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Vibration Simulation Using MATLAB and ANSYS - Michael R. Hatch

Vibration Simulation Using MATLAB and ANSYS

Hardcover Published: 21st September 2000
ISBN: 9781584882053
Number Of Pages: 676

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Transfer function form, zpk, state space, modal, and state space modal forms. For someone learning dynamics for the first time or for engineers who use the tools infrequently, the options available for constructing and representing dynamic mechanical models can be daunting. It is important to find a way to put them all in perspective and have them available for quick reference.

It is also important to have a strong understanding of modal analysis, from which the total response of a system can be constructed. Finally, it helps to know how to take the results of large dynamic finite element models and build small MATLABĀ® state space models.

Vibration Simulation Using MATLAB and ANSYS answers all those needs. Using a three degree-of-freedom (DOF) system as a unifying theme, it presents all the methods in one book. Each chapter provides the background theory to support its example, and each chapter contains both a closed form solution to the problem-shown in its entirety-and detailed MATLAB code for solving the problem.

Bridging the gap between introductory vibration courses and the techniques used in actual practice, Vibration Simulation Using MATLAB and ANSYS builds the foundation that allows you to simulate your own real-life problems.


  • Demonstrates how to solve real problems, covering the vibration of systems from single DOF to finite element models with thousands of DOF
  • Illustrates the differences and similarities between different models by tracking a single example throughout the book
  • Includes the complete, closed-form solution and the MATLAB code used to solve each problem
  • Shows explicitly how to take the results of a realistic ANSYS finite element model and develop a small MATLAB state-space model
  • Provides a solid grounding in how individual modes of vibration combine for overall system response
  • Introductionp. 1
    Representing Dynamic Mechanical Systemsp. 1
    Modal Analysisp. 1
    Model Size Reductionp. 4
    Transfer Function Analysisp. 7
    Introductionp. 7
    Deriving Matrix Equations of Motionp. 8
    Three Degree of Freedom (tdof) System, Identifying Components and Degrees of Freedomp. 8
    Defining the Stiffness, Damping and Mass Matricesp. 8
    Checks on Equations of Motion for Linear Mechanical Systemsp. 11
    Six Degree of Freedom (6dof) Model--Stiffness Matrixp. 11
    Rotary Actuator Model--Stiffness and Mass Matricesp. 12
    Single Degree of Freedom (sdof) System Transfer Function and Frequency Responsep. 15
    sdof System Definition, Equations of Motionp. 15
    Transfer Functionp. 15
    Frequency Responsep. 17
    MATLAB Code sdofxfer.m Descriptionp. 21
    MATLAB Code sdofxfer.m Listingp. 21
    tdof Laplace Transform, Transfer Functions, Characteristic Equation, Poles, Zerosp. 23
    Laplace Transforms with Zero Initial Conditionsp. 23
    Solving for Transfer Functionsp. 24
    Transfer Function Matrix for Undamped Modelp. 27
    Four Distinct Transfer Functionsp. 28
    Polesp. 29
    Zerosp. 30
    Summarizing Poles and Zeros, Matrix Formatp. 32
    Matlab Code tdofpz3x3.m--Plot Poles and Zerosp. 32
    Code Descriptionp. 32
    Code Listingp. 32
    Code Output--Pole/Zero Plots in Complex Planep. 38
    Problemsp. 49
    Frequency Response Analysisp. 51
    Introductionp. 51
    Low and High Frequency Asymptotic Behaviorp. 52
    Hand Sketching Frequency Responsesp. 57
    Interpreting Frequency Response Graphically in Complex Planep. 58
    MATLAB Code tdofxfer.m--Plot Frequency Responsesp. 61
    Code Descriptionp. 61
    Polynomial Form, For-Loop Calculation, Code Listingp. 62
    Polynomial Form, Vector Calculation, Code Listingp. 64
    Transfer Function Form--Bode Calculation, Code Listingp. 65
    Transfer Function Form, Bode Calculation with Frequency, Code Listingp. 67
    Zero/Pole/Gain Function Form, Bode Calculation with Frequency, Code Listingp. 70
    Code Output--Frequency Response Magnitude and Phase Plotsp. 72
    Other Forms of Frequency Response Plotsp. 73
    Log Magnitude versus Log Frequencyp. 74
    db Magnitude versus Log Frequencyp. 75
    db Magnitude versus Linear Frequencyp. 76
    Linear Magnitude versus Linear Frequencyp. 77
    Real and Imaginary Magnitudes versus Log and Linear Frequencyp. 78
    Real versus Imaginary (Nyquist)p. 79
    Solving for Eigenvectors (Mode Shapes) Using the Transfer Function Matrixp. 80
    Problemsp. 85
    Zeros in Siso Mechanical Systemsp. 87
    Introductionp. 87
    "n" dof Examplep. 88
    MATLAB Code ndof_numzeros.m, Usage Instructionsp. 89
    Seven dof Model--z7/F1 Frequency Responsep. 89
    Seven dof Model--z3/F4 Frequency Responsep. 91
    Seven dof Model--z3/F3, Driving Point Frequency Responsep. 92
    Cantilever Model--ANSYSp. 94
    Introductionp. 94
    ANSYS Code cantfem.inp Description and Listingp. 95
    ANSYS Code cantzero.inp Description and Listingp. 99
    ANSYS Results, cantzero.mp. 102
    Problemp. 104
    State Space Analysisp. 105
    Introductionp. 105
    State Space Formulationp. 106
    Definition of State Space Equations of Motionp. 108
    Input Matrix Formsp. 109
    Output Matrix Formsp. 111
    Complex Eigenvalues and Eigenvectors--State Space Formp. 113
    MATLAB Code tdof_non_prop_damped.m: Methodology, Model Setup, Eigenvalue Calculation Listingp. 115
    Eigenvectors--Normalized to Unityp. 119
    Eigenvectors--Magnitude and Phase Angle Representationp. 121
    Complex Eigenvectors Combining to Give Real Motionsp. 122
    Argand Diagram Introductionp. 124
    Calculating [zeta], Plotting Eigenvalues in Complex Plane, Frequency Responsep. 126
    Initial Condition Responses of Individual Modesp. 128
    Plotting Initial Condition Response, Listingp. 130
    Plotted Results: Argand and Initial Condition Responsesp. 132
    Argand Diagram, Mode 2p. 133
    Time Domain Responses, Mode 2p. 134
    Argand Diagram, Mode 3p. 135
    Time Domain Responses, Mode 3p. 136
    Problemsp. 137
    State Space: Frequency Response, Time Domainp. 139
    Introduction--Frequency Responsep. 139
    Solving for Transfer Functions in State Space Form Using Laplace Transformsp. 139
    Transfer Function Matrixp. 142
    MATLAB Code tdofss.m--Frequency Response Using State Spacep. 144
    Code Description, Plotp. 144
    Code Listingp. 144
    Introduction--Time Domainp. 148
    Matrix Laplace Transform--with Initial Conditionsp. 148
    Inverse Matrix Laplace Transform, Matrix Exponentialp. 149
    Back-Transforming to Time Domainp. 149
    Single Degree of Freedom System--Calculating Matrix Exponential in Closed Formp. 150
    Equations of Motion, Laplace Transformp. 150
    Defining the Matrix Exponential--Taking Inverse Laplace Transformp. 151
    Defining the Matrix Exponential--Using Series Expansionp. 152
    Solving for Time Domain Responsep. 152
    MATLAB Code tdof_ss_time_ode45_slnk.m--Time Domain Response of tdof Modelp. 153
    Equation of Motion Reviewp. 153
    Code Descriptionp. 155
    Code Results--Time Domain Responsesp. 156
    Code Listingp. 157
    MATLAB Function tdofssfun.m--Called by tdof_ss_time_ode45_slnk.mp. 159
    Simulink Model tdofss_simulink.mdlp. 160
    Problemsp. 161
    Modal Analysisp. 163
    Introductionp. 163
    Eigenvalue Problemp. 164
    Equations of Motionp. 164
    Principal (Normal) Mode Definitionp. 165
    Eigenvalues / Characteristic Equationp. 165
    Eigenvectorsp. 168
    Interpreting Eigenvectorsp. 172
    Modal Matrixp. 172
    Uncoupling the Equations of Motionp. 173
    Normalizing Eigenvectorsp. 177
    Normalizing with Respect to Unityp. 177
    Normalizing with Respect to Massp. 178
    Reviewing Equations of Motion in Principal Coordinates--Mass Normalizationp. 182
    Equations of Motion in Physical Coordinate Systemp. 182
    Equations of Motion in Principal Coordinate Systemp. 183
    Expanding Matrix Equations of Motion in Both Coordinate Systemsp. 183
    Transforming Initial Conditions and Forcesp. 184
    Summarizing Equations of Motion in Both Coordinate Systemsp. 185
    Back-Transforming from Principal to Physical Coordinatesp. 186
    Reducing the Model Size When Only Selected Degrees of Freedom are Requiredp. 187
    Damping in Systems with Principal Modesp. 189
    Overviewp. 189
    Conditions Necessary for Existence of Principal Modes in Damped Systemp. 190
    Different Types of Dampingp. 192
    Defining Damping Matrix When Proportional Damping is Assumedp. 195
    Problemsp. 200
    Frequency Response: Modal Formp. 201
    Introductionp. 201
    Review from Previous Resultsp. 202
    Transfer Functions--Laplace Transforms in Principal Coordinatesp. 204
    Back-Transforming Mode Contributions to Transfer Functions in Physical Coordinatesp. 206
    Partial Fraction Expansion and the Modal Formp. 209
    Forcing Function Combinations to Excite Single Modep. 211
    How Modes Combine to Create Transfer Functionsp. 213
    Plotting Individual Mode Contributionsp. 217
    MATLAB Code tdof_modal_xfer.m--Plotting Frequency Responses, Modal Contributionsp. 226
    Code Overviewp. 226
    Code Listing, Partialp. 226
    tdof Eigenvalue Problem Using ANSYSp. 230
    ANSYS Code threedof.inp Descriptionp. 230
    ANSYS Code Listingp. 230
    ANSYS Resultsp. 234
    Problemsp. 237
    Transient Response: Modal Formp. 239
    Introductionp. 239
    Review of Previous Resultsp. 239
    Transforming Initial Conditions and Forcesp. 240
    Transforming Initial Conditionsp. 240
    Transforming Forcesp. 241
    Complete Equations of Motion in Principal Coordinatesp. 241
    Solving Equations of Motion Using Laplace Transformp. 243
    MATLAB Code tdof_modal_time.m--Time Domain Displacements in Physical/Principal Coordinatesp. 247
    Code Descriptionp. 247
    Code Resultsp. 248
    Code Listingp. 251
    Problemsp. 254
    Modal Analysis: State Space Formp. 255
    Introductionp. 255
    Eigenvalue Problemp. 256
    Eigenvalue Problem--Laplace Transformp. 257
    Eigenvalue Problem--Eigenvectorsp. 259
    Modal Matrixp. 262
    MATLAB Code tdofss_eig.m: Solving for Eigenvalues and Eigenvectorsp. 262
    Code Descriptionp. 262
    Eigenvalue Calculationp. 262
    Eigenvector Calculationp. 264
    MATLAB Eigenvectors--Real and Imaginary Valuesp. 265
    Sorting Eigenvalues / Eigenvectorsp. 266
    Normalizing Eigenvectorsp. 269
    Writing Homogeneous Equations of Motionp. 273
    Individual Mode Contributions, Modal State Space Formp. 277
    Real Modes--Argand Diagrams, Initial Condition Responses of Individual Modesp. 279
    Undamped Model, Eigenvectors, Real Modesp. 280
    Principal Coordinate Eigenvalue Problemp. 283
    Damping Calculation, Eigenvalue Complex Plane Plotp. 284
    Principal Displacement Calculationsp. 286
    Transformation to Physical Coordinatesp. 287
    Plotting Resultsp. 288
    Undamped/Proportionally Damped Argand Diagram, Mode 2p. 290
    Undamped/Proportionally Damped Argand Diagram, Mode 3p. 292
    Proportionally Damped Initial Condition Response, Mode 2p. 293
    Proportionally Damped Initial Condition Response, Mode 3p. 295
    Problemsp. 298
    Frequency Response: Modal State Space Formp. 301
    Introductionp. 301
    Modal State Space Setup, tdofss_modal_xfer_modes.m Listingp. 301
    Frequency Response Calculationp. 303
    Frequency Response Plottingp. 305
    Code Results--Frequency Response Plots, 2% of Critical Dampingp. 309
    Forms of Frequency Response Plottingp. 311
    Problemp. 315
    Time Domain: Modal State Space Formp. 317
    Introductionp. 317
    Equations of Motion--Modal Formp. 317
    Solving Equations of Motion Using Laplace Transformsp. 319
    MATLAB Code tdofss_modal_time_ode45.m--Time Domain Modal Contributionsp. 322
    Modal State Space Model Setup, Code Listingp. 322
    Problem Setup, Initial Conditions, Code Listingp. 324
    Solving Equations Using ode45, Code Listingp. 325
    Plotting, Code Listingp. 326
    Functions Called: tdofssmodalfun.m, tdofssmodal1fun.m, tdofssmodal2fun.m, tdofssmodal3fun.mp. 327
    Plotted Resultsp. 329
    Problemp. 332
    Finite Elements: Stiffness Matricesp. 333
    Introductionp. 333
    Six dof Model--Element and Global Stiffness Matricesp. 333
    Overviewp. 334
    Element Stiffness Matrixp. 334
    Building Global Stiffness Matrix Using Element Stiffness Matricesp. 335
    Two-Element Cantilever Beamp. 339
    Element Stiffness Matrixp. 340
    Degree of Freedom Definition--Beam Stiffness Matrixp. 341
    Building Global Stiffness Matrix Using Element Stiffness Matricesp. 342
    Eliminating Constraint Degrees of Freedom from Stiffness Matrixp. 344
    Static Solution: Force Applied at Tipp. 345
    Static Condensationp. 346
    Derivationp. 346
    Solving Two-Element Cantilever Beam Static Problemp. 349
    Problemsp. 352
    Finite Elements: Dynamicsp. 353
    Introductionp. 353
    Six dof Global Mass Matrixp. 353
    Cantilever Dynamicsp. 354
    Overview--Mass Matrix Formsp. 354
    Lumped Massp. 354
    Consistent Massp. 355
    Dynamics of Two-Element Cantilever--Consistent Mass Matrixp. 356
    Guyan Reductionp. 358
    Guyan Reduction Derivationp. 358
    Two-Element Cantilever Eigenvalues Closed Form Solution Using Guyan Reductionp. 361
    Eigenvalues of Reduced Equations for Two-Element Cantilever, State Space Formp. 363
    MATLAB Code cant_2el_guyan.m--Two-Element Cantilever Eigenvalues/Eigenvectorsp. 366
    Code Descriptionp. 366
    Code Resultsp. 366
    MATLAB Code cantbeam_guyan.m--User-Defined Cantilever Eigenvalues/Eigenvectorsp. 367
    ANSYS Code cantbeam.inp, Code Descriptionp. 367
    MATLAB cantbeam_guyan.m / ANSYS cantbeam.inp Results Summaryp. 367
    10-Element Beam Frequency Comparisonp. 367
    20-Element Beam Mode Shape Plots, Modes 1 to 5p. 368
    MATLAB Code cantbeam_guyan.m Listingp. 373
    ANSYS Code cantbeam.inp Listingp. 383
    Problemsp. 386
    Siso State Space Matlab Model from ANSYS Modelp. 387
    Introductionp. 387
    ANSYS Eigenvalue Extraction Methodsp. 389
    Cantilever Model, ANSYS Code cantbeam_ss.inp, MATLAB Code cantbeam_ss_freq.mp. 389
    ANSYS 10-Element Model Eigenvalue/Eigenvector Summaryp. 391
    Modal Matrixp. 393
    MATLAB State Space Model from ANSYS Eigenvalue Run--cantbeam_ss_modred.mp. 394
    Inputp. 395
    Defining Degrees of Freedom and Number of Modesp. 396
    Sorting Modes by dc Gain and Peak Gain, Selecting Modes Usedp. 396
    Damping, Defining Reduced Frequencies and Modal Matricesp. 402
    Setting up System Matrix "a"p. 403
    Setting up Input Matrix "b"p. 405
    Setting up Output Matrix "c" and Direct Transmission Matrix "d"p. 407
    Frequency Range, "ss" Setup, Bode Calculationsp. 409
    Full Model--Plotting Frequency Response, Step Responsep. 410
    Reduced Models--Plotting Frequency Response, Step Responsep. 413
    Reduced Models--Plotted Results--Four Modes Usedp. 416
    Modred Descriptionp. 417
    Defining Sorted or Unsorted Modes to be Usedp. 420
    Defining System for Reductionp. 421
    Modred Calculations--"mdc" and "del"p. 423
    Reduced Modred Models--Plotting Commandsp. 424
    Plotting Unsorted Modred Reduced Results--Eliminating High Frequency Modesp. 427
    Plotting Sorted Modred Reduced Results--Eliminating Lower dc Gain Modesp. 429
    Modred Summaryp. 430
    ANSYS Code cantbeam_ss.inp Listingp. 431
    Ground Acceleration MATLAB Model from ANSYS Modelp. 435
    Introductionp. 435
    Model Descriptionp. 435
    Initial ANSYS Model Comparison--Constrained-Tip and Spring-Tip Frequencies/Mode Shapesp. 436
    MATLAB State Space Model from ANSYS Eigenvalue Run--cantbeam_ss_shkr_modred.mp. 440
    Inputp. 440
    Shaker, Spring, Gram Force Definitionsp. 441
    Defining Degrees of Freedom and Number of Modesp. 441
    Frequency Range, Sorting Modes by dc Gain and Plotting, Selecting Modes Usedp. 442
    Damping, Defining Reduced Frequencies and Modal Matricesp. 446
    Setting Up System Matrix "a"p. 446
    Setting Up Matrices "b," "c" and "d"p. 449
    "ss" Setup, Bode Calculationsp. 451
    Full Model--Plotting Frequency Response, Shock Responsep. 452
    Reduced Models--Plotting Frequency Response, Shock Responsep. 456
    Reduced Models--Plotted Results, Four Modes Usedp. 459
    Modred--Setting up, "mdc" and "del" Reduction, Bode Calculationp. 461
    Reduced Modred Models--Plotting Commandsp. 464
    Plotting Unsorted Modred Reduced Results--Eliminating High Frequency Modesp. 466
    Plotting Sorted Modred Reduced Results--Eliminating Lower dc Gain Modesp. 468
    Model Reduction Summaryp. 471
    ANSYS Code cantbeam_ss_spring_shkr.inp Listingp. 472
    SISO Disk Drive Actuator Modelp. 477
    Introductionp. 477
    Actuator Descriptionp. 478
    ANSYS Suspension Model Descriptionp. 479
    ANSYS Suspension Model Resultsp. 481
    Frequency Responsep. 482
    Mode Shape Plotsp. 482
    ANSYS Actuator/Suspension Model Descriptionp. 485
    ANSYS Actuator/Suspension Model Resultsp. 487
    Eigenvalues, Frequency Responsesp. 488
    Mode Shape Plotsp. 489
    Mode Shape Discussionp. 495
    ANSYS Output Example Listingp. 496
    MATLAB Model, MATLAB Code act8.m Listing and Resultsp. 499
    Code Descriptionp. 499
    Input, dof Definitionp. 500
    Forcing Function Definition, dc Gain Calculationp. 501
    Ranking Resultsp. 506
    Building State Space Matricesp. 509
    Define State Space Systems, Original and Reducedp. 513
    Plotting of Resultsp. 515
    Uniform and Non-Uniform Damping Comparisonp. 518
    Sample Rate and Aliasing Effectsp. 521
    Reduced Truncation and Matched dc Gain Resultsp. 522
    Balanced Reductionp. 527
    Introductionp. 527
    Reviewing dc Gain Ranking, MATLAB Code balred.mp. 528
    Controllability, Observabilityp. 530
    Controllability, Observability Gramiansp. 535
    Ranking Using Controllability/Observabilityp. 541
    Balanced Reductionp. 542
    Balanced and dc Gain Ranking Frequency Response Comparisonp. 546
    Balanced and dc Gain Ranking Impulse Response Comparisonp. 552
    MIMO Two-Stage Actuator Modelp. 561
    Introductionp. 561
    Actuator Descriptionp. 562
    ANSYS Model Descriptionp. 563
    ANSYS Piezo Actuator/Suspension Model Resultsp. 565
    Eigenvalues, Frequency Responsep. 565
    Mode Shape Plotsp. 567
    Mode Shape Discussionp. 574
    ANSYS Output Listingp. 575
    MATLAB Model, MATLAB Code act8pz.m Listing and Resultsp. 578
    Input, dof Definitionp. 578
    Forcing Function Definition, dc Gain Calculationsp. 580
    Building State Space Matricesp. 592
    Balancing, Reductionp. 596
    Frequency Responses for Different Numbers of Retained Statesp. 607
    "del" and "mdc" Frequency Response Comparisonp. 614
    Impulse Responsep. 616
    MIMO Summaryp. 623
    Problemsp. 624
    MATLAB and ANSYS Programsp. 625
    Laplace Transformsp. 631
    Definitionsp. 631
    Examples, Laplace Transform Tablep. 633
    Dualityp. 635
    Differentiation and Integrationp. 635
    Applying Laplace Transforms to LODE's with Zero Initial Conditionsp. 636
    Transfer Function Definitionp. 637
    Frequency Response Definitionp. 637
    Applying Laplace Transforms to LODE's with Initial Conditionsp. 637
    Applying Laplace Transform to State Spacep. 638
    Referencesp. 641
    Indexp. 643
    Table of Contents provided by Syndetics. All Rights Reserved.

    ISBN: 9781584882053
    ISBN-10: 1584882050
    Audience: Professional
    Format: Hardcover
    Language: English
    Number Of Pages: 676
    Published: 21st September 2000
    Publisher: CRC PR INC
    Country of Publication: US
    Dimensions (cm): 23.5 x 15.24  x 3.81
    Weight (kg): 1.04
    Edition Number: 1

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