basic dynamic analysis

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MSC.Nastran Version 68

Basic Dynamic Analysis

User’s Guide

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C O N T E N T SMSC.Nastran Basic Dynamics User’s Guide

Preface About this Book, xvi

Acknowledgements, xvii

List of MSC.Nastran Books, xviii

Technical Support, xix

Internet Resources, xxii

Permission to Copy and Distribute MSC Documentation, xxiii

1sFundamentals ofDynamic Analysis

Overview, 2

Equations of Motion, 3

Dynamic Analysis Process, 12

Dynamic Analysis Types, 14

2Finite ElementInput Data

Overview, 16

Mass Input, 17

Damping Input, 23

Units in Dynamic Analysis, 27

Direct Matrix Input, 29

❑ Direct Matrix Input, 29❑ DMIG Bulk Data User Interface, 30❑

DMIG Case Control User Interface, 31

3Real EigenvalueAnalysis

Overview, 34

Reasons to Compute Normal Modes, 36

Overview of Normal Modes Analysis, 37

Methods of Computation, 43

❑ Lanczos Method, 43❑

Givens and Householder Methods, 44❑ Modified Givens and Modified Householder Methods, 44

MSC.Nastran Basic DynamicsUser’s Guide

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❑ Automatic Givens and Automatic Householder Methods, 44❑ Inverse Power Method, 45❑ Sturm Modified Inverse Power Method, 45

Comparison of Methods, 46

User Interface for Real Eigenvalue Analysis, 48

❑ User Interface for the Lanczos Method, 48❑ User Interface for the Other Methods, 50

Solution Control for Normal Modes Analysis, 54

❑ Executive Control Section, 54❑ Case Control Section, 54❑ Bulk Data Section, 55

Examples, 56

❑ Two-DOF Model, 57❑ Cantilever Beam Model, 60❑ Bracket Model, 66❑ Car Frame Model, 71❑ Test Fixture Model, 76❑ Quarter Plate Model, 78

❑ DMIG Example, 81

4Rigid-body Modes Overview, 88

SUPORT Entry, 90

❑ Treatment of SUPORT by Eigenvalue Analysis Methods, 90❑ Theoretical Considerations, 91❑ Modeling Considerations, 94

Examples, 96

❑ Unconstrained Beam Model, 96❑ Unconstrained Bracket Example, 101

5FrequencyResponseAnalysis

Overview, 104

Direct Frequency Response Analysis, 106

❑ Damping in Direct Frequency Response, 106

Modal Frequency Response Analysis, 108

❑ Damping in Modal Frequency Response, 109❑ Mode Truncation in Modal Frequency Response Analysis, 112❑ Dynamic Data Recovery in Modal Frequency Response Analysis, 112

Modal Versus Direct Frequency Response, 114

Frequency-Dependent Excitation Definition, 115

❑ Frequency-Dependent Loads – RLOAD1 Entry, 115❑ Frequency-Dependent Loads – RLOAD2 Entry, 116❑ Spatial Distribution of Loading -- DAREA Entry, 117❑ Time Delay – DELAY Entry, 117❑

Phase Lead – DPHASE Entry, 118in Index


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❑ Dynamic Load Tabular Function -- TABLEDi Entries, 118❑ DAREA Example, 121❑ Static Load Sets – LSEQ Entry, 122❑ LSEQ Example, 123❑ Dynamic Load Set Combination – DLOAD, 124

Solution Frequencies, 125

❑ FREQ, 125❑ FREQ1, 126❑ FREQ2, 126❑ FREQ3, 126❑ FREQ4, 127❑ FREQ5, 128

Frequency Response Considerations, 129

Solution Control for Frequency Response Analysis, 130

Examples, 133

❑ Two-DOF Model, 133❑ Cantilever Beam Model, 139❑ Bracket Model, 145

6TransientResponseAnalysis

Overview, 150

Direct Transient Response Analysis, 151

❑ Damping in Direct Transient Response, 152❑ Initial Conditions in Direct Transient Response, 154

Modal Transient Response Analysis, 156

❑ Damping in Modal Transient Response Analysis, 157❑ Mode Truncation in Modal Transient Response Analysis, 160❑ Dynamic Data Recovery in Modal Transient Response Analysis, 161

Modal Versus Direct Transient Response, 162

Transient Excitation Definition, 163

❑ Time-Dependent Loads -- TLOAD1 Entry, 163❑ Time-Dependent Loads – TLOAD2 Entry, 165❑ Spatial Distribution of Loading – DAREA Entry, 166❑ Time Delay -- DELAY Entry, 166❑ Dynamic Load Tabular Function – TABLEDi Entries, 166❑ DAREA Example, 169❑

Static Load Sets -- LSEQ Entry, 170❑ LSEQ Example, 171❑ Dynamic Load Set Combination -- DLOAD, 172

Integration Time Step, 174

Transient Excitation Considerations, 175

Solution Control for Transient Response Analysis, 176

Examples, 179

❑ Two-DOF Model, 179❑ Cantilever Beam Model, 182❑

Bracket Model, 190in Index


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7Enforced Motion Overview, 196

The Large Mass Method in Direct Transient and Direct FrequencyResponse, 197

The Large Mass Method in Modal Transient and Modal FrequencyResponse, 199

User Interface for the Large Mass Method, 201

❑ Frequency Response, 202❑ Transient Response, 203

Examples, 204

❑ Two-DOF Model, 204❑ Cantilever Beam Model, 211

8Restarts InDynamic Analysis

Overview, 218

Automatic Restarts, 219 Structure of the Input File, 220

User Interface, 221

❑ Cold Start Run, 221❑ Restart Run, 221

Determining the Version for a Restart, 226

Examples, 228

9Plotted Output Overview, 242 Structure Plotting, 243

X-Y Plotting, 249

10Guidelines forEffective Dynamic

Analysis

Overview, 288

Overall Analysis Strategy, 289

Units, 292

Mass, 293

Damping, 294

Boundary Conditions, 297

Loads, 298

Meshing, 299

Eigenvalue Analysis, 301

Frequency Response Analysis, 302in Index


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❑ Number of Retained Modes, 302❑ Size of the Frequency Increment, 302❑ Relationship of Damping to the Frequency Increment, 302❑ Verification of the Applied Load, 303

Transient Response Analysis, 304

❑ Number of Retained Modes, 304❑ Size of the Integration Time Step, 304❑ Duration of the Computed Response, 305❑ Value of Damping, 305❑ Verification of the Applied Load, 305

Results Interpretation and Verification, 306

Computer Resource Requirements, 308

11AdvancedDynamic AnalysisCapabilities

Overview, 312

Dynamic Reduction, 313

Complex Eigenvalue Analysis, 314 Response Spectrum Analysis, 315

Random Vibration Analysis, 317

Mode Acceleration Method, 318

Fluid Structure Interaction, 319

❑ Hydroelastic Analysis, 319❑ Virtual Fluid Mass, 319❑ Coupled Acoustics, 319❑ Uncoupled Acoustics, 320

Nonlinear Transient Response Analysis, 321❑ Geometric Nonlinearity, 321❑ Material Nonlinearity, 321❑ Contact, 322❑ Nonlinear-Elastic Transient Response Analysis, 322❑ Nonlinear Normal Modes Analysis, 324

Superelement Analysis, 325

❑ Component Mode Synthesis, 325

Design Optimization and Sensitivity, 326

Control System Analysis, 328 Aeroelastic Analysis, 329

❑ Aerodynamic Flutter, 329

DMAP, 331

AGlossary of Terms Glossary of Terms, 334

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BNomenclature forDynamic Analysis

Nomenclature for Dynamic Analysis, 338

❑ General, 338❑ Structural Properties, 339❑ Multiple Degree-of-Freedom System, 340

CThe Set NotationSystem Used inDynamic Analysis

Overview, 342

❑ Displacement Vector Sets, 342

DSolutionSequences forDynamic Analysis

Overview, 348

❑ Structured Solution Sequences for Basic Dynamic Analysis, 348❑ Rigid Formats for Basic Dynamic Analysis, 348

ECase ControlCommands forDynamic Analysis

Overview, 350

❑ Input Specification, 350❑ Analysis Specification, 350❑ Output Specification, 350

ECase ControlCommands forDynamic Analysis

ACCELERATION 352B2GG 354

BC 354

DISPLACEMENT 355

DLOAD 357

FREQUENCY 358

IC 358

K2GG 360

M2GG 361

METHOD 361

MODES 362

OFREQUENCY 363OLOAD 365

OTIME 367

SACCELERATION 368

SDAMPING 369

SDISPLACEMENT 370

SUPORT1 371

SVECTOR 371

SVELOCITY 372

TSTEP 373

VELOCITY 374

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FBulk Data Entriesfor DynamicAnalysis

Overview, 378

FBulk Data Entriesfor DynamicAnalysis

CDAMP1 381

CDAMP2 381

CDAMP3 382

CDAMP4 383

CMASS1 384

CMASS2 385

CMASS3 386

CMASS4 386

CONM1 387

CONM2 388

CVISC 389

DAREA 390

DELAY 391

DLOAD 392

DMIG 393

DPHASE 396

EIGR 396

EIGRL 400

FREQ 404

FREQ1 404

FREQ2 405

FREQ3 406

FREQ4 408

FREQ5 410

LSEQ 411

PDAMP 412

PMASS 413

PVISC 413

RLOAD1 414

RLOAD2 416

SUPORT 418

SUPORT1 419

TABDMP1 420

TABLED1 422TABLED2 423

TABLED3 424

TABLED4 426

TIC 426

TLOAD1 427

TLOAD2 429

TSTEP 432

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GParameters forDynamic Analysis

Overview, 436

HFile ManagementSection

Overview, 444

Definitions, 445 MSC.Nastran Database, 446

File Management Commands, 447

❑ INIT, 447❑ ASSIGN, 448❑ EXPAND, 450❑ INCLUDE, 452❑ Summary, 452

IGrid Point WeightGenerator

Overview, 454

Commonly Used Features, 455

Example with Direction Dependent Masses, 458

JNumericalAccuracy

Considerations

Overview, 470

❑ Linear Equation Solution, 470

❑ Eigenvalue Analysis, 470❑ Matrix Conditioning, 471❑ Definiteness of Matrices, 472❑ Numerical Accuracy Issues, 472❑ Sources of Mechanisms, 473❑ Sources of Nonpositive Definite Matrices, 474❑ Detection and Avoidance of Numerical Problems, 474

KDiagnosticMessages forDynamic Analysis

Overview, 480

LReferences andBibliography

Overview, 500

General References, 501

Bibliography, 502

❑ DYNAMICS – GENERAL, 502

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❑ DYNAMICS – ANALYSIS / TEST CORRELATION, 513❑ DYNAMICS – COMPONENT MODE SYNTHESIS, 518❑ DYNAMICS – DAMPING, 522❑ DYNAMICS – FREQUENCY RESPONSE, 523❑ DYNAMICS – MODES, FREQUENCIES, AND VIBRATIONS, 524❑ DYNAMICS – RANDOM RESPONSE, 536❑ DYNAMICS – REDUCTION METHODS, 537❑ DYNAMICS – RESPONSE SPECTRUM, 538❑ DYNAMICS – SEISMIC, 538❑

DYNAMICS – TRANSIENT ANALYSIS, 540

INDEX MSC.Nastran Basic Dynamics User’s Guide, 543 543

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Page

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MSC.Nastran Basic Dynamics User’s Guide

Preface

About this Book

List of MSC.Nastran Books

Technical Support

Internet Resources

Permission to Copy and Distribute MSC Documentation

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About this Book

The MSC.Nast ran Basic Dynami cs User’ s Gui de is a guide to the proper use of MSC.Nastran fo

solving various dynamic analysis problems. This guide serves as both an introduction to

dynamic analysis for the new user and a reference for the experienced user. The major emphas

focuses on understanding the physical processes in dynamics and properly applying

MSC.Nastran to model dynamic processes while restricting mathematical derivations to a

minimum.

The basic types of dynamic analysis capabilities available in MSC.Nastran are described in th

guide. These common dynamic analysis capabilities include normal modes analysis, transien

response analysis, frequency response analysis, and enforced motion. These capabilities are

described and illustrative examples are presented. Theoretical derivations of the mathematic

used in dynamic analysis are presented only as they pertain to the proper understanding of th

use of each capability.

To effectively use this guide, it is important for you to be familiar with MSC.Nastran’s static

analysis capability and the principles of dynamic analysis. Basic finite element modeling andanalysis techniques are covered only as they pertain to MSC.Nastran dynamic analysis. For

more information on static analysis and modeling, refer to the MSC.Nastran Linear Static Analys

User’s Guide and to the Get t i ng St art ed w i t h M SC.Nast ran User’s Guide .

This guide is an update to theMSC.Nastran Basi c Dynam i cs User’ s Gui de for Version 68, whic

borrowed much material from the MSC.Nastran Handbook for Dynamic Analysis. However, no

all topics covered in that handbook are covered here. Dynamic reduction, response spectrum

analysis, random response analysis, complex eigenvalue analysis, nonlinear analysis, control

systems, fluid-structure coupling and the Lagrange Multiplyer Method will be covered in the

MSC.Nastr an Adv anced D ynami c Analy si s User’s Guide .

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http://../linear/linear.pdfhttp://../linear/linear.pdfhttp://../linear/linear.pdfhttp://../getstart/getstart.pdfhttp://../adv_dynamics/advdyn.pdfhttp://../adv_dynamics/advdyn.pdfhttp://../getstart/getstart.pdfhttp://../linear/linear.pdfhttp://../linear/linear.pdf


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Acknowledgements

Much of the basis of this guide was established by Michael Gockel with the MSC.Nastran

Handbook for Dynamic Analysis. This guide has used that information as a starting point. John

Muskivitch began the guide and was the primary contributor. Other major contributors

included David Bella, Franz Brandhuber, Michael Gockel, and John Lee. Other technical

contributors included Dean Bellinger, John Caffrey, Louis Komzsik, Ted Rose, and Candace

Hoecker. Ken Ranger and Sue Rice provided the bracket and test fixture models, respectively

and John Furno helped to run those models.

This guide benefitted from intense technical scrutiny by Brandon Eby, Douglas Ferg,

John Halcomb, David Herting, Wai Ho, Erwin Johnson, Kevin Kilroy, Mark Miller, and Willia

Rodden. Gert Lundgren of LAPCAD Engineering provided the car model. Customer feedbac

was received from Mohan Barbela of Martin-Marietta Astro-Space, Dr. Robert Norton of Jet

Propulsion Laboratory, Alwar Parthasarathy of SPAR Aerospace Limited, Canada, and Manfre

Wamsler of Mercedes-Benz AG. The efforts of all have made this guide possible, and their

contributions are gratefully acknowledged.

Grant Sitton

June 1997

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List of MSC.Nastran Books

Below is a list of some of the MSC.Nastran documents. You may order any of these documen

from the MSC.Software BooksMart site at www.engineering-e.com.

Installation and Release Guides

❏ Installation and Operations Guide

❏ Release Guide

Reference Books

❏ Quick Reference Guide

❏ DMAP Programmer’s Guide

❏ Reference Manual

User’s Guides

❏ Getting Started

❏ Linear Static Analysis

❏ Basic Dynamic Analysis

❏ Advanced Dynamic Analysis

❏ Design Sensitivity and Optimization

❏ Thermal Analysis

❏ Numerical Methods

❏ Aeroelastic Analysis

❏ Superelement

❏ User Modifiable

❏ Toolkit

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Technical Support

For help with installing or using an MSC.Software product, contact your local technical suppo

services. Our technical support provides the following services:

Resolution of installation problems

• Advice on specific analysis capabilities

• Advice on modeling techniques

• Resolution of specific analysis problems (e.g., fatal messages)

• Verification of code error.

If you have concerns about an analysis, we suggest that you contact us at an early stage.

You can reach technical support services on the web, by telephone, or e-mail:

Web Go to the MSC.Software website at www.mscsoftware.com, and click on Support. Here, you ca

find a wide variety of support resources including application examples, technical applicationnotes, available training courses, and documentation updates at the MSC.Software Training,

Technical Support, and Documentation web page.

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Madrid, SpainTelephone: (34) (91) 5560919

Fax: (34) (91) 5567280

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Email Send a detailed description of the problem to the email address below that corresponds to the

product you are using. You should receive an acknowledgement that your message was

received, followed by an email from one of our Technical Support Engineers.

Training

The MSC Institute of Technology is the world's largest global supplier of

CAD/CAM/CAE/PDM training products and services for the product design, analysis and

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The MSC Institute of Technology is located at:

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The Institute maintains state-of-the-art classroom facilities and individual computer graphicslaboratories at training centers throughout the world. All of our courses emphasize hands-on

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We specialize in customized training based on our evaluation of your design and simulation

processes, which yields courses that are geared to your business.

In addition to traditional instructor-led classes, we also offer video and DVD courses, interactiv

multimedia training, web-based training, and a specialized instructor's program.

Table 0-1

MSC.Patran SupportMSC.Nastran SupportMSC.Nastran for Windows SupportMSC.visualNastran Desktop 2D SupportMSC.visualNastran Desktop 4D SupportMSC.Abaqus SupportMSC.Dytran SupportMSC.Fatigue SupportMSC.Interactive Physics SupportMSC.Marc SupportMSC.Mvision SupportMSC.SuperForge Support

MSC Institute Course Information

[email protected]@[email protected]@mscsoftware.comvndesktop.support@mscsoftware.commscabaqus.support@mscsoftware.commscdytran.support@[email protected]@[email protected]@[email protected]

[email protected]

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Course Information and Registration. For detailed course descriptions, schedule

information, and registration call the Training Specialist at (800) 732-7211 or visit

www.mscsoftware.com.

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Internet Resources

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MSC.Software corporate site with information on the latest events, products and services for th

CAD/CAE/CAM marketplace.

Simulation Center (simulate.engineering-e.com)

Simulate Online. The Simulation Center provides all your simulation, FEA, and other

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CHAPTER

1Fundamentals of Dynamic Analysis

Overview

Equations of Motion

Dynamic Analysis Process

Dynamic Analysis Types

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1.1 Overview

In static structural analysis, it is possible to describe the operation of MSC.Nastran without a

detailed discussion of the fundamental equations. Due to the several types of dynamic analyse

and the different mathematical form of each, some knowledge of both the physics of dynamic

and the manner in which the physics is represented is important to using MSC.Nastran

effectively and efficiently for dynamic analysis.

You should become familiar with the notation and terminology covered in this chapter. This

knowledge will be valuable to understand the meaning of the symbols and the reasons for th

procedures employed in later chapters. “References and Bibliography” on page 499 provide

a list of references for structural dynamic analysis.

Dynamic Analysis Versus Static Analysis. Two basic aspects of dynamic analysis differ

from static analysis. First, dynamic loads are applied as a function of time. Second, this

time-varying load application induces time-varying response (displacements, velocities,

accelerations, forces, and stresses). These time-varying characteristics make dynamic analysi

more complicated and more realistic than static analysis.

This chapter introduces the equations of motion for a single degree-of-freedom dynamic syste

(see “Equations of Motion” on page 3), illustrates the dynamic analysis process (see “Dynam

Analysis Process” on page 12), and characterizes the types of dynamic analyses described in th

guide (see “Dynamic Analysis Types” on page 14). Those who are familiar with these topics

may want to skip to subsequent chapters.

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1.2 Equations of Motion

The basic types of motion in a dynamic system are displacement u and the first and second

derivatives of displacement with respect to time. These derivatives are velocity and

acceleration, respectively, given below:

Eq. 1

Velocity and Acceleration. Velocity is the rate of change in the displacement with respect to

time. Velocity can also be described as the slope of the displacement curve. Similarly,

acceleration is the rate of change of the velocity with respect to time, or the slope of the velocit

curve.

Single Degree-of-Freedom System. The most simple representation of a dynamic system is

single degree-of-freedom (SDOF) system (see Figure 1-1). In an SDOF system, the time-varyin

displacement of the structure is defined by one component of motion . Velocity and

acceleration are derived from the displacement.

Figure 1-1 Single Degree-of-Freedom (SDOF) System

Dynamic and Static Degrees-of-Freedom. Mass and damping are associated with the motio

of a dynamic system. Degrees-of-freedom with mass or damping are often called dynamicdegrees-of-freedom; degrees-of-freedom with stiffness are called static degrees-of-freedom. It

possible (and often desirable) in models of complex systems to have fewer dynamic

degrees-of-freedom than static degrees-of-freedom.

The four basic components of a dynamic system are mass, energy dissipation (damper),

resistance (spring), and applied load. As the structure moves in response to an applied load,

forces are induced that are a function of both the applied load and the motion in the individu

components. The equilibrium equation representing the dynamic motion of the system is

known as the equation of motion.

u· du

dt ------ v

velocity= = =

u·· d

2u

dt 2

--------- dv

dt ------ a acceleration= = = =

u t ( ) u·

t ( )

u··

t ( )

m = mass (inertia)

b = damping (energy dissipation

k = stiffness (restoring force)

p = applied force

u = displacement of mass

= velocity of mass

= acceleration of mass

u ·

u ··

p t ( )

u t ( )

bk

m

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Equation of Motion. This equation, which defines the equilibrium condition of the system a

each point in time, is represented as

Eq. 1

The equation of motion accounts for the forces acting on the structure at each instant in time.

Typically, these forces are separated into internal forces and external forces. Internal forces a

found on the left-hand side of the equation, and external forces are specified on the right-hanside. The resulting equation is a second-order linear differential equation representing the

motion of the system as a function of displacement and higher-order derivatives of the

displacement.

Inertia Force. An accelerated mass induces a force that is proportional to the mass and the

acceleration. This force is called the inertia force .

Viscous Damping. The energy dissipation mechanism induces a force that is a function of a

dissipation constant and the velocity. This force is known as the viscous damping force

The damping force transforms the kinetic energy into another form of energy, typically heat,which tends to reduce the vibration.

Elastic Force. The final induced force in the dynamic system is due to the elastic resistance i

the system and is a function of the displacement and stiffness of the system. This force is calle

the elastic force or occasionally the spring force .

Applied Load. The applied load on the right-hand side of Eq. 1-2 is defined as a function

of time. This load is independent of the structure to which it is applied (e.g., an earthquake is

the same earthquake whether it is applied to a house, office building, or bridge), yet its effect o

different structures can be very different.

Solution of the Equation of Motion. The solution of the equation of motion for quantities

such as displacements, velocities, accelerations, and/or stresses—all as a function of time—is th

objective of a dynamic analysis. The primary task for the dynamic analyst is to determine the

type of analysis to be performed. The nature of the dynamic analysis in many cases governs th

choice of the appropriate mathematical approach. The extent of the information required from

a dynamic analysis also dictates the necessary solution approach and steps.

Dynamic analysis can be divided into two basic classifications: free vibrations and forced

vibrations. Free vibration analysis is used to determine the basic dynamic characteristics of th

system with the right-hand side of Eq. 1-2 set to zero (i.e., no applied load). If damping is

neglected, the solution is known as undamped free vibration analysis.

Free Vibration Analysis. In undamped free vibration analysis, the SDOF equation of motion

reduces to

Eq. 1

Eq. 1-3 has a solution of the form

Eq. 1

mu··

t ( ) bu·

t ( ) ku t ( )+ + p t ( )=

mu··

t ( )

bu·

t ( )

ku t ( )

p t ( )

mu··

t ( ) ku t ( )+ 0=

u t ( ) A ωn t sin B ωn t cos+=in Index


27/545

The quantity is the solution for the displacement as a function of time . As shown in Eq.

4, the response is cyclic in nature.

Circular Natural Frequency. One property of the system is termed the circular natural

frequency of the structure . The subscript indicates the “natural” for the SDOF system. I

systems having more than one mass degree of freedom and more than one natural frequency

the subscript may indicate a frequency number. For an SDOF system, the circular natural

frequency is given by

Eq. 1

The circular natural frequency is specified in units of radians per unit time.

Natural Frequency. The natural frequency is defined by

Eq. 1

The natural frequency is often specified in terms of cycles per unit time, commonly cycles per

second (cps), which is more commonly known as Hertz (Hz). This characteristic indicates the

number of sine or cosine response waves that occur in a given time period (typically one second

The reciprocal of the natural frequency is termed the period of response given by

Eq. 1

The period of the response defines the length of time needed to complete one full cycle ofresponse.

In the solution of Eq. 1-4, and are the integration constants. These constants are determine

by considering the initial conditions in the system. Since the initial displacement of the system

and the initial velocity of the system are known, and are evaluated by

substituting their values into the solution of the equation for displacement and its first derivativ

(velocity), resulting in

Eq. 1

These initial value constants are substituted into the solution, resulting in

Eq. 1

Eq. 1-9 is the solution for the free vibration of an undamped SDOF system as a function of its

initial displacement and velocity. Graphically, the response of an undamped SDOF system is

sinusoidal wave whose position in time is determined by its initial displacement and velocity a

shown in Figure 1-2.

u t ( ) t

ωn n

ωn

k

m----=

f n

f

n

ωn

2π

------=

T n

T n

1 f

n

----2πω

n

------= =

A B

u t 0=( ) u·

t 0=( ) A B

B u t 0=( )= and A u

·t 0=( )ω

n

----------------------=

u t ( ) u

·0( )

ωn

----------- ωn t sin u 0( ) ωn t cos+=

in Index


28/545

Figure 1-2 SDOF System -- Undamped Free Vibrations

If damping is included, the damped free vibration problem is solved. If viscous damping is

assumed, the equation of motion becomes

Eq. 1-

Damping Types. The solution form in this case is more involved because the amount of

damping determines the form of the solution. The three possible cases for positive values of

are

• Critically damped

• Overdamped

• Underdamped

Critical damping occurs when the value of damping is equal to a term called critical damping

. The critical damping is defined as

Eq. 1-

For the critically damped case, the solution becomes

Eq. 1-Under this condition, the system returns to rest following an exponential decay curve with no

oscillation.

A system is overdamped when and no oscillatory motion occurs as the structure return

to its undisplaced position.

Underdamped System. The most common damping case is the underdamped case where

. In this case, the solution has the form

Eq. 1-

Time t

A m p l i t u d e u

t ( )

mu··

t ( ) bu·

t ( ) ku t ( )+ + 0=

bcr

bcr

2 km 2mωn

= =

u t ( )

A Bt +( )

e bt – 2m ⁄

=

b bcr >

b bcr <

u t ( ) e bt – 2m ⁄

A ωd t sin B ωd t cos+( )=in Index


29/545

Again, and are the constants of integration based on the initial conditions of the system. Th

new term represents the damped circular natural frequency of the system. This term is

related to the undamped circular natural frequency by the following expression:

Eq. 1-

The term is called the damping ratio and is defined by

Eq. 1-

The damping ratio is commonly used to specify the amount of damping as a percentage of th

critical damping.

In the underdamped case, the amplitude of the vibration reduces from one cycle to the next

following an exponentially decaying envelope. This behavior is shown in Figure 1-3. The

amplitude change from one cycle to the next is a direct function of the damping. Vibration is

more quickly dissipated in systems with more damping.

Figure 1-3 Damped Oscillation, Free Vibration

The damping discussion may indicate that all structures with damping require damped free

vibration analysis. In fact, most structures have critical damping values in the 0 to 10% range

with values of 1 to 5% as the typical range. If you assume 10% critical damping, Eq. 1-4 indicat

that the damped and undamped natural frequencies are nearly identical. This result is

significant because it avoids the computation of damped natural frequencies, which can involv

a considerable computational effort for most practical problems. Therefore, solutions for

undamped natural frequencies are most commonly used to determine the dynamic

characteristics of the system (see “Real Eigenvalue Analysis” on page 33). However, this doe

not imply that damping is neglected in dynamic response analysis. Damping can be included

other phases of the analysis as presented later for frequency and transient response (see

“Frequency Response Analysis” on page 103 and “Transient Response Analysis” on

page 149).

A B

ωd

ωd

ωn

1 ζ2

–=

ζ

ζ b

bcr

-------=

Time t

A m p l i t u d e u

t (

)

in Index


30/545

Forced Vibration Analysis. Forced vibration analysis considers the effect of an applied load

on the response of the system. Forced vibrations analyses can be damped or undamped. Sinc

most structures exhibit damping, damped forced vibration problems are the most common

analysis types.

The type of dynamic loading determines the mathematical solution approach. From a numeric

viewpoint, the simplest loading is simple harmonic (sinusoidal) loading. In the undamped

form, the equation of motion becomes

Eq. 1-

In this equation the circular frequency of the applied loading is denoted by . This loading

frequency is entirely independent of the structural natural frequency , although similar

notation is used.

This equation of motion is solved to obtain

Eq. 1-

where:

Again, and are the constants of integration based on the initial conditions. The third term

in Eq. 1-17 is the steady-state solution. This portion of the solution is a function of the applie

loading and the ratio of the frequency of the applied loading to the natural frequency of the

structure.

The numerator and denominator of the third term demonstrate the importance of the

relationship of the structural characteristics to the response. The numerator is the static

displacement of the system. In other words, if the amplitude of the sinusoidal loading is applie

as a static load, the resulting static displacement is . In addition, to obtain the steady stasolution, the static displacement is scaled by the denominator.

The denominator of the steady-state solution contains the ratio between the applied loading

frequency and the natural frequency of the structure.

Dynamic Amplification Factor for No Damping. The term

A =

B =

mu··

t ( ) ku t ( )+ p ω t sin=

ω

ωn

u t ( ) A ωn t sin B ωn t cos+=

p k ⁄

1 ω2– ωn2 ⁄ ----------------------------- ωt sin+

Initial ConditionSolution

Steady-StateSolution

u·

t 0=( )ω

n

---------------------- ω p k ⁄

1 ω2

– ωn

2 ⁄ ( )ω

n

------------------------------------------–

u t 0=( )

A B

p k ⁄

u p k ⁄

1

1 ω2

– ωn

2 ⁄

-----------------------------

in Index


31/545

is called the dynamic amplification (load) factor. This term scales the static response to create a

amplitude for the steady state component of response. The response occurs at the same

frequency as the loading and in phase with the load (i.e., the peak displacement occurs at the

time of peak loading). As the applied loading frequency becomes approximately equal to the

structural natural frequency, the ratio approaches unity and the denominator goes to zero

Numerically, this condition results in an infinite (or undefined) dynamic amplification factor

Physically, as this condition is reached, the dynamic response is strongly amplified relative tothe static response. This condition is known as resonance. The resonant buildup of response

shown in Figure 1-4.

Figure 1-4 Harmonic Forced Response with No Damping

It is important to remember that resonant response is a function of the natural frequency and th

loading frequency. Resonant response can damage and even destroy structures. The dynam

analyst is typically assigned the responsibility to ensure that a resonance condition is controlle

or does not occur.

Solving the same basic harmonically loaded system with damping makes the numerical solutio

more complicated but limits resonant behavior. With damping, the equation of motion becom

Eq. 1-

ω ωn ⁄

Time t

A m p l i t u d e u

t ( )

mu··

t ( ) bu·

t ( ) ku t ( )+ + p ωt sin=

in Index


32/545

In this case, the effect of the initial conditions decays rapidly and may be ignored in the solution

The solution for the steady-state response is

Eq. 1-

The numerator of the above solution contains a term that represents the phasing of thedisplacement response with respect to the applied loading. In the presence of damping, the pea

loading and peak response do not occur at the same time. Instead, the loading and response a

separated by an interval of time measured in terms of a phase angle as shown below:

Eq. 1-

The phase angle is called the phase lead, which describes the amount that the response lead

the applied force.

Dynamic Amplification Factor with Damping. The dynamic amplification factor for the

damped case is

Eq. 1-

The interrelationship among the natural frequency, the applied load frequency, and the phas

angle can be used to identify important dynamic characteristics. If is much less than 1, th

dynamic amplification factor approaches 1 and a static solution is represented with the

displacement response in phase with the loading. If is much greater than 1, the dynam

amplification factor approaches zero, yielding very little displacement response. In this case, th

structure does not respond to the loading because the loading is changing too fast for the

structure to respond. In addition, any measurable displacement response will be 180 degrees

out of phase with the loading (i.e., the displacement response will have the opposite sign from

the force). Finally if , resonance occurs. In this case, the magnification factor is

and the phase angle is 270 degrees. The dynamic amplification factor and phase lead are show

in Figure 1-5 and are plotted as functions of forcing frequency.

Note: Some texts define as the phase lag, or the amount that the response lags the applie

force. To convert from phase lag to phase lead, change the sign of in Eq. 1-19 and

Eq. 1-20.

u t ( ) p k ⁄ ω t θ+( )sin

1 ω2

ωn

2 ⁄ –( )

22ζω ω

n ⁄ ( )

2+

----------------------------------------------------------------------------------=

θ

θ tan1– 2ζω ωn ⁄

1 ω2

– ωn

2 ⁄ ( )

-----------------------------------–=

θ

θ

θ

1

1 ω2 ωn2 ⁄ –( )2

2ζω ωn ⁄ ( )2+

----------------------------------------------------------------------------------

ω ωn ⁄

ω ωn ⁄

ω ωn ⁄ 1= 1 2ζ( ⁄

in Index


33/545

Figure 1-5 Harmonic Forced Response with Damping

In contrast to harmonic loadings, the more general forms of loading (impulses and general

transient loading) require a numerical approach to solving the equations of motion. This

technique, known as numerical integration, is applied to dynamic solutions either with or

without damping. Numerical integration is described in “Transient Response Analysis” on

page 149.

360°

180°

1

ωn

Forcing Frequency ω

P h a s e L e a d θ (

D e g r e e s )

A m p l i f i c a t i o n F a c t o r

in Index


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1.3 Dynamic Analysis Process

Before conducting a dynamic analysis, it is important to define the goal of the analysis prior t

the formulation of the finite element model. Consider the dynamic analysis process to be

represented by the stepsLK in Figure 1-6. The analyst must evaluate the finite element model

terms of the type of dynamic loading to be applied to the structure. This dynamic load is know

as the dynamic environment. The dynamic environment governs the solution approach (i.e.,

normal modes, transient response, frequency response, etc.). This environment also indicates

the dominant behavior that must be included in the analysis (i.e., contact, large displacement

etc.). Proper assessment of the dynamic environment leads to the creation of a more refined

finite element model and more meaningful results.

Figure 1-6 Overview of Dynamic Analysis Process

An overall system design is formulated by considering the dynamic environment. As part of th

evaluation process, a finite element model is created. This model should take into account th

characteristics of the system design and, just as importantly, the nature of the dynamic loadin(type and frequency) and any interacting media (fluids, adjacent structures, etc.). At this poin

the first step in many dynamic analyses is a modal analysis to determine the structure’s natur

frequencies and mode shapes (see “Real Eigenvalue Analysis” on page 33).

In many cases the natural frequencies and mode shapes of a structure provide enough

information to make design decisions. For example, in designing the supporting structure for

rotating fan, the design requirements may require that the natural frequency of the supportin

structure have a natural frequency either less than 85% or greater than 110% of the operating

speed of the fan. Specific knowledge of quantities such as displacements and stresses are not

required to evaluate the design.

No

Yes

Yes

No

Finite ElementModel

DynamicEnvironment

ModalAnalysis?

ResultsSatisfactory?

ResultsSatisfactory? Forced-ResponseAnalysis

End

Yes

No

in Index


35/545

Forced response is the next step in the dynamic evaluation process. The solution process reflec

the nature of the applied dynamic loading. A structure can be subjected to a number of differe

dynamic loads with each dictating a particular solution approach. The results of a

forced-response analysis are evaluated in terms of the system design. Necessary modification

are made to the system design. These changes are then applied to the model and analysis

parameters to perform another iteration on the design. The process is repeated until an

acceptable design is determined, which completes the design process.The primary steps in performing a dynamic analysis are summarized as follows:

1. Define the dynamic environment (loading).

2. Formulate the proper finite element model.

3. Select and apply the appropriate analysis approach(es) to determine the behavior of th

structure.

4. Evaluate the results.

in Index


36/545

1.4 Dynamic Analysis Types

This guide describes the following types of dynamic analysis that can be performed with

MSC.Nastran:

• Real eigenvalue analysis (undamped free vibrations).

• Linear frequency response analysis (steady-state response of linear structures to loadthat vary as a function of frequency).

• Linear transient response analysis (response of linear structures to loads that vary asfunction of time).

The above list comprises the basic types of dynamic analysis. Many MSC.Nastran advanced

dynamic analysis capabilities, such as shock/response spectrum analysis, random response

analysis, design sensitivity, design optimization, aeroelasticity, and component mode synthesi

can be used in conjunction with the above analyses.

In practice, very few engineers use all of the dynamic analysis types in their work. Therefore,

may not be important for you to become familiar with all of the types. Each type can beconsidered independently, although there may be many aspects common to many of the

analyses.

Real eigenvalue analysis is used to determine the basic dynamic characteristics of a structure.

The results of an eigenvalue analysis indicate the frequencies and shapes at which a structure

naturally tends to vibrate. Although the results of an eigenvalue analysis are not based on a

specific loading, they can be used to predict the effects of applying various dynamic loads. Re

eigenvalue analysis is described in “Real Eigenvalue Analysis” on page 33.

Frequency response analysis is an efficient method for finding the steady-state response tosinusoidal excitation. In frequency response analysis, the loading is a sine wave for which th

frequency, amplitude, and phase are specified. Frequency response analysis is limited to linea

elastic structures. Frequency response analysis is described in “Frequency Response Analysis

on page 103.

Transient response analysis is the most general method of computing the response to

time-varying loads. The loading in a transient analysis can be of an arbitrary nature, but is

explicitly defined (i.e., known) at every point in time. The time-varying (transient) loading ca

also include nonlinear effects that are a function of displacement or velocity. Transient respon

analysis is most commonly applied to structures with linear elastic behavior. Transient responanalysis is described in “Transient Response Analysis” on page 149.

Additional types of dynamic analysis are available with MSC.Nastran. These types are

described briefly in “Advanced Dynamic Analysis Capabilities” on page 311 and will be

described fully in the M SC.Nast ran A dvanced D ynami c Analy sis U ser’s Gui de .

in Index

http://../adv_dynamics/advdyn.pdfhttp://../adv_dynamics/advdyn.pdf


37/545


CHAPTER

2Finite Element Input Data

Overview

Mass Input

Damping Input

Units in Dynamic Analysis

Direct Matrix Input

in Index


38/545


39/545

Finite Element Inpu

2.2 Mass Input

Mass input is one of the major entries in a dynamic analysis. Mass can be represented in a

number of ways in MSC.Nastran. The mass matrix is automatically computed when mass

density or nonstructural mass is specified for any of the standard finite elements (CBAR,

CQUAD4, etc.) in MSC.Nastran, when concentrated mass elements are entered, and/or when

full or partial mass matrices are entered.

Lumped and Coupled Mass. Mass is formulated as either lumped mass or coupled mass.

Lumped mass matrices contain uncoupled, translational components of mass. Coupled mass

matrices contain translational components of mass with coupling between the components. Th

CBAR, CBEAM, and CBEND elements contain rotational masses in their coupled formulation

although torsional inertias are not considered for the CBAR element. Coupled mass can be mo

accurate than lumped mass. However, lumped mass is more efficient and is preferred for its

computational speed in dynamic analysis.

The mass matrix formulation is a user-selectable option in MSC.Nastran. The default mass

formulation is lumped mass for most MSC.Nastran finite elements. The coupled mass matrixformulation is selected using PARAM,COUPMASS,1 in the Bulk Data. Table 2-1 shows the

mass options available for each element type.

Table 2-1 Element Mass Types

Element Type Lumped Mass Coupled Mass*

CBAR X X

CBEAM X X

CBEND X

CONEAX X

CONMi X X

CONROD X X

CRAC2D X X

CRAC3D X X

CHEXA X X

CMASSi X

CPENTA X X

CQUAD4 X X

CQUAD8 X X

CQUADR X X

CROD X X

CSHEAR X

CTETRA X Xin Index


40/545

The MSC.Nastran coupled mass formulation is a modified approach to the classical consisten

mass formulation found in most finite element texts. The MSC.Nastran lumped mass is identic

to the classical lumped mass approach. The various formulations of mass matrices can be

compared using the CROD element. Assume the following properties:

CROD Element Stiffness Matrix. The CROD element’s stiffness matrix is given by:

Eq. 2

The zero entries in the matrix create independent (uncoupled) translational and rotational

behavior for the CROD element, although for most other elements these degrees-of-freedom a

coupled.

CROD Lumped Mass Matrix. The CROD element classical lumped mass matrix is the same a

the MSC.Nastran lumped mass matrix. This lumped mass matrix is

CTRIA3 X X

CTRIA6 X X

CTRIAR X X

CTRIAX6 X X

CTUBE X X

*Couple mass is selected by PARAM,COUPMASS,1.

Table 2-1 Element Mass Types (continued)

Element Type Lumped Mass Coupled Mass*

Length LArea ATorsional Constant JYoung’s Modulus EShear Modulus GMass Density ρPolar Moment of Inertia I ρ1-4 are Degrees-of-Freedom

L

2 (Torsion)

1 (Translation)

4 (Torsion)

3 (Translation)

K [ ]

K [ ]

AE

L------- 0

AE –

L----------- 0

0 GJ

L------- 0

GJ –

L----------

AE –

L----------- 0

AE

L------- 0

0 GJ –

L

---------- 0 GJ

0

-------

=

1

2

3

4

1 2 3 4

in Index


41/545

Finite Element Inpu

Eq. 2

The lumped mass matrix is formed by distributing one-half of the total rod mass to each of th

translational degrees-of-freedom. These degrees-of-freedom are uncoupled and there are no

torsional mass terms calculated.

The CROD element classical consistent mass matrix is

Eq. 2

This classical mass matrix is similar in form to the stiffness matrix because it has both

translational and rotational masses. Translational masses may be coupled to other translation

masses, and rotational masses may be coupled to other rotational masses. However,

translational masses may not be coupled to rotational masses.

CROD Coupled Mass Matrix. The CROD element MSC.Nastran coupled mass matrix is

Eq. 2

The axial terms in the CROD element coupled mass matrix represent the average of lumped

mass and classical consistent mass. This average is found to yield the best results for the CRO

element as described below. The mass matrix terms in the directions transverse to the elemen

axes are lumped mass, even when the coupled mass option is selected. Note that the torsiona

inertia is not included in the CROD element mass matrix.

Lumped Mass Versus Coupled Mass Example. The difference in the axial mass

formulations can be demonstrated by considering a fixed-free rod modeled with a single CRO

element (Figure 2-1). The exact quarter-wave natural frequency for the first axial mode is

M [ ] ρ AL

12--- 0 0 0

0 0 0 0

0 012--- 0

0 0 0 0

=

M [ ] ρ AL

13--- 0

16--- 0

0

I ρ

3 A------- 0

I ρ

6 A-------16--- 0

13--- 0

0 I ρ

6 A------- 0

I ρ

3 A-------

=

M [ ] ρ AL

512------ 0

112------ 0

0 0 0 0

112------ 0

512------ 0

0 0 0 0

=

in Index


42/545

Using the lumped mass formulation for the CROD element, the first frequency is predicted to

which underestimates the frequency by 10%. Using a classical consistent mass approach, the

predicted frequency

is overestimated by 10%. Using the coupled mass formulation in MSC.Nastran, the frequenc

is underestimated by 1.4%. The purpose of this example is to demonstrate the possible effects

the different mass formulations on the results of a simple problem. Remember that not all

dynamics problems have such a dramatic difference. Also, as the model’s m

esh becomes finer, the difference in mass formulations becomes negligible.

Figure 2-1 Comparison of Mass Formulations for a ROD

1.5708 E ρ ⁄

l------------

1.414 E ρ ⁄

l------------

1.732 E ρ ⁄

l------------

1.549 E ρ ⁄

l------------

MSC.Nastran Lumped Mass:

Classical Consistent Mass:

MSC.Nastran Coupled Mass:

Theoretical Natural Frequency:

ωn 2 E ρ ⁄

l---------------- 1.414 E ρ ⁄

l----------------= =

ωn

3 E ρ ⁄

l---------------- 1.732

E ρ ⁄ l

----------------= =

ωn

12 5 ⁄ E ρ ⁄

l---------------- 1.549

E ρ ⁄ l

----------------= =

u t ( )

1

Single Element Model

2

l

ωnπ2---

E ρ ⁄ l

---------------- 1.5708 E ρ ⁄

l----------------= =

in Index


43/545

Finite Element Inpu

CBAR, CBEAM Lumped Mass. The CBAR element lumped mass matrix is identical to the

CROD element lumped mass matrix. The CBEAM element lumped mass matrix is identical t

that of the CROD and CBAR mass matrices with the exception that torsional inertia is include

CBAR, CBEAM Coupled Mass. The CBAR element coupled mass matrix is identical to the

classical consistent mass formulation except for two terms: (1) the mass in the axial direction

the average of the lumped and classical consistent masses, as explained for the CROD elemen

and (2) there is no torsional inertia. The CBEAM element coupled mass matrix is also identic

to the classical consistent mass formulation except for two terms: (1) the mass in the axial

direction is the lumped mass; and (2) the torsional inertia is the lumped inertia.

Another important aspect of defining mass is the units of measure associated with the mass

definition. MSC.Nastran assumes that consistent units are used in all contexts. You must be

careful to specify structural dimensions, loads, material properties, and physical properties in

consistent set of units.

All mass entries should be entered in mass consistent units. Weight units may be input instea

of mass units, if this is more convenient. However, you must convert the weight to mass bydividing the weight by the acceleration of gravity defined in consistent units:

Eq. 2

where:

The parameter

PARAM,WTMASS,factor

performs this conversion. The value of the factor should be entered as . The default valu

for the factor is 1.0. Hence, the default value for WTMASS assumes that mass (and mass densit

is entered, instead of weight (and weight density).

When using English units if the weight density of steel is entered as , using

PARAM,WTMASS,0.002588 converts the weight density to mass density for the acceleration o

gravity . The mass density, therefore, becomes . If the weigh

density of steel is entered as when using metric units, then using

PARAM,WTMASS,0.102 converts the weight density to mass density for the acceleration of

gravity . The mass density, therefore, becomes .

PARAM,WTMASS is used once per run, and it multiplies all weight/mass input (including

CMASSi, CONMi, and nonstructural mass input). Therefore, do not mix input type; use all ma

(and mass density) input or all weight (or weight density) input. PARAM,WTMASS does no

affect direct input matrices M2GG or M2PP (see “Direct Matrix Input” on page 29).

PARAM,CM2 can be used to scale M2GG; there is no parameter scaling for M2PP. PARAM,CM

= mass or mass density

= acceleration of gravity

= weight or weight density

ρm

1 g ⁄ ( )ρw

=

ρm

g

ρw

1 g ⁄

RHO 0.3 lb/in3

=

g 386.4 in/sec2= 7.765E-4 lbf -sec2/in4

RHO 80000 N/m3

=

g 9.8 m/sec2

= 8160 kg/m3

in Index


44/545

is similar to PARAM,WTMASS since CM1 scales all weight/mass input (except for M2GG an

M2PP), but it is active only when M2GG is also used. In other words, PARAM,CM1 is used i

addition to PARAM,WTMASS if M2GG is used.

MSC.Nastran Mass Input. Mass is input to MSC.Nastran via a number of different entries.

The most common method to enter mass is using the RHO field on the MATi entry. This field

assumed to be defined in terms of mass density (mass/unit volume). To determine the total

mass of the element, the mass density is multiplied by the element volume (determined from th

geometry and physical properties). For a MAT1 entry, a mass density for steel of

is entered as follows:

Grid point masses can be entered using the CONM1, CONM2, and CMASSi entries. The

CONM1 entry allows input of a fully coupled 6 x 6 mass matrix. You define half of the terms

and symmetry is assumed. The CONM2 entry defines mass and mass moments of inertia for

rigid body. The CMASSi entries define scalar masses.

Nonstructural Mass. An additional way to input mass is to use nonstructural mass, which i

mass not associated with the geometric cross-sectional properties of an element. Examples of

nonstructural mass are insulation, roofing material, and special coating materials.

Nonstructural mass is input as mass/length for line elements and mass/area for elements wit

two-dimensional geometry. Nonstructural mass is defined on the element property entry

(PBAR, for example).

1 2 3 4 5 6 7 8 9 10

$MAT1 MID E G NU RHO A TREF GE

MAT1 2 30.0E6 0.3 7.76E-4

7.76E-4 lbf -sec2/in

4

in Index


45/545

Finite Element Inpu

2.3 Damping Input

Damping is a mathematical approximation used to represent the energy dissipation observed i

structures. Damping is difficult to model accurately since it is caused by many mechanisms

including

• Viscous effects (dashpot, shock absorber)

• External friction (slippage in structural joints)

• Internal friction (characteristic of the material type)

• Structural nonlinearities (plasticity, gaps)

Because these effects are difficult to quantify, damping values are often computed based on th

results of a dynamic test. Simple approximations are often justified because the damping valu

are low.

Viscous and Structural Damping. Two types of damping are generally used for linear-elast

materials: viscous and structural. The viscous damping force is proportional to velocity, andthe structural damping force is proportional to displacement. Which type to use depends on th

physics of the energy dissipation mechanism(s) and is sometimes dictated by regulatory

standards.

The viscous damping force is proportional to velocity and is given by

Eq. 2

where:

The structural damping force is proportional to displacement and is given by

Eq. 2

where:

For a sinusoidal displacement response of constant amplitude, the structural damping force i

constant, and the viscous damping force is proportional to the forcing frequency. Figure 2-2

depicts this and also shows that for constant amplitude sinusoidal motion the two damping

forces are equal at a single frequency.

= viscous damping coefficient

= velocity

= structural damping coefficient

= stiffness

= displacement

= (phase change of 90 degrees)

f v

f v

bu·

=

b

u·

f s

f s

i G k u⋅ ⋅ ⋅=

G

k

u

i 1–

in Index


46/545

At this frequency,

Eq. 2

where is the frequency at which the structural and viscous damping forces are equal for a

constant amplitude of sinusoidal motion.

Figure 2-2 Structural Damping and Viscous Damping Forcesfor Constant Amplitude Sinusoidal Displacement

If the frequency is the circular natural frequency , Eq. 2-8 becomes

Eq. 2

Recall the definition of critical damping from Eq. 1-11

Eq. 2-

Some equalities that are true at resonance ( ) for constant amplitude sinusoidal displacemen

are

Eq. 2-

and Eq. 2-

where is the quality or dynamic magnification factor, which is inversely proportional to th

energy dissipated per cycle of vibration.

The Effect of Damping. Damping is the result of many complicated mechanisms. The effect

damping on computed response depends on the type and loading duration of the dynamic

analysis. Damping can often be ignored for short duration loadings, such as those resulting fro

G k b ω∗ or b Gk

ω∗-------= =

ω∗

f

Forcing Frequency

Structural Damping

Viscous Damping f

v bu

·i b ω u= =

f s

i G k u=

ω∗

ω

D

a m p i n g F o r c e

ω∗ ωn

b G k

ωn

--------- G ωn m= =

bcr

2 km 2mωn

= =

ωn

b

bcr ------- ζ

G

2----= =

Q1

2ζ------

1G----= =

Q

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a crash impulse or a shock blast, because the structure reaches its peak response before

significant energy has had time to dissipate. Damping is important for long duration loading

(such as earthquakes), and is critical for loadings (such as rotating machinery) that continuall

add energy to the structure. The proper specification of the damping coefficients can be

obtained from structural tests or from published literature that provides damping values for

structures similar to yours.

As is discussed in detail in “Frequency Response Analysis” on page 103 and “TransientResponse Analysis” on page 149, certain solution methods allow specific forms of damping t

be defined. The type of damping used in the analysis is controlled by both the solution being

performed and the MSC.Nastran data entries. In transient response analysis, for example,

structural damping must be converted to equivalent viscous damping.

Structural Damping Specification. Structural damping is specified on the MATi and

PARAM,G Bulk Data entries. The GE field on the MATi entry is used to specify overall

structural damping for the elements that reference this material entry. This definition is via th

structural damping coefficient GE.

For example, the MAT1 entry:

specifies a structural damping coefficient of 0.1.

An alternate method for defining structural damping is through PARAM,G,r where r is the

structural damping coefficient. This parameter multiplies the stiffness matrix to obtain the

structural damping matrix. The default value for PARAM,G is 0.0. The default value causes th

source of structural damping to be ignored. Two additional parameters are used in transient

response analysis to convert structural damping to equivalent viscous damping: PARAM,W

and PARAM,W4.

PARAM,G and GE can both be specified in the same analysis.

Viscous Damping Specification. Viscous damping is defined by the following elements:

1 2 3 4 5 6 7 8 9 10

$MAT1 MID E G NU RHO A TREF GE

MAT1 2 30.0E6 0.3 7.764E-4 0.10

CDAMP1 entry Scalar damper between two degrees-of-freedom (DOFs) with referento a PDAMP property entry.

CDAMP2 entry Scalar damper between two DOFs without reference to a property entr

CDAMP3 entry Scalar damper between two scalar points (SPOINTs) with reference to

PDAMP property entry.

CDAMP4 entry Scalar damper between two scalar points (SPOINTs) without referenc

to a property entry.

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Viscous damping for modal transient response and modal frequency response is specified wit

the TABDMP1 entry.

Note that GE and G by themselves are dimensionless; they are multipliers of the stiffness. Th

CDAMPi and CVISC entries, however, have damping units.

Damping is further described in “Frequency Response Analysis” on page 103 and “Transien

Response Analysis” on page 149 as it pertains to frequency and transient response analyses.

CVISC entry Element damper between two grid points with reference to a PVISC

property entry.

CBUSH entry A generalized spring-and-damper structural element that may be

nonlinear or frequency dependent. It references a PBUSH entry.

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2.4 Units in Dynamic Analysis

Because MSC.Nastran does not assume a particular set of units, you must ensure that the uni

in your MSC.Nastran model are consistent. Because there is more input in dynamic analysis

than in static analysis, it is easier to make a mistake in units when performing a dynamic

analysis. The most frequent source of error in dynamic analysis is incorrect specification of th

units, especially for mass and damping.

Table 2-2 shows typical dynamic analysis variables, fundamental and derived units, and

common English and metric units. Note that for English units all “lb” designations are . Th

use of “lb” for mass (i.e., ) is avoided.

Table 2-2 Engineering Units for Common Variables

Variable Dimensions*Common

English UnitsCommon

Metric Units

Length L in m

Mass M kg

Time T sec sec

Area

Volume

Velocity in / sec m / sec

Acceleration

Rotation -- rad rad

Rotational Velocity rad / sec rad / sec

Rotational Acceleration

Circular Frequency rad / sec rad / sec

Frequency cps; Hz cps; Hz

Eigenvalue

Phase Angle -- deg deg

Force lb N

Weight lb N

Moment in-lb N-m

lbf lbm

lb-sec2

in ⁄

L2

n2

m2

L3

in3

m3

LT1–

LT 2– in sec2 ⁄ m sec2 ⁄

T1–

T2–

rad sec2

⁄ rad sec2

⁄

T1–

T1–

T2–

rad2

sec2

⁄ rad2

sec2

⁄

MLT2–

MLT2–

ML2

T2–

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Mass Density

Young’s Modulus

Poisson’s Ratio -- -- --

Shear Modulus

Area Moment of Inertia

Torsional Constant

Mass Moment of Inertia

Stiffness N / m

Viscous Damping Coefficient lb-sec / in N-sec / m

Stress

Strain -- -- --

* L Denotes length

M Denotes massT Denotes time -- Denotes dimensionless

Table 2-2 Engineering Units for Common Variables (continued)

Variable Dimensions*Common

English UnitsCommon

Metric Units

ML3–

lb-sec3

in4

⁄ kg m3

⁄

ML 1–

T 2–

lb in2 ⁄ Pa; N m2 ⁄

ML1–

T2–

lb in2

⁄ Pa; N m2

⁄

L4

in4

m4

L4

in4

m4

ML2

in-lb-sec2

kg-m2

MT2– lb in ⁄

MT1–

ML1–

T2–

lb in2

⁄ Pa; N m2

⁄

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2.5 Direct Matrix Input

The finite element approach simulates the structural properties with mathematical equations

written in matrix format. The structural behavior is then obtained by solving these equations

Usually, all of the structural matrices are generated internally based on the information that yo

provide in your MSC.Nastran model. Once you provide the grid point locations, element

connectivities, element properties, and material properties, MSC.Nastran generates the

appropriate structural matrices.

External Matrices. If structural matrices are available externally, you can input the matrices

directly into MSC.Nastran without providing all the modeling information. Normally this is n

a recommended procedure since it requires additional work on your part. However, there ar

occasions where the availability of this feature is very useful and in some cases crucial. Some

possible applications are listed below:

• Suppose you are a subcontractor on a classified project. The substructure that you aanalyzing is attached to the main structure built by the main contractor. The stiffne

and mass effects of this main structure are crucial to the response of your componen but geometry of the main structure is classified. The main contractor, however, can

provide you with the stiffness and mass matrices of the classified structure. By readin

these stiffness and mass matrices and adding them to your MSC.Nastran model, you

can account for the effect of the attached structure without compromising security.

• Perhaps you are investigating a series of design options on a component attached to aaircraft bulkhead. Your component consists of 500 DOFs and the aircraft model

consists of 100,000 DOFs. The flexibility of the backup structure is somewhat

important. You can certainly analyze your component by including the full aircraft

model (100,500 DOFs). However, as an approximation, you can reduce the matrices fthe entire aircraft down to a manageable size using dynamic reduction (see “Advance

Dynamic Analysis Capabilities” on page 311). These reduced mass and stiffness

matrices can then be read and added to your various component models. In this cas

you may be analyzing a 2000-DOF system, instead of a 100,500-DOF system.

• The same concept can be extended to a component attached to a test fixture. If the finielement model of the fixture is available, then the reduced mass and stiffness matrice

of the fixture can be input. Furthermore, there are times whereby the flexibility of th

test fixture at the attachment points can be measured experimentally. The

experimental stiffness matrix is the inverse of the measured flexibility matrix. In thiinstance, this experimental stiffness matrix can be input to your model.

One way of reading these external matrices is through the use of the direct matrix input featur

in MSC.Nastran.

Direct Matrix Input

The direct matrix input feature can be used to input stiffness, mass, damping, and load matric

attached to the grid and/or scalar points in dynamic analysis. These matrices are referenced

terms of their external grid IDs and are input via DMIG Bulk Data entries. As shown in Table

3, there are seven standard kinds of DMIG matrices available in dynamic analysis.in Index


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The symbols for g-type matrices in mathematical format are , , , and { }. The

three matrices K2GG, M2GG, and B2GG must be real and symmetric. These matrices are

implemented at the g-set level (see “The Set Notation System Used in Dynamic Analysis” o

page 341 for a description of the set notation for dynamic analysis). In other words, these term

are added to the corresponding structural matrices at the specified DOFs prior to the applicatio

of constraints (MPCs, SPCs, etc.).

The symbols for p-type matrices in standard mathematical format are , , and .The p-set is a union of the g-set and extra points. These matrices need not be real or symmetric

The p-type matrices are used in applications such as control systems. Only the g-type DMIG

input matrices are covered in this guide.

DMIG Bulk Data User Interface

In the Bulk Data Section, the DMIG matrix is defined by a single DMIG header entry followed

by a series of DMIG data entries. Each of these DMIG data entries contains a column of nonze

terms for the matrix.

Header Entry Format:

Column Entry Format:

Example:

Table 2-3 Types of DMIG Matrices in Dynamics

Matrix G Type P Type

Stiffness K2GG K2PP

Mass M2GG M2PP

Damping B2GG B2PPLoad P2G –

1 2 3 4 5 6 7 8 9 10

DMIG NAME “0" IFO TIN TOUT POLAR NCOL

DMIG NAME GJ CJ G1 C1 A1 B1

G2 C2 A2 B2 -etc.-

DMIG STIF 0 6 1

DMIG STIF 5 3 5 3 250.

5 5 -125. 6 3 -150.

K gg2

[ ] M gg2

[ ] Bgg2

[ ] Pg2

K pp2[ ] M pp

2[ ] B pp2[ ]

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DMIG Case Control User Interface

In order to include these matrices, the Case Control must contain the appropriate K2GG, M2GG

or B2GG command. (Once again, only the g-type DMIG input matrices are included in this

guide.)

Examples

1. K2GG = mystiff

The above Case Control command adds terms that are defined by the DMIG Bulk Da

entries with the name “mystiff” to the g-set stiffness matrix.

Field Contents

NAME Name of the matrix.

IFO Form of matrix input:

1 = Square

9 or 2 = Rectangular

6 = Symmetric (input only the upper or lower half)

TIN Type of matrix being input:

1 = Real, single precision (one field is used per element)

2 = Real, double precision (one field per element)

3 = Complex, single precision (two fields are used per element)

4 = Complex, double precision (two fields per element)

TOUT Type of matrix to be created:

0 = Set by precision system cell (default)

1 = Real, single precision

2 = Real, double precision

3 = Complex, single precision

4 = Complex, double precision

POLAR Input format of Ai, Bi. (Integer = blank or 0 indicates real, imaginary format;

integer > 0 indicates amplitude, phase format.)

NCOL Number of columns in a rectangular matrix. Used only for IFO = 9.

GJ Grid, scalar, or extra point identification number for the column index or colum

number for IFO = 9.

CJ Component number for GJ for a grid point.

Gi Grid, scalar, or extra point identification number for the row index.

Ci Component number for Gi for a grid point.

Ai, Bi Real and imaginary (or amplitude and phase) parts of a matrix element. If the

matrix is real (TIN = 1 or 2), then Bi must be blank.

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2. M2GG = yourmass


entries with the name “yourmass” to the g-set mass matrix.

3. B2GG = ourdamp


entries with the name “ourdamp” to the g-set damping matrix.Use of the DMIG entry for inputting mass and stiffness is illustrated in one of the examples in

“Real Eigenvalue Analysis” on page 33.

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CHAPTER

3Real Eigenvalue Analysis

Overview

Reasons to Compute Normal Modes

Overview of Normal Modes Analysis

Methods of Computation

Comparison of Methods

User Interface for Real Eigenvalue Analysis

Solution Control for Normal Modes Analysis

Examples

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3.1 Overview

The usual first step in performing a dynamic analysis is determining the natural frequencies an

mode shapes of the structure with damping neglected. These results characterize the basic

dynamic behavior of the structure and are an indication of how the structure will respond to

dynamic loading.

Natural Frequencies. The natural frequencies of a structure are the frequencies at which the

structure naturally tends to vibrate if it is subjected to a disturbance. For example, the strings

a piano are each tuned to vibrate at a specific frequency. Some alternate terms for the natura

frequency are characteristic frequency, fundamental frequency, resonance frequency, and

normal frequency.

Mode Shapes. The deformed shape of the structure at a specific natural frequency of vibratio

is termed its normal mode of vibration. Some other terms used to describe the normal mode a

mode shape, characteristic shape, and fundamental shape. Each mode shape is associated wi

a specific natural frequency.

Natural frequencies and mode shapes are functions of the structural properties and boundary

conditions. A cantilever beam has a set of natural frequencies and associated mode shapes

(Figure 3-1). If the structural properties change, the natural frequencies change, but the mod

shapes may not necessarily change. For example, if the elastic modulus of the cantilever beam

is changed, the natural frequencies change but the mode shapes remain the same. If the

boundary conditions change, then the natural frequencies and mode shapes both change. Fo

example, if the cantilever beam is changed so that it is pinned at both ends, the natural

frequencies and mode shapes change (see Figure 3-2).

Figure 3-1 The First Four Mode Shapes of a Cantilever Beam

x

y

z4

x

y

z1

x

y

z2

x

y

z3

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Figure 3-2 The First Four Mode Shapes of a Simply Supported Beam

Computation of the natural frequencies and mode shapes is performed by solving an eigenvalu

problem as described in “Rigid-Body Mode of a Simple Structure” on page 40. Next, we solv

for the eigenvalues (natural frequencies) and eigenvectors (mode shapes). Because damping

neglected in the analysis, the eigenvalues are real numbers. (The inclusion of damping make

the eigenvalues complex numbers; see “Advanced Dynamic Analysis Capabilities” on

page 311.) The solution for undamped natural frequencies and mode shapes is called real

eigenvalue analysis or normal modes analysis.

The remainder of this chapter describes the various eigensolution methods for computingnatural frequencies and mode shapes, and it conclud

basic dynamic analysis

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