Vibration simulation using MATLAB and ANSYS /

Bibliographic Details
Main Author:	Hatch, Michael R (Author)
Format:	Book
Language:	English
Published:	Boca Raton : Chapman and Hall/CRC, [2001]
Subjects:	MATLAB. Vibration > Computer simulation. ANSYS (Computer system) Vibration > Simulation par ordinateur. ANSYS (Système informatique) ANSYS Computersimulation MATLAB Schwingung Mechanische Schwingung VIBRATION. COMPUTERIZED SIMULATION. COMPUTER SYSTEMS SIMULATION. COMPUTER PROGRAMS. ANALYSIS (MATHEMATICS)
Online Access:	CRCnetBASE Publisher description

Table of Contents:

1.1 Representing Dynamic Mechanical Systems 1
1.2 Modal Analysis 1
1.3 Model Size Reduction 4
Chapter 2 Transfer Function Analysis 7
2.2 Deriving Matrix Equations of Motion 8
2.2.1 Three Degree of Freedom (tdof) System, Identifying Components and Degrees of Freedom 8
2.2.2 Defining the Stiffness, Damping and Mass Matrices 8
2.2.3 Checks on Equations of Motion for Linear Mechanical Systems 11
2.2.4 Six Degree of Freedom (6dof) Model--Stiffness Matrix 11
2.2.5 Rotary Actuator Model--Stiffness and Mass Matrices 12
2.3 Single Degree of Freedom (sdof) System Transfer Function and Frequency Response 15
2.3.1 sdof System Definition, Equations of Motion 15
2.3.2 Transfer Function 15
2.3.3 Frequency Response 17
2.3.4 MATLAB Code sdofxfer.m Description 21
2.3.5 MATLAB Code sdofxfer.m Listing 21
2.4 tdof Laplace Transform, Transfer Functions, Characteristic Equation, Poles, Zeros 23
2.4.1 Laplace Transforms with Zero Initial Conditions 23
2.4.2 Solving for Transfer Functions 24
2.4.3 Transfer Function Matrix for Undamped Model 27
2.4.4 Four Distinct Transfer Functions 28
2.4.5 Poles 29
2.4.6 Zeros 30
2.4.7 Summarizing Poles and Zeros, Matrix Format 32
2.5 Matlab Code tdofpz3x3.m--Plot Poles and Zeros 32
2.5.1 Code Description 32
2.5.2 Code Listing 32
2.5.3 Code Output--Pole/Zero Plots in Complex Plane 38
Chapter 3 Frequency Response Analysis 51
3.2 Low and High Frequency Asymptotic Behavior 52
3.3 Hand Sketching Frequency Responses 57
3.4 Interpreting Frequency Response Graphically in Complex Plane 58
3.5 MATLAB Code tdofxfer.m--Plot Frequency Responses 61
3.5.1 Code Description 61
3.5.2 Polynomial Form, For-Loop Calculation, Code Listing 62
3.5.3 Polynomial Form, Vector Calculation, Code Listing 64
3.5.4 Transfer Function Form--Bode Calculation, Code Listing 65
3.5.5 Transfer Function Form, Bode Calculation with Frequency, Code Listing 67
3.5.6 Zero/Pole/Gain Function Form, Bode Calculation with Frequency, Code Listing 70
3.5.7 Code Output--Frequency Response Magnitude and Phase Plots 72
3.6 Other Forms of Frequency Response Plots 73
3.6.1 Log Magnitude versus Log Frequency 74
3.6.2 db Magnitude versus Log Frequency 75
3.6.3 db Magnitude versus Linear Frequency 76
3.6.4 Linear Magnitude versus Linear Frequency 77
3.6.5 Real and Imaginary Magnitudes versus Log and Linear Frequency 78
3.6.6 Real versus Imaginary (Nyquist) 79
3.7 Solving for Eigenvectors (Mode Shapes) Using the Transfer Function Matrix 80
Chapter 4 Zeros in Siso Mechanical Systems 87
4.2 "n" dof Example 88
4.2.1 MATLAB Code ndof_numzeros.m, Usage Instructions 89
4.2.2 Seven dof Model--z7/F1 Frequency Response 89
4.2.3 Seven dof Model--z3/F4 Frequency Response 91
4.2.4 Seven dof Model--z3/F3, Driving Point Frequency Response 92
4.3 Cantilever Model--ANSYS 94
4.3.2 ANSYS Code cantfem.inp Description and Listing 95
4.3.3 ANSYS Code cantzero.inp Description and Listing 99
4.3.4 ANSYS Results, cantzero.m 102
Chapter 5 State Space Analysis 105
5.2 State Space Formulation 106
5.3 Definition of State Space Equations of Motion 108
5.4 Input Matrix Forms 109
5.5 Output Matrix Forms 111
5.6 Complex Eigenvalues and Eigenvectors--State Space Form 113
5.7 MATLAB Code tdof_non_prop_damped.m: Methodology, Model Setup, Eigenvalue Calculation Listing 115
5.8 Eigenvectors--Normalized to Unity 119
5.9 Eigenvectors--Magnitude and Phase Angle Representation 121
5.10 Complex Eigenvectors Combining to Give Real Motions 122
5.11 Argand Diagram Introduction 124
5.12 Calculating [zeta], Plotting Eigenvalues in Complex Plane, Frequency Response 126
5.13 Initial Condition Responses of Individual Modes 128
5.14 Plotting Initial Condition Response, Listing 130
5.15 Plotted Results: Argand and Initial Condition Responses 132
5.15.1 Argand Diagram, Mode 2 133
5.15.2 Time Domain Responses, Mode 2 134
5.15.3 Argand Diagram, Mode 3 135
5.15.4 Time Domain Responses, Mode 3 136
Chapter 6 State Space: Frequency Response, Time Domain 139
6.1 Introduction--Frequency Response 139
6.2 Solving for Transfer Functions in State Space Form Using Laplace Transforms 139
6.3 Transfer Function Matrix 142
6.4 MATLAB Code tdofss.m--Frequency Response Using State Space 144
6.4.1 Code Description, Plot 144
6.4.2 Code Listing 144
6.5 Introduction--Time Domain 148
6.6 Matrix Laplace Transform--with Initial Conditions 148
6.7 Inverse Matrix Laplace Transform, Matrix Exponential 149
6.8 Back-Transforming to Time Domain 149
6.9 Single Degree of Freedom System--Calculating Matrix Exponential in Closed Form 150
6.9.1 Equations of Motion, Laplace Transform 150
6.9.2 Defining the Matrix Exponential--Taking Inverse Laplace Transform 151
6.9.3 Defining the Matrix Exponential--Using Series Expansion 152
6.9.4 Solving for Time Domain Response 152
6.10 MATLAB Code tdof_ss_time_ode45_slnk.m--Time Domain Response of tdof Model 153
6.10.1 Equation of Motion Review 153
6.10.2 Code Description 155
6.10.3 Code Results--Time Domain Responses 156
6.10.4 Code Listing 157
6.10.5 MATLAB Function tdofssfun.m--Called by tdof_ss_time_ode45_slnk.m 159
6.10.6 Simulink Model tdofss_simulink.mdl 160
Chapter 7 Modal Analysis 163
7.2 Eigenvalue Problem 164
7.2.1 Equations of Motion 164
7.2.2 Principal (Normal) Mode Definition 165
7.2.3 Eigenvalues / Characteristic Equation 165
7.2.4 Eigenvectors 168
7.2.5 Interpreting Eigenvectors 172
7.2.6 Modal Matrix 172
7.3 Uncoupling the Equations of Motion 173
7.4 Normalizing Eigenvectors 177
7.4.1 Normalizing with Respect to Unity 177
7.4.2 Normalizing with Respect to Mass 178
7.5 Reviewing Equations of Motion in Principal Coordinates--Mass Normalization 182
7.5.1 Equations of Motion in Physical Coordinate System 182
7.5.2 Equations of Motion in Principal Coordinate System 183
7.5.3 Expanding Matrix Equations of Motion in Both Coordinate Systems 183
7.6 Transforming Initial Conditions and Forces 184
7.7 Summarizing Equations of Motion in Both Coordinate Systems 185
7.8 Back-Transforming from Principal to Physical Coordinates 186
7.9 Reducing the Model Size When Only Selected Degrees of Freedom are Required 187
7.10 Damping in Systems with Principal Modes 189
7.10.2 Conditions Necessary for Existence of Principal Modes in Damped System 190
7.10.3 Different Types of Damping 192
7.10.4 Defining Damping Matrix When Proportional Damping is Assumed 195
Chapter 8 Frequency Response: Modal Form 201
8.2 Review from Previous Results 202
8.3 Transfer Functions--Laplace Transforms in Principal Coordinates 204
8.4 Back-Transforming Mode Contributions to Transfer Functions in Physical Coordinates 206
8.5 Partial Fraction Expansion and the Modal Form 209
8.6 Forcing Function Combinations to Excite Single Mode 211
8.7 How Modes Combine to Create Transfer Functions 213
8.8 Plotting Individual Mode Contributions 217
8.9 MATLAB Code tdof_modal_xfer.m--Plotting Frequency Responses, Modal Contributions 226
8.9.1 Code Overview 226
8.9.2 Code Listing, Partial 226
8.10 tdof Eigenvalue Problem Using ANSYS 230
8.10.1 ANSYS Code threedof.inp Description 230
8.10.2 ANSYS Code Listing 230
8.10.3 ANSYS Results 234
Chapter 9 Transient Response: Modal Form 239
9.2 Review of Previous Results 239
9.3 Transforming Initial Conditions ands Forces 240
9.3.1 Transforming Initial Conditions 240
9.3.2 Transforming Forces 241
9.4 Complete Equations of Motion in Principal Coordinates 241
9.5 Solving Equations of Motion Using Laplace Transform 243
9.6 MATLAB Code tdof_modal_time.m--Time Domain Displacements in Physical/Principal Coordinates 247
9.6.1 Code Description 247
9.6.2 Code Results 248
9.6.3 Code Listing 251
Chapter 10 Modal Analysis: State Space Form 255
-- 10.2 Eigenvalue Problem 256
10.3 Eigenvalue Problem--Laplace Transform 257
10.4 Eigenvalue Problem--Eigenvectors 259
10.5 Modal Matrix 262
10.6 MATLAB Code tdofss_eig.m: Solving for Eigenvalues and Eigenvectors 262
10.6.1 Code Description 262
10.6.2 Eigenvalue Calculation 262
10.6.3 Eigenvector Calculation 264
10.6.4 MATLAB Eigenvectors--Real and Imaginary Values 265
10.6.5 Sorting Eigenvalues / Eigenvectors 266
10.6.6 Normalizing Eigenvectors 269
10.6.7 Writing Homogeneous Equations of Motion 273
10.6.8 Individual Mode Contributions, Modal State Space Form 277
10.7 Real Modes--Argand Diagrams, Initial Condition Responses of Individual Modes 279
10.7.1 Undamped Model, Eigenvectors, Real Modes 280
10.7.2 Principal Coordinate Eigenvalue Problem 283
10.7.3 Damping Calculation, Eigenvalue Complex Plane Plot 284
10.7.4 Principal Displacement Calculations 286
10.7.5 Transformation to Physical Coordinates 287
10.7.6 Plotting Results 288
10.7.7 Undamped/Proportionally Damped Argand Diagram, Mode 2 290
10.7.8 Undamped/Proportionally Damped Argand Diagram, Mode 3 292
10.7.9 Proportionally Damped Initial Condition Response, Mode 2 293
10.7.10 Proportionally Damped Initial Condition Response, Mode 3 295
Chapter 11 Frequency Response: Modal State Space Form 301
11.2 Modal State Space Setup, tdofss_modal_xfer_modes.m Listing 301.