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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 .
in Index
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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.
PhoneandFax
United StatesTelephone: (800) 732-7284Fax: (714) 784-4343
Frimley, CamberleySurrey, United KingdomTelephone: (44) (1276) 67 10 00Fax: (44) (1276) 69 11 11
Munich, GermanyTelephone: (49) (89) 43 19 87 0
Fax: (49) (89) 43 61 71 6
Tokyo, JapanTelephone: (81) (3) 3505 02 66
Fax: (81) (3) 3505 09 14Rome, ItalyTelephone: (390) (6) 5 91 64 50Fax: (390) (6) 5 91 25 05
Paris, FranceTelephone: (33) (1) 69 36 69 36Fax: (33) (1) 69 36 45 17
Moscow, RussiaTelephone: (7) (095) 236 6177Fax: (7) (095) 236 9762
Gouda, The NetherlandsTelephone: (31) (18) 2543700Fax: (31) (18) 2543707
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
manufacturing market. We offer over 100 courses through a global network of education
centers. The Institute is uniquely positioned to optimize your investment in design and
simulation software tools.
Our industry experienced expert staff is available to customize our course offerings to meet youunique training requirements. For the most effective training, The Institute also offers many
our courses at our customer's facilities.
The MSC Institute of Technology is located at:
2 MacArthur Place
Santa Ana, CA 92707
Phone: (800) 732-7211
Fax: (714) 784-4028
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
computer laboratory work to facility skills development.
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]
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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
MSC.Software (www.mscsoftware.com)
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
engineering tools over the Internet.
Engineering-e.com (www.engineering-e.com)
Engineering-e.com is the first virtual marketplace where clients can find engineering expertis
and engineers can find the goods and services they need to do their job
CATIASOURCE (plm.mscsoftware.com)
Your SOURCE for Total Product Lifecycle Management Solutions.
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Permission to Copy and Distribute MSC Documentation
If you wish to make copies of this documentation for distribution to co-workers, complete thi
form and send it to MSC.Software. MSC will grant written permission if the following condition
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MSC.Nastran Basic Dynamics User’s Guide
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
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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+=
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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
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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 (
)
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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 ⁄
-----------------------------
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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=
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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( ⁄
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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
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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
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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.
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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 .
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MSC.Nastran Basic Dynamics User’s Guide
CHAPTER
2Finite Element Input Data
Overview
Mass Input
Damping Input
Units in Dynamic Analysis
Direct Matrix Input
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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
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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
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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
=
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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----------------= =
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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
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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
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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–
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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
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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
The above Case Control command adds terms that are defined by the DMIG Bulk Da
entries with the name “yourmass” to the g-set mass matrix.
3. B2GG = ourdamp
The above Case Control command adds terms that are defined by the DMIG Bulk Da
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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MSC.Nastran Basic Dynamics User’s Guide
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
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