modal v7n4 - an advanced model for stinger force, motion

-

The International Journal of Analytical and Experimental Modal Analysis v 7 n 4 p 271-283 Oct 1 992

by X Hu and K. G_ McConnell, Iowa State University

ABSTRACT A longitudinal bar model that includes both stinger elastic and inertia properties is used to analyze the stinger's

axial dynamics. Stinger motion and force transmissibilities, axial resonance, and excitation energy transfer problems

are discussed. Results show that the peaks of the force and motion transmissibilities are closely related to the

resonances and anti resonances of the test structure and the stinger-structure systems. The ability of the stinger to

transfer excitation energy is mainly dependent on the ratio of the stinger's stiffness to the structure's driving point

dynamic stiffness. Two structures that consist of a rigid mass and a double cantilever beam are chosen for numerical examples to illustrate the stinger's axial transmissibilities.

A cross-section area a acceleration c longitudinal wave propagation velocity E Young's modulus

E* complex Young's modulus EI flexural rigidity of the beam F force

H,(m) structure's driving point accelerance H,,(m) stinger-structure system accelerance

Kr = AE!l, stinger's axial stiffness K,(m) = ( -m)2/H/OJ), structure's dynamic

stiffness

l stinger's length . ' mr stmger s mass

M/m) = l /H5(0J), structure's apparent mass Ta stinger's motion transmissibility T1 stinger's force transmissibility

Greek a =mile dimensionless system parameter p density TJ structural damping factor OJ angular frequency

Mechanical mobility tests require that both the structure's input force and output acceleration be measured. When a shaker is used as an exciting source, it is necessary to connect the shaker's driving platform to the structure through a force transducer. The connection of the exciter to the structure inherently

Dr. Ximing Hu (SEM member), Visiting Scientist, and Kenneth G. McConnell, (Fellow ofSEM), Professor, Department of Aerospace

Engineering and Engineering Mechanics, Iowa State University, Ames, Iowa 50011.

Final manuscript receil'ed: July 14, 1992

271

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creates a problem, however, because the dynamic characteristics of the exciter become combined with those of the test structure, a situation that can obviously lead to contamination of the test's results. The exciter's annature (moving element) is very mobile along its intended motion axis but is very stiff to motion in any other direction. Thus, any structure's response motions other than in the direction of the exciter's action will cause resisting forces or moments that contribute to the measured response in the fonn of a secondary excitation. The response transducers pick up the total responses that are caused not only by the driving force but also by the secondary and unknown forces. Also the input force transducers are sensitive to these secondary shear and bending moments in an unknown manner. Thus, errors are introduced through the forces measured as well as the additional structural constraint. A practical solution to this problem is to introduce a special drive rod between the shaker and the structure. This rod is stiff in the direction of excitation but flexible in all other directions so that the exciter's constraint is eliminated from the test. This drive rod is called a stinger. It is usually a thin circular beam that has high transverse flexibility and high axial stiffness. The result of this need for a stinger is part of the proposed standard mobility test arrangement [1].

Recently some researchers[2-4] have developed an interesting insight into the stinger problem. Mitchell [3] presented an approximate method for sizing the stinger that takes the transverse freedom of the exciter into account. The system analysis used the transfer matrix method and gave two sets of design guidelines for the selection of stinger bending and exciter support. Hieber [ 4] basically used the same method to analyze stinger transverse dynamics. The effects of the rotary inertia of the shaker and the load cell moment sensitivity were considered. Both Mitchell [3] and Hieber [4] concentrated on the stinger's transverse properties.

In order to understand the transfer of the force and motion through the stinger from the exciter end to the structure end, the effects of the stinger's mass and compliance on the structure's frequency response function (FRF) measurement, the stinger's axial dynamic behavior needs to be theoretically investigated. In previous works, the stinger is modeled axially in two different ways. When the compliance of the stinger is considered, the stinger is treated as a massless spring; however, when the stinger mass compensation is considered, the whole stinger is assumed to have rigid body motion. In this paper the stinger is treated as a distributed mass elastic system represented as a longitudinal bar. This approach includes both of the stinger's compliance and mass effects at the same time. This approach is shown to be more suitable for high frequency.

Consider a long, slender bar of uniform cross section as shown in Fig. 1. Let u(x,t) be the displacement in the x direction. Then the longitudinal vibration of the rod may be written as the wave equation [5] in the following manner

(1)

The wave propagation velocity c is given by c2=Eip where E is the rod's Young's modulus and pis the rod's density. The standing-wave solution to Eq. (1) has the fonn

272 October 1 992

u(x,t) = cl sin + c2 cos ej(J)f c c (2)

where m is the angular frequency and C 1 and C 2 are constants that are dependent on the rod's boundary conditions. For this case, the boundary condition� are

u x=O = "• = -a. I (1)2

u x=l = u2 = -a2 I m2

where a1 and a2 are the stinger's end accelerations. The force acting on the rod's ends are given by

atx=O

atx= 1

The relationship between the stinger's stiffness and mass is given by

where

0

2 K =

AE = mr� r I a

c

(3)

(4)

is a dimensionless system parameter. Here a can be interpreted as 27t times the ratio of tiT where t is the time required for a longitudinal stress wave to travel the length of the stinger (t =lie) and T is the test

frequency period (27t/ m). Also, a can be interpreted as a frequency ratio of OJ/ m,. where m,. = � K,. I m,. is

a reference frequency. This reference frequency is related to the fundamental natural frequency of a longitudinal free-free rod by m1 = ncor and a free-fixed rod by m1 = ( rcl2) m,.. Thus, co,. is a convenient reference frequency to judge the behavior of a stinger. The value of cor is large so the value of a is considerably less than unity under most practical situations.

From Eq. (2) and the stinger's boundary conditions, the stinger's transfer matrix equation can be derived as

•

cos a -mr stna

F2 Fi a -- •

a2 astna a I cos a mr

F, ----�· ��-----------------------------------------�

u,

Fig. 1 The stinger's lonltudinal bar model

u2

. I

(5)

The International Journal of Analytical and Experimental Modal Analysis 273

(a) Formulas

•

The excitation energy is transmitted to the structure through a stinger. Investigating the force and the motion relationships between the stinger's two ends is instructive. Consider the coupling of the stinger and the structure under test as shown in Fig. 2. The force and acceleration at the stinger's driven end (vibration source attachment) are represented by F1 and a1• F2 and a2 represent the force and acceleration at the stinger's structure end. Transducer and mounting hardware masses are not included here.

Equations for two components (stinger and test structure) connected together may be related by virtue of the fact that at the connection point their respective accelerations and forces must be equal so that

If the structure's driving point accelerance is H 5( m), then .

a a H (m) = s

= 2 s Fs F2

Substituting Eq. (7) into Eq. (5) and solving for the force and acceleration ratios gives

p: 1 T = 2 = ----------

! Fi H s (m)mr (sin a / a)+ cos a

Test Structure

H!J

F,

F. 2

,.

Stinger

Exciter

Fig. 2 Coupling of the stinger and the test structure

27 4 October 1992 .

•

a,

(6)

(7)

(8)

(9)

Equations (8) and (9) give the force and acceleration transmissibilities across the stinger where subscripts f and a denote force and acceleration, respectively. Obviously, the transmissibilities depend on the stinger's properties, test structure's accelerance, and test frequency.

(b) Numerical Examples

Two structures, a rigid mass and a double cantilever beam with central clamping, are chosen for the numerical examples. The longitudinal bar model of the stinger is used.

( 1) Rigid mass specimen (M)

This is the simplest case where the structure's accelerance H5( m) is a constant 1 /M. Computations have been made for T1and T0 by substituting H5( m) = 1/M into Eqs. (8) and (9) and assuming the ratio Mlmr has values of 10 and 100. The T1and T0 calculation results are presented in Figs. 3 and 4 and have been plotted in tenns of the dimensionless parameter a (a= mil c). These curves are discussed in detail in Section (c).

(2) Beam specimen

The aluminum double cantilever beam specimen is shown in Fig. 5. The single beam is 10 in. long x

0. 12 in. deep x 0. 75 in. wide with head mass mh = 0.087 lb. The driving-point accelerance at the center of the beam is given by

1.2,..-----------------

.._-b

·-:g 1.1 ·-

en en

·-E Cl)

! 1-

0 -8 .a ·-

c wo.9 ::E

Mlm,= 100

0.1

a= mile

Mlm,= 10

0.4

Fig. 3 Force transmissibility of the rigid mass

specimen

0.5

( 10)

...... 1()3

� ·--·-

.0 · -

U)

U) · -E U) c l!

1-

� 0 LL 0 -8 a ·-

c i» �

1()2

101

100

10-1

.. ... -

• • • I I I • • II II II .. •• I I t I I I I I I I I I I I I J , ' , ' a �J. ,: \ mtm,= 10

I ' , ' , ;'

'

• '

' ' '

' ' '

.. .. ' .. ..

.. ... ..... ...... .. .. ·�J. ..... m1m, = 100 - - - ...... � .. ... - ... .. .. ----

........ � ... ...

1 o-2 ..._,., ......... -�-..-.....-........_..___........_....._...__....._ ................. 0.0 0.1 0.2 0.3 0.4 0. 5

a� mile

Fig. 4 Motion transmissibility of the rigid mass

specimen


The Euler beam equation [5] describes the transverse or bending vibration of a uniform beam without damping as follows

dx (1 1 )

where f34=pm2/EI, p is the mass per unit length of the beam, and El is the flexural rigidity of the beam. The structural material damping, which is proportional to the strain but is in phase with the strain

velocity, can be introduced by replacing the elastic modulus E by a complex modulus E* = E( 1 + jf1) so that {3 can be replaced by (/f)4 = par I(E* l) [ 6]. From knowledge of the boundary conditions, we can determine V0• Then, substituting the V0 expression into Eq. (10), we can obtain the driving-point accelerance at the center of the beam with damping [7].

The aluminum beam, assumed to have a structural damping factor 11 = 0.01 and a 3.65 in. long by 0.15 in. diameter steel stinger are used for this example. The corresponding force and acceleration transmissibilities are calculated and plo�ted in Figs. 6 and 7.

(c) Results and Discussion

( 1) Peaks of transmissibility

Ideally, the exciter's force and motion is transmitted by the stinger to the structure under test without any change; that is, T1and T0 are equal to unity. However, examination of Figs. 3 and 4 indicates that there is variation in the T1 plot and a single peak in the T0 plot for the mass-specimen. Furthermore, Figs. 6 and 7 show a series of peaks and valleys occurring in T1and T0 plots for the beam specimen. Clearly it is not acceptable to predict the structure's FRF by measuring the force and motion at the stinger's exciter end as recommended by Ref. [ 1].

To fully comprehend these results, we note that two systems are involved: the structure under test and the combined stinger-structure system. The resonant and antiresonant frequencies for these two systems,

Fig. 5 Double cantilever beam specimen

276 October 1992

v.

y

a,

F,

v.

Clamp Block (mass= mJ

I

X

although different, are related. Referring to Fig. 2 and using the T1, Ta expressions [Eqs. (8), (9)], we obtain the stinger-structure system accelerance expression

at F2 a a2 Tf H (m ) = = · 1 • = H (m ) · ......;._ ss Fi Fi a2 F2 s Ta

(12)

or

Hss (m ) = Hs (m ) cosa- a sin aIm/'

Hs( m)m,.(sin a I a)+ cos a (13)

where the subscript ss designates the stinger-structure system. Equation (12) reveals that the force and acceleration transmissibilities are closely related to the FRFs of both structure and stinger-structure systems. Inspection of T1, Ta and Hss<w) expressions [Eqs.(8), (9) and (13)] reveals two facts:

I. Force transmissibility peaks at the stinger-structure system resonant frequencies and notches at the structure's resonant frequencies.

2. Acceleration transmissibility peaks at the stinger-structure system antiresonant frequencies and notches at the structure's an tire sonant frequencies.

This behavior can be explained as follows: At stinger-structure system resonances, the force at the stinger's exciter end (F 1) tends to become much smaller than the force at the stinger's structure end (F 2) so that T1 becomes a peak. On the other hand, when the structure is at resonance, the force at the stinger's structure end becomes very small compared to the force at the stinger's exciter end since this force includes the stinger's inertia loading. Thus, T1has a notch. The same reason applies to the Ta behavior because the motions at the structure end and the exciter end become very small when Hs(ro) and Hss(w) are near antiresonances, respectively.

20

......... r � 15

... ·-·-

.J:J

·-

c:

lo t-r

-5 0.0

I I

0.00

. I 0.5

I _l 1.0 1.5

Frequency (Hz) r lt r&el 1 1 t' L J.J f t 0.05 0.10 0.15 0.20

a= colic

I 2.0

I I

'

0.25

2.5 *103

, I

Fig. 6 Force transmissibility of the beam specimen

10�--------------------------------�

..._. 8 1-� ... r · -

-· -

.ll r · -

"' "'

· - 6 E ,_ "' t-c: l!

..... t-B 4 ,_ ._ 0 u. � � 0 -8 2 -.a -·-

c: J .,) i I I

::E 0 """

t-

r -2 ��-�--��--��-�-��---��

0.0 0.5 1.0 1.5 2.0 Frequency (Hz)

I I 'a I I I t -b v I I - d d-- J ... • I I I ' ' '

0.00 0.05 0.10 0.15 0.20

a= colic

Fig. 7 Motion transmissibility of the beam specimen

0.25

2.5 *103

, I


In the rigid mass example (Figs. 3 and 4), on the other hand, there is no structure resonance and anti resonance and no stinger-structure resonance; thus there is no notch in the T0 or r1 and no peak in r1. However, since there is one stinger-structure antiresonance, one peak occurs in T0 which can be confirmed

from Fig. 4. The stinger-structure system antiresonance frequency approximates to mn = � K,. I M , where

K,. = AEI l is the stiffness of the stinger and M is the specimen mass.

(2) Comparison of three mechanical models of stingers

It is of interest to compare the stinger's longitudinal rod model with the massless spring model and the rigid body (lumped) mass model since these two simpler models have been used in the past. These two models are shown in Fig. 8.

From Fig. 8, we can readily obtain the following results.

• Massless spring model

• Rigid body model

F T -

2 - I f- -�

I Tf =

I+ m,.Hs(m)

T = 1 a

(I4)

(15)

(16)

(17)

For the longitudinal rod model, if the frequency is not very high so that a= mil c is considerably less than unity, then cos a= 1; sin a= a and Eqs. (8) and (9) can be approximated by

1<,= AE/1

F1 F2 � 4

a, (F1 = F2) •

(a) Massless Spring Model

Fig. 8 Massless spring and rigid body stinger models

278 October 1992

•

F, F2 m, I • ...

a, 82 • ..

(b) Rigid Body Model

(18)

and

(19)

A comparison of Eqs. ( 18) with ( 16) and Eq. ( 19) with ( 15) shows that the low frequency vibrations give sufficient time for stress waves to travel in the stinger's length so that lie<< 1/m. The massless spring model is a good model of motion transmissibility but a poor model of force transmissibility, while the rigid body model predicts the force ratio but is inadequate to predict the acceleration ratio. When the test frequency increases, the errors predicted by the massless spring and rigid body models are not correct and the longitudinal rod model must be used.

. . .. .. . . · · .

. .·

..

. .

·. :

(a) Stinger Axial Resonance

Several research workers [2,4,8] mention stinger axial resonance but define the axial resonance frequency and its effects in different ways. Ewins [2] mentioned that it is always necessary to check for the existence of an internal resonance of the drive rod either axially or in flexure because this can introduce spurious effects on the measured mobility properties. Furthermore, in the case of an axial resonance, very little excitation force will be delivered to the test structure at frequencies above the first mode. Hieber [ 4]

suggests that the stinger axial resonant frequency should be mn= .J K,. I M where K,.= AE/1 is the stinger's stiffness and M is the mass of the exciter armature. He also declared that if this axial resonance is too low, there may be difficulty transferring enough force to the specimen at high frequencies. Anderson [8] gave the same expression for the fundamental longitudinal resonance frequency as Hieber [4] but suggests that M should be the force transducer's seismic mass and mounting hardware mass at the stinger's exciter end. At the resonant frequency, the force transmissibility of the stinger (T1) will be a peale

In the resonant frequency calculation both Refs. [4] and [8] assumed that the structure's accelerance is zero. Unfortunately, this is true only when the structure is near antiresonances. Generally, the structure's accelerance cannot be neglected compared to that of the stinger's exciter end.

Traditionally researchers think that the stinger's axial resonances occur when the motion of the exciter table is considerably greater or smaller than the motion of the test specimen. This will happen when T0 notches or peaks, which will always occur when the test structure or the stinger-structure system approach an antiresonance. This kind of stinger axial resonance is closely related to the accelerance of the structure under test. Each time the test structure approaches an antiresonant frequency, it is inevitable that this kind of axial resonance happens. On the other hand, a common problem encountered in high frequency testing is that the stinger's axial vibration influences the transfer of force and motion from the exciter to the structure. This problem is discussed in the following section.

(b) Maximum Force and Motion Applied to the Test Structure With and Without a Stinger

It is customary to control an exciter while using level control on a feedback signal. The feedback signal is usually the input force, the exciter's motion, or the structure's input motion. An electrodynamic exciter


-

is limited by the maximum force that is available and the maximum acceleration it can develop. These limits are influenced by the dynamic characteristics of the structure under test as well as those of the exciter and its power amplifier.

• The shaker motion is controlled at a constant acceleration level of a First, consider that the structure is directly mounted on the shaker table. The force that can be applied on the structure becomes

where the superscript d represents the structure mounted to the shaker directly without a stinger.

Second, the test structure is connected to the exciter through a stinger. The force acting on the structure is

or

ps -d =T F a (20)

where superscript s represents the case of the structure being connected to the shaker through a stinger. Therefore T0 not only reflects the motion transmissibility between the stinger's two ends during experiments but also reflects the ratio of the force acting on the structure with and without the stinger when the exciter table is controlled to have the same input acceleration level a.

• The shaker is controlled at a constant force level ofF When a similar approach is applied, the following results are obtained

Without stinger:

With stinger in place:

or

(21)

where F is the shaker's output force level, as is the structure's acceleration excited by F through the stinger, and ad is the structure's acceleration excited by F without a stinger. Therefore T1not only represents the force transmissibility between the stinger's two ends but also represents the ratio of the structure's response motions with and without a stinger when the exciter is controlled to produce the same exciting force level F.

280 October 1 992

(c) Stinger's Excitation Energy Transfer Ability is Related toK (m)IK s r

to As mentioned before, when a= mile is significantly less than unity, then Ta and T1can be simplified

T .

1 f= 1 + m,.H s ( m)

1 = ----

1+ m,. ms ( (J))

(22)

(23)

An evaluation of Eqs. (22) and (23) suggests that T1is controlled by the stinger's inertia property mr and the structure's driving point apparent mass M5(m), while Ta is controlled by the stinger's stiffness K,. and the structure's driving point dynamic stiffness K5(m). In most testing cases, the stinger's mass is seldom much larger than the structure's driving point apparent mass except near resonances. On the other hand, the structure's driving point dynamic stiffness can be significant larger than the stinger's stiffness. Therefore, Ta is more important than r1 in excitation energy transfer considerations.

When the test structure is mass-like, the stinger-structure system is similiar to the one-degree massspring vibration isolation problem [5]. Ta is less than unity when m/mn > -J2 and decreases quickly with increasing frequency, which means that less and less exciter table motion can be transfered to the test structure through the stinger. For general systems with multiple degrees of freedom, Eq. (23) shows that Ta is controlled by the K5(m)/K,. term. As Ewins [9] points out, the general level of the structure's driving point dynamic stiffness [K5( m)] increases with increasing frequency with a result that indicates a progressive localizing of the vibration at higher modes. Therefore, as the test frequency increases, Ks(m)IK,. becomes larger and larger and Ta becomes smaller and smaller, which means that more and more the exciter table's motion is absorbed by the stinger's compliance and less and less exciting force can be applied to the structure. The stinger's excitation energy transfer ability is controlled by the ratio of Ks(m)/Kr. Thus we must select a stinger with sufficient stiffness compared to the structure's driving point dynamic stiffness in order to overcome the exciting energy transfer problem.

A numerical example is given to show the influence of the stinger's stiffness on Ta. A 40 in. long x 2 in. wide x 0.5 in. deep aluminum double cantilever beam is used as the test structure. Two different sizes of steel stingers are chosen for comparison. Stinger 1 is 6 in. long and 0.06 in. in diameter and has a stiffness of 13,700 lb/in. Stinger 2 is 4 in. long and 0. 12 in. in diameter and has a stiffness of 82,000 lb/in. The calculation results of T0 are plotted in Fig. 9, which shows that Ta is seriously influenced by the stinger's stiffness. The stiffness of Stinger 1 is insufficient for the test structure beyond 550Hz since T0 << 1 for most of the frequency region above 550 Hz. The significance of the stinger's stiffness relative to the structure's driving point dynamic stiffness in the energy transfer problem is obvious. Figure 9 also shows that the notches of Ta for both stingers occur near the same frequencies while peaks occur at the different


frequencies. As expected, the notches of Ta are near the structure's antiresonances, while peaks are near stinger-structure system antiresonances. For different stingers, the structure's antiresonance will not change, but the stinger-structure's antiresonance will change. Therefore, for the same test structure, different stingers have the same T0 valley frequencies but have the different peak frequencies.

The longitudinal bar model of stinger axial dynamics shows the following. I. When the test frequency is low enough so that a = mil c is considerably less than unity, then the

traditional stinger massless spring model is suitable for motion transmissibility but is a poor model of force transmissibility. The rigid mass model, on the other hand, is a good model of force transmissibility but a poor model of motion transmissibility. The longitudinal bar model is best since it includes both elastic and inertia properties of the stinger. When frequency is high, a is no longer far less than one so that only the longitudinal bar model is acceptable in predicting the stinger's responses.

2. Stinger force transmissibility has valleys when the test structure approaches resonances and has peaks when the stinger-structure system approaches resonances. Stinger motion transmissibility has valleys when the structure approaches antiresonances and has peaks when the stinger-structure system approaches antiresonances.

3. The difficulty in transferring sufficient exciting energy from the shaker to the test structure is mainly due to the stinger's compliance. The motion transmissibility Ta, which is given by Eq. (8), also reflects the ratio of the applied forces to the test structure with and without a stinger when the exciter table has the same acceleration. The basic principle in selecting stinger size to satisfy the energy transfer consideration is that the stinger must be axially stiff when compared to the structure's driving point dynamic stiffness.

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II I I I I : I

� I I I

I I I � I I I I II r I � II I I

I I I I

I 1 I I I � I I I I I

I I I I I

101

I I I I '

' : I ' I ' I I

-8 100 .a ·-

10•1

�·'

,

� �

1-

�

' ' , ,.

I I I : 1 ,,

I I I I II ,, II

' �

, I

I I

I I ,

, ...

I 10-2 0.0 0.5

Fig. 9 Effects of the stinger's stiffness on T8

282 October 1992

I I , I I ' I

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I ' I , I I I ' I ' ' ,

I ' I I I I

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Frequency (Hz)

I I ' I I I I I I I I I

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Stinger 1 Stinger 2

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\ \ \ \

\ \

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.

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2.0 *1()3

The authors wish to thank the Iowa State University Engineering Research Institute and the Department of Aerospace Engineering and Engineering Mechanics for support of this research.

1. Standards Secretariat, Acoustical Society of America. Methods for the Experimental Determination

ofM echanical M ability, Part Two: Measurements Using Single-Point Translational Excitation. (ANSI

Standard S2.32-19S2). New York: Acoustical Society of America; 1982. 2. Ewins, D.J. Model Testing: Theory and Practice. Letchworth, Hertfordshire, England: Research

Studies Press Ltd.; 1 985. 3. Mitchell., L.D.; Elliott, K.B. ''A method for designing stingers for use in mobility test." Proceedings

of the 2nd International Modal Analysis Conference, Orlando, FL, Feb 6-9, 1984. v 2 p 872-876 4. Hieber, G. M. "Non-toxic stingers." Proceedings of the 61h International Modal Analysis Confer

ence, Orlando, FL, Feb 1 -4, 1 988. v 2 p 1 371 - 1379 5. Thomson., W. T. Theory ofVibration with Applications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall,

Inc.; 1988. 6. Snowdon, J.C. "Transverse vibration of free-free beams." J Acoust Soc Am v 35 n 1 p 47-52 Jan

1963. 7. Hu, X. Effects of Stinger Axial Dynamics and Mass Compensation Methods on Experimental Modal

Analysis. Ph.D. Dissertation, Iowa State University, 1 99 1 . 8. Anderson, I.A. "Avoiding stinger rod resonance on small structures." Proceedings of the 8'h

International Modal Analysis Conference, Kissimmee, FL, Jan 29-Feb 1 , 1989. v 1 p 673-678. 9. Ewins, D.J. "Measurement and application of mechanical impedance data (Part 3)." J Soc Environ

Eng v 15 n 2 p 7- 17 Jun 1976.


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Arkansas, Fayetteville, AR 72701 Managing Editor (Nom� and Compl�t� MailittR Addr?s.f)

Leanne Mitchell, Scholarly Communications Project, Virginia Tech, University Libraries, P.O. �x 90001; Black�m1�g, VA 24062-9001 7. Owner (I/ OWIIt>d by a corporation. il.f nam� and tuldrtss maul IN SlUt� and also imm�diuttl)' th�reurtd�r tht ftQIII�S DNI Gddr�ss�s of stodtJaold�n owrtin' or ltoldinl

I ptrctllt or mnrtt of total omount of stoc-k .. If not O'M'Ift'd b.v a corporation, thtt namts and addnss�s of thl' individ11al nwrttrs rruul bt 1ivm. If ownftl by t1 pGrtn�nhip or othtr uniru·nrpnrattd firm. its IIQm(' und oddrcss, as �II as that of tach indi•·idua/ must b� giwn. If th� publiclltion is publishtd by a IIIHipro/it orranit.lltion, its nanrt and addrt'.u m11st bt stat�d.) (lt�m must bt compltttd.)

Full Name Complete Addreu

8. Known Bondholders, Mortgagees, and Other Security Holders Owning or Holding 1 Percent or More of Total Amount of Bonds, Mortgages or Other Securities rq lhtrr flre nont, so SIIJlt}

Full Name Complete Maling Addreu

9. For Completion by Nonprofit Organizations Authorized To Mail at Special Rates (DMM �ai011 424.12 Ottly) The purpose. function, and nonprofit status of this organization and the exempt status for Federal income tax purposes (Cirtdc on�)

111 (2) GJ Has Not Changed During 0 Has Changed During (1/ dttlngtd. publislttr IJIIISt .rllbmit tzp/IJittltitltf of Preceding 12 Months Preceding 12 Months chan� with tltir stDttmmt.)

10. Extent •nd Nature of Circulation (&� instfllctinns 011 rtwnt' .fid('}

A. Total No. Copies (N�t Prrss Ibm) .

8. Paid and/or Requested Circulation 1. Sales through dealers •nd carriers. street vendors and counter sales

2. Mail Subscription (PQid andlnr r�qwst�d)

c. Total Paid and/or Requested Circulation (Sum of /OBI and /082)

D. Free Distribution by Mail, Carrier or Other Means Samples, Complimentary. end Other Free Copies

E. Total Distribution (SMm of C tutti DJ

F. Copies Not Distributed 1. Office use. left over, unaccounted. spoiled after printing

2. Return from News Agents

G. TOTAl (Sum nf E. FJ and 2-should �I nrt prt.ts "'"shown ;, A)

Average No. Copies Each Issue During Aetu•l No. Copiet of Single l11ue Preceding 12 Months Published Nearest to Filing D•te

• 1,136 882 0 0

610 668 610 668 372 39 982 707 154 175

1,136 882 11. I certify that the statements made by Signat�1pe of Editor. Publisher, Business M•n•ger, or Owner

/i"' /: me above are correct and complete PS Form 3526. January 1991

� tfl-e< - - ,S,t. ttl( (S�e itJstruction.s on rtt:�rs�)

modal v7n4 - an advanced model for stinger force, motion

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