlation in that respect. This is the only

tion on the 600 degrees of freedom? way that we can have any confid ence in the lineariz-ation of the array.

Mr. Weinberger: Yes. We did It to get the fre-quencies and the mode shapes and the generalizedmasses. Mr. Clevensen (Langley liesearch Center): If I

understood you correctly, you used NASTRAN forMr. Mains How do you have any handle on the verifying some of your results?

reliability of an elgenvalue solution of that size?Mr. Weinberger: That's right.

Mr. Wr¢nberger: I think the reliability of struc-tural models of that size is fairly well documented, Mr. Clevensen: Why could not you have usedfor example, in the NASTRAN program and users NASTRAN exclusively and saved considerable work?manual.

Mr. Weinberger: Of course one of the problemsMr. Mains: You might be surprised if you would that we looked at was the coupling of the control sys-

check the orthogonality of the vectors sometime. tem. I am not familiar with the NASTRAN programwith regard to the demap instructions and the auxil-iary useage of NASTRAN, other than for structural

Mr. Zudans: I would like to comment more on and vibration analysis. I assume that you might bethese questions, because 600 degrees of freedom dy- able to code subroutines which represent the orient-namically today Is nothing. You can handle 3,000 and ation control system and the rigid body mechanics.there is a perfcct orthogonality. The NASTRAN pro- One thing that comes to mind immediately is the sizegram and many other programs use the invrse iter- of NASTRAN. We have been able to modularize thisation routine with spectral shapes and it is. very, very program in such a way that it uses much less digitalreliable. However, I wanted to ask a question. How computer time in core than the NASTRAN programdid you account for obvious nonlinear atti'.udes in would use. We felt it was more efficient in develop-your solar array? It is such a flexible st,ucture that ing our own program, rather than to resort to theit could not be handled as a linear one. NASTRAN program.

152

PARAMETRICALLY EXCITED COLUMN WITH HYSTERETIC MATERIAL PROPERTIES

. T. MozerIBM Corporation

East Fishkill, New York

and

R. M. Evan-Iwanowski, ProfessorSyracuse UniversitySyracuse, New York

An investigation is performed to determine the effects of hysteretic material l.properties on-the dynamic response of parametrically excited systems. Expres-sions are developed for stress distributions, forces and moments for a-para-metrically excited column. Column response is detemined for-the stationarycase with-additional assumptions on material properties and stress distributions.

INTRODUCTION

This paper is related to the description, Pit) P0 + P1(t) come(t)simulation and means of minimizing the effectsof mechanical vibration on technical hardware bymeans of a theoretical investigation of the ef-fects of hysteretic material properties on theresponse of parametrically excited systems. I

If an initially straight column, Fig. 1, issubjected to a periodic axial load, P(t) = Po + u(x,t)PI(t) cose(t), and if the maximum amplitude ofthe load is less than that of the static buck-ling (Euler) load, PE, then the column experi- xences only longitudinal vibrations. There are,however, certain relationships between the dis-turbing frequency 6(t) = v(t) and the naturalfrequency n of transverse vibrations of the col-umn for which the straight column becomes dynam-ically unstable and lateral vibrations occur. uFor a sufficiently small value of Pl(t) one suchrelationship between b(t) and n is 0 - 2n. Theresponse which occurs under these conditions is Fig. 1 - Column Configurationcalled parametric resonance. This phenomenon ofparametric resonance may be expressed more gen-erally as v(t)/2l = 1/.n where m = 1,2,3,-.. obeys Hooke's law. It is well known that few

solids actually obey Hooke's law very closely.For increasing values of P1 (t) the points Of these few solids, many are used in structural

of instability mentioned above cover instability members. The principal reasons for their useregions. That is, there exist continuous ranges are: (1) they display nearly Hookean behaviorof the values of v(t) for which the column will to a relatively high stress level, (2) over-allbe unstable, in (v/24, P1/2(PE-Po)) space. structure cost, (3) over-all structure weight.

The largest of these instability regions, In many instances, however, it is desiredand thus the most important region, Is the first to have machine or structural members performregion characterized by v = 2o. Consequently multiple functions. For example it may be de-this paper is concerned with the nature of vi- sirable to have a column perform both as a col-bratory motion of the column related to this umn and as an electrical or thermal conductor.region. In this case the selection of the material will

Our discussion thus far has implied, among be based on some compromise between its loadother things, that the material of the column carrying capabilities and its electrical or

153

thermal conductivity. Such situations may pre-sent the necessity to use materials whose me-chanical properties are non-Hookean over-all butthe smallest stress levels. Once we leave thedomain of the Hookean solid elastic column, alarge field opens up even to the most casual ob-server. The number and complexity of the mater-ial properties, along with their relativelycrude rheological models becomes overwhelming if -coone attempts to apply them with rigor to theproblem of the parametrically excited column, co c(an exhaustive discussion is given in Ref. 1).

Several authors have considered material

properties other than the simple Hookean case inthe analysis of the arametrically excited col-umn. K. K. Stevens [1), for example, solves forthe cases of the Maxwell Element and the Three-Parameter Model as material properties for the Fig. 2 - DaVidenkov's Modelstationary case. The case of a simple viscousdamper in parallel-with a spring (Voigt Element) cross section at any time will be developed fromis shown in references treated for stationary the stress distributions. On other instancesand nonstationary modes. Later K. K. Stevens expressions for the temporal part of the motionand R. M. Evan-lwanows'd [2] introduced the corn- of the column are derived.plex modulus material property representation tothe stationary response of the parametrically Analysis of the system configuration yieldsexcited column. It is important to note here the partial differential equationthat in this analysis energy dissipation occursonly due to the bending of the column and not a2 t 2 t a2 Udue to its axial compression. V. V. Bolotin (3] x JR) + Pit)2u+m...ZO (1.2)considers amplitude dependent damping forces due a xa x a t-

to viscous and dry friction at an end support ofthe column, but these forces do not arise from where M is the moment at any cross section andthe material properties of the column itself. M = fA a y d A. We have made the usual Bernoul-Detailed discussion is also given by Mozer (4]. li-Euler assumptions, and the deflection u is re-

garded as small.This paper deals with determination of re-

gions of stability and instability as well as The strain dt any point isthe lateral amplitude response of the axially 22 Uexcited column whose material properties are of C = C + Y2 c + y K = E + y 2 (1.3)the pointed hysteretic loop variety. The Davx- a Xdenkov model is used to represent this materialproperty. So far the treatment is for any material proper-

1. BASIC RELATIONSHIPS ty.

Consider now the stresses in the columnDavidenkov (5) developed relations to rep- whose material property is such that it is char-

resent the behavior of metals in the form acterized by a pointed hysteresis loop. ..e knowSnalready that in the principal region of insta-

a E(c t (o + C)n - cnol o (1.1) bility the frequency of lateral response of then Ccolumn is half that of the excitation frequency.

Thus the period of the bending stresses will beThese relations depend only on the amplitude of twice the period of che periodic portion of thethe strain. It is graphically represented in axial stresses. In addition these two stressesFig. 2. will in general be out of phase. A typical

stress strain diagram for a fiber at some dis-The Davidenkov expressions have been used tance from the neutral axis is presented in

extensively by many authors, e.g. [6] to solve Fig. 3.several classes of problems in which the rater-ial is assumed to have hysteresis loops of the In Fig. 3 the small loop is due to the per-form represented by (l.1). iodic axial load and the larger loop is due to

the slower periodic bending. The period of theStress Distributions in Parametrically Excited total loop is the same as the period of lateralColumn. motion of the column. The loop contains four

distinct branches. If the bending is reducedIt is the objective of this section to de- the picture is altered to look somewhat like

velop stress or strain distributions for the that shown in Fig. 4. If the bending is re-parametrically excited column. In some instan- moved completely, the two loops converge into aces the expression for the moment on any column single two branch loop, the period of which is

154

-j

the same as the period of P(t).0

AlI

Fig. 3 - Stress strain in a fiber Fig. 5 - Stress strain in a crossof a column with hyster- section of a column withetic material. Large hysteretic mterial.moment.

The stresses on a cross section of the col-umn are such that at certain times three branch-es of the four branches of the loop are present.Thus for a material having a pointed hysteresisloop, we require several equations to representthe stress across the column at different times.Each equation contains two or three terms cor-responding to the different branches of the hy-steresis loop present in a cross section.

The use of the Davidenkov relations in thecase of the parametrically excited column whereaxial stresses are taken into account is in gen-

- eral not possible since the constants in the re-4 lations will not always allow the loop to close

after a complete cycle of the column. However,if one assumes that the strains due to bendingand the strains due to the axial load are inphase, then it is possible to write equationsfor a four branch loop based on the Davidenkovrelations. The evaluation of the constants inthe resultant expressions is difficult sincethey depend on the distance from the neutralaxis, y.

2. RESPONSE OF COLUMN WITH POINTEDLOOP MATERIAL HYSTERESIS

We make the additional assumption that theFig. 4 - Stress strain in a fiber axial stresses are small and will not contribute

of a column with hysteretic appreciably to the column response. This assump-material. Small moment. tion is a fairly good one for the slender col-

um. Under the assumptions made, moment actingThe stress strain distribution at a cross on a cross-section is

section of the column will be somewhat like that a2ushown in Fig. 5. M E I 32u+ () (2.1)

3 X2

155

where E is a small parameter such that E > 0,E<< 1 and where i(lnd her ) 2n-l(a2 u)n yn+l d A (2.7)

(a, X) x2 maxa X

is the dissipation functional depending only on Comparing (2.7) with (2.1) we deduce thatthe curvature and material constants. The ar-rows above t represent the branches correspond- E f [( _U a2 u n -

ing to increasing and decreasing curvature. n [(- u) .Substituting (2.1) into (1.2), we get A x max a x

4.t

E I " u + Z -I-" (; ) " +- 2n-l (12u n )y n+l d A (2.8)a x a x ax2 ax max

+ P(t) ( ) + 0 Substituting the value for u from (2.3) in-a x2 t 2 to (2.8) and differentiating twice with respectto X, we get

We seek a solution to equation (2.2) of the form

L-~~~~~a - E nl in2Iu(x,t) - f(t) Sin (2.3) 2 = f._ n 2ax A L

Substituting these values into (2.2) we get- n Sin n [(f(t)m 4 f(t))n -

E 12-4Sn Z-E(t) + Z -2-2 (3 2 3] "-

a x2 . 2n-I (.l)n f(t)n ]} yn . y d A (2.9)m(

- P(t)f(t)!gin x+ m i(t) Sin tx- 0 (2.4) where fm(t) is the maximum value of f(t). Itcan be shown that the expression + Ey/n{ )yn(in (2.9) is always odd about y = 0 regardless ofIn order to make any headway toward solving the value of n. Thus the total integrand of

(2.4) we need to know more about the functional (2.9) is always even. Due to this property we,. We rewrite the equation for Davidenkov's may integrate the above expression from zero to-model h/2 and multiply by two instead of integrating

from -h/2 to +h/2. This removes the necessity-[(co - )n - 2n-I on)) (2.5) to keep track of the signs of y and c. Thus

__2____ h/2 2 -n4

and the definition for the moment , ; 2 EYi W" 2 na X2 = 0 L2 n [(n-1) Sin n '

". ydA ax2 L

and also noting that .nSin n ix (I)n+2 [(-f(t)m f(t)n

a2u . 2n-l (.1 )n f(t)nm)) yn+l d yY -y y dax

where W indicates the width of the column. In-in the case where the axial strains are neglec- tegrating the above, we obtainted. We may now substitute the above value forc from (2.5) and substitute the resulting ex-pression for the stress into M we get a2 4 2 E W n+4

C 32 2 ( Wn IT [(n-1) Sinn 2 n -a x2 1. 1?

E 2y2 ~ U[,2U +a x, d A + x2 max - n Sinn x [(-f(t)m ;f)) n - 2

+ 3 u)n yn+l d A (2.6) f(t)nmn] (h)n n (2.10)a x a X2a max

We consider the special case where n 2

=E I 2 -u -:EX [(,2 2U)M +. 2 u n

2 n 2 - =2a x a x max ax

156

4form

3.2 -Ey W W6 h4' 21rX x- 2)C 2 6 {[l 2 Sin--- + n2 f t [f2 t 2ff .m+

2 + Z2n2 u Cos o f (2.14)E(f(t) + 2 f(t) fit) + f(t)-

- 2f(t)m]} (2.1l) where Es 96 nm 96 L6 m

Equation (2.11) represents the distributed lat- and the term 2n2p has been replaced by i2n2p.eral load due to dissipative material properties. Note that if (2.14) were expanded we would getWe now substitute (2.11) into (2.4) and apply the term + c B 2fmf. It would appear at firstGalerkin's method by multiplying the resultant glance thit the coefficient + Z02fm could beby sin vx/L and integrating, we get combined with n2 on the left hand side of (2.14).

It must be noted, however, that this term is ac-, L .2 L tually a variable coefficient of f due to the al-2El 24L f(t) - 2 P(t) f(t) Z- + ternating signs and we are thus justified inL 1 leaving it on the right hand side of the equa-

tion. We represent (2.14) in the form+ 2m)it ! TEy W h4 r5 . [ f(t)248 Ls f + n2 f = ;F (f,o,r) (2.15)

i2f(t)2 f(t) + f(t) ] 0 (2.12) where

Remembering that P(t) P0 + P, cos and f F m.2 + 2f12 lpf Cos 0

E L 2We seek a solution to (2.15) in the formis Eulers' buckling load and fit) =_ f = aft) Cos (0 + (t)) (2.16)

w 2 El

L'2V( We have assumed analysis of the first instabil-ity region only by using the ang,,lar displace-

is the transverse natural frequency of the col- ment term o/2 in (2.16). The terms a(t) andumn without axial load and the transverse natu- t(t) are to be determined from the usual rela-ral frequency of the column loaded by Po is tions,

a = EA1(r,a, ) (2.17)

1 E 41 v(t) + cBl(Tpa,, ) (2.18)

We also denote the loading parameter v as After some calculation we find:

P1(t) E y W h 2 6 (4a _a2)

= 2 (E - P) 9 E" L6 m 2n a

Substituting these relationships in (2.12) andrearranging, we get,E

y W h - P aCos2 (2.20)

+ 2(1-2 Cos O) f Ey W h4S . 96 L6 ma96 L6 m Stationary Response

2[f2 !2 f- f 0 (2.13) We now proceed to determine the stationary

response of the column by investigating (2.19)Equation (2.13) represents the temporal equation and (2.20) in a special case. Stationary re-of the motion of the column, and is the subject sponse is defined as the case where no systemof the following asymptotic solution, parameter changes with time. Explicitly we

write v(t) = constant, u(t) - constant whichAsymptotic Solution implies that a = 0 0. Under these conditions(2.19) and (2.20) become

Only the first approximation will be soughtin this analysis. We rearrange (2.13) in the

157

: 2The region of stability is distinguished fromT 2 S_0 (2 the region of instability by the curve repre-

-ITO( 4 am -a2) - Sin 2¢u.O (2.21) sented by

V. n B~nam n-- Cos 2p- 0 (2.22) a=n - ).¢v (2.28)

where called "stability curve". The stable and un-

EyWh4 W6 stable branches of the phase angle ' are found

S L6 (2.23) to be96 L m (n-V)/d < 0 (2.29)

Eliminating € from these equations while noting

I that am a a constant for the stationary case, Hence the solution stable for v ; 20 and d/dv >Swe get the following 0, otherwise it is unstable.

a 2r2 p {2- [(2n- v) 2 3. RESULTS AND CONCLUSIONS

81(4 + 9) Discussion of Material Properties and Stress

2)204 U2)p) Distributions.

4 2 2 (2.4)The significance of the harmonic axial loadon column response is demonstrated. Stress dis-

The t sign denotes the possibility of two solu- tributions in the column are also illustratedtions for a. for the pointed loop hysteretic material.

We may now determine the stationary phbse Stationary response curv for the parame-angle o from (2.21) a using the value of a from trically excited column whose mwiterial may be(2.24) represented by the Davidenkov Model with the

special assumption that the axial stresses are1 [ negligible are plotted in Figs. 6-10. Fig. 6

= Sin- [2A - v and Fig. 7 show the effect of varying the load2nP(4 f2 + 9) parameter P on the column response, Fig.8 shows

the effect of varying the Davidenkov model pa-

- 2 4 V2 + 9 rameter y, while Fig. 9 shows the change of the[(2 f-V

2 - 2 ) - V)2 - first instability region with amplitude of re-sponse. Fig. 10 shows the instability region inthree dimensions.

- P-t ] ] (2.25) It is seen from Fig. 6 and Fig. 8 that the

point defined by da/dv = 0 separates the regionfor which non-trivial stable solution can exist

The so-called "backbone curve" of the sta- to the right and the region for which no non-tionary response is that curve lying halfway be- trivial stable solutions can exist to the left,tween the two values of a obtained from (2.24). n, u, P0 and y being constant. This means thatIt is no non-trivial solutions exist for y/2aless than

2___f?(2ov - Eywh4V 6 that at which da/dv = 0. This result has nota, Z 6 = E wh been frequently observed in the literature, al-

si(4n2+9) 96L m though it can be shown that the response curveof a parametrically excited column having non-

The boundaries of instability zone are: linear damping of the form dff 2 where d is thenonlinear damping coefficient and structural

V [1 11 2p] nonlinear elasticity does close at the backbonecurve in a similar manner to the present case.

Applying well known steps in determination of It is further noted that as long as y is posi-,saly n sable bnownhes, we oteratn heof- tive the backbone curve leans toward decreasingstable or unstable branches, we obtain the fol- .frequencies, characterizing soft systems.lowing: For d ao/dv > 0 the solution is stable (Negative y implies that the material propertyif is such that it generates energy over a cycle).

Pisarenko [6] arrives at dynamic response curves8, ao for various problems Using the Davidenkov rela-

- V 2 - 4 - > 0 (2.26) tions which have also soft characteristics, butbeyond this, comparison of results loses most of

and for d ao/dv < 0 the solution is stable if its meaning since Pisarenko analyzes systemsnear dynamic resonance, and the present work is

a, ao 2 concerned with systems near parametric reson-2a - v - 2 - 4 < 0 (2.27) ance.V3

158

It is seen from Fig. 9 that the curve de- second frequency to the response of the modelfining the region of instability for zero ampli- does not undully complic2tesubsequent use intude of vibration is the same as that for the the governing equations then such work wouldperfectly elastic case, The authors believe constitute a valuable 6ontribution. It may alsothat this is due to the fact that the axial be possible to incorporate into.this model'thestresses were considered small. The coupling of nonlinear elastic effects without difficulty.the bending and the axial stress probably would Such nonlinear elasticity could be obtained fromhave provided a more significant dissipation experimental data on the maximum point locusterm for a 1 0 and thus the curve would have curve.been shifted slightly to the right from u - 0;v/2n - 1 and would have been rounded somewhat, REFERENCESsimilar to the case where linear velocity de-pendent damping is included. In the case con- 1. K. K. Stevens, "On the Parametric Excitationsdered the energy dissipation per cycle is pro- of a Viscoelast'ic Column," A.I.A.A, Journal,portional to y a3 and so the curves in Fig. 9 Vol. 4, No. 12, Dec. 1966.defining the region of instability shift to theright and become rounded with increasing ampli- 2. K. K. Stevens, R. M. Evan-Iwanowski, "Para-

tude. metric Resonance of Viscoelastic Columns,"int. J. Solids Structures, Vol. 5, pp. 755-•The downward shift in the curves defining 765, 1969.

the region of instability with increasing ampli-tude may be due to the nature of the maximum 3. V. V. Bolotin, The Dynamic Stability ofpoint locus curve for the Davidenkov model: Elastic Systems, Holden-Day, Inc., San

Francisco, London, Amterdam, 1964.

tE co n 01 4. D. T. Mozer, "Parametrically Excited Column

with Dissipative Material Properties," M.S.This equation contains a nonlinear component Thesis, Syracuse University, Syracuse, N.Y.,proportional to co2 which is symmetric, due to 1969.the alternati:g sign. Lazan [7J indicates thatthe maximum point locus curve may be an appro- 5. N. N. Davidenkov, "Energy Dissipation in Vi-priate one to use for the calculation of stored brations," J.-Tech. Phys., 8, No. 6, p. 483,energy in material. In our case this "nonlin- 1938.ear elastic" term is probably the most appropri-ate one to use in explaining the nonlinear char- 6. G. S. Pisarenko, Dissipation of Energy inacter of the column response. From Fig. 9 one Mechanical Vibrations, (in Russian), Kiev,can further note that in the present case re- Izd-Vo Akad Nauk, USSR, 1962.gions of instability exist for large values of awhich do not exist in the linear elastic damped 7. B. J. Lazan, Damping of Materials and Mem-case. For example, in Fig. 9 let our system be bers in Structural Mechanics, Pergamondescribed by state v/2a - 0.9 and u = 0.13 (for Press, Inc., 1968.the elastic case, our system is stable). Letthe system now be perturbed such that ah/L > 8. T. J. Mentel, C. C. Fu, "Analytical Formula-.0013. Under these conditions the system finds tion of Damped Stress-Strain Relations Baseditself in a region of instability and a large on Experimental Data with Applications toamplitude results. Vibrating Structures," ASD Technical Report,

One must approach the problem of selecting p. 61-63, 1961.

a mathematical model to represent the real ma- LIST OF SYMBOLSterial of the parametrically excited column (orin fact any vibratory system) with extreme cau- A Area of cross section of columntion. If one attempts to use a strictly linear E Young's Modulusmathematical relationship to represent a materi- h Depth of column in y directional whose characteristics are essentially nonlin- I Moment of inertia about Z axisear, large errors may arise in the prediction of K Curvature I/system vibratory response even though care is L Length of columntaken to see that energy dissipation per cycle m Mass per unit lengthis the same as the real material; see Mentel r..o Material constants

and Fu (8], for example. R Euler's LoadP t) Axial load Po + PI(t) Cos 0(t)

Biot's linear hysteretic model is perhaps Po Static compressive loadthe best linear hysteretic model to extend for PI(t) Amplitude of dynamic loaduse with multi-frequency excitations as encoun- W Width of columntered in the parametrically excited column or x Axial coordinate endplate. It possesses the characteristic of amp- y Distance from neutral axis.litude depe;odent damping found in most engin- Z Coordinateeering materials and it results in a well de- c, Lo Total normal strain, Maximum strainfined mathematical problem at least for the Ca Normal strain due to Po 4 P1 Cos 0single frequency case. If the addition of a Small positive quantity

159

Radius of curvature C1f~j

O4 Anua dslcmetla

All Frqunc .1.0;,(Loa Oraet /(P P

FTig. 8 Effect of Varying on Amplitude '

Fig. 6 - Efftaictt oft Vayn -n-mltd

Fig. - Cangeof Intabiity i icreasing

Fig. 7 -Effect of Varying uon Phase Shift -p

Fig,1 l0 - Three Dimrensional Representa2tion ofinstability Region

160

DISCUSSION

Mr. Zudans (Franklin Inetitute): You surprised Hertz. It was in this general frequency range that weme with very poor orthogonality. Was that mainly be- were concerned. We had a number of modes which,cause of the use of experimental modes wihich hadnot through increased growth of the spacecraft, had grad-been orthogonalized before their usage? ually crept down into the pogo frequency range. We

were also concerned with some of the large amplitudeMr. Stahle: The orthogonality referred to the modes like the first longitudinal which actually went

abbreviated model test. The criterion that we had set as high as 55 Hertz.up was that the measured experimental mode wouldcheck within 10 percent -no' the analytical modes. The Mr. Schrantz (Comsat Labs): Did you coupleproblem is the very limited amount of instrumenta- your model with the Thor Delta to check out thetion used on the solar array panels. I think our main responses?confidenco was gained from the fact that the frequen-cies matched up very well, and that the main struc-tural modes agreed relative to the modal admittancethrough the base shear. Mr. Stable: This is done by Douglas personnel.

The model that we have been using is the model thatMr. Prause (Battelle Institute): What are some I presented here. Essentially it Is a modal model us-

of the important frequencies? We saw a Ioc of natural ing modal coupling techniques to marry the spacecraftfrequencies in the presentation but what rre the con- back to the launch vehicle. It follows the basic iner-trol system frequencles and what are the pogo fre- tial coupling procedures of component synthesis dis-quencies for this type of space stations? cussed in the literature to couple this analytical rep-

resentation of the spacecraft back to the luanch ve-Mr. Stable: This is the Earth Resources Tech- hicle.

nology Satellite which is a fairly small, 2,000 pound,space craft going up on the Thor Delta. The main po- *The paper was presented and discussed by C. V.go frequency varies somewhere between 17 and 23 Stahle for the authors.

161

DYNAMIC INTERACTION BETWEEN

VIBRATING CONVEYORS AND SUPPORTING STRUCTURE

Mario PazProfessor, Civil Engineering Department

University of LouisvilleLouisville, Kentucky

and

Oscar MathisDesign Engineer, Rex Chainbelt Inc.

Louisville, Kentucky

The dynamic analysis of the conveyor-structure system is pre-sented using the stiffness method and an iterative scheme inwhich the structure and the vibrating conveyor are analyzedsuccessively taking into account interacting effects. Anexample of a structural truss supporting a vibrating conveyoris given.

INTRODUCTION the supporting structure is rigid, thedynamic analysis of the conveyor is

Vibrating conveyors are widely performed. Then the structure is ana-used in industry for conveying granular- lyzed under the effect of the reactivetype material (1]. Depending on spe- forces at the points of support of thecific applications, these conveyors conveyor. From this analysis the mo-could weigh from a few hundred pounds tion of the points of support is deter-to several tons. The main components mined and used as boundary conditionsare the pan or trough where the mate- in repeating the analysis of the convey-rial is conveyed, a driving mechanism or. This iterative scheme cintinues,which produces rectilinear motion at an and the new reactive forces are appliedangle with regard to the horizontal di- in repeating the analysis of the struc-rection, and a system of springs which ture, and after a few cycles, the finalconnects the trough to a base, absorb- results are reached.ers, or boosters, depending on the typeof conveyor (2). ANALYSIS

Usually the conveyors are support- The vibrating conveyor and theed to a rigid foundation through isola- supporting structure are analyzed inde-tion springs, but, in some cases, they pendently by the dynamic stiffness meth-are suspended from or supported by a od where the system is divided into astructural system. When vibrating con- number of members, each having knownveyors are supported by a frame or elastic and inertial properties.truss, the dynamic forces which aretransmitted to the supporting structure Dynamic forces and moments at theresult in vibration of and interaction ends of each member are expressed inbetween the structure and the conveyors, terms of the displacements and rotations

at the ends, giving the so-called ecl-The dynamic analysis of the con- ment dynamic stiffness matrix [31.

veyor-structure system is presented Basically, this result can be achievedusing an iterativr scheme where the by solving the appropriate equation ofthree dimensional structure and the two motion. The nystem dynamic stiffnessdimensional conveyor are analyzed suc- matrix is assembled from element matri-cessively taking into account inter- ces using the conditions of continuityacting effects. First, by assuming t' between elements and equilibrium at the


joints; the latter will include any The drive unit houses two motors, oneapplied external force and the forces above the other, with double extendedfrom all the elements forming the shafts to which eccentric masses arejoint. Equilibrium conditions at the affixed. The motors rotate in oppositejoints result in n equations relating directions producing a horizontdl hat-the applied external forces to the monic force.independent joint displacements of thesystem. The n equations may be written The dynamic boosters are tuned atas a natural frequency in the neighborhood

of the operating frequency of the mo-[S]{uJ = {FJ tors. The boosters function as dynamic

(1) absorbers (4) for the component of theimpressed force along the direction of

[S] is an nxn symmetric matrix com- the boosters. As a consequence of thisposed of terms derived from the dynamic absorbing action, the trough vibratesstiffness matrices of the component harmonically along a direction approxi-elements of the system,(uJis a vector mately normal to the orientation of theof the n independent joint displace- boosters.ment in the system, and{F is a vectorof the external forces which are excit- In the analysis of the conveyoring the structure at frequency w. the boosters are treated as rigid

bodies connected elastically to theVibrating conveyors are manufac- trough. The trough is assumed to be

tured in a variety of types, ranging composed of continuously connectedfrom a single moving deck to three or beams with distributed and concentratedmore vibrating masses. In this pre- masses. The drive unit as well as thesentation the so-called "dynamic boost- connecting leaf springs is considereder conveyor" is described in relation to be a special beam element, and theto the dynamic problem originated by isolation springs are treated as mass-the interaction between vibrating con- less elastic members.veyors and aupporting structures. Asshown in Fig. 1, the dynamic booster Diagrams for the basic elementsconveyor consists of a trough supported of the booster conveyor are shown inby isolation springs, a drive unit con- Figs. 2 through 4, and the correspond-nected through leaf springs to the ing dynamic stiffness matrices are giventrough, and a series of spri ng-mass in the appendix. The coordinates indi-assemblies known as dynamic boosters. cating joint displacements at the ends

A A A A A

C B B a B B

Oe G G G( G NG

'/ D GLOBAL x

COORDINATES

Fig.l - Dynamic booster cenveyor

164

of the elements are numbered consecu- In general, any member of thetively. truss, although assumed to be ideally

pin connected at its ends, may undergo,under the action of inertial forces,

El flexural deformation in each of the twoA 5 principal planes in addition to the

2 A extensional deformations along the Ion-mgitudinal axis. It is assumed that for

each individual element of the system-1 4 these three deformations are uncoupled;

thus, the dynamic stiffness matrix forthe element of the truss is obtained

L independently for the two flexural de-formations in each of the principalplanes and for the axial deformations.

The dynamic stiffness matrix for auniform pin-jointed bar element shownin Fig.5 is obtained by solving the

5 corresponding Bernoulli-Euler differen-tial equation for flexural deformationand the wave equation for the axial andby introducing the appropriate boundaryconditions. The dynamic stiffness

6 matrix for the pin-jointed bar elementis given in the appendix.

2 5

ELATOER -4

9 L

1~

Fig.3 - Dynamic booster element Fig.5 - Pin-jointed bar element

EXAMPLE

A dynamic booster conveyor support-5 ed by a truss type bridge between two

64 buildings is presented to illustratethe interaction analysis.

Fig. 6 shows the schematic diagram2- | of the space truss supporting the con-

3veyor. As explained above, the convey-or is analyzed initially under theaction of the driving force and thecondition of zero displacements at thesupporting points. Then, the reactive

Fig.4 -Isolation spring element forces are applied in performing theanalysis of the truss to obtain thefirst approximation for the displace-

SUPPORTING STRUCTURE ments at the points of support of theconveyor. In the next cycle these

i nstallations of conveyor support- dynamic displacements are imposed as

insionaltres anr complex three external actions on the conveyor. The

di m in fu r e s a n d omtr u s s e s to s i m - f i r s t f o u r c y c l e s o f t h e i n t e r a c t i n g

ple building floor systems. A space effects between the conveyor and theple bui din fl or yst ms. exs a m- supporting structure are shown in

truss is used in the interaction exam- Table I.

ple presented.

165

Ph, 4-1--. -------- 7: ,-A j -, -t-*I

BWOTTO DIAGONALS

Plan View

BOOSTERSRO-

SPRIN To,/

' IxSPIG "TOOL STAGGERED

' 7 /2' 2 End View8 Spaces at 30-240"

Elevation View

Fig.6 - Example space truss supporting vibrating conveyor

TABLE IInteraction Results

(a) Amplitude of truss vibration at conveyor suports fin.)Suppot Support 2 support suport4

Cycle Supo -t 3 - upy lorizontal Vertical Horizontal Vertical Horizontal Vertical Horizontal Vertical1 0 0 0 0 - 0 0 0 02 -0.0077 -0.0370 0.0022 -0.0830 0.0082 -0.0998 0.0389 -0.0480

3 -0.0365 -0.0710 -0.0198 -0.1560 -0.0045 -0.1890 0.0600 -0.07704 -0.0390 -0.0730 -0.0200 -0.i590 -0.0053 -0.1940 0.0620 -0.08005 -0.0390 -0.0730 -0.0200 -0.1580 -0.0052 -0.1940 0.0620 -0.0800

(b) Amplitude of conveyor reactive forces at supports (lbs.)

Cycle Supp ort I Support 2 Support 3 Support 4iorizontal Vertical Horizontal Vertical Horizontal Vertical Rorizonta Vertical

1 339 -172 203 -T6 175 -33 1392 153 -123 162 -111 178 -62 2043 133 -132 153 -110 173 -48 2374 136 -132 153 -110 173 j -48 239

166

*'4

NOTES ON COMPUTATIONAL METHOD CONCLUSIONS

A computer program in Fortran IV An iterative method for the analy-is developed for the analysis of the sis for vibrating conveyors mounted onspace truss, the dynamic booster con- supporting structure has been presented.veyor, and the initerative procedure The two systems are analyzed separatelyas described in this paper. The flow using as boundary conditions the de-diagram of the computer program follows. flections and forces developed at the

points where the conveyor is supportedFlow Diagram by the structure.

Machine-Structure InteractionSeveral testing problems as well

as actual cases of installation of con-START veyors supported by structural systems

were analyzed by the iterative method.1(l) READ AND PRINT INPUT DATA This study of the interaction of a vi-

brating conveyor and the supporting2) SET DEFLECTION AT CONVEYOR structure indicates that the methodSUPPORT POINTS EQUAL TO ZERO presented requires four to eight cycles

4to converge to the final solution.(3) LOOP I = 1 CALL

CONVEYOR ANALYSIS PROGRAM APPENDIX

(4) SET REACTION ON TRUSS EQUAL The dynamic stiffness matrices forTO FORCES DETERMINED IN (3) the basic elements of the booster con-

iveyor and for the pin-jointed bar ele-(5) CALL TRUSS ANALYSIS PROGRAM] ment of the truss shown in Figs. 2

TO DETERMINE DEFLECTION through 5 may be written as follows:AT CONVEYOR SUPPORTS

Ia.- Isolation Spring1 (6) IS THERE A SIGNIFICANT CHANGEIN DEFLECTIONS AT CONVEYOR SUPPORTSI Kt 0 -Kv -Kt 0 Kv

NO YES 0 Ka 0 0 -Ka 0

vSET NEW -K 0 K Kv 0 KR/2DEFLECTION VALUES

-K t 0 K v K t 0 KrCALL CONVEYOR PROGRAM TO t v t vIDETERMINE MOTION AND FORCES 0 -K 0 -K 0a a

CALL TRUSS POGRAM TO -K 0 Kv 0 KRDETERMINE MOTION AND FORCES R

IPRINT OUTPUT MOTION AND FORCES~where:STOP Ka = Axial spring constant

The prescription of imposed displace- Kv = 1/2 KtLments at the points of conveyor supportmay be programmed for a digital com- KR = 1/3 KtL2puter such that there results a reduc-tion in the total number of equations L = Length of the springto be solved. In order to avoid re-arrangement of computer storage, how- Kt = Transverse Spring Constantever, it is often more convenient toproceed with the direct solution byimplementing a computatiunal device dueto Payne and Iron and referred to byZienkiewicz(5). The diagonal coefficientof the dynamic stiffness matrix cor-responding to the equation with imposeddisplacement is multiplied by a verylarge number (say 1021); at the sametime the corresponding force term ofthis equation is replaced by the productof the nevly formed diagonal element andthe prescribed displacement.

167

A

b.- Conveyor Beam Element

E Nb iw 2

o DaQ-Mit2 Symmetric

o DS DP-J.W2-RA0UDaSaHsa-EABUb 0 0 EABNb-afW 2

o "Da (S+Hsa) D (Ca-Hca) 0 Da2Q-mIW2

-he: Da (Hca-Ca) D(Ha-Sa) 0.aSaHsa DP-Jf 2

where:

E = Modulus of elasticity Ha = eosh(aL)A = Cross-sectional area Hs sinh(aL)B =(m2/EA)

D -Ea/(l-C a ca)w Angular velocity of forcing S - SIN(BL)frequency

bm Mass per unit length Cb = COS(BL)L a Length of beam

Ub - COSEC (EL)

Nb = COT(BL) Q a Hsa+Salca

Mi - Concentrated mass at left end of P Salca-CaHsabeam Mf = Concentrated mass at right end of

a = (mw/EI) 1/4 beamC s= Concentrated

mass moment ofI Cross-sectional moment of inertia inertia at left end of beama = COS(aL) Jf Concentrated mass moment of

Sa = SIN(aL) inertia at right end of beam

c.- Booster Element

C2K +21 M2c s-

SC(Kc-Ks) SSKe+CCKsmW2 Symmetric

-Cx c - i K, (1+e/l 2)-j

SC(K-K) CK 1C 2K+S2Xs Cse+ScSC(K _- (SI +C2KCS

SC(K-K) (SKcC s)S1cl SC(K-Ks) S2 Kc.C2Kssc c s c (cK8)S +I s

0 0 "ejc/12 0 0 e 2 / 2

where:

C = coso m = mass of booster

S = sinO j= mass moment of inertia about

Kc = Spring modulus-compression Point 0 (iq. 3)

,= Spring moduluse. 0 , (See Fig. 3)Es =Sprig moulu-she~r

18

d.- Pin-jointed Bar REFERENCES

(1) Paz, M. - Conveying Speed ofA1 0 0 A2 0 0 Vibrating Equipment

Publication No. 64-WA/MH-I -

0 Q(Z) 0 0 P(Z) 0 ASME 1965

0 0 Q(Y) 0 0 PY) (2) Hinkle, T. Rolland - Design ofMachines

A2 0 0 Al 0 0 Prentice-Hall, Inc. New Jersey1957

0 P(Z) 0 0 Q(Z) 0(3) Przemieniecki, J. S. - Theory of

0 0 P(Y) 0 0 Q(Y) Matrix-Structural AnalysisMcGraw-Hill, New York 1968

where: (4) DenHartog, J. P. - MechanicalVibratioas

Al f EA~cot0L McGraw-Hill Book Company, Inc.1956

A2 -EAcosec8L 1(5) Zienkiewiez, 0. C. and Cheung,Y.K.

a1EIr The Finite Element Method inQ(r) = -- r (cOtr L-cotha rL) Structural and Continuous Mechanics

McGraw-Hill Company, London 1967

acEIP(r) - (coeecharL-CosecrL)

2r - (r -y, Z)

Ur r DISCUSSION

S= M-- 2Mr. Zudans (Franklin Institute): I do not think IAE understand your model. My impression is that dy-

namically you only considered the conveyer and thatNOMENCLATURE the truss was taken as a simple static structurewith-

out any dynamics considered In the process of analy-A = Area sis. Is that correct?

E = Modulus of elasticity

L, l,e,h = dimension Mr. Paz: No, sir. The forces coming from theconveyer to the truss were dynamic forces and the

G = Modulus of elasticity in truss was analyzed as a dynamic problem. The trussshear members had distributed mass. The paper shows the

dynamic stiffness matrix for a truss which, althoughI,yIy, I z = Cross-sectional moment of it is pin-connected at the ends, still has bending due

inertia to the inertia effect.

JiJf = Mass moment of inertia Mr. Zudans: Did you have your truss represent-

J'Jx = Polar moment of inertia ed as a lumped mass system with dynamic degrees ofk kfreedom and not only static degrees of freedom?

kaikc,k s = Spring constantMr. Paz: Actually this was done by the co-author,

Mimi'm j = Mass but this is not the case presented here. This is thedistributed mass case It has a finite number of de-grees of freedom because of the masses of the ma-

0 = Angle trix method of structural analysis. But the equation todetermine the dynamic stiffness for each element

w = Angular velocity takes Into account the distributed mass and elasticity.

169

RESPONSE OF A SIMPLY SUPPORTED CIRCULAR PLATE

EXPOSED TO THERMAL AND PRESSURE LOADING

J.E.KochNorth Eastern Research Associates, Upper Montclair,N.J.

and

M.L.CohenNorth Eastern Research Associates, Upper Montclair,N.J.,and

Stevens Institute of Technology, Hoboken,N.J.

Using classical plate theory, equations are derived forthe response of a simply supported circular plate exposed totime dependent pressure and thermal loading. Results basedon these equations are presented for several loading condi-tions and plate'geometries. One interesting effect is thepresence of thermally induced vibrations having rather sub-stantial amplitude in some cases. In addition it was ob-served that the- thermal stresses may be compressive at boththe front and rear surfaces, and tensile in the center.

INTRODUCTION are time dependent but assumed not tovary in the in-plane directions. Regard-

There has been, in the recent past, ing the heat conduction problem as un-a sustained interest in the consequences coupled, the temperature distributionof exposing a structure to time-varying and history are computed using a finitethermal as well as mechanical loading difference technique.conditions. One may readily envisionsituations in which this type of loading Results computed on the basis ofmight occur. For example, in the envi- these equations are presented forronment of a nuclear event, a period of several loading conditions and platerapid and intense heating would soon be geometries. One interesting result isfollowed by a shock wave. Whereas the that in some cases the rapid heating ofeffect of a shock wave on various struc- such a plate by exposure to thermaltures has recieved considerable study, radiation induces vibrations of substan-both analytical as well as experimental, tial amplitude compared to the quasi-the effect of rapid heating has been static displacements. Another result ofgivin less attention, the computations is the observation that

the thermal stresses may be compressiveThe object of this study is to on both the front and rear surfaces and

determine the response induced in a tensile at the center. In effect therecircular plate simply supported on its appear to be two stress-free surfaces.boundary, when exposed to such loadingconditions. The regimes in which the GENERAL FORMULATIONdynamic effect and the thermal loadingbecome important may then be identified. We wish to determine the displace-

ments in a simply supported circularThe analysis is based on the plate subjected to a uniform time-

thermbelastic equation of motion for dependent pressure p(t) and a time-flexure of a circular plate subjected to dependent temperature field T(z,t) whichtime dependent lateral loads and temper- varies depth-wise through the plate. Theature distribution. Classical modal problem shall be formulated along clas-series methods are used together with sical plate-theory lines wherein, briefly,the Mindlin-Goodman procedure for the displacement w must satisfy the welltreating time dependent boundary condi- known [I] thermoelastic-dynamic platetions. The elastic properties have been equation together with the appropriateassumed to be uniform and not temperature boundary and initial conditions.dependent. The applied pressure loadingand the temperatures within the plate


I

kv , 117P-

Furthermore, we impose the conditionthat the resulting displacements and

4 stresses be finite at r-0, the center ofthe plate.

Since no generality is lost if theeffects of arbitrary initial conditions

" are not included, we choose the followingconditions to apply at t-O for all values

- of r:T W=0

DYNAMIC DISPLACEMENTSWe seek a solution to the governing

equations together with the boundary andinitial conditions in the followinggeneral form

t= j,)T* ,, a ,.()%4 M21 6 (7)

Figure 1 - Typical Circular Plate

in which Rm(r) are the normal functionsarising from the free vibration problem,

Consider the thin circular plate of i.e.;the homogeneous solution. gl(r)thickness h and radius a, as shown in and g2(r) are to be selected to satisfyFig.l, with its median plane in the r-B the inhomogeneous boundary conditions.plane and with z denoting the distance fl(t) and f (t) represent the timefrom this plane. Within this plate the dependency 8f the boundary conditions (2].displacement w must satisfy the govern-ing differential equation 1. Free Vibrations

( The displacements of a plate vibrat--a r () ing freely satisy the homogeneous form of

where e -- the governing equation (3).

A 1 Choosing a product solution of the formNMT = ) T* (2) we•?Rr) e

Since the temperatures as well as tbepressures are independent of r and l, we find that the normal function Rn(r)this must be true of the thermal param- is composed of Bessel functions of theeter MT as well. Thus we may say that first and second kinds [3,41:T: VIMT=O in the above equation and omitderivatives with respect to) :

,)r- rdr where:,, 4.-Dycr,

For the simply supported case we require 4 ff%that the moment Mr and the displacement w (9)vanish on the boundary of the plate.Thus, on r=a for all t>0: The condition of finiteness of displace-

/- t) a) 0 (4) ments at the center of the plate (at r=0)requires that the coefficients Cn and Dn

r& r )] 0r vanish. Applying the homogeneous formOr .of boundary conditions (4) and (5) we

or, alternatively: obtain the normal function

,- d (5) (10)

and the frequency ecuation

172

,7

The terms on the right side of this equa-' ,t4) ',) tion are expressed in infinite series

2-. (11) form:I-L V

c( being the successive eigenvalues of £thys equation. .

_____________________ -~t fjc ,(4) ",,-) (16)2. Orthogonality Of Normal Modes

The orthogonality condition among R.-C') Ra(r)normal modes may be deduced from Clebsch's qthereom E5], stated below for the rth These series expressions may be sub-and sth distinct normal modes: stituted into equation (15) to obtain:

(12), -G.)~ R Q,(+In 'e case of a circular plate, if u, v where: Q() P(0t m(t) -4 (t)and w are the radial, circumferential, Equating the coefficients of Rm on theand lateral displacements, respectively, left and right sides of this equation, wethen we may write find that Tm must satisfy

from which the solution for Tm is readily

Ws,;: Rs e- it found to be

(As -~ o57 - r Boundary Conditions

- r= 0 The technique of Mindlin andGoodman [21 is used to treat the time

Substituting these displacements into dependent boundary conditions which appearequation (12) we obtain the desired in this case in connection with the ther-

condition of orthogonality among normal mal loading. This is done by selecting

modes: the arbitrary functions gl(r) and g2(r)in such a way as to transform them into6 - R, Rj' rd 0- = (13) homogeneous conditions.

4L S Substituting equation (70 into theboundary conditions (4) and (5) yields,

3. Response To Thermal And Mechanical at r=a:

Making use of the homogeneous solutionand the orthogonality conditions we maynow turn to the response of the plate to .0 j- f&)the actual time dependent thermal andmechanical loads. Substituting into thegoverning equation of motion (3) themodal series representation of the dis- Noting that the normal function Rnplacement w given by equation (7), we satisfiei the homogeneous boundaryobtain:

obtain: - conditionh: these equations are equiva-lent to the following set of conditions

SR(14) on gl and g2 at r=a:-Pa~ t atI /• -_ -DEV"',,(. - fkl , 3z;'

From the homogeneous equation (8) we see ( Ithat DV7R, - pA'q9,'?m . Making use of orthis to eliminate V , equation (14) r . -becomes: r 0

173

Initial Conditions

Making use of the assumed dis- (Ihd, i .. 1placement form (7) we find that the T -1 4))o/'L' Trinitial conditions (6) at t-0, for all -,values of r, become O&

(19)0 Thus the coeficient may be computed

R-'+Te according to the equationThe second term of each of these equationsmay be expanded in infinite series form,thereby simplifying these equations: Yr -)f k

Similarly

Thus i we note from equation (17) that 10 ,) "14 "[drJ"i"- ~ TM(0)-A m and ar,,,/,l-'=-., and substitute WoL _,Jr, 0A".

equatioiis (20) into (19) we may ej-ulre. i1 , , ,

to zero the coefficients of Rm toobtain-the values of Am and N: &17 61 R-. T r , ]

4 2

Am =Em (22)

S* whereBm EmIf-II i' L ' L r.

The formal solution for w may now be -- ILwritten: Finally, using the actual values p=p(t),

f =0, f = "Ir/(,-4), gl=l, and g =r0 9A(,,L,)/ (d.)1 " z.) I,) .m) aAd carrying out the integratigns these

become:

,¢ ... ,-

-'?' (21)t-1.

an Evaluation Of Coefficients E' E M (23)and Qm, etc. P , 50-0

The coefficients contained within - _n 14T(o)equation (21) may be computed in detail D.0= 3,) DT,-O)by applying the orthogonality condition where(13) to the infinite series expansions(16) and (19). For example, from the K I ',/ -_ J(rL)rfirst of equations (16) the expression K, ]rka') Jr

for p is

to both sides of this equation, inte--grating with respect to r and making use

of the orthogonality condition: J(- 2 -• "1}

[7121 Or i J Using these displacements w may now be

174

wr

written as: RESULTS

Making use of equations (24) for the-" dynamic case and (25) for the static case,

-. [the displacements were computed for theM,_ Lo w., center of an aluminum-magnesium alloy

.-LM(C) Los&4 -..1

4K) I ob plate exposed to two radiant heatingpulses. These pulses rapidly rose to

-- ,, -..e-)d peak intensities, which occured at 0.1trJ a seconds and 0.34 seconds. The t~tal

thermal exposures were 71 cal/cm and145 cal/cm2 respectively. PlAtes of anumber of thicknesses and diameters werestudied.

fp(r) I AnuW -)dtJ The displacement-time history ofthose plates having the greatest funda-

Pk kL mental periods (ranging from 0.62 secondsz(8*O i L) to 0.16 seconds) are presented in Figures

2 and 4, in which we may clearly seeThis may be further simplified by thermally induced vibrations of a sub-carrying out the integration of T and stantial magnitude. Dynamic displacementsobserving that were observed to range as high as 1.7

times the corresponding maximum static2 2 displacements. We see also that foris the series representation of (r - a2) pulses having a rise time fairly close

deduced in the first of equations (20): to the fundamental period, the magnifica-cc) tion is greatest.

Figures 3 and 5 summerize thej_ - Jiau.I_ r/ results of these calculations as well as

f. (v),44-O1zIC (24) those for one other thickness. Strictlyspeaking, these other plates may not be

.,l compared simply on a fundamental period, - ) basis because the variation in thickness

l) does not merely change the period, but0 Pr),4)t-u,/ renders the plates somewhat d±simildr

in thermal response as well. Note,QUASI-STialC DISPLACEMENTS however, that for a given thickness, thedynamic effects diminish as the periodThe quasi-static counterpart of becomes substantially smaller than the

equation (24) in which inertia terms are heating time.neglected is readily shown to be: v,

,, (.rL flr 2"On the basis of some preliminary411- 6755 3 L 471 .;,1 stress calculations, another interesting

(25) phenomenon may be observed. The stresses26 Q within the plate may be compressive atp" ,1. -- ") both the front and rear surfaces, while

they are tensile in the central region.where{, are the quccessive roots of the This is clearly seen in Figure 6, whichequation ( . presents the static and dynamic stress

distribution in a thin plate shortlySTRESSES after the thermal radiation peak has

occured. The static distribution isalmost precisely that described above,

the displacements (24) or (25) by sub- and we see that there are now twostituting into the thermo-elastic stress- "neutral" surfaces for stress. From thedisplacement relations (1): nearly symmetrical stress distribution

_ .7 Av/"r _e in this case we also see that the static./ - J' 4- s ) ,, stresses induce no net moment in the

plate. The dynamic case is somewhat(26) different. While still satisfying the.- I;_ AlL Yr r condition of no net axial force in the

,3L---' plate, a substantial moment is induced.Where Clearly this unusual stress distribution

arises because of the sharp thermalH . /gradients and should be expected whenNT . rapid heating is involved, or during the

2.

175

early stages of the heating period.

ACKNOWLEDGEMENT

This work has been supported by theUnited States Navy under ContractsN00140-70-C-0019 and N60921-71-C-0197monitored by the Ntval Applied ScienceLaboratory, Brooklyn, New York, and theNaval Ordnance Laboratory, Silver Spring,Maryland.

REFERENCES a

I - Boley, B.A. and Weiner, J.g., Theory ,of Thermal Stresses, Wiley, New"York,

2 - Mindlin, R.D. and Goodman, L.E.,"Beam Vibrations With Time DependentBoundary Conditions", J. App. Mech.,

Vol.72, 1950, p.377

3- Jahnke, Emde, and Losch, Tables ofHigher Functions, McGraw-Hil,NewYork, 1960

4 -Watson, G.N., Theory of BesselFucin, CambridgeU-niv. Press,Lodn 966

5 Love, A.E.H., Theory of Elasticity,Cambridge Univ.Press, London, 1927,p.189

176

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71 cal/cmiRadiant Puls

177I',

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l-fki iiI 1 i

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-,2t0 -CC( -'it M o ~T4C'~ C''Ffue6 Srs itiu i t Cne Of /8 i lat At !i l im 0 0 e

Afe Expsur 1!!o* AI 71 cal/cm Rad n Pulse1

;M wl Ii .

179.

DISCUSSION

Mr. Mercurio (Sperry Gyroscope Company): I Mr. Zudans (Franklin Institute): I would like towould like to commend you on a very interesting refer to your last slide. Something confused me, andpaper. I noticed that you bad fluxes of 71, and I be- I want to clarify it. You showed the static stressesaslieve, and 145. being perfectly symmetrical.

Mr. Koch: Yes, these happen to be particular Mr. Koch: No, not quite --- almost.weapon pulses.

Mr. Zudans: How can that be explained? WasMr. Mercurio: What were the associated pres- the heat flux coming to one side of the plate or both

sure loadings? sides of the plate?

Mr. Koch: I think they were 10 psi. They would Mr. Koch: One side of the plate only. It happen-have wiped out these plates compietely. I had to get ed, I suspect, that the temperature distribution wasrather flexible plates in order to show the dynamics, such that, if you just turned the plate a bit, you getand I was constrained to use particular pulses by my something that looked symmetrical. It was not actu-contract, ally, it just looked like it.

Mr. Mercurio: You have not shown the pressure Mr. Zudans: How could the temperature distri-stresses on the plate. Is that correct? bution be anything near symmetrical If you bad heat

flux from one side only. This is what I do not under-Mr. Koch: Yes stand.

Mr. Mercurio: Have you done any work in this Mr Koch: If you shifted the distribution curve

area? I would like to know, because some of te com- you would have something that would appear to bemon materials that we are dealing with get into nearly symmetrical by coincidence. Accidentally thisproblems when you apply both the thermal and pres- looked symmetrical, but the actual numerical valuesure loads simultaneously, was such that it was not actually symmetrical.

Mr Koch: We are getting into it right now. In Mr. Yang (University of Maryland): I have twofact in about a months time I am due to have a report questions. On one of your curves you show variouson this subject. We are also attempting to answer the diameters of 1.5, 2 and 3 meters. Since the rise timequestion: "What actually happens if the properties is dependent upon the diameter, I wonder why the risevary with temperature, let's say, through the thick- time for a diameter of 3 meters is the slowest, where-nesis?" The frequency changes and the modulus as the time for 1.5 meters Is in the middle and for 2changes, so the whole dynamic picture might change. meters is the fastest?

Mr. Koch: I think I may have flashed the curveMr. Mercurio: There is also the problem of the too fast. I do not think that actually happened. The 1.5

properties changing over very short time durations meter one was quickest.where very little irlormation is available.

Mr. Yang: The second question is, lave youMr. Koch: I have access to some data which is done any work in the thermal stresses of composite

in a range less than one second -- perhaps in the materials?millisecond range. I have not looked at it, but I waspromised the data. I agree there is some lack of data Mr, Koch: No, however I hace done some non-on materials in the short time range. thermal work in the vibration of layered materials.

180+ 1.

WHIRL FLUTTER ANALYSIS OF PROPELLER-NACELLE-PYLON

SYSTEM ON LARGE SURFACE EFFECT VEHICLES

Yuan-Ning Liu

Naval Ship Research and Development Center

Washingtonr D.C. 20034

A typical propeller-nacelle-pylon configuration installed

on large surface effect vehicles is analyzed for dynamic sta-bility in normal operations. A whirl-flutter theory deveAopedby the aircraft industry for propeller-rotor dynamics is usedto establish stability boundaries for variations of physicalparameters of a propeller-nacelle-pylon system. The stabilityboundaries are presented in chart forms which can be used asdesign guideb. The uppermost and lowest bounds for stabilityare shown on these charts. Classical whirl-flutter theory,which considers ideally rigid blades, defines the most con-servative design and the uppermost bound. The lowest boundis found when the blade fundamental bending frequency is 4/10of the propeller rotating speed. This is considered to be anoptimum value in the sense of greatest freedom of choice ofnacelle mounting stiffness for dynamic stability.

Although direct comparisons of experimental data withthese analytical results are not possible in the present anal-ysis, reported observations on the investigation of nacellevibration on existing hovercrafto indicates that whirl-fluttercould properly explain the results. Further studies arerecommended.

I. INTRODUCTION the understanding of this phenomenon.

This is attested to by the fact that on

In the aircraft industry, the pea- some V/STOL configurations, propeller

sible occurrence of whirling-type insta- whirl-flutter stability is one of the

bility on propeller-driven aitcrafts was main considerations in design [3-10].

recognized as early as 1938 by Taylor

and Browne [1]. This type of instabil- On a large existing surface effect

ity received a great deal of attentions vehicle (SEV) such as SR.N4 [11], the

in 1960 with the impetus of two early propeller-nacelle-pylon system is very

failures on the Lockheed "Electra" air- similar to the propeller-rotor system

crafts [2]. Suboequent investigations on a typical V/S'OL [3,10]. It seems

revealed that propeller whirl flutter logical then, for a future large SEV

could cause total destruction of the equipped with a similar propulsion

propeller-nacelle system if the nacelle unit, that studies be made to determine

was not mounted with sufficient rigid- the controlling parameters of whirl-

ity. In the past decade, large hseli- flutter to assure an adequate design.

copters and vertical and short takeoff The prdsent analysis was under-

and landing aircrafts (V/STOL) came taken for the purpose of gaining a

into existence, at least partly, due to general knowledge of the propeller

181

whirl-flutter phenomenon as related to k8 Effective blade bendingthe design of the propeller-rotor sys- stiffness constant attern for a SEV. Design criteria may be hub-center 1developed in order to define a stable Equivalent mass matrixpropeller-nacelle-pylon system ae func-

mb Blade mass per unit span-tions of design parameters, such as wise lengthnacelle mounting stiffness, nacelle

m Nacelle (include shaft)inerti~s, etc., and blade vibration mass per unit lengthcharacteristics. The overall design N Number of bladesfor SEV must also consider the vibra-tion characteristics of other mcchiner- n Blade designation numberies and the platform, (n -1,2,3, . . . , N)

R Propeller radiusRe(s) Real point of s

II. NOIENCLATJRE

oa r sr Spanwise distance alonga Local lift curve slope of a the blade measured fromblade section

hub-center to a bladeACJ Equivalent damping matrix

sections Laplace transform variableCdC Sectional blade airfoil

drag and lift coefficients, t TimerespectivelyBlade chord T Kinetic energy y

U Resultant air velocity onc ,c Effective viscous dampingAx y constants on nacelle pitch- a blade elementing and yawing motions, V Proveller advance speedrespectively

fucinx Normalized spanvise dis-4Dissipating tance, r/R

D, dL Blade sectional drag an d 4 41lift, respectively Xb, s Position vectors for ablade sectin and aE Potential energy nacelle se'tion,respectivelyh Nacelle length ea Instantaneous

blade sec-I Blade moment of inertia tion angle of attackabout hub-center

A'

a o Initial angle of attackIm(s) Imaginary part of s at a blade section

1I11 Nacelle moment of inertid an Effective flapping angle iA including the mass of pro- of the nth blade at itspeller in pitch and yaw, first bending mode,respectively

referred to the plane ofrotation

[K] Equivalent stiffness , Propeller disc pitch andEualet

yaw angles, respectively,referred to the plane ofrotati onKdol% ,Kd Drag factors

1 2 Y Blade lock number,k k Effective nacelle spring acR 4/bx y constants in pitch andyaw, respectively

Exponential decay factor

182

IA- -~. A-: A~A ~AAA .~ ~ AAA __-__________________4_

Nacelle damping ratio For a forward moving propeller-rotor in

A Propeller inflow ratio, level flight, the resulting flow of air

V/RO normal to the propeller plane is gen-

p Air density erally symmetrical and produces a sym-

metrically distributed propeller thrust.4X, y Nacelle instantaneous The pitching and yawing rotations of the

pitching and yawingangles, respectively propeller disc due to the nacelle flex-

Azimuth position angle of ibility destroy the central symmetry of

the nth blade the flow in a continuously varying man-ner and create additional undesirable

$1 Propeller rotating speedmoments which may lead to unstable

Nacelle natural frequency nacelle circular motion in the direc-

W Nacelle natural frequen- tion opposite to that of the propellerx y cies in pitch and yaw, rotation, i.e., the backward whirling.

respectively r i

The aerodynamic source of this insta-W Non-rotating blade funda- bility accounts for the usual designa-

mental bending frequency a

tion "whirl flutter," which is to be,q n Nacelle frequency ratios distinguished from the purelyx y 04 /a) and (w y/),

X y mechanically-induced whirling motionrespectively in a rotating shaft s)stem.

nB Blade frequency ratio

(.) Dot over a quantity indi- A. DYNAMIC STABILITYcates differentiation withrespect to time t, or, dotbetween two vector quan- When considering the dynamic sta-tities indicates dot- bility of a propeller-rotor system aproduct

irore realistic representation of the

(') Superscript indicates dif- system would be to take into accountferentiation with respectto the time variable Qt the flexible motion of the blades as

well as that of the rotor. The coupled('b) Tilt over a quantity indi-

cates Laplace transforma- motions of the blades and rotor form the

tion theoretical background of this analysis.

(4) Arrow over a quantity

indicates a vector Fig. I is a sketch which Illus-

trates the configuration of the

III. THEORETICAL CONSIDERATIONS propeller-nacelle-pylon system on

SR.N4. A schematic representation of

The classical whirl-flutter phe- this system or any other propeller-

nomenun, in which the propeller blades nacelle-pylon configuration for the

are considered infinitely stiff, is mathematical model is shown in Fig. 2.

briefly explained as follows. The The mathematical model used tn this

motion of the rigid-bladed propeller on analysis is essentially that of a four-

a flexibly mounted nacelle consists of degree-of-freedom system. The steady

thd pitching and yawing rotations of aerodynamic loads and vibration theory

ti e nacelle combined with the corre- used in this analysis are considered to

sponding motions of the propeller plane, be linear. It was demonstrated by

183

I.{

FLAPPING HANGE

. AXIS OF LADE

C -- SHAFT SOTATING AWI

• L.6'96AD[ MEJINT

47

PIT01 AXIS YAWARlS

Fig. 2a - Propeller-Xotor StructuralMode!

JL

Fig. 1 - Propeller-Nacelle-Pylon System PARAtLTO I[ELANEOOfATOPon the SR.N4 Kovercraft

Fig. 2b - Aerodynamic Forces on a BladeElement

Lytwyn [7] that the nonlinear effects Fig. 2 - Schematic Representation of

were small on the variance of stabil- Mathematical Model for a Propeller-

ity boundaries. It is felt that omit- Rotor System

ting higher order effects is justifiedand the stability boundaries resulting dssptopteiawrenedadfrod p this analysis are valid.

were formulated as follows.

The nacelle is allowed to move in T T + T

pitch and yaw directions and its flex- (shaft)

ibility is assumed to be concentrated fR

at the root of the nacelle. The indi- E (bb n d (la)2' I bb dm b s

vidual propel,-r blade is allowed to n-l J0flap in its fundamental bending mode_1

and an equivalent system is f rmed such + (- s( x'8s ) dmathat the blade Is considered rigid but 0

with an elastically restrained flapping

hinge located at the center of pro- E - k k 2 + 2 ky

peller disk (see Fig. 2). y (lb)

+ E k 82nThe variational principle was used n- n

to derive the governing differential

equations. Hence, expressions for D c ;2 + Ic (2c

kinetic energy, potential energy and 2 x 2 4y y

184

Utilizing the Lagrangian equations, the

equations of motions of the dynamic

system shown ia Fig. 2 are given by

d ' E + (2) 8" c

dt(~ q aqj C j

oillSilaerodynamic + ilexternal if + (5

is the generalized aerodynamic or the y yexternal applied force associated with

the generalized coordinates qi" The

generalized coordinates used in this

analysis were 8c' 8, #,xf and *y. A The elements in the matrices [H], [C],constant rotating force is on and [K] are formed from the coefficients

the shaft so as to produce a constant in the governing differential equations

driving torque and this was the base in of motions, again, they may be found inthe derivation of e rnalRef. [71. By taking Laplace transform

th deiato Eof Q5) itxtrl Theomeaerodynamic forces Qi aerodynamic on Eq. (5), it becomes

involved in the above derivations were

based on the quasi-static aerodynamics. foci

The sectional lift and drag acting on

the blade element are expressed in the 2

form (see Fig. 2b) (s[H] + s[C] + [K - 01 (6)

dL iPcU C~dr (3a) y

This represents a typical complex eigen-1 2dD - cU Cddr (3b) value and eigenvector problem. The

characteristic roots a are those which

where C£ and Cd are assumed in the form make the determinant of the coefficient

metrix in Eq. (6) vanish. Each elgen-

[Z . a a (4a) value s, or the characteristic root,a corresponds to a mode of oscillation

CK (2 and can be expressed in the form- Cd K Kd + K d a + K d2a (4b)

d0 d1 2a - Re(s) + jIm(s) (7)

The variations of Re(s) and Im(s) of

The detailed analytical derivations all the modes as a function of differ-

for the mathematical model shown in ent configurations may be expressed in

Fig. 2 may be found in Ref. [7]. The root-locus plots on a complex plane.

governing equations from the derivations Instability is considered to occur

may be expressed in matrix forms, whenever any of the characteristic

185

_______________ ~ ~ ~ ~ 4 .- -7-T.p r, 771- 7,17- -,.

roots enters the positive real axis *A_______

plane. In physical terms, the damping //MMI .value associated with a particular mode 0.05 OARHRLN

hsbecome negative. The roots which VNIMM/ MSCKVRD11INare located on the imaginary axis arecorresponding to neutral stable con- 0.0) X-"s ~ / /

variation of Re(s) alone assa function N

of different configurations is enough Z YK ''~/ ~ ""~c ~~/to define the stability boundaries. aK YON1'1/"""01/1

However, occasionally the variatio~n of IACILLS -0.01 NCLLPIH

Im(s) is also needed in order to define FREQUENCYU#,4R

whether the stability is a dynamic or 00

astatic divergence case, i.e., when____ ___

both Re(s) and Im(s) become zero.STBL

Fig. 3 shows a typical root-locus plot

for blade bending frequency ratio _O.O11 1. Furthermore, sufficient vari- j00

ations in parameter for structural con-

figurations are necessary in order to Fig. 4 -Exponential Decay Factor 6define accurate stability boundaries. as a Function of Nacelle Pitch

Frequency for Various BladeThese are shown in Figs. 4 and 5 for Bending Frequenciesdifferent n1 values for illustrating

* purpose. Any particular configurationthen may be checked for stability in

the preliminary design stage by uti-

lizing these results. 0.10_________________

0.0FORWARD WHIRLING0.

0.00 BACKWARD) WHIRLING________0.0

0 0046

~ Is 000

02s1

STA LE o UNSTABLE ___

04 0.1 0.001 .4

-. Bending Fequency Rati PITCH FREONCY 0orjir0 u

Fig. 3 -Root-Locus Plot for Blade Fig. 5 -Nacelle Frequency as a Function

(W0 /In) -1.0BldBednFrqncs

18j

A digital computer program 112] zero or by omitting time-dependent terms

developed for the Naval Ship Research in Eq. (5). One obtains

and Development Center (NSRDC) was used

to solve Er,. (6).

B. SPECIAL CASE - IDEALLY RIGID BLADES [K] S f)(8)10)

Ideally speaking, an infinitely y

rigid blade is not possible in practice.

However, this condition provides us the

uppermost bound in defining whirl-flutter

stability boundaries. A necessary and sufficient condition to

have a nontrivial solution for the above

Referring to the set of governing equation is that the determinant of the

differential Eq. (5) one may reduce these coefficient matrix [K] is identically

four equations to a special set of two equal to zero, i.e.,

equations for the case of ideally rigid

blades. This may be accomplished by I[K]I = 0 (9)

assuming that the blade root constraint

stiffness or the blade fundamental bend- Eq. (9) was utilized in the process of

ing frequency is infinite. These two mapping the static divergence bounds-

equations of motion represent the "clas- ries.

sical" whirl-flutter phenomenon of the

two-degree-of-freedom propeller-nacelle

combinations. By studying the character- IV. PARAMETRIC STUDIES

istic roots of these equations, explicit

expressions for stability boundaries may A practical design for propeller-

be obtained. This was done by assuming rotor systems would have physical prop-

that the mass moment of inertia of erties such that their parameters were

nacelle was the same in both pitch and in the region bounded by the uppermost

yaw directions. and lowest boundaries in the stability

plot. The limiting boundaries have

already been discussed as "Ideally Rigid

C. SIATIC DIVERGENCE Blades" and "Static Divergence." There-

fore, as a design guide, parametric

Static divergence of a propeller- studies were made In order to define

rotor system may be considered as a these whirl-flutter stability boundaries.

limiting case of whirl-flutter phenome- The ratio of natural blade bending fra-

non by conaidering only the static quency to propeller rotational speed,

restoring ability from the structure. i.e., n,, and its variations form a

This may be visualized by letting the family of stability curves that are the

"natural frequency" of the system become results of such studies.

1

~187

The propeller-roto'r-system on SR.N4 ,1 one can evaluate the variation of

represents a large high-soced SEV'system, eigenvalues of different configurations

and its structural configuration will and hence define the stability bounda-

probably resemble one of an anticipated ries. The results are shown in Fig. 6.

design. It is believed that design

values for the propeller-nacelle-pylon Isystem on a larger and faster vehicle .-- ,

would not vary too much from those of SIATIC VIIGCCOOUNARI$ /

SR.N4. Therefore, a set of typical input CA.-...,._.._.

values resembling the propeller-nacelle- STABLEitTABLE

pylon configuration on SR.N4 is used in 0.s -this report to define whirl-flutter sta- i

bility boundaries and are summarized in leyT a b l e 1 . 1 .4 T )

TABLE 1 0. -- 0Nacelle and Blade Parameters Used in

Defining Stability Boundaries -

Parameter Value "UNSTABLE STATC DIVERGNC 8W~ * -RO

T1 Variable 0.: N T1EI~i IC6N(atN~tC

Variable z

0 01 0.2 CO. 04 0.) 0.A 01?

Ia ye l NACELLE PITCH FREQUEFNCY ftATIOq.~

N4h/R 0.83 Fig. 6 - Propelle..-Rotor Whirl Flutter

Stability Boundaries for Various0.2 Blade Bending Frequencies

" 0.3

'1 3K 0.0087

The stability boundaries for the

K -0.0216 case of ideally rigid blade and staticd1 divergence were also evaluated and pre-

K d 0.4 sented in Fig. 6. By examining the

a 5.73 result, an optimur. value for bladea 5.73bending frequency ratio n is found to

a 4 deg be approximately 0.4. In a similar

0 propeller-rotor configuration with a

value of ng less than 0.4, a sudden

Note: Assume 1 -1 -I reversal from the backward whirl insta-X Y bility to a forward whirling would be

developed. The stability boundary curveUsing the input values from with n, . 0.4 encloses almost the larg-

Table 1, one may form the matrix equa- eat sti-ble region and allows a designer

rion shown in Eq. (5). Solving this thp .ayvmum freedom to fulfill nacelle-

equation as functions of n8 , r , and sup, .in& 3tiffneas requirements.x

188

-I l yrk '1 -

V. TRANSIENT RESPONSE ANALYSIS VI. COMPARISON OF THEORY WITH AVAILABLE

EXPERIMENTAL INFORMATION ON SURFACE

A physical interpretation of a EFFECT VEHICLES

propeller-rotor configuration for dynamic

stability is the transient response due Since test results and structural

to some initial disturbances. For a sta- information on existing SEV are not

ble system, the amplitude of response readily available, it is very difficult

would damp out in time, but in an unsta- to make any rational precise experimental

ble system the dmplitude increases with evaluation of the applicability of the

time. A special case is the neutral theory to propeller-rotor systems. How-

stable condition, i.e., the response due ever, with the limited structural infor-

to any initial disturbance could be rep- mation available for SR.N2, SR.N3 and

resented by some harmonic functions with SK.5 al- cushion vehicles, the following

constant amplitudes. Numerical evalu- evaluations are made. The propeller-

ations were performed on a PACE nacelle-pylon on SR.N2 is essentially the

Model-2312 active analog computer for same as the one on SR.N3. The values of

five different propeller-rotor config- nacelle pitch and yaw natural frequen-

urations to illustrate the above cdes, and the propeller blade funda-

phenomena. mental beniing frequency onf SR.N2 or

SR.N3 were obtained from Ref. [I]. The

Referring to Fig. 6, use n, 1 to values of the same parameters on SK.5

define the stability boundary and pick were obtci!,ed from a vibration shake

five points along the n /,no = I line. test performed by NSRDC.*x y

In practice, this line represents anThe stability boundaries of the

isotropic mounted nacelle. These five

pointspropeller-rotor configurations on SR.N2,y) SR.N3, and SK.5, shown in Fig. 8, were

0.4, 0.3, 0.2825, 0.25, and 0.2. The obtained from the interpolation of those

first two cases are in the stable shown in Fig. 6. This was based on the

region, the third one is on the neutral assumption that the nondimensional param-

stable boundary, and the last two are in ters used for stability apalysis for

the unstable region. An initial angular SR.N2, SR.N3, and SK.5 were the same as

velocity of 0.01 in the pitch direction those shown in Table 1. This may not

was assumed throughout. The results necessarily be true; however, for the

were expressed through the usage of an purposes of preliminary evaluation, the

X-Y plotter and are shown in Figs. 7a-7e. results should be close e:..ugh to give a

These figures clearly show the backward designer a rough guidance. Nevertheless,

whirling phenomena as well as the degree experimental confirmation of the vklidityof stability for each configuration. In of the stability boundaries shown in

these plots, the X-axis represents the Fig. 8 as well as those shown in Fig. 6

pitch response and the Y-axis represents is still lacking. The only experimental

the yaw response. Heuce, these figures

actually represent the locus of the * A ]lagcn at NSRDC performed the

motion of a Propeller hub. SK-5 vibration shake test inMarch 1971.

189

4C

- 0.

- -- - - - - - - 4 I

$ .1 ~ (44

4-- _ __ ~0

I 14.0 1

SPaY Ma I ION CC

A4 0

__ _ 44 a

-4 u

4w 4.

a 4

v 190

T -and the condition is not serious . . .STAIC DYEUN~CII UIS 1 (Ref'. (111, p. 219.)

uSTABLE A

toe ALLY RIMP KIAN. ~e . The above quoted statement clearly

/ describes the propeller-rotor flutterwhirling phenomenon and the actual occur-

rence on some SEV. Fig. 8 shows the

same behavior, i.e., SK.5 has a very sta-

02 __ \.?L hise propeller-rotor conilguration and

." SR.2 or SR.N3 has a ilightly stable one.UNSTABLE* II0S.T TDIVIRCCSCE -AR 1$7 How stable a system io can be judged by

~Ithe distance between the data point and_ its corresponding stability boundary

a . 02 03 . S CA .

NAC PITCHREQUENYRAIO icurve. As one can see from Fig. 8, the

data point associated with SR.N2 or SR.N3

is very close to the neutral stable lineFig. 8 - Propeller-Rotor Whirl FlutterStability Boundaries for SK.5, SR.N2, and nacelle vibratory motion on SR.N2 and

and SR.N3 Air Cushion Vehicles SR.N3 were observed.

results the author could find were some

observations on propeller-rotor vibra- In order to have a better feeling

tion in Ref. 111]. of the actual physical behavior of the

propeller-rotor system on SR.N2 or SR.N3,

an evaluation of transient responses due

The pylon situation is fur- to an initial disturbance and the per-

ther complicated to the extent that not formance ot a response analysis due to

only will it have modes in the lateral some random excitations were undertaken

plane but also in the vertical plane. In through the use of analog computer simu-

some of both the lateral and vertical lation. Fig. 9 shows the transient

modes the angular deflection at the pro- responses in both pitch and yaw direc-

peller is considerable. As the propeller tions due to an initial angular veloc-

is rotating at high npeed,, angular ity 0.01. Fig. 10 shows the responses

motion, or precessio, in one plane will in the same directions due to some ran-

induce a force in the plane at right dom excitations. Here the random exci-

angles, due to the gyroscope effect . . . tations were generated by attaching a

Gaussian noise generator to the analog

The propeller is obviously the main circuit at the point associated with the

source of forcing in these modes . . . nacelle pitching angular velocity.

Fig. 10 also includes this particular

Experience so far has disclosed noise output which has a rus value of

appreciable vibration only in the low- 3.16, frequency range 0 to 50 Rz, and

eat modes. Both SR.N2 and SR.N3 pylons 3 JB cutoff. Fig. 9 shows the rela-

can be seen to nod gently when running tively small amount of damping in the

at idling conditions. The forcing at system and Fig. 10 shows the actual

these conditions is very low, however, response due to some random excitations

191i

NACELLE PITCH FREO'JEOCaz q#0.1

MACILL I YAW FREQUENCYsII AA^

Il-

Fig. lOa - Random Exciting Forces

10 to 0 30 4 0 0 i I F -10

Fig. 9a- Nacelle Pitch Response _..1 tIj

0.04

ili- 0 t0 20 0 0 0 to 10 go to

-0.011 Fig. lOb - Nacelle Pitch Response

Fig. 9b - Nacelle Yaw Response 0.

Fig. 9 - Analog Simulation of Propelleriub Motion due to In4.tial Disturbance 0 10 20 0 30 60 ?0 lb go 10

for SR.N2 and SR.N3 Air Cushion , .mUI0,,i-.,i'a[4.4.t'.,,,,

VehiclesIi1 L L iwhich always exist in reality. Both Fig. 10c - Nacelle Yaw Response

Figs. 9 and 10 clearly illustrate the

relatively large oscillatory motion in Fig. 10 - Analog Simulation of PropellerHub Motion due to Random Excita-

the nacelle pitch direction in compari- tion for SR.N2 and SR.N3 Air

son with yaw. These results agree with Cushion Vehicles

the observations in Ref. [11] quoted

above, confirm the theoretical computations;

however, some observations on SR.N2 and

VII. DISCUSSION SR.N3 tend to agree with theoretical

predictions.

A propeller-rotor whirl flutter

theory has been applied to pylon-necelle- As indicated in the evaluation of

propeller configurations on surface stability boundaries, blade fundamental

effect vehicles. Stability boundaries bending frequency plays an important

based on a typical propeller-rotor system role since it affects the stability

on large SEV were evaluated for the pur- mapping. The classical whirl-flutter

pose of preliminary design and guidance. theory in which blades are assumed to be

No experimental data were available to infinitely rigid gives the most rigid and

192

NI.. Wt..1~tt~ N'~fl ~ l.N..

overly conservative stability criteria, design guidance. Direct experimental

leading to over-design and possibly confirmation is not possible at this

uneconomical construction. An optimum time, however, tht following statemenhts

value n, a 0.4 was found, and when the can be based on this analysis.

blade bending frequency is below this 1. Observations of the nacelle

value, forward whirling as well as motions on existing SEV tend to confirm

backward whirling'could develop. How- the theoretical analysis used in this

ever, the optimum value for blade bend- report.

ing frequency is restricted in the sense 2. Optimum fundamental blade bending

that it would allow a designer the most frequency with respect to propeller

freedom to choose nacelle mounting rotating speed should have e, value of

stiffness in order to avoid whirl- 0.4 in order to impose the winimum

flutter instability. Other dynamic con- requirements on nacelle supporting

siderations should be also included in stiffness for a stable proptller-rotor

overall design. As implemented in configuration.

Fig. 6, increasing nacelle mounting 3. Classical whirl-flutter theory

stiffness to increase nacelle natural gives overconservative stability cri-

frequencies is another way to move a teria which impose the most severe

particular propeller-rotor configuration design requirements on the naceile

from the unstable region into the stable supporting stiffness.

zone. 4. Caution should be taken in the

consideration of fatigue failure even

when the propeller-rotor system isTransient analysis by analog com- operating i the stable region.

puter simulation gives a direct physical

interpretation of the actual whirl-

flutter phenomenon. Transient responses IX. RECOMMENDATIONS

due to initial disturbance and responses

due to random excitations can be con- In view of the importance of a sta-

sidered as the results of gusts and tur- ble propeller-nacelle-pylon system on

bulence in the atmosphere or due to the large high-speed SEV, and insufficient

irregularity in the ground/water surface. experimental data to substantiate the

They are in connection with dynamic loads theoretical analysis of such a system,

and fatigue failures. These loads may be the following efforts are recommended.

aignificant for fatigue failure criteria I. Perform experimental evaluations

even though the system is operating in by using a model with the same nondimen-

the stable region. Observations on SR.N2 sional parameters used in this report.

and SR.N3 tend to strengthen this 2. Perform a vibration shake test and

consideration. underway test on a large SEV, and compare

the test results with the theoretical

computations in this report.

VIII. CONCLUSIONS 3. Extend the existing theory by

including blade first torsion mode,

Analysis performed by applying blade second bending mode, and unsteady

the whirl-flutter theory on a typical aerodynamic loadings.

propeller-rotor system of a large SEV 4. Study the effects on the change of

resulted in a parametric chart for stability boundaries as functions of

193

important nonClmensional parameters such 6. Niblett, T., "A Graphical Repre-as nacelle damping, nacelle length, sentation of the Binary Flutter Equa-tions in Normal Coordinates," Royalblade inertia, propeller power setting, Aircraft Establishment Tech. Rept. 66001,

Jan 1966etc.

5. Study the effects on the change of 7. Lytwyn, R. T., "Propeller-Rotorstability boundaries due to the overall Dynamic Stability," The Boeing Co.,

Vertol Div., Tech. Rept. D8-0L95, 1966structural dynamic characteristics of

surface effect vehicles. 8. Edenborough, H. K., "Investigationof Tilt-Rotor VTOL Aircraft Rotor-PylonStability," Journal of Aircraft, Vol. 5,No. 6, Mar-Apr 1969

REFERENCES 9. Brindt, E. E., "Aeroelastic Prob-lems of Flexible V/STOL Rotors,' paperpresented at 34th AGARD Flight Mechanics

1. Taylor, E. S. and Browne, K. A., Panel Meeting, 21-24 Apr 1969

"Vibration Isolation of Aircraft PowerPlahts," J. Aero. Sci., Vol. 6, No. 2, 10. Gaffey, T. M., Yen, J. G., andpp. 43-49, Dec 1938 Kvaternik, R. G., "Analysis and Model

Teats of the Proprotor Dynamics of a2. oubolt, J.C. and Reed, W.H, III, Tilt-Propotor VTOL Aircraft," Paper

"Propeller-Nacelle Whirl Flutter," presented at the Air Force V/STOL Tech-I.A.S., pp. 333-347, Mar 1962 nology and Planning Conference,Las Vegas, Nevada, Sep 23-25, 1969

3. Reed, W. H., III and Bennett, R.M.,"Propeller Whirl Considerations for 11. Elsley, G. H. and Devereus, A. J.,V/STOL Aircraft," CAL-TRECOM Symposium on Hovercraft Design and Construction,Dynamic Loads Problems Associated with Cornell Maritime Press, Inc., 1968the Helicopters and V/STOL Aircraft,Buffalo, N.Y., Jun 26-27, 1963 12. Peterson, L., "SADSAM V User's

Manual," MacNeal-Schwendler Corporation4. Reed, W. H., III, "Propeller Rotor Project Report, 1970

Whirl Flutter: A State-of-the-ArtReview," Symposium on the Noise and Load-ing Actions on Helicopter, V/STOL andGround Effect Machines, Southampton,England, Aug 30 - Sep 3, 1965

5. Hall, W. E., Jr., "Prop-Rotor Sta-bility at High Advance Ratiou," J.A.H.S.,Apr 1966

DISCUSSION

Mr. Gayman (Jet Propulsion LaboratorX): I ask Mr. Liu: Perhaps I did not make the point veryfor a point of clarification. Early in your presenta- clear. We do consider the blade ae an elastic bladetion you discussed the degrees of freedom you were but restrict it to the first bending mode. I made anadmitting to the problem in reference to blade bend- equlvelant system by considering the blade as rigid,ing. Did you not mean blade flapping as a ridgid mo- but hinged at the hub. The system also included antion? equivalent rotation spring at the hub, and the bending

or flexing frequency was the same frequency as theMr. Llu: Yes, I meant the flapping motion of the first bending mode.

blade, but only restricted to the first bending mode.

Mr. Zudans: (rranklin Institute): Because of theyaw and pitch of the nacelle, the blade plane moves asa rigid body. Was M'at motion included in your hydro-

Mr. Gayman: That is associated with the oscil- dynamic forces?lation of the plane ofthe propeller disk, Is it not? Theblades themselves are treated as rigid, are they not? Mr. Liu: Yes.

194

THE DYNAHIC RESPONSE OF STRUCTURES SUBJECTED TO TINE-DEPENDENT

BOUNDARY CONDITIONS USING THE FINITE ELEMENT METHOD

George H. Workman

Battelle, Columbus LaboratoriesColumbus, Ohio

The dynamic-matrix equation of motion characteristic of structures modeledby the finite element method of analysis is vritten in general form. Thismatrix equation is then rearranged and partitioned to separate constrainedand unconstrained displacement degrees of freedom.

Once the general matrix equation has been properly partitioned, then bystandard matrix manipulations, the original mixed boundary value problemis transformed to a modified force motion problem.

The dynamic response of a bellows subjected to dynamic edge displacementsand internal pressure, as determined by this approach, is presented. Thisexample is used as a vehicle to demonstrate the versatility and effective-ness of this solution technique.

INTRODUCTION discretized continuum elastic undamped dynamicproblem becomes the solution of the hiatrix

Over the past decade and a half, with equation characterized as:the advent of high-capacity, high-speed digitalcomputers and the increasing needs of the aero- [N) (A(t)) + [K- (A(t)] - (F(t)) (1)space industry, the finite element method hasemerged as a powerful tool for the structural whereanalysis of large, complex structures. Evidenceof this is the large number of general purpose IN) = mass matrixfinite element computer programs in use today.

roC - stiffness mtrixThe capability of this approach to

obtain the dynamic response of compcC struc- (F(t)) - nodal time-dependent forcestures to known forcing functions and base mo.ionproblems is well documented. This paper pre- (A(t)) - nodal displacementssents a straightforward extension of the finiteelement approach, to solve dynamic response prob- -At) nodal accelerations.

lems having time-dependent displacement. TThe theoretical background and &ssufnptions

MErHOD OF ANALYSIS leading to the development of Matrix Equation(I) through the use of the finite element meth-

In the finite element method, the odology is well known and not presented herein.continuum is separated by imaginary lines or Two excellent books describing the finite ele-surfaces into a number of "finite or discrete ment method have been authored by Zienkiewicz(1 )

elements". These elements are assumed to be and Przemieniecki( 2f.

interconnected only at a discrete number ofnodal points situated on their boundaries. The Each row of Matrix Equation (1) repre-displacements of these nodal points are then the aents a particular degree of freedom that, forbasic unknown parameters of the problem. The this study, can be described as being associatedform that these discrete elements takes depends with an unconstrained or constrained displace-on the type of structural behavior assumed and ment. Displacements as used herein representthe form of the approxication to that behavior, both translational and rotational motions.

Constrained displacements are those degrees ofOnce nodal points and structural freedom for which the motion is known as a

elements have been defined, the solution of this function of time. This includes both zero and

'195

iC,

nonzero motions. 'Therefore, accelerations and BELLOWS ANALYSISvelocities of theme consuriined degrees offreedom are also known. Unconstrained displace- A dynamic anilysis of a bellows wasments are those degrees of freedom for which conducted utilizing the method of analysisthe applied forces are known functions of time given above. The bellows serves asa pressureand whose resulting displacements are not seal between a vessel and a piston. The topprescribed. edge of the bellows is attached to the station-

ary vessel and the bottom edge is attached toMatrix Equation (1) can be rearranged the moving piston used to generate a pressure

such that those degrees of freedom associated pulsation in the vessel.with the unconstrained displaceQents and thoseassociated with the constrained time-dependent Figure I shows the bellows crossdisplacements are partitioned as shown below: section. This shape was modeled by seven

toroidal parts. Each part was described by its1 -1 major and minor radii and the angular coordi-11 J nates, tp, so that the exact shape of the bellows

+ in its neutral position was reproduced quite

Lii ~J accurately. The bellows is of uniform thick-ness, t - 0.062 inch, and is constructed of

K IK Fsteel sheet -: E a 29 x 106 bin and

(2) (2) poisson's ratio - 0.3.L;]i J] J

K iswhere subscript i is associated with the uncon-strained displacements and subscript J isassociated with the constrained displacements,. -- '

atrx Eqution (2) can then be Lexpanded by rows and rearranged to yield[Mid ] (Ai ) + [K id [L) . 4P

[Fi) -Mj [ - [Kij3 (A J). (3)

andIh

Fj = I + [M 3 j + &W* k[Kj1) [Ai + CKj) IAj1 (4) N'ID

Matrix Equation (3) is in the standard formassociat.d with the elastic undamped multi-degree-of-fre-dom dynamic problem with a slight-ly modified right-hand side to include theeffect of the constrained time-dependent dis-placements. This system of equations can be- -

solved by a variety of methods depending on the-requirements of the problem. Three excellentbooks giving a number of these teihniques havebeen authored by Hurty and Rubinstein(3),Biggs(4 ) and Bisplinghoff, Ashley, andllalfman().

Matrix Equation (4) yields the re- Fig. I - Cross section of bellows used foractive forces at the constrained displacement. vibration analysis (neutral position)

Once the t -history solution of Three finite element models of thetiatrix Equation (3) been accomplished the bellows were created. These models containednodal displacements of the discretized problem 20, 30, and 40 conical elements, respectively.are known. The resulting time history of the The basic conical element used in this analysisstresses and strains in the individual elements was similar to thet developed by Grafton andcan be determined from the assumed behavior Strome(6 ) except numerical integration was em-pattern of the individual elements. ployed to generate the element stiffness matrix.

An earlier study of this bellows was conductedutilizing a continuum axisymmetric shell programas the basic analysis tool. This program uti-

196

I

lized numerical integration of the governing TABL9 2continuum thin-shell ecustions. Unfortunately, Natural Frequencies of the 30-Element Modelthe modulus gf elasticity was taken to be 27.4x 106 lb/in.4 for ;he bellws material and was Mode No. Frequency, cpaassumed to 29 x 100 lb/in.4 for this study. 44.96

The meas of the bellows was luIsped atthe nodal points for the finite element model 2 318.38and was evenly distributed along the length forthe continuum model. The two lowest axisym- 3 354.20metric natural frequencies of the bellows werecalculated for the four models assuming both 4 376.17boundaries fixed. Table I gives the results ofthe natural frequency prediction along with the 5 398.58frequency determined experimentally for thefirst mode. The finite element models predicted 6 440.96natural frequencies slightly higher duc to thedifference in the assumed modulus of elasticity.

TABLE INatural Frequencies of Bellows Model

Finite Element Muel

20-Element 30-Element 40-Elemen]t Continuum Model Experiment

Mode 1 45.03 44.96 44.95 43.2 44.14

Mode 2 318.44 318.38 318.49 311

Utilizing these natural frequency datait was decided to use the 30-element model forthe dynamic analysis of the bellows. eigure 2shows the node locations and elements for the30-element model. Figures 3 and 4 give modeshapes for the first and second natural fre-quency, respectively. Also, the next four natu-ral frequencies of the 30-element model weredetermined and are given in Table 2.

4 # :

R4Cor,*.. \h

2197

004~

S, \30

21

204

24 4 4 ? 24- 0 I 7 7 4 7

Fig. 3 -Modal deflection shape for firstFig. 2 -30 element model of bellows . axisymmetric mode of bellows

197

Two different techniques were emp!oy-ed for the solution of Matrix Equation (3).

to One was the straight numerical integration ofMatrix Equation (3) by a f urth-order single-step Runge-Kutta method.(7 The other was the

, modal acceleration method.(5 ) Within the modal

"- ' acceleration method only the first mode wasemployed. The static displacement component

~~4 wasn determined utilizing standard static. finite/3 4 element techniques. By comparing the numerical

:s 0 " integration solution and modal acceleration% /solution with only the first mode employed, it\.14 was clear that, for engineering purposes, only4 % Ithe first vibration mode makes any appreciable15 s contribution to the dynamic solution.

116 Figure 6 shows the time history ofI the maximum stress occurring at the top aide otI7 the bellows; this stress ocurred at Node 5 ir

/ the 30-element model. Figure 7 shows the tim,gel, 4, history of the maximum stress occurring at th/ bottom side of the bellows; that occurred at12 Node 77 of the model. The numerical integra-

tion solution is-plo2.ed ".n Figures 6 and 7

2o along with staLic stressesi. The modal accel-21/ eration solution was not plotted because It

. 20 o_ er notediscrnibl om te nuecaeltN.__ 2Z-" /0 __ould not be discernible from the numericalA.S.. , integration solution.25 2

24Z The dynamic magnification of thestatically calculated stresses is quite clearwhen examining the plots given in Figures 6and 7.

Fig. 4 - Modal deflection shape for second Figure 8 shows the time history ofaxisymmetric mode of bellows the vertical displacement of Nodes 8, 16, 24,

The dynamic problem was formulated and 31.

along standard finite element techniques for theforming of Matrix Equation (1). The constraineddisplacements are that the radial and vertical This paper has given and demonstrateddisplacements and rotation of Node I are zero,the radial displacement and rotation of Node 31 a very straightforward and effective method of

are zero, and the vertical displacement of Node determining the dynamic response of structures31 is constrained to move downward as a veraine subjected to time-dependent boundary condi-31 s cnsraied o ovedowwad a a erin suete d toime-ependnty bounariy acod-

with a period of 33 milliseconds and a maximum tions. This capability can be easily incor-porated into an existing finite element con-

amplitude of 0.80 inch. The internal pressure pote proa eady p iie emn dnmi)f the bellows, initially at 80 psi, is in phase puter program already progrotoed fo dynamic%*ith the edge displacement, dropping to 10 psi force motion problems.as a versine. Figure 5 shows the internal pi.s-sure and edge displacement as a function of REFERENCEStime. The matrix rearrangement and partitioninggoing from Matrix '"quation (1) to Matrix Equa- 1. Zienkiewicz, 0. C., The Finite Elementtion (3) is very straightforward when accom- Method in Structural and Continuumplished on the digital computer. The additional Mechanics, McGraw-Hill, London, 1967.time, both engineering and computer, required toincorporate the dynamic mixed boundary value 2. Przemieniecki, J. S., Theory of Matrixproblem as formulated herein as compared with Structural Analysis, Mcrraw-llill, New York,that for a typical force motion problem for a 1968.similar application is negligible.

3. lurty, W.C., and Rubinstein, M. F.,One pulse of the piston was examined Dynamics of Structures, Prentice-llall,

and no damping was included in this analysis. Englewood Cliffs, New Jersey, 1965.Experience has shown that, for cnalysis of thistype, modal .-itical damping in the range of 0.5 4. Biggs, J. H., Introduction to Structuralto 1.0 percent gives realistic results. This Dynamics, McGraw-Hill, New York, 1964.small amount of damping has a minimal effect forthe first few cycles and therefore was neglected 5. Bisplinghoff, R. L., Ashley, H., andin this preliminary study. llalfman, P. L., Aeroelasticity, Addison-

198

Wesley, Reading, Hassachusetts, 1955 2342-2347, 1963.

6. Grafton, P. E., and Strome, D. R., "Analysis 7. Ralston, A., A First Course in Numericalof Axi-Symmetric Shells by the Direct Stiff- Analysis, pp 191-202, McGraw-Hill, New York,neas Method", Journal A.l.A.A., Vol. 1, pp 1965.

leo

000

g-0

i5

Tim milliseconds10 / 39 40 so s

0. I

Fig. 5 - Time history of piston displacement and vessel pressure

40-

/ Tkne,nilsecods

"X_ ' D-ynomlc

-40

Fig. 6 Time history of maximum stress at Node 5

199

- '! ..¢.' "".',-'. ."'' -ia' , -' , , '" -,* **" " '' ~d "" !'" , ,t . l" L' ' ''' '' ' ' "'" '" ... '' *' :Y'rv :/d;t'/"d r

Fig. 7 - Time h~stor fmaiu trs t oe2

ICI

D.-€-

Time, miliseconds .

\% \-,-DiNe 1 R

-% Node 16-a-Nod 24-

Fig. 8 - Time history of vertical displacement4of Nodes 8,16,24 and 31

200

DISCUSSION

Mr. Zudans (Franklin Institute): You mentioned Mr. Schrantz (Comsat Laboratory): You saidcomparison between modal solutions and you aid that you used conical elements to define the bellows?the response was entirely in the first mode. Yet themode you showed was incompatible with boundrydis- Mr. Workman: Rightl I used a series of conicalplacement, so you must have done something in add- elements.ition to that.

Mr. Workman: No, the third slids ,,tA.ed theequation after It is rearranged t, get it into the form Voice: Is this a toroidal bellows? What formthat is solved. If you use the modal method on that is it?equation, that boundary appears to be fixed. Then thedisplacement is used in the static solution. That mode Mr. Workman: Yes, actually it is a torold a-finds the dynamic component, not the static displace- round a circular piston rather than a flat bellows.ment. It is a shell of revolution.

201

I VIBRATION ANALYSIS AND TEST OF THE

EARTH RESOURCES TECHNOLOGY SATELLITE

T. 3. Cokonis and G. Sardella

General Electric CompanySpace Division

Philadelphia, Pennsylvania

This paper presents a unique approach used for the launch vibraticn analysisof the Earth Resources Technology Satellite (ERTS) and compares the analy-tical results with experimental measurements. The ERTS is basically a mod-ification of the Nimbus vehicle with solar arrA7 paddles unchanged. The com-plex paddle system could best be represented by measured data obtained fromprevious Nimbus modal testing. The successful extraction and subsequent re-coupling on ERTS of the solar array paddle modes from the original Nimbusexperimental mode shapes is given. The analytical model is described alongwith its verification by an abbreviated modal test. Good correlation betweentest and analysis was evidenced by frequency and mode shape comparisons.Some areas of discrepancy in the analytical model were uncovered which weresubsequently modified to improve the analytical representation of the spacecraft.

INTRODUCTION in mounted on an adapter which is bolted to thelaunch vehicle adapter ring. Spacecraft adapterThe Earth Resources Technology Satellite separation occurs at the lower ring flange of

(ERTS) system (Figure 1) is being developed by the sensory ring.the Goddard Space Flight Center (NASA) for ob-taining data from near earth orbit immediately appli- During the evolution of the Nimbus space-cable to the determination of earth resources craft from a 900-pound configuration to the t

and their management. The spacecraft config- 2000-pound ERTS configuration, the dynamicuration will be launched on a Thor/Delta N-9 representation used for loads and clearancebooster. The structural subsystem is based on studies was based on an extrapolation ofthe flight-proven Nimbus satellite using the same modal test data on Nimbus A. Analytical c" m-philosophy of design. The greater weight of the parisons with test results indicated tbat the

ERTS spacecraft and specific ERTS mission extrapolated modal model no longer predictedrequirements necessitated structural modifica- the dynamic behavior of the spacecraft with thetions and some redesign. required degree of accuracy. This was evi-

denced in the flight loads analysis for the POGOThe ERTS spacecraft structural segments condition using the extrapolated model which

are shown in Figure 2. These segments are: predicted extremely high loads due to high1. The Attitude Control Subsystem (ACS); 2. coupling between the lateral modes and theThe Solar Array Paddles; 3. The A-Truss axial excitation. To overcome these deficien-Structure; 4. The Sensory Ring; and S. The cies, a new analytical model was developed.Launch Vehicle Adapter. Structural differenceswith respect to the Nimbus were primarily in For an analytical model c the spacecraftthe center section of the sensory ring assembly. the representation of the solar array becomesNew structure was designed to accommodate critical. Thir complex solar array was foundthe ERTS payload and other new equipment, to dominate all of the spacecraft modes. IfSensory ring and truss structure was modified finite element representation was used for thefor increased stiffness and strength. In the solar array, it would be difficult to derive andlaunch configuration, the solar array is folded its accuracy would be questionable. This con-along its longitudinal axis to fit within the sensory sideration was avoided, however, with a modalring envelope and is secured to the sensory ring coupling approach using measured mode shapes.structure by a latch mechanism. The spacecraft This required the extraction of 19 solar array


L a

complete spacecraft (less solar paddles) matrix.To complete the model, the paddles were addedby the modal coupling technique. Nineteenpaddle modes and 11 spacecraft modes (lesspaddles) were coupled to produce the final setof complete spacecraft modes and frequencies.

I. Spacecraft. The launch vehicle adapterk' structure was represented using over 160 in-

ternal joints in the MASS digital computer pro-gram (reference 1). These joints connected theS b-am and panel elements used to represent theadapter structure. The resulting stiffnessmatrix was reduced to nine external joints whichwere selected to coincide with the sensory ringexternal joints. The sensor-, ring structure in

Figure 3 was modeled also using the MASS pro-gram. This representation consisted of over200 joints connecting the beam and plate elements.The resulting stiffness matrix was reduced to 18external joints common to the adapter, the truss,and other mass points of interest.

In a similar manner, the stiffness matrices of thetruss and the ACS structures were also devel-oped. The complete spacecraft structure, lesspaddles, was then assembled by the stiffnesscoupling of each segment. The fixed-free natu-

Figure 1. ERT8 Spacecraft Vibra, in Test ral frequencies and mode shapes were calculatedusing this stiffness matrix and the mass matrix

normal vibration modes from Nimbus spacecraft given by the distribution shown in Figure 4.modal data. Such an approach is valid sinceERTS uses a solar array which is structurally 2. Solar Array. A very accurate representa-unchanged from Nimbus. Paddle frequencies tion of the solar array paddles was possible(starting at 13 hz. ) and mode shapes having 154 since measured mode ohapes could be utilized.coordinates were utilized for the new ERTS ana- These measurements were taken from a previouslytical model. Other major substructures were modal survey of NIMBUS A (4) according to thederived by a conventional finite element stiffness pattern given in Figure 5. Since the ERTS space-routine known as MASS (1). These substructures craft uses the NIMBUS solar array with minorwere then combined by modal coupling through modifications, these data were considered to bea statically determinant interface in a manner appropriate for the ERTS model,similar to that used by Hurty and Hou (5, 8).

The extraction of the 19 paddle modes from theThe following sections of the paper describe NIMBUS modal survey started with the formula-

the formulation of the ERTS spacecraft analy- tion of the characteristic matrix vibration equa-tical modal and Its verification and evaluation by tion:modal testing. Much of the modal dynamic analy-sis and testing activity conducted on ERTS!Nim- n x n n x mbus programs parallels the well-documented [K) ] -

Mariner Spacecraft Programs (5, 6, 7) and have n x n n x m m x (had a similar degree of success.

DESCRIPTION OF ANALYTICAL MODEL with [01 normalized to,

A, The analytical nodel of the spacecraft was [9] [H) (9] = [tJ

developed by considering the major spacecraftsegments separately. These structural seg- wherements are identified in Figure 2. Stiffness (K) is a n x n square symmetric matrix ofmatrices were developed for the launch vehicle stiffness coefficientsadapter, the sensory ring, truss, and ACS, and [H) is a n x n square symmetric matrix ofeach segment was then combined to form the inertial mass items

204

ACSso: :.:: .. L AC

ERTS/ACS

INTERFACE

ASSE 4BLT

L/V IhTERFACE

Figure 2 E'ITS Structures Subsystem

A is a scalar of resonant frequency squared - 2

[9] is a n x m modal matrix of eigenvectors (mode shapes) listed columnwise tiaere m < n[A3 is a m x m diagonal spectral matrix of eigenvalues (A, resonant frequencies squared) listed

diagonally where T [

jIJ is a m x m identity matrixm refers to the number of modal degrees of freedom (modal courdinates)n refers to the number of generalized displacement coordinate degrees of freedom

Equation I can be manipulated to derive the stiffness matrix if the modal, spectral and inertial ma-trices are known (2, 3).

Post multiplying by: T

C*] [14]

[K] [9] (4- T [M) - [M4] [ 4 [Aj (M) (2)

and utilizing the normalization

(9 I" [t [I] (9JJ - [9] 11 - [4] (3)

regrouping

205

2 i

[] 1 T [H]J it)I 1#1

gives the identity

!.] []T [H] E [I]

which simplifies equation 2 to

n x n n xn nxm mxm mxn nxn[K] = (J [*] [,HA( [MJT (H] (4)

Utilizing equation 4 nineteen constrained solar array paddle modes were determined from the originalmeasured modes of the Nimbus Spacecraft by applying constraints at all non solar array coordinates.

The derived stiffness matrix was partitioned as folloit:

KI K2 XI 0 M O XI

[21 K22 X2 [ 22 12X1 ,Let = Y4where: K11 is the desired cantilevered paddle stiffness matrix

X is the generalized physical displacement vector( )1 coordinates being retained; i.e., solar array

)2 coordinates being restrained, i.e., spacecraftI 1 ,1 2 (I ] s] ,T T rl-o I 1 I [1 T 2 T I o (6)

K21 K22] [ 2'2 ] [ 0 2J

= N 01 [1 A * 0 A 1*T~. 01F0 M2j .t2 -; *1 1 ¢ A 0T~ M2 2 (7)M2 1 h io T H 1 2l 1 h 2 T MM2

T --- (8)2 H 2 A " 2

Transforming Equation 5 in expanded form

[ A] 1 i A 0 T i

1 A1 Ml H1 A * 2 [112 2 A 1T MIV2 42 A 02 T 2 0

# *T N1 * A ~T m1 *Jy I M ht A1 (9)~j

solving the above elgenvalue problem yields Y , and transforming

back to obtain the paddle e'ode rhapes I#ade I I# Y1j'~ (10)

206'Llak 4L.- Y O 4, 1 -o

... . .. ,-. , , ,= ,.f ........... ,,

n ,z

-- I',

'NI

I "tI I

I * I

FIGURE 3 - MASS COMPUTER MODEL -SIMULATION OF SENSORY

RING STRUCTURE (Cross-beam omitted for clarity)

3. Modal Coupling. Having obtained the con-strained modes, the statically determinant inter-face consisting of three translations and two ang-ular reactions at the Solar Array Paddle Shaft "and a latch line vertical tie to the sensory ringwas released in the modal coordinate. This ob-tained the free modes. The free solar array andanalytically derived ERTS spacecraft structures FIGURE 4 - SPACECRAFT ANALYTICAL MODELwere then ready to be modally coupled to obtainthe desired ERTS spacecraft vibration modeshapes and resonant frequencies.

part of the structural verification test. The

Basically, a simplified Hurty (5, 8) attachment primary objective of this test was to obtaintechnique was used which entailed free mode fixed-free modal vibration data suitable forcoupling at statically determinate interfaces, evaluating the analytical model of the space- IThe resulting eigenvalue equation form consisted craft for use in the flight loads analysis. Thisof a coupled generall -,,d mass matrix and dia- required the measurement of data sufficientlygonal generalized stiffness matrix with 30 de- detailed to enable the response in the funda-grees of freedom. Eleven modes of the struc- mental pitch, roll and yaw modes and the lateralture and nin, een modes nf the solar array modes in the frequency range of the POGO exci-were utilized in developing the complete system tation to be evaluated.modes. The first twenty-four complete space-craft modes ranging from 13 to 85 hz were sub- Sine sweeps for modal definition weresequently correlated by vibration testing. performed with low level base excitation at a

rate of 8 minutes per octave. A total of ninetyMODAL TEST DESCRIPTION accelerometer channels were recorded during

the resonant dwells. These were located onModal testing of the ERTS full-scale the spacecraft structure in triaxial and biaxial

structural dynamic model was performed as a groups at points corresponding to mass points

4 207

........ ........... ,

10161 TaftS pool$ W SMOM

4'4

- p~il .I I Illel

4 J-- I

aIl l~~* I

4 I + 4 1

I A I

I I I

14ACJ4 V5W"A.4.

FIGUR 5 - ACCELEROMETER LOCATIONS FOR PADDLEMDAL SURVEY -NIMBUS

Kof the analytical model. The paddles were interms for the nine spacecraft modes was 9. 4strumented at its extremities only since modal percent. The test plan set a goal of + 1076 maxl-data for this solar array had previously been mum coupling as evidenced by the off-diagonalobtained. (Previous Nimbus modal testing had terms for successful mode shape measurement.recorded ZOO channels of data with over 100 data The greatest deviation from the criterion waspoints on the solar paddles), In-phase and quad- found in the higher paddle modes. This resultrature response plots referenced to the input was anticipated since very few accelerometersacceleration were obtained according to the were available for paddle response measurement.phase separation technique of Reference (9) for This became significant in the higher paddleparticular accelerometer locations. These lo- modes where the necessary coarse mass distri-cations were selected based on analytical pre- bution was inadequate.dictions of the paddle latch, ACS and sensoryring responses. Frequencies of interest were A comparison of test and analytical rca-identificd for resonant dwells. Since the test- onant frequencies is given in Table 1. In general,ing was limited to the fundamental mode in each the measured resonant frequencies are in goodaxis and ihle cross-axis modes in particular agreement with the corresponding calculated fre-frequency tanea. not all resonances had modal quencies. This is particularly true for the funda-dwells. Modal identification was established by mental pitch and yaw modes, the ACS roll mode,plots of the quadrature comppaent of accelera- and the second paddle roll mode at 18.81 hz.tion. These plots were of the type given in Other frequencies were considered to be reason-Figure 6 and they were made for all of ,.jp dwell ably close to the corresponding calculated values.frequencies. From this group, nine significant Resonant frequencies for which dwells were notmode shapes were determined. These modes made were estimated from total response plotswere then used for direct comparison with the and are noted with an asterisk in the tabulation.analytical mode shape.

Structural damping was determined by theThe measured responses for the nine method of Reference (9). This required the

modes obtained were considcred to behave as measurement of the frequency of the peak in-natural modes. This was bared on resonably phase component of acceleration occurring justgood orthogonality shown by the evaluation of before and just after the resonance. The dampingthe generalized mass matrix which was calcu- values obtained ranged from g =0. 032 to 0. 12.lated using the normalized experimental mode This corresponded to the range of magnificationshapes. The average value of the off-diagonal factors obtained in the high level vibration test.

4 E 208

4'Vg

•n i2 4 1 i 2" T -''in 1 '6r ,,,

A constant damping coefficient of g 2 C/Cc0. 10 was assumed for all modes in the analysis,

ANALYTICAL MODEL EVALUATION

The analytical model was evaluated on thebasis of frequency, mode shape, and amplitudecorrelation with the analysis. Considering eacharea of comparison it was concluded that theanalytical model was a good representation ofthe ERTS spacecraft. The basis of this conclu-sion is summarized in Table 2. Confidence inthe model was supported by the good frequencyagreement and the orthogonality checks.

In addition to frequency comparisons andorthogonality checks, another evaluation ofmode shape was made. This was accomplishedby the calculation of the modal shear for bothtest and analysis. The results here showed theagreement between test and analysis predictionsto be within + 29 percent for the primary re-sponse axis. There were larger deviations forthose modes where the quantity of paddle mea-surements is critical. No criterion for thisevaluation had been previously set, but the re-sults were considered reasonable.

Improvements in the amplitude responsewere suggested by the damping evaluation wherea range of values were measured for individualmodes. A need for variation of the dampingcoefficient was indicated, especially in thelower frequency modes. Other possible areasof improvement considered as a result of the FIGURE 6 - ERTS MODAL TEST PLOT

modal testing were in the mass distribution of FREQUENCY: 15.29 Hz

the sensory ring and the representation of thelaunch vehicle adapter stiffness. Refinementof the analytical model was obtained without made during the modal dwells restricted thesignificant changes in the frequency correspon- scope of the analytical model evaluation. Wheredence. This resulted in better modal shear deviations from the criterion were large, as incorrelation with test within + 16 percent. Also, the orthogonality check and modal shear coin-better amplitude correlation'was obtained by parison, they could be traced to lack of p-,llevarying modal damping coefficients, measurements and not lack of validity of the

mode. It was possible, however, to establishCONCLUSIONS confidence in the model and to uncover some

r.-as of discrepancy. This resulted in subse-Analytically derived and test correlated quent modification and improvements in the

mathematical models providc the most desirable model. Improvement was made in the modedata for use in ascertaining structural integrity shapes which was reflected in better correla-of a spacecraft. Limitations in schedule and tion of the modal shear. Also, better responsecost, however, frequently force the dynamicist amplitudes were obtained by the selection ofto resort to abbreviated test and analysis pro- damping for individual modes. The adjustnemtscedures. The ERTS/NIMBUS experience has in the model were accomplished with insignifi-shown that satisfactory vibration models of sub- cant changes in the frequency correspondence.systems or complete spacecraft can be estab-lished by analysis, test or a combination of both. REFERENCESThe extraction technique was successfully dem-onstrated by the correlation with the modal test. I. Beitch, L., "MASS System - The Computer

Program for General Redundant StructuresThe limited quantity of measurements With Vibratory and General Static Loading, I

209

TABLE 1.

COMPARISON OF TEST AND CALCULATED NATURAL FREQUENCIES

CALCULATED TESTNATURAL FREQUENCY NATURAL FREQUENCY

,NU.]BER CPS- MAJOR RESPONSE REGION CPS

1 13.39 PADDLES X-AXIS 13.722 14.13 PADDLES X-AXIS 15.293 18.48 PADDLES X, Y-AXIS 18.814 19.57 ACS X, Y-AXIS 17.47

5 21.81 PADDLES X, Y-AXIS 21.00'

6 24.03 ACS" Y, X-AXIS 24.257 29.29 ACS Y-AXIS 28.50*8 31.09 PADDLES Z-AXIS 30.22

9 33.26 SENS.RY, PADDLE Y, X-AXIS 34.50*10 35.77 PADOLL, SENSORY X, Y-AXIS 39.2311 38.53 PADDLE,T,IST Y, X-AXIS 37.00*12 40.24 SENSORY, PADDLES Y, X-AXIS 35.27

13 43.71 SENSORY, PADDLES X, Y-AXIS 39.00k14 47.42 SENSORY X, Y-AXIS 41.00'1s 48.92 SENSORY Y-AXIS 42.50*16 53.57 PADDLES, SENSORY Y-AXIS 50.00*

17 55.34 SENSORY, PADDLES Z-AXIS 53.00*18 58.24 PADDLES Z-AXIS 55.00*19 58.76 ACS Z-AXIS 59.3520 62.38 SENSORY, PADLdLES Z, X-AXIS 63.00*

21 63.54 PADDLES, SENSORY Z-AXIS 65.00*22 66.24 ACS Z-AXIS 68.00*23 76.68 PNEUIATICS Y-AXIS 79.50*24 85.29 PADDLES Z-AXIS 90.00'

* APPROXIMATION FRO4 TOTAL RESPONSE PLOT; ALL OTHERS DETERNIfiED PRECISELYFRO1M QUADRATURE PLOTS.

TABLE 2 - ANALYTICAL MODEL EVALUATION

BASIS ANAL. & TEST REMARKS

a. Frequency Very good Correlation within less than 12% in most modes.

b. Orthogonality Good 50% of all off-diagonal elements meet +10%criterion; 80% meet 415% criterion; overallaverage is 9.4%.

c. Modal Shear Good Primary axis of response less than 16% deviation.Those in excess lack paddle representation.

d. Amplitude Fair Measured daping ranges from 0.03 to 0.12 accord-ing to mode; analysis used 0.10 for all modes.Better correlation obtained by varying modaldamping coefficients.

210

- - ...- ~ .IVII

General Electric Co., TIS R66FPD172, Sep-tember 13, 1966.

2. Cokonis, T.J., - "Structural DynamicsModal Matrix Methods for the Coupling of Space-craft/Launch Vehicle Systems, " General Elec-tric Company, TIS 68SD325, September, 1968.

3. Berman, A. and Flannelly, W. G. -"Theory of Incomplete Models of DynamicStructures," AIAA Journal, Vol. 9, No. 8,pp. 1481-1487, August, 1971.

4. Smith, F., Freelin, T.R., Romano, R.,and Hutton, F., - Nimbus Spacecraft ModalSurvey Vibration Test Final Report, GeneralElectric Company, TIS Report 68SD281, June,1968.

5. Hurty, W. C. - "Dynamic Analysis ofStructural Systems by Component Mode Synthe-sis," JPL Technical Report 32-530, Jan., 1964.

6. Holbeck, H.J., Arthurs, T.D., andGaugh, J.J. - "Structural Dynamic Analysisof the Mariner Mars '69 Spacecraft, " 38thShock and Vibration Bulletin, Part 2, 1968.

7. Freeland, R.E. and Gaugh, W.J. -"Modal Survey Results from the Mariner Mars1969 Spacecraft, " Shock and Vibration Bulle-tin 39, February, 1969.

8. Hou, S., "Review of Modal SynthesisTechniques and a New Approach, " Shock andVibration Bulletihj No. 40, Part 4, Dec., 1969.

9. Stahle, C. V., Jr., "Phase Sepax.tionTechnique for Ground Vibration Testing,"Aerospace Engineering, Vol. 21, No. 7, July,

i 1962.

211

2W,

FINITE AMPLITUDE SHOCK WAVES IN INTERVERTEBRAL PISCS

William F. HartmanThe Johns Hopkins University

Bal timore, Maryland

The nonlinear deformation of intervertebral discs is discussed. Theupward turning stress-strain curve implies that the discs will tendto shape pulses having sub-millisecond rise-times into shock waves andthat shock inputs will propagate as shocks. These implications areexplored for axial compressive impact of the spine, such-as is incurredduring aircraft-pilot ejection or during-a fall onto the buttocks.Correlation with experimental results suggests that the application offinite amplitude wave theory to the shock loading of the spine shouldbe further investigated.

INTRODUCTION structure consisting of an elastic fibrocarti-laginous envelope, the annulus fibrosus, and a

A knowledge of the mechanical properties of fibrogelatinous core, the nucleus pulposus,human vertebrae and intervertebral discs is fun- which contains water in a mucopolysaccharidedamental to understanding body mobility and collagen framework[2]. The disc is cappedspinal injuries and to designing protectors and above and below by bony end-plates which areprotheses for these spinal components. The dy- distinct from the vertebral bodies. Thisnamic response of an intervertebral disc must be structure of the disc suggests a possible appro-studied in order to predict the vertebral priate mechanical model to be a sealed elasticstresses that are caused by rapid motions or cylinder containing a viscous fluid. However,impact. Most measurements of the dynamic mechani- not enough is known about the geometry or thecal properties of the disc have been restricted material properties to warrant such modelling.to small-amplitude deformations which are as- The disc seems mechanically symmetrical aboutsumed appropriate for the perspective of linear- its vertical axis, as can be inferred from theized response. I here describe some of the compression measurements of Brown et al.[3].large amplitude non-linear characteristics of Since the only data, which I shall discuss here,the intervertebral discs and discuss the impli- pertains to compressive loads and the resultingcation of these on the dynamic response of the average strain, I shall assume the disc to bespine, uniform, having the average mass and size of the

actual disc-complex.The human spine is a nonuniformly curved

column normally consisting of 7 cervical, 12 Herniation of the intervertebral discthoracic, 5 lumbar vertebrae and the sacrum- results in a loss of material fro the nucleuscoccyx structure. Each vertebra has a cylin- pulposus. This can be caused by trauma ordrical anterior part called the vertebral body disease and is one of the frequent etiologicaland posterior arch-like parts called pedicles factors associated with back-pain and, inand facets. The latter guide and restrict rela- severe cases, sciatica[4,5). Furthermore, 60%tive displacements between the vertebrae for of all disc protrusions are caused by injury;bending and torsion deformations but take up one-third of these by a fall onto one's feetonly 20% of axial loads[l]. The vertebrae are or buttocks, resulting mostly in injuries tosoft, cancelluus bones, decreasing in height the lumbar spine[6]. During static compressionupwards, from about 1.2 in. for the fifth lumbar of the lumbar spine the vertebral endplateto about 0.5 in. for the third cervical vertebra. frequently fractures, allowing the nucleusAdjacent vertebral bodies are separated by an pulposus to protrude into the spongiosa ofintervertebral disc which in the lumbar region the vertebral body[7. Often this causes theis approximately 0.4 the height of the cancellous bone to collapse and continuedvertebrae. loading produces a comminuted fracture of the

vertebral body.The intervertebral disc is a nonuniform

Preceding page blarnI2.

Hyptheses about Ehe type and mechanism STRESS-STRAIN CURVES FOR INTERVERTEBRAL DISCSof dyn.mI1callY Induced injuries could resultfrom impact experiments or they may be formu- Load-deflection data for the compression oflated from an extension of the available lumbar intervertebral discs are given in [3,10,static data. Of course, viscous effects are 11]. The compression tests of both Brown et al.naturally anticipated for the deformation of a [31 and Virgin[lO] employei conventional testingstructure having the material composition of mechines and used specimens which had thethe..disc. Although the viscous properties of posterior facets and pedicles removes. Thevertebral bodies and intervertebral discs have dimensions of specimenfs were not reported, so Inot been documented for the large non-linear calculate stress by dividing axial load valuesdeformations which precede failures, the damp- by the average area of the appropriate lumbarIng properties have been measured for small disc as given by Perey[7]. Strain is calculateddeformations. Fitzgerald and Freeland[8J have by dividing the deflection by the nominalshown that the damping of fresh canine discs height[12] of the corresponding disc. For thedecreases with increasing frequency, while data of Virgin and Brown, this results in theMarkolf and Steidel[9] report the damping of stress-strain curve-shown in Fig. 1. Virgin'sthe entire intervertebral joint is small for data is reproduced from his "typical" load-the compression-tension mode of deformation. deflection curve while Brown's is computed usingThese results suggest neglecting viscous effects the averaged values of four compression testsin a first analysis of the stresses caused by which are each described in [3]. The two curvescompressive impact. It would be overly pre- are more than similar for both are proportionalsumptuous to expect that accurate results could to the-squared strain and the proportionalitybe obtained for a large. range of impact speeds; constant Is the same for each. This suggestsbut any-analysis using only-the data which is that the non-linear response of the disc can beavailable is worthwhile if only to point out well defined, for these two sources of data werethe necessity for specific additional experi- obtained by-different investigators in differentmental programs. In what follows I shall countries using different testing machines andestimate the impact response of intervertebral probably different loading rates. The tests ofdiscs using static stress-strain data obtained Virgin were conducted 20 years ago, those offrom the literature. Brown et al. 14 years ago and yet, to my best

knowledge, I present here the first correlationof any independently performed tests of this

0.30-

E 9 kg/mme e- - *

II II A

E

E0

0'0

*0

00 0.05 Strain 0.150 0.005 (Strain) 2 0.015

Fig. 1. Stress-strain curves for lumbar discs obtained from the data of Virgin, o,and Brown et al.,A. Filled-in symbols are stress plotted against squared strain.

214

Y4,

0.05

i: ?[ I+ ' ft 0.04 1

Imm14 kig/mm 2,E,

I20.02- /9 kg/mm2

0.01 .- z:T.- kg/mm2

00O0 ,'"i ,i i

0.01 0.02 Strain 0.04 0.05 0.06I I I I I

0 0.001 0.002 (StraIn) 2 0.004

Fig. 2. Stress versus strain, o, and squared strain s - from fourth lumbar disc data ofHirsch[ll]. The near-zero modulus for canine disc[8] is shown as ---.

kind. Furthermore, since neither [3] nor [10] specimens contained half the upper and half thereport the cross-sectional areas of the speci- lower vertebral body and the correspondingmens, the shift in the curves of Fig. 1 might intervertebral joints. The posterior pediclesbe partially due to the use of inappropriate and facets have been shown to take up approxi-areas for one or both sets of data. mately 20% of the axial load[l]. Therefore in

calculating the stress on the disc from Hirsch'sReproducibility and consistency can be data, I use 80% of the load values. The

found in other mechanical properties of spinal longitudinal strain is calculated from thecomponents. For example, the average vertebral reported lateral bulge of the disc, assumingfracture stress calculated from the data of uniform incompressible deformation there. ThisPerey[7J is .34 kg/mm2, which is the same value is a poor assumption for large values of stressobtained from averaging compressive fracture because Nachemson[13] has shown that the lateralstrengths given in [3]. Of course, it is bulging varies nonlinearly with axial strain andreasonable to assume that a general material the disc's volume is known to decrease[3]. How-description will aoply to vertebrae and discs ever, it should hold approximately for smallonly for the averaged data of several specimens values of stress and give at least an estimatefrom several bodies. Variations are due to for moderate stress. This p)-codure results indifferences in age, size, sex, disease, injury, the stress-strain curve of Fig. 2. The initialand specimen preservation. Normal deteriorations nonlinearity is precisely the same as that ofare not yet well documented and certain patholo- Fig. 1. The change in the slope of the stress-gical disorders certainly go undetected. Never- squared strain plot might correlate with thetheless, the degree of consistency noted above inapplicability of the incompressibilitysuggests that gross material characterization assumption. A qualitative correction, based onis both sensible and ultimately useful. Nachemson's measurements, would increase the

strains for increasing stress, thereby makingThe compressive deformation of interverte- the slope more akin to the initial behavior

bral discs is also reported by Hirsch[l1]. His which agrees with the data of Fig. 1.

215

4 4.

WAVE PROPAGATION IN THE SPINE

A distinguishing feature of the stress- fracture stress -" -"

strain curve for the compression of interverte-

bral discs is its concavity away from the strain 0.30axis. Such material behavior is typical of manybiological materials but here it has interesting 4implications regarding stress-wave propagation --

along the spine. E

The one-dimensional finite amplitude wave b Etheory as developed by Karman and Taylor isgiven in (14]. Some pertinent results of thattheory, which is applicable for nonlinear kstrain-rate independent elastic or plastic 2deformations, are as follows. If the stress- 0.1strain relation is written as 0.10-

a= *() (1)

where a is the longitudinal stress, c the longi-tudinal strain,'and #the response function, 0then the speed of propagation, C(c), of any 00strain level is 0.05

C(C) (/p)l/ 2 (2) Fig. 3. Stress versus strain for a lumbarvertebral body - from the data of [7].

where p is the density of the solid. Theparticle velocity, V(c), associated with any eor-p) can be written distinctions. The vertebrae, especially for

large stresses, behave linearly. Also, thestrength of the disc is generally assumad to be

V(e) = c !dc' (3) greater than that of the vertebrae. A per-spective retaining the individual materialproperties of these components will result in a

If 0(c) is an upward turning function then rather complicated picture for impact problems,*'(c) increases with strain and so the wave especially since the sizes of the discs are suchspeeds given by Eq. (2) increase with strain, that individually they are not rods, their dia-

An interpretation of this result is that if a meters being larger than their lengths. Sinceunit-step shock input is applied to the one- I have already indicated that I will ignore thedimensional material then it will remain nonuniform structure of the discs, I do notshock-like since the speed of the maximum hesitate to assume that one-dimensional wave

stress is the greatest speed. On the other propagation can be used to obtain reasonablehand, if the material is subjected to a estimates.gradually risig pulse, then this pulse as ittravels through the material will become con- Using the data described thus far, andtinuously steeper since the larger strains Eqs. (2) and (3), I consider: (1) the pulse-propagate faster. If the material is suffi- shaping performed by the discs, (2) theciently long, every pulse regardless of its prediction of the impact response of a preloadedshape will eventually develop into a shock. vertebrae-disc-vertebrae specimen and a com-parison with the experimental results given in

Although the spine has the appearance of a (11), and (3) the prediction of the impact speedcurved rod, its axial dynamic response could be which would produce fracture in the fifth lumbarproperly described as one-dimensional only if vertebra. In all cases only compressive axialits major components, the vertebral bodies and deformations are considered, the spinal curva- 4

the intervertebral discs, were dynamically ture, bending, twisting and viscosity are ignored.similar. The vertebrae are sometimes assumedas rigid bodies in comparison with the discs for The gross weights and dimensions of fresh

dynamic analysis[12]. Such an assumption does canine tntervertebral discs are given in [8].not square with the stress-strain relation shown Using these, a bulk density for the disc isin Fig. 3. This was constructed using the data calculated to be 2.22 g/cm . Taking the averageon (7] for the compression of vertebral bodies. curve of Fig. 1 and the above value for P,Both the deformation and the slope of this Eqn. (2) results in the curve of Fig. 4.stress-strain curve are comparable with those of Nachemson and Morris[15] have reported in vivoFigs. 1 and 2. Therefore, since the densities measurements of intradiscal pressures foFi' aldiffer only by a factor of 2, the spinal com- loads such as those occurring during standingponents must be considered dynamically similar. and sitting. The corresponding axial stressesNevertheless, there are important subtle range from .04 to .08 kg/mm2. The average

216

stress is applied sufficiently rapidly, such asa stress pulse whose length is less than theaverage pore size, then the effective cross-sectional area is reduced and failure will occurat a lower nominal stress. Thus the shock waveformation as described here could account for

100 the fact that vertebral fractures occir in-thethoracic vertebrae during pilot ejection evenwhen no damage is sustained by the lumbar spine[12].

Some impact experiments on spinal segmentswere performed by Hirsch[1l]. A lumbar disc andEhalf of each adjacent vertebra including theposterior processes were preloaded in compressionand then subjected to impact by allowing a weight

50 . to fall through a known height onto the loading50 lever of the testing machine. Since Hirsch didnot report the dimensions or the raterials of

.10 a .30 the testing apparatus, the effective impact speed(kg/mm 2 ) at the vertebra cannot be calculated. Neverthe-

mless, his recorded values of the dynamic bulgingof the disc together with the stress strain

Fig. 4. Variation of wave speed with stress. curves of Figs. 2 and 3 and Eqn. (3) permit acheck on the consistency of the present mechani-cal perspective. Assuming continuity of stress

X-L5 X=Li XaT! and particle velocity at the vertebra-discinterface the following Jump equations apply for

- ~~a shock input.+(R (]- [a ll + [ RI - 1 T]T(4)

.08 i1 + [VR i VTJ.8-"10.5 m sec 1-

where 1, R label the incident and reflected shockFig. 5. Example of shock formation by discs. in the vertebra and T labels -the transmitted

shock in the disc. Using a constant vertebraeimpedance, the linear shock equations, the

vertebral fracture stress, which cin be obtained stress-strain curve of Fig. 2 and Eqn. (3), thefrom the data of [7], is .35 kg/mW . Therefore, response curve of Fig. 6 is obtained. TheFig. 4 shows that the wave speeds double over experimental points shown in the figure arethe range of probable dynamic stresses. plotted taking the equivalent impact speed to be

twice that of the drop speed. The agreement withFig. 5 illustrates the pulse shaping which the theory further suggests that it might be

can occur as a stress pulse propagates along the rewarding to pursue experimental studies of wavespine. In this case a linearly rising pulse propagation in the spine.increasing from the normal sitting stress to avalue which is less than the fracture stress Using Eqns. (4), the speed sufficient tosteepens from a 0.5 irsec rise time et the fifth cause vertebral fracture from impact onto anlumbar vertebra to a partial shock an 0.1 msec infinite impedance may be calculated. Thetail at the first thoracic vertebra. Of course resulting expression is:any pulse whose initial rise time is less than0.5 msec would become a more fully developedshock; while pulses whose rise times are longer 2[Vii = [aD] + $C(C')dcthan several msec would not be noticeably shaped. +z--(Rise times less than.1 msec seem plausible for c

dynamic stressing such as those occurring during!iaircraft-pilot ejection. where Z. is the impedance of the -.rtebral body.

The porous structure of the soft cancellous In the above, let [a n * 0.26 andbon2 of the vertebrae make them inherently c, c = 0.17, 0.05, respectiely. Thisweaker in tension than in compression. During orr sornds to a dynamic jump from an tnilial

sufficiently gradual application of compressive stress of .04 kg/mm2 to a valie,.30 kg/mmz.forces the deforming vertebrae uniformly distri- which is 90% of the fracture stress. Thebute the load effectively over the entire croEs remaining stress is easily accomplished throughsection due to the collapse of voids and the the multiple reflections dt the infinite m-reduction of porosity. However, if compressive pedance. Using the curve of Fig. 4, this gives

217

j

6- Calculated6-

A Measured (-Hirsch 1955)

E4

0.04 0.05 0.06Strain increment

Fig. 6. The calculated additional strain increment due to impact is nonlinear with the speed.The case shown is for a lumbar disc-vertebrae segment prestre-sed 0.005 kgr ia2 .

an impact speed in the vertebra of 12.2 m/sec, Surg., Vol. 39-A, No. 5, pp. 1135-1164,which corresponds to a free terrestrial fall Oct., 1957.from approximately 7.5 meters. Since buttockand pelvic elasticity have been ignored, this (4] D. C. Keyes and E. L. Compere, "The Normalvalue is surprisingly large and certainly does and Pathological Physiology of the Nucleusnot seem to agree with ordinary experience. On Pulposus of the Intervertebral Disc,"the other hand, impacts which are truly axial J. Bone and Joint Surg., ol. 14, pp. 897-are seldom achieved and the consideration of 938, 1932.bending and shear would reduce the injury speed.

(5] S. Fribert, "Low Back and Sciatic Painc ile mch of what I have discussed will Caused by Intervertebral Disc Herniation,"be considered speculative because it is based Acta Ortho. Scand., Suppl. 25, 1957.

upon minimal appropriate data, the existenceof finite amplitude shock waves in nonlinear (6] J. E. A. O'Connell, "Protrusions of theintervertebral discs has been shown to be Lumbar Intervertebral Discs," J. Bone andplausible and its study is potentially Joint Surgery, Vol. 33-8, pp. 8-30, 1951.important in understanding dynamicallyinduced spinal injuries. [7] 0. Perey, "Fracture of the Vertebral End

Plate In the Lumbar Spine," Acta Ortho.REFERENCES Scand., Suppl. 25, 1957.

(8] Edwin R. Fitzgerald and Alan E. Freeland,[1) Af. Nachemson, "Lumbar Intradiscal "Viscoelastic Response of intervertebralPressure," Acta. Ortho. Scand., Suppl. 43, Disks at Audiofrequencies," to appear in1960. J. Ned. and Biol. Eng.

(2] M. B. Coventry, "Anatomy of the Interverte- [9] Keith L. Harkolf and Robert F. Steidel,bral Disk," Clin. Ortho. and Rel. Res., "The Dynamic Characteristics of the HumanVol. 67, pp. 9-15, 1969. Intervertebral Joint," ASME publication

[3] Thornton Brown, R. J. Hansen, A. J. Yorra, 70-WA/BHF-6, 1970.

"Some Mechanical Tests on the LumbosacralSpine with Particular Reference to theIntervertebral Discs," J. Bone and Joint

218

[10) W. J. Virgin, "Experimental Investigationsinto the Physical Properties of theIntervertebral Disc," J. Bone and JointSurg., Vol. 33-8, pp.607-611, 1951.

[11] Carl Hirsch, "The Reaction of Interverte-bral Discs to Compression Forces," J. Uoneand Joint Surg., Vol. 37-A, pp.1188-1196,1955.

[12) David Orne and Y. King Liu, "A Mathemati-cal Model of Spinal Response to Impact,"J. Biomechanics, Vol. 4, pp.49-71, 1971.

(13) Alf Nachemson, "Some Mechanical Propertiesof the Lumbar Intervertebral Discs," Bull.Hosp. Joint Disease, Vol. 23, pp.130-143,1962.

(14) H. Kolsky, "Stress Waves in Solids,"Oxford University Press, 1953.

(15) Alf Nachemson and James M. Morris, "InVivo Measurements of Intradiscal Pressure,"J. Bone and Joint Surg., Vol. 46-A,pp. 1077-1092, 1964.

219

219

a't

a'4~I

ACCELERATION RESPONSE OF A BLAST-LOADED PLATE

Lawrence W. FagelBell Telephone Laboratories, Inc.

Whippany, New Jersey

A solution for a simply supported plate loaded by a step-functionpressure is closely examined to determine contributions to acceleration

the plate's higher modes of vibration. Plate dimensions are as-sumed to be such that classical bending equations apply, and it is estab-lished that the peak acceleration response can be as much as 2.6 timesthe peak resnse of a one-degree-of-freedom analog. When damping isincorporated in the solution, the peak values are attenuated and much ofthe very-high-frequency response appears to dissipate rapidly. In prac-*.ical situations where the damping ratio will be at least 1 percent, ninemodes should adequately represent the plate's true response; however,computed accelerations may be nonconservative b up to 40 percent ifonly four modes are considered and by more than 100 percent if only one

INTRODUCTION MOTIVATION

A commonly used technique for calculating Structures which are designed to withstandthe approximate response of a plate subjected nuclear-weapon effects are often either shell-to blast loading is to consider the plate to be like structures or are encapsulated within pro-equivalenit to a one-degree-of-freedom spring- tective shell-like structurei whibh, for designmass system, the frequency of which corre- and analytical purposes, are sometimes re-sponds to the fundamental frequencyof the plate. garded as composites of plate elements. More-

simply supported plate acted upon by a uni- over, for structural-motion-response studies,form step-function pressure is analyzed here to it is a usual inherent requirement to considerdetermine the validity of this approximation. these plate elements to be externally loaded byClassical bending equations for plates, that is, blast-induced overpressures* which may inplane-stress equations, are assumed to be ap- some cases be approximated by a step functionplicable and these lead to a time-dependent, for determining early-time responses. In thisdouble infinitetrigonometric series solution in regard, a prevalent practice for analysts is towhich each term of the series represents the construct a mathematical model of the structuretransverse vibrational response of a different assuming that an entire panel can be repre-mode. Comparison of the relative amplitudes sented by a single degree of freedom.t A directof each modal response demonstrates that the consequence of this approach is that the calcu-first mode predominates for the dynamic related peak acceleration response at the centersponses of displacement and stress; hence the of a square plate element exposed to a stepone-degree-of-freedom approximation is ap- overpressure ispropriate for these quantities. However, theamplitudes of some of the higher modes of ac-celeration response are significant compared Peak Peak Overpressureto those of the first mode, indicating that these Acceleration 5Mass per Unit Area of Panelshould not be ignored. Because the solution in-

volves an infinite series in time, the maximum (1)amplitude of response is not obvious from thesolution expression. Acceleration responsesare plotted as functions of time to determinethe contributions associated with frequencies ;Pressures in excess of ambient.higher than the fundamental. Peak accelerationresponse for an undamped plate appears to be tThis practice is recommended in "Design ofabout 2.6 times the first mode response i.e., Structures to Resist the Effects of Atomicwhat would be calculated using the one-degree- Weapons," U.S. Corps of Engineers Manualof-freedom analog. EMl110-345-110, 15 March 1957.


(1.5 is approximate to about *10%; the value w = -- ^' 4 wof this coefficient depends on edge-support NxAcxoy +Yconditions.) An examination of a plate-vibration-problem solution that includes modes gowhich are higher than the fundamental, indi-

cates that Eq. (1) characterizes the response in ij(t)the first mode; however, the amplitudes of ac- i= j=Iceleration responses in modes other than thefirst, although also significant, have been sin I= sin (4b)ignored. a b

In situations where structural responses arecomputed for the sole purpose of specifying Expanding q(x,y,t) in a double Fourier seriesshock environments for acceleration-sensitive involving x and y,components within the structure, it is of coursenecessary that all modes which contribute sig-nificantly to acceleration response be accounted 0 40for. This apparently reasonable requirement Is j-runfortunately somewhat ambiguous, and theee- q(x,y,t) = L1 Pij(t) sin -i1* sin L-T-.

fore there is perhaps a need for a more quanti- J=I a btative definition of the number of modes which,for the stated practical purpose, should be con-sidered. The intentions of this study are to de- The Fourier coefficient Pr(t) is evaluated astermine an upper limit of acceleration responsevalues resulting from the consideration ofhigher modes, to ascertain the number of modes b awhich Significantly contribute to accelerationi ! f q(x,y,t) sin i. x sin 15 ddresponse, and to establish a quantitative corn- Pi(t) dxdy.

parison between one-mode and many-mode ac- 0 0celeration response solutions by considering asimply supported plate loaded by a step-functionpressure. For uniform-loading cases where q(x,y,t) varies

with time but is constant with respect to x andANALYSIS OF A SIMPLY SUPPORTED PLATE y, i.e., q(x,y,t) = q(t),

The response solutions for an undamped sim- {ply supported plate acted upon by a step-function b aoverpressure are directly derived as follows. (t) 4q(t) I sin sin dxdyAssuming that the classical bendings, i.e., plane ii ab J f a bstress equations apply, the governing differen- 0 0tial equation [11 is

Pi(t) = q(t) ocs

Vh w + Eh V4 w =q(x,y,t). (2) ab a 0 , J T 0

g 12(1- v2 ) (Therefore,

A deflection which conforms with the simple-support boundary conditions at x = 0 and a, y =0 Pi (t) = 0 for even values of I or J, andand b is

Pij(t) = -6- q(t) for odd values of i and J.w = E E sin'a'snJT. (3

1= Jla bi**1 J=l If q(t) is a Heaviside step function of amplitude

Po, i.e., q(t) = PoU(t), then Pij(t) = 16 PoU(t)/The derivatives of w are Iij f for odd values of I and J. The expression

for q(x,y,t) then becomes

t)Sil)in Sin Ly 16P Ut '0 sin sin).! y

w J ; j a b (xv y t) 16P 0 .t (5)

(4a) i=1 j=l

222

Eqs. (3), (4), and (5) substitute into Eq. (2) to The initial conditions are 0i(0) 0.

form Solving Eq. (8),

( i_ t) sin sin i 2 2N

g " a b S + W(9)

Eh 121- (ifr)2 + (ig)2]2 t1i(t) =2(I - Cos coj t)

sin--sinsi a b

Eq.0~t (9 susiue intE._3)yild

n in nsin I for odd-1=1 a b values 2 (10)

j1of I l

and j,

and each model differential equation becomes Using the stress-str'an relations for plates:

Ii 2 2,2 Ez, ( 2w 1 , 2w\

w .L6 o sin k inI

0 for even values of i and j, (6a) 2 W/g - a2 b

- 16PoU(t)

2 a b (11)

for odd value ofi and e 2,(b i-i 1Y~-ow

with all initial conditions being zero. Modaldisplacements the for even values of I and jare trivially equal to zero by Eq. (6a). For oddvalues of i and J the equations are in the form Ez 2w 2w

Mryy +.U Eh-- ++ IL-6I

2 16 o U(t)= AijU(t) (7)

ij-16 'o Ez +V

The Laplace transform of Eq. (7) is (l 7), (Lb

for o Salu(0s i( ) + d ij (S ) bi sin irx sin ! y I - co s W ijt) (

ij2 2(a b t

W 2

wihaliiilcniiosbigzr.Mdltdispacemnts ~j fr evn vaues f i223

are rivallyequl tozer byEq. 6a) Forodd2 Lvalus o i ad jthe quaionsarein te frm a Ez +

------------------------------------- --------------- - - -

Ez. 2 w where I and j can only be odd values. The rela-._ ... tive amplitudes of higher-modal values com-

xy = MY -pared to the respective amplitudes associatedwith the fundamental mode are plotted on Fig. 1.Information from these graphs concurs with the

00 ( O0 philosophy that displacements and stresses re-1G Po Ez 2 sulting from mode shapes other than the funda-

mental are negligible compared to-displacements",2W/g1+v I +and stresses associated with the fundamental

I=1 jl mode. On the other hand, the contributions toacceleration from some higher modes appear tobe significant. Just how the higher-mode accel-

({I- Cos it) eration amplitudes supplement the accelerationCOS cos jXy . (13) response of the first mode is not obvious from

a b 2 Fig. I nor from Eq. (14); therefore this phe-a j nomenon will now be more closely examined.

Interest is focused at the center of the plateThe acceleration response is where the single-degree-of-freedom analog had

been presumed to be applicable. At the midpointsin Go the acceleration is

16 o a b COS Wijt . 0 0,r2 6/go16 sin -sinW/g~ J4-I Wmidpoint 2 W/g L4 i

(14) I=1 1=

In Eqs. (10) through (14) 1 and J are odd nd Cos Wit (15)

ff #h 3ij\ -(. i + where l and j are oddJ. At t 0, ros wit I1

a2 v 2) \2 2 for all i and J, and Eq. (15) degenerates to

Comparing higher-mode amplitudes to " 16 2o E E 2fundamental-mode amplitudes for displacement, wmidpoint - W/"g - i Jstresses, and acceleration, we have: t=1 j=1

Displacement: where i and j are odd. Note that

S wi, 00 o in

,(first mode) 2' 2 2 (-I)k 1i (2k + 1)

izl,3,5,etc, k--0O

Stresses:

tan-1I4 j j4

_ , xxtljax (first mode) -/ 2

j : (+j) (I + ) so thata t ,

, (first mode) 1 )2 , 16 PO (g (_)r

k n~p IF 2 W/ig k r (2k +I' (2r + 1)

Accelerg.tion:

_i 16 P'O IT2 P0

w(first mode) iJ' - W g -" =W'

224

~ £

'. 1 .7.,7//777/7T.

. . . . . . . . . . . . . . . . .. . .. . . . . . .

•~~~~~~~~~~~~ . . . . . . . . ."

9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..* " " ' " '

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II .. . . . . . . . . . . . . . . . 1. . . . . . . . . . . . .

ISl . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . .

I17 .. . . . . . . . . . . . . . . . IT ..

. . . . . . . . . . . . . ..

19 . . . . . . . . . . . . . . . .. it . . . . . . . . . . . . . . . . .

21 . . . . . . . . . . . . . . . . al . . . . . . . . . . . . . . . .

2$ . . . . . . . . . . . . . . . . 3- . . . . . . . . . . . . . . . .J

25S .. . . . . . . . . . . . . . . . 113 . . . . . . . . . . . . . . . ..

27 .. . . . . . . . . . . . . . . . T- .. . . . . . . . . . . ..

.S . . .. .. . . . . . . . . . . . .. . . . . .' . . . . . . . . . ..

1 o ~O .• • • • • •I . . . . . . . . . . . . . . . ."

RELATIlVE AMiPLITUDErS OF MODAL. DlISPLACEMENTS RErLATIVE AMPiLITUDESl Of MOAL NORMAL STRE~SrS

Ai

7-.4IT-

5-. .

, ........ . . .. .. . . . . ........... . . . . . .. . . . . . . . . . . . .

. . . . . . . . . .7 , . .. .. . . . . .'I . .. . . .. .. . . .

.. . . . . . . . . . . .. .I- . . . . . . . . . . . . . . .

1.. . . . . . . . . . . . . . .. . II.J

. . . . . . . . . . . . . . .

I.. . . . . . . . . . . . . . . . 19 - j

. .. . . . . . . . . . . . .

21 .. . . . . . . . . ... i2, -

_j . . . . . . . . . . . . . .23 . . . . . . . . . . . . . ... . -J . ... . . . . . . . . . ....

9 ........................................ 1 ' . . .. . . .

I ................................................. I

27 .. . .. ....... ..... . ......... . .............................I- ................. 2P

- J . ..............

S.?...... . . . . . . ........ .. . .. . . I . . . . ......

RELATIVE AM/PLITUDES OF MO&.AL $HlEAM S."lESS RELATIVE AMdPLITUDES[€

OF MW AL ACCELEPAITIONS

Fig. I Comparisons of the Relative Amplitudes of Modal Responses

225

By-comparing this with the first-mode contri- they are based on damping forces defined'In abutton at t =0 [(16/) Po/(W/g) 1.6 Po/(W/g) manner which preserves the-linear nature of

the modal differential equations of motioi bywhich approximately agrees.with Eq. (1)], it ntrodicing a viscous damping term. Dampingappears that the one-mode-only representation is thereby defined so that Eq. (7) is repFaced byof response Is reasonable and conservative forthe initial value of acceleration. At t> 0 how-ever, it is not'obvious that the absolute value 2 16' Po (t)of-the-sum-.of-the contributlonsafromuall middeb 4'j + 2(3 w 4, + W4'=1 0 t U~should remain bounded by the amplitude ;,Iof the Mi I Ii I Ii i - i ,response of the first mode, and in fact, plots of (17)normalized acceleration versus normalizedtime show a contrary trend. That is, theequation where 2i w-0 is the modal viscous damping

force,and iJ is the p~rcent-of-critical viscousm n sin Lr sin Jr damping ratio. When the governing partial

Wmidpoin 6 2 2 differential equation i. definedso that Eq. (17)Po/(W/g r2 I j results the total response w and its time-

P( ) * J=l derivatives are summations of the modal values

oij and time-derivatives of OiJ respectively.

cos wijt (I and j are odd) (16) The solution to Eq. (17) in

plotted versus wot/2v on Fig. 2, indicates that A1 1 e (jithe acceleration response increases substanti- ij'1 (t) = I - sin 1ol I3 t + ,ally when higher modes are included in the 2o 2 icalculation. These time-varying plots of Eq. (16)for increasingly larger values of m and ndemonstrate the effect of considering higherfrequencies in the acceleration response. Thepeak values from each of these graphs are 0 tan"I "plotted in Fig. 3 as a function of the number of 9I1modes, and this curve appears to be asymptotic lito about 4.12 Po/(W/g) as m and n becomelarge, that Is about 2.6 times larger than the and the modal acceleration isone-m~ode-only representation of response. Up anthmolacertonito and including 225 modes are considered incomputing these values even though it is recog-t i nized that Eq. (2), which neglects shear and ro- e-lij=oijt, tary inertia is probably not appropriate for the 4'iJ A= A iji + (4very-high-frequency mode shapes. Nonetheless, 2this simplifying assumption permits the calcula- ition of an approximate upper value, and as willbe shown, th consideration of damnping cangreatly reduce the contributional effect of these in iJ t + 0very-high-frequency modes.

DAMPED RESPONSE / '

The previous discussion pertained only to - Wicos kj -3 1 t+0undamped vibration; therefore, computed accel-erations must be conservative since all me-chanical vibrations, certainly all structural For small values of PiJ 0 v/2 and the acceler-vibrations, are impeded by damping. An effortto consider structural damping is presented be- ation can be approximated bylow; however, some discussion regarding thistopic is appropriate.

The phenomena Involved in damping of struc- Ajt el-ijwijCt 16 P 1tural vibrations is complex and to the author's =A(t) ij e coB - - -knowledge, no unified mathematical treatment V gof the problem exists which exactly defines thephysics of such damping. It Is generally con-ceded to be a combination of air damping andctructural damping and although approximate c 1 1 1ij CO tsolutions exist which incorporate these effects,

226

PLATE VIBRATION, 1 MODE CONSIDERED

MI n 1 0 PERCENT DAMPING

S 6.000 -

a.0o

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 2.0001 CT/FUNDAMENTAL- PERIOD)

PLATE VIBRATION,9 MODES CONSIDEREDMz5 nlz 5 0 PERCENT DAMPING

S6.000

00

.00

~V-

0.000.0.200 0.400 0.600 0.8010 1.000 1,200 1.400 1.600 LOW0 2.000(TI FUNDAMENTAL PEPIOD)

PLATE VIBRATION, 8I MODES CONSIDEREDinn17 n 17 0 PERCENT DAMPING

600

40000 . ..0 0. 0.60 .0 . .0 .0 .0 .0 . .0

(TFUDAETA0PRID

Fig. 2 - oprsnotceeainRsosCosierin One and Many Mode

4 2.220

'.5 S w,4 ~s. . ~ '..4~. '~4(5, s~ '. s~x. ,.sYU.-i s~s'

PAKACCEM[RATION AT THE MIDPOINT OF ANUI09D SIMPLY-SUPPORTED PLATE SUBECTED TO ASTEP-FUNCTION PRES*UK. VALUiS ARE CALCULATEDFROM- TN[E -[XACT 9OUyION:.

ACcL[ERATION . isf11 S IN F 14 SW I

Po/(WI) 711 .I Jul i i €O*4 lIt'

WHERE THE SUMMATION LIMITS ARE REPLACED BYFINITE VALUES, OF m AND a FOR I AND j REAiCVILY.

t i n e x ep tI' It it is e l bn is w i gINDE[X OF m AND

1 4 9 16 29 3 49 44 IN 10 111 144 149 196 22NUtM[M OF MODES$ €ONS1IIO RAr

tFig. 3 -Peak Acceleration at the Midpoint of an Undamped Simply SupportedPlate Subjected to a Step-Function Pressure

Thus a damped modal vibration can be con- damping in real structural vibrations has beensidered the same as an undamped modal vibra- the subject of extensive experimentation, andtion except that it Is enveloped by an exponen- there is apparent world-wide disagreement in

Ualdeay e i w i t. tisepszdthths the interpretations of results from such experi-reslt eca ared aJ't i s emofhadsie thtohi ments asevidenced by a sample of some of theresult is arrived at because of a desire for literature on the subject. While testing build-mathematical simplicity rather than from an ing structures Alford and Housner [21 found thatexact definition of the physical phenomena for the damping ratios for the higher modes wereenergy dissipation. The damping forces in the same as for the fundamental mode; andstructural vibrations are not, in fact, known to Sesan, Crongradi Diaconu and Strat [31 con-be viscous; however, this simplifying assump- curred with this finding in a separate studytion provides a means to an end in that it yields On the other hand, Kawasumi and Kmi[41approximately correct dynamic responses, and claim to have experienced proportion:l dampingit is a standard method for evaluating damped by doing similar experiments. Nielson's [51multimodal vibration responses. experiments showed that the same damping co-

efficient was evidenced at different modes duringRegarding values for modal viscous damping tests on one structure; however proportional

ratio iJ, the literature on structural damping damping was experienced while testing a differ-contains different opinions regarding this sub- ent structure. Kimball [61 found that internalject. If the viscous-damped assumption were damping is neither proportional to frequencycompletely true, a Kelvin (alIso known as a nor constant, but rather that for most materialsStokes) model could be assumed for the mate- it increases to a maximum value at some fre-rial (stress is proportional to strain rate as quency and then decreases for all subsequentwell as strain), the governing biharmonic equa- frequencies. Mindlin, Stubner and Cooper.[7[

recommend using a constant damping ratio.tion would be modified by adding a (/ t) V4w Adamson's [8) observations are similar toterm, and modal Eq. (17) would evolve where Kimball's; however, Adamson has a compro-P J is proportional to wij. Whether or not the mise recommendation to use proportional damp-modal damping ratio is indeed proportional to ing if only a few modes are inherent but a

228

'I * .,a %j.,t - 1 Adt3,'.~~~~xstkvt~ c A c-. - - ~ ~- 'ir. .! .

constant ratio if many modes of vibration exist sidered, most materials (even steel plates)in the response solution, probably have minimum equivalent damping

ratios of about ['percent. The nine-mode curveThere exists, therefore, a variety of opinions on Fig. 5 indicates only about a 15 peicentd is-

regarding how equivalent modal viscous damp- agreement with the 225-mode response at 1 per-ing ratios vary as a function of modal frequency. cent damping; therefore nine-modes should, forThe advice of Mindlin and others who recom- practical purposes usually be adequate for dy-mend the use of a constant damping ratio, inde- namic response calculations involving platespendent of frequency is used in the investiga- subjected to impulsive loads.tion below to-determine the effect that dampinghas on the acceleration response of a vibrating If fewer than nine modes are used to repre-plate subjected to a step-functi'n overpressure. sent plate-responses in a dynamic-analysis

If Pij = 3 for all values of I and j and j << 1, study, the calculated peak acceleration re-sponses will possibly be nonconservative by an

the acceleration response at the midpoint be- amount dependent on the percent of criticalcomes damping and the number of modes considered.

If the excitation is impulsive (blast loading),the curves on Fig. 5 are indicative of the possi-m n sin ir sin--! ble amount of nonconservatism which should beW16 - expected. For example, the peak acceleration

midpoint L L I J response calculated for a 2-percent-of-criticallyP W i=1 j=1 damped plate that is approximated by using only

one mode and considered to be subjected to animpulsive load, could be nonconservative by a

e iJt CosWfactor of two.If the excitation is oscillatory the above

recommendation is not intended to apply. If

dominant frequencies in the forcing functionwhere I and j are odd. The effect that P has coincide with frequencies of the plate's higheron response is demonstrated in Fig. 4 which modes, these frequencies will contribute moreshows the midspan response considering 225 significantly to acceleration responses (dis-modes with and .without damping. The very- placements and stresses also) than if the struc-high-frequenicy characteristics quickly dis- ture were impulsively loaded. For these situ-appear and the peak responses are attenuated. ations it most certainly would be prudent toOf particular interest is the comparison of the have the mathematical model of the plate containamplitude of the damped-acceleration peak re- modes which at least include the dominant fre-sponse considering few and many modes of vi- quencies of the forcing function.bration. This is shown on Fig. 5 where it isinferred from the asymptotic nature of the CONCLUSIONScurve on Fig. 3 that the response considering225 modes is equivalent to the response con- The contribution to peak acceleration re-sidering an infinite number of modes. sponse of a blast-loaded plate from modes

other than the fundamental can be significant;The curves on Fig. 5 suggest that larger therefore modal values associated with frequen-

values of critical damping ratio produce two cies above the fundamental should be incorpo-desirable effects from a structural analysis rated in blast-response computations. Forviewpoint. They of course reduce the peak ac- engineering purposes where at least 1 percentceleration responses; however, the error intro- of critical damping may be assumed, it isduced by using only a few rather than many recommended that the mathematical represen-modes to represent dynamic response is also tation of the plate contain at least nine modesreduced. For materials which have three or if blast-loading conditions exist and accelera-more percent damping, the difference between tion responses are desired. Computed acceler-the peak responses using 9 and 225 modes is ations may be nonconservative by up to 40 per-indistinguishable. From a practical standpoint, cent if only four modes are used and by morewhen both air and structural damping are con- than 100 percent if only one mode is used.

229

-f1

PLATE VIBRATION, 225 MODES CONSIDERED

mx29 n:29 0 PERCENT DAMPING

S6.000. . . . . . . . . . . . . . . . . .°

3R° 4.000. . . . .--

-00

0 0.000

Z -4.000 .......K. ..... .

0.000 0.200 0.400 0.600 0.00 1.000 1.200 1.400 1.600 1.600 2.000(T/FUNDAMENTAL PERIOD)

PLATE VIBRATION,225 MODES CONSIDEREDmz29 nz29 I PERCENT DAMPING

6.000

-...... ...............2 -. 000 -

0.000 inA 1A : A / - -- I-

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.900 2.000(T/FUNDAMENTAL PERIOD)

,PLATE VIBRATION, 225 MODES CONSIDEREDm s 29 n 29 3 PERCENT DAMPING

6.000

4.002.000

S-2.000 .. o V* .

-4.0001 . . . H.... --6000 , i

0000 0.200 0.400 0.600 0.800 1.000 1200 1.400 1.600 1.800 2.000(T/FUNDAMENTAL PERIOD)

Fig. 4 - Comparison of Normalized Acceleration Response Withot jid With Damping

230

U )

to NOmsE CONSINNRED

9 AIOSS CONSINERI

i i ! I I i I I

0 I 2 3 4 1 6 7 SPIERCENT OF CRITICAL DAMPIIiN

Fig. 5 - Ratio of Peak Acceleration Response Considering225 Modes to Peak Acceleration Response

Considering Fewer Modes

231

APPENDIX I - REFERENCES

1. Timoshenko, Vibration Problems in Engi- 5. N. N. Nielson, "Damping in Multistory Build-neering, Van Nostrand, January -1955. lngs Determined from Steady-State Vibration

Tests," ASCE Structural Engineering Ccn-2. B. J. Alford and G. W. Housner, "A Dynamic ference, January 31, 1966.

lTest Cf a Four-StoryReinforcedCo ncrete 6. A. L. Kimball, "Vibration Problems, Part V -Building," Bulletin of Seismological Soiciety, Fito n ienl ~ JunloJ 1Applied Mechanics, Vol. 8, 1941.

3. A. Sesan, I. Crongradi, D. Diaconu, and 7. R. D. Mindlin, F. W. Stubner, and H. L.L. Strat, "Experimental Determinations of Cooper, "Response of Damped Elastic Sys-Natural Periods and Dami in Buildings," tems to Transient Disturbances," Proceed-Buletinul Institutului Polite cniciTomul X ings of the Society for Experimental Stress(XIV), Fasc. 3-4, 1964. Analysis, Vol. 5, No. 2, 1548.

4. H. Kawasumi and K. Kana, "Small Amplitude 8. B. Adamson, "A Method for Measuring Damp-Vibrations of Actual Buildings," Proceedings ing and Frequencies of High Modes of Vibra-World Conference on Earthquake Engineer- tion of Beams," Publication of Internationaling, 1956. Association for Bridges and Structural En-

g.nering, 1955.

2

APPENDIX II - NOTATION

ab = plate dimensions in x,y directions w = transverse displacement

Alj = modal coefficient of forcing function w = transverse acceleration

E = Young's modulus W = weight per unit area =yh

g = gravitational constant x,y,z = plate coordinates

h = plate thickmess y = density

i,j = modal indices V = del operator

Pi (t) = Fourier coefficient = jjthiJ frequency

Po = overpresmure 0o--- fundamental frequencyq = pressure function

= Poisson's ratioS = Laplace transform variablet = timeq = normal stresses

U(t) = Heaviside unit step function y = shear stress

xy

233

EFFECT OF CORRELATION IN HIGH-INTENSITY

NOISE TESTING AS INDICATED BY

THE RESPONSE OF AN INFINITE STRIP (U)

Charles T. MorrowV Advanced Technology Center, Inc., j Dallas, Texas

(U) A narrow strip (bar or ribbon) is taken as a theoretical test case forthe realism of high-intensity no4se testing in much the same way that a simplemechanical resonator is commonly taken as a theoretical test case for moreconventional shock and vibration testing. It is shown that in an effort todesign a realistic test, one must consider tile point-to-point correlation ofthe applied field as well as the sound pressure level, even when the damping ofthe strip is large enough to prevent significant retu:-n reflections from theends. Three types of correlation are investigated in particular--complete cor-relation at the coincidence angle for the given frequency, and independent ex-citations at the different antinodal regions. With minor reinterpretation, theformulas remain applicable when, in addition, an exponential decay of correla-tion with distance in either direction along the strip is introduced.

(U) This type of analysis is also potentially useful in the prediction ofresponses. The prediction method that has received the most intensive devel-opment, Statistical Energy Analysis, is conceptually suited primarily to thereverberant field, since normal modes are assumed in the exciting field aswell as in the structure. It may eventually be possible to calculate responsecorrections for reverberant versus flight fields.

INTRODUCTION will necessarily do the same in any item of morecomplicated internal dynamics. Yet, if the

(U) Environmental testing is intended to pro- responses were not similar in the simple res-vide an evaluation, with a minimum of computa- onator, they would have almost no rhance of be-tion, of whether a test item would survive and ing similar in the more complicated item offunction as intended, in practical use. To ac- equipment. In spite of the limitations of thecomplish this instead by theory, for an item of simple test case cnu the compromises that areequipment subject to shock or vibration, would necessary to achieve a practical environmentalbe time consuming and inaccurate, and some modes specification, it is generally possible to pre-of failure might be overlooked. The same would scribe a useful test.be true for a complete space vehicle excited byturbulence or by high-intensity noise. (U) Turbulence and rocket noise are distributed

excitationc. So is the high-intensity noise(U) No environmental simulation is ever com- usel frequently now to simulate them. It is notpletely realistic. Nevertheless, shLck and possible to obtain vaid criteria for realismvibration testing can be a useful engineering by a test case that involves only a single-tool when it is planned intelligently on the point input. The simple mechanical resonator,basis of suitable criteria for realism. These even if the excitation iq conceived as a forcecriteria are obtained by calculation of the re- acting on the mass, i-" inadequate.sponses of a simple structure as a test case.The time-honored test case, more than any other, (U) The test case ,ilized in this paper isfor vibration and shock, is the simple mechan- a simple strip of structure, narrow by compar-Ica] resonator. For vibration, the amount, of ison with the lateral correlation distonces ofdazmp'ng is an important parameter. For shuck, the exciting field. The velocity of transverseU .age potential is less critically depen- vibration along the strip is assumed to varydent n damping and the resonator is frequently with frequency, possibly according to the squareassumeL to be undamped. It is not true that root, but the precise relationship and thetwo shocks or two vibrations that produce prac- fundamental theory behind it. are not criticaQtically identical .'espon3es in a simple resont, or lo the discussion. The strip could Le general-

Preceding page blank

ized to two dimesinns, but the ultimate purpose (U) It is well knovn that the power spectralJs inference about practical situations. As density for the sum or difference of two randonwith shock and vibration, the simplest test case signals is given byyields mucb of the information one needs to knowfor-this. For the present study the strip willbe assumed infinite in both directions. Damping, wA+B = wA + wB : 2wABor, more properly, attenuation with propagationdistance, will be an important parameter. As w 1!2 we are concerned only with relative responses WA

+ B WB) AB (1)

to differez,t fields, the mechanical impedance ofthe strip on ais absolute scale is unimportant.The important field characteristics will turn where wA and wB are the individual power spec-out to be not only the power spectral density of tral densities,the sound pressure as a function of position and

frequency but the point-to-point correlation as correlation density, co-spectrum or real part ofwell. the cross-power spectrum, and cAB 03 the nar-

row-band correlation coefficient.

(U) At one time, I carried out a simplified (U) That c is of magnitude not greater than

analysis of the infinite strip by assuming that unity msy be proved by squaring the sum and dif-* excitation took place only at the antinodes, as ference of two random signals A and B, normal-

in Figure 1. This is a convenient simplified ized to unity variance by division by the re-* model to visualize and to use for inferences in spective standard deviations oA and OB, and

advance of any calculation. For a completely averaging over time

correlated wave at normal incidence, the trans-verse waves propagating in either direction fromalternate antinodes should be of opposite phase - -

and tend to cancel each other. The lower the 2 A 2 2damping, the more complete the cancellation. At (A/ I B/OB 2

= A2/O 2 2AB/OA B + B =the other extreme, for a completely correlatedwave incident at the coincidence angle so that 2(1 > 0. (2)its trace velocity equals the velocity of pro- -AB

pagation of transverse waves in the strip, thetransverse waves propagating from the antinodes Henceshould all be directly additive. The lower thedamping, the more antinodes contribute signif-

B i

icantly to the summation. For completely un- IcAB- (3)correlated excitations exhibiting no dominantphase angle, the waves propagating from theantinodes should combine as the square root of (U) In the analysis to follow, an expression

the sum of the squares, producing an inter- analogous to equation (1) must be derived formediate behaviour. the sum of an infinite number of random signals,

subject to a phase reversal according to whether

(U) Although the analysis based on this aim- the integer numbering a particular antinode isplified model does not permit accurate prediction even or odd, and subject to attenuation of theof the response of the strip to continuously dis- transverse waves on their way to the observationtributed excitation, it does provide insights point, which will be chosen as the origin.into the nature and effect of correlation. Itis therefore worth summarizing here as a pre- (U) Let a sinusoidal pressure p ut the i'thliminary exercise. antinode result in a transverse 4elocity

vi = 8 i P" *|V, pi

i=-2 i=-1 0 i=1 i=2

k=-2 k=-1 0 k=l k 2

FIGURE 1 EXCITATION ONLY AT THE ANTINODES OF AN INFINITE STRIP.

236

at, the origin, with a real quantity. Simi-

larly, for the k in antinode, wI = (

Vk = Ok Pk' K5 1 (10)

(U) Now assume random pressures p if(t) and

Pkf(t) within a narrow bandwidth Af. The and

corresponding total velocity at the origin is K (l)ke Jk l$ (11)

v Af(t) SpiAf(t) 0k PkAf(t) (6) where K is a constant of proportionality, 6ii=-M k=-w a decay constant, and the choice of sign for the

exponents i and k has no numerical effect and

(U) The square is will merely produce a form more similar to thatof a derivation to come. Equation (6) becomes

V f2(t) = IP~ (t ) " p8 Pf (t )

wf) =K 2w (f i -k e-"- IiI+IkI)

pci(f). (12)I Sl k PiAf (t) P kAf(t) (T)

I- k=-w (U) For complete correlation at normal inci-dence, all cp(f) = 1. By expressing Equation

pik(U) Average over time and let Af approach zero (10) as the product of two summations in i andto obtain the power spectral density of the k and applying the expression for the sum of atotal response binomial series and the definitions for the hy-

perbolic functions, we obtain

wv(f) I L k wpik(f) wvn(f) = 2w p(f) tanh 2 (-/2) (13)

i=-- k=--1/2 (U) For incidence at the coincidence angle,

YQ [w (f) (f) (f)Ic M (8) i-kFik pi Wpk pik c pik(f) = (-I) ,and we obtain

i=- k=--

at the origin. All possible power spectral Wvcf = 2p(f)/tanh 2(=12)densities (ik) appear In the summation, andall possible co-spectra. The latter are shown (U) Finally, if the excitations are uncorrela-also as geometric means of all possible pairs ted Fill t f hicions r out, bu

of power spectral density, multiplied by the Cp (f) =f I when I = k. We obtaincorresponding narrow band correlation coeffi- pikcients. w(f) = K2 w p(f) tanh (15)(U) If all i = 0k = 1, this would be ., direct

generalization of Equation (1), taken witi, the (U) These results are consistent with thepositive sign for the power spectral density of trends predicted in advance but are not accu-a sum. If, for any pair of I and k, 8 8k = -1, rate for a strip excited all along its length.

indicating a phase reversal before combinationat the origin, it would be a generalization of EXCITATION AT ALL POSITIONS

the power spectral density of a difference. (U) We will now turn our attention from this

(U) For the problem at hand, we will assume preliminary exercise to the analysis of a more

a constant power spectral density w (f), a realistic model excited continuously as ap function of position. Except in special cases,

phase reversal when the integer i or k is odd, the summations become integrals. As phaseindicating an odd number of half wavelengths shifts in propagation are not limited to 0spacing from the c;ign, and a transverse wave and 1800, the quad spectrum or imaginary partpropagating in either lirection from each anti- of the cross-power spectrum of the sound pres-node and decaying exponentially. In short, sure has an effect.

237

(U) In Appendix 1, the derivatior of a gcneral (U) For a completely correlated field incidentequation for the response of the continuous strip at the coincidence angle such that the traceand expressions for three specific cases is car- velocity equals the velocity of propagation ofried out in detail. transverse vibration in the strip, the final# result is

(U) The equation corresponding to Equation (8)

~is) i cf. X2H2W(r) a2+T

2 2 p ~v 2(02+h,2) (8X(f) Hwp (f) d a(Igl+lhla)

(U) Up to this point, the fields considered0)dh, (16) have been completely correlated except for phase

(f)cos(Igj-Ihj- effects. We would like an indication of what

Lppens to the response as the correlation dis-tance of the pressure field approaches zero.

where 0 and C (f) describe the correlation Let Cpg(f) = +1 and 0 IgI - i for g -Ag

charactelistlg of the incident field, H is a I h < g + Ag, and C pxy(f) w 0 for all other

constant, and g and h are continuous variables pairs of x and y.along the strip, not confined to the integralvalues i and k, and A is the wavelength of A2 +12 g (f)transverse vibration in the strip. The express- w (f) - -P dgion v qIhd g+Ag

(")akc i c( Igl+lhl )dh, (19)

is replaced by the continuous expression which approaches zero as Ag approaches zero.

(U) On the other hand, If we take the antinodes

C (f)cos(ig-jhj-0) to be the centers of half wavelength segments ofpg h the strip, as in Figure 2, and excite each sag-

ment by an independent normally incident wavo no

again dependent on both the transmission char- that the excitations of different segmantgP aro

acteristic and the correlation of the incident uncorrelated, the final result is

fiela. dlI2wp(f) 2 U .3

(U) For a completely correlated field at normal wvu M P '

incidence (e.g. incident normally from a single a2+ 2 1-u GU)

distant source), the final result is

X2 H2w(f) low, to weigh the realism of exiLting a lti'1i1-_ ture by a field of one charuIcterintie at, I1

= 2 imulation of tile effect of field (l' it 111'"vn a (17) ferent characteristic, we aru partiulaPrly In

-2 -1 0 1 2

0 0 0 0 1 o 0 ° I

-2 -1 0 1 211 - 2yfl

FIGURE 2 ANTINODES AS CENTERS OF HALF-WAVELENGTH SEGMENTS,

238

i.

terested in ratios of response. It follows CONCLUSIONimmediately that

W(f) 2a (U) The effect of the type of correlation of

vn l-e " the correlation of the pressure field onRn =- 2 ctructural response is by no means negligible

Wvu(f) 2+3e'a-e3 , (21) even for infinite structures of typical Q's or

structures long enough so that return reflect-

ions are negligible. Introducing an exponential

is the ratio of normal incidence to uncorrelated decay of correlation along the strip has the

response. The ratio of coincidence to un- same effect on response as increasing the inter-

correlated response is nal attenuation. If the analysis given-herewere extended to cover a strip of finite length,

it would also show a marked difference in re-W (ff le.2a sponse to the different fields according to

vc • (a (22) whether the number of antinodes is odd or even.

w VU(f) a2 (a2+4V2 ) 2+3e'o-e'3a (U) It follows that the correlation of the

pressure field as well as its sound pressure

(U) We can find an equivalent Q corresponding level must be considered in establishing a

to the decay constant a in the following way. simulation. As usual, however, realism must

Consider a finite strip starting at g=O and be a compromise limited by practical constaints.

ending at gun, an integer. The return wave When it is not possible to coritr0l the correla-

from the m'th reflection at the far end, after tion closely, it may be desirable to introduce

stoppage of the excitation, is proportional to corrections in the SPL to compensate. Analyses

such as the one given here may be useful insuggesting the amount of correction to apply.

ACKNOWLEDGMENT

is The time of arrival of the m'th return wave I am indebted to my colleague,Warren A. Meyer, for verifying the derivation

t = mn/f, of Equation (20).

so that

mn=ft.

(U) A simple resonant system decays according

to

(U) Consequently, the equivalent Q is

Q = /2u (23)

(U) The two ratios are plotted against both a

and Q in Figure 3, bracketing the possible re-

sponses of an infinite strip or a strip with

enough internal attenuation tc make returns from

,.nd reflections negligible. Quite large effectsof the type of correlation occur for Q's thatare typical of airframe structure.

239

- ,- ' - - 2' ."" ... ... . , - .... .... .., .... , . . , , o- '

11. 0.2 0.4 0.6 0.8 1.0 a50 -

10

S1 " Rc FOR COMPLETELY CORRELATED|1- - . EXCITAlION AT THE COINCIDENCE ANGLE

20 10 5 0 2

o.1 7 € ". EXCITATION AT NORMAL INCIDENCE

0.10 0.2 0.4 0.6 0.8 .

FIGURE 3 RATIO OF RESPONSE P.S.O. TO RESPONSE P.S.D. FOR EXCITATIONCORRELATED ONLY WITHIN HALF WAVELENOTH SEGMENTS.

240

- V

APPenIDIX I

ANALYSIS OF All INFINITE STRIP

EXCITED AT ALL POINTS

(U) The equation very simply.

+ (U) The analysis to follow will be expenditede =

2nf t cos 21ft + J sin 2nft, (24) by using the complex exponential.

(U) Imagine a loop of magnetic tape, with awhere j is a unit vector along the axis of sample of random pressure signal, played backimaginaries, expresses a rotating unit vector continuously so as to generate an arTificiallyas a complex sum of projections along two periodic function, with energy at the funda-orthogonal axes. The symbol e is the base of mental frequency and at each harmonic. Let usnatural .logarithms. It follows that examine the n th harmonic, which is among those

within a small bandwidth Vf. Let the pressureat position x

eJ 2 11f t f+ eJ2sft j (27ff- )

2 (25) p MP= P (30)

at the k th antinode result in a transverseor more simply, the cosine function can be velocity responseextracted from Equation (24) by taking the realpart. J (2rfm -lxm)x(U) A phase shift 0 can be expressed as com- &vm a P e AX (31)

plex multiplier:

e.Jo j2 ft e(2tft-0) at the origin. The y XM is taken to be complex,e e for all possible phase shifts can occur in

propogation Prom various x to the origin, butit is assumed constant over Af, equal to

cos(2,nft-0) + J sin(21ft-0) Yxfo

(U) The velocity produced by all pxn within

Af isor

f-I" AvJ(2nf -0)o 0 e J2ift . ej(2,7ft-) mAV mxm

Af

cos(2ft-0) - J sln(2dft-0) (27)

• x f J2nf -x e 2JI x~ ( 2

so that e P xm e Ax (32)

t~

J(2nf-0) +Letcos(2nft-0) e j(2nt- ) + e- J (21ft - 0)

2 (28) "Jexm (33)

Note also that (U) The total complex velocity at the originis

d j2nft r J2nf j2nft

dt ' (29)

241

/V

F-i1

the transmission characteristic of 'the strip,J (2i~f 0 ) are alwy simple functions of frequency andYxm P m dx position and way be assumed constant over AfVxf Af xm as follows:

j (20 2021 XI * J(2f"-Q-0=)d

f xn xn e dx G G (38)

Af(34)

x=eX (39)

(U) The total real velocity at the origin Is eyn 0y (40)

VA f G P c o s (2 1vf -e -0 )d x eI " x y eJ(" -' +)

"J" Gxmui xn Pyn e 0 dy (4

-j (211fro" Ox0-+xC) ] dx,+e-XW E • dy+x •M (3)A xn Pyn (l

xm~d (35)

which can be written in terms of y as the in- 2 f Jdependent variable and n as the subscript. Ob- f "=tain the square by multiplying the integrals inx and y together, and average over time

P P OS 0 -0 yn)

+ ( sin Iy

+ 2fV 2 f dx f Gx G P PAfxf m yn xm yn r-j 2

+ ___ As( -0y)- ffe xy mon+ e xw yn xn yn Idy P Pexm yn sInn(mxm-y

n dy h3)

• dx P P

dx G Pxn 2yn

Pxn Pyn

(U) 11ow let the length of the sample increasecoS(xm.Oxn+0n )dy" (36) beyond limit and Af approach zero, more slowlyy(36) so that the number of spectral lines within Af

also increases beyond limit. The power spectral(U) Now, thle P and Pn are random quantities, density of transverse vibration at the origin is(U)No, te xn a nnaerdo

The 0 anid 0 are also random, although theirx n y n adj W M + j

difference may or may not be a simple function Wv(f) dx GxGy e0O (f) jqp (f)jof frequency and ponition, depending on the de- JJ.L P Y PXYgree of correlation of the incident wave. flow-ever, Gxm Gyn n 0 and 0 , characterizing the (hi)

y xn yn

242

+ eJ e [Wpxy(f) - Jqpxy ()]1 dy1

where w is. the co-spectrum or real part of J(0-) dy f d x GXY

the cross-power spectrum and x is the quad- "/

spectrum or imaginary part. It follows that 1w/() Wp(f)I2 c o"d

L PC PY J .PY COSW(0)dyk(f ) w u(f ) - f - '€ 0 ,

xy (45) (5)

where tow letw ( r) [ x z ( ) + x ( ] l/ g = 2 x / A s a

Wx 6f, G(j f 2wf.2 f)]/2)52

PXY PX +q XY(46) and

adh - 2y/A (53)

where A is the wavelength of transverse vibrationqpxy(f ) in the strip.=

(f = (fk- n d9)Wvf

(U) Further, the magnitude W y(f) is obtained 1/h2g~) ] Cxyfcs(~id

by mutltiplying out the complex conjugates and 9 hW9ph CpYf)o(OOd

taking the square'root. (5")

Wpx(f) " P xM P o~lmn Let

WpXyf - Cos(00 ) Lt = He-Igl (55)

___ Xyf COfl h0)+E Px nG Ieah

xm yo(9f P (56)sln((xm. yn)0 = l2o:(19l-lhl) (57)

yn(D"-' 2 Sin(0 m-0 andfW ((f)

9f) = wplf), (58)

(U) It is important to note that within etchproduct the m's and n's of one summation are a constant with distance.independent of those of the other. Without ac-tually assigning new symbols to one pair of sub-scripts and multiplying out, we can see that the A2112w (f)result would reduce to a single summation of w ( d I e-a(Igl+lhl)terms each involving four pressure amplitudes v 4 . .f

multiplied by the squared cosine of a conponddifference angle, and, if all these differenceangles should be zero, reaches a maximum value Cpgh~fes( gl.l -)h

Max W (f) Wpx (f) w (f) W f) 1/2 , (19) which is the same as Euation (16).

w (U) Consider first a field completely correlatedwhen Af approaches zero, which can also be in- so that Cpxy(f) =1 for all x and y, and normallyferred from Equation (3). Xincident so that 0 o.

(U) If we define a new coefficient

Cpx(f) *1(f)/ Wpx(fW Mv/2f) - A2CW P f ea(Igl+lhl)

p Wpxy ,px (OW y M ," (50)

243

- ,A

P e-0 d9H~ +f -(. g+2rgdgr 2 -aO)0 Jc(g+h)4 Iffo)2 2w (f M 4 f&5h-Jguhdh]

A2H2W M xc-~i ~zJu2 U0 5120 1

A2~gh)+ (f) ,g9h)J Finally cosie a4on il uhta

X2H2W (f)which~~5 is the 12n~ .~ Eqaton(1).p~

poiie2 in 2 2

A2H~w (f I ''gjIhI)A24 22 f

we )-ajrhd3 -i - '- Jdg e~ Boh)r4o whic istesaea qato 1)

A2HHVw (f) 1_gj d e1d U ialcnie on il uhta

k4 k+-2 PX

excep fo haeefetHs ht w (f) 2 1

24

X2H2W (f

cos~gj--jhjh~d +2 _c(j~jjhj+J,,j~jjhj8 gf ed

k- k-

[ -, ~j j)Jv 'j -jh~ h - 2 r~ii , ma~ja aS

e - 9.chl+jwilhlh )22H2W -2a) - 3c -2m +e:c

+f dik . . , - * 2+2e-2t e +j~1 2 e2 +0 +02 2 1a 2 2

2 [1- 3 2a

f ) 2H2w (fM- ~~2H2w (f +e~ 3a

I .1 (. 2 2) l~e-2aedg 2 ejIhIJ dg eI dh

kf-ki2uw k-l which is the same as Equation (20).

2 , 2

-2H2W ef~ 5-cg+Jirg 2-ah-juh

Mdg f e dl'

,k k4+ I. fT IF C-ag+J1Tg d +2 -ah-ith dh

2 2 _g e +2~

I a- I

2 2 4- j (k+-i-)

+~~~~ + [.ky- e

p +C -A2[J /2 + j k4)]

k-a + C-0 jJ

- AH~wV)~ + e - + e/ +~I/ e

~2+~.2

12

245

shock and vibration 1972

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