EBOOK - Random Vibration Mechanical, Structural and Earthquake Engineering Applications (Zach Liang & George C. Lee)


The editor takes pride in presenting another well-developed manuscript in the series. This excellent book is a result of the authors’ more than two decades of extensive research  and  teaching  on  the  subject  of  random  vibrations  related  to  earthquake structural response and multiple hazard mitigations of structural engineering. Since natural hazards rarely occur, the associated solutions must be based on probability criteria in the random process. Current random vibration books, however, are focused on conventional engineering problems, and consequently the authors coupled the traditional solution techniques with their research and teaching experiences to shape up this 11-chapter textbook.
They intend to assist the reader in utilizing the traditional mathematical logic and then comprehensively and effectively solving more complex problems.
In earthquake engineering applications, the authors include some difficult and currently  in  vogue  structural  problems,  such  as  load-and-resistance  factor  design under multiple hazard load effects, nonlinear vibration with random excitation, etc. These  advanced  topics  are  valuable  and  are  believed  to  lead  the  researchers  and practitioners to pursue further in the constructed facility community.
The book is aimed at one semester graduate class; in order for the reader to clearly grasp the concepts and mathematical formulation, the authors developed extensive homework problems for individual chapters accompanied by detailed solutions. The editor  strongly  suggests  that  the  reader  should  patiently  and  gradually  digest  the materials in the book with the assistance of the solution manual. Note that a comprehensive solution manual is seldom available for other books of this nature that reflects the authors’ admirable objective of preparing this manuscript, and the book is believed to be useful for years to come.

Section i  Basic Probability theory
Chapter 1  Introduction ..........................................................................................3
1.1  Background of Random Vibration ............................................3
1.1.1  General Description .....................................................3
1.1.2  General Theory of Vibration ........................................4
1.1.2.1  Concept of Vibration.....................................4
1.1.3  Arrangement of Chapters .............................................9
1.2  Fundamental Concept of Probability Theory .......................... 10
1.2.1  Set Theory .................................................................. 10
1.2.1.1  Basic Relationship (Operation) ................... 11
1.2.2  Axioms of Probability ................................................ 15
1.2.2.1  Random Tests and Classic Probability ....... 15
1.2.2.2  Axiom of Probability .................................. 17
1.2.3  Conditional Probability and Independence ................ 18
1.2.3.1  Conditional Probability............................... 18
1.2.3.2  Multiplicative Rules .................................... 19
1.2.3.3  Independency ..............................................20
1.2.3.4  Total Probability and Bayes’ Formula ........ 21
1.2.3.5  Bayes’ Formula ...........................................23
1.2.4  Engineering Examples ...............................................23
1.2.4.1  Additive Rules ............................................23
1.2.4.2  Multiplication Rules....................................25
1.2.4.3  Independent Series ......................................25
1.2.4.4  Return Period of Extreme Load ..................26
1.3  Random Variables ...................................................................29
1.3.1  Discrete Random Variables and PMF ........................29
1.3.1.1  Single Random Variables ...........................29
1.3.1.2  “Two-Dimensional” Approach ...................30
1.3.1.3  Probability Mass Function ..........................30
1.3.1.4  Bernoulli Distribution (0–1 Distribution) ...30
1.3.1.5  Binomial Distribution ................................. 31
1.3.1.6  Poisson Distribution .................................... 32
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1.3.1.7  Poisson Approximation ............................... 33
1.3.1.8  Summary of PMF PN
(n) .............................. 35
1.3.2  Continuous Random Variables and PDF .................... 35
1.3.2.1  Continuous Random Variables ................... 35
1.3.2.2  Probability Density Function ......................36
1.3.2.3  Uniform Distribution ..................................38
1.3.2.4  Exponential Distribution .............................39
1.3.2.5  Rayleigh Distribution .................................. 42
1.3.3  Cumulative Distribution Functions ............................ 42
1.3.3.1  Probability of Cumulative Event ................ 42
1.3.3.2  Cumulative Distribution Function (CDF) ... 43
1.3.3.3  Certain Applications of PDF and CDF .......44
1.3.4  Central Tendency and Dispersion .............................. 45
1.3.4.1  Statistical Expectations and Moments ........ 45
1.3.4.2  Central Tendency, Mean Value ................... 45
1.3.4.3  Variation, Variance, Standard
Deviation, and Coefficient of Variation ......46
1.3.4.4  Expected Values ......................................... 47
1.3.4.5  Linearity of Expected Values ..................... 47
1.3.5  Normal Random Distributions ...................................48
1.3.5.1  Standardized Variables Z ............................48
1.3.5.2  Gaussian (Normal) Random Variables .......48
1.3.5.3  PDF of Normal Distribution .......................48
1.3.5.4  Cumulative Distribution Function of
Normal Distribution ....................................49
1.3.6  Engineering Applications ........................................... 51
1.3.6.1  Probability-Based Design ........................... 51
1.3.6.2  Lognormal Distributions ............................ 53
1.3.6.3  Further Discussion of ProbabilityBased Design .............................................. 55
Chapter 2  Functions of Random Variables ......................................................... 59
2.1  Systems and Functions ............................................................ 59
2.1.1  Dynamic Systems ....................................................... 59
2.1.1.1  Order of Systems ........................................ 59
2.1.1.2  Simple Systems ...........................................60
2.1.2  Jointly Distributed Variables ...................................... 61
2.1.2.1  Joint and Marginal Distributions of
Discrete Variables ....................................... 61
2.1.2.2  Joint and Marginal Distributions of
Continuous Variables ..................................63
2.1.3  Conditional Distribution and Independence...............66
2.1.3.1  Discrete Variables .......................................66
2.1.3.2  Continuous Variables ..................................68
2.1.3.3  Variable Independence ...............................68
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2.1.4  Expected Value, Variance, Covariance, and
Correlation ..................................................................70
2.1.4.1  Expected Value of g(X,Y) ...........................70
2.1.4.2  Conditional Expected Value ....................... 71
2.1.4.3  Variance ...................................................... 71
2.1.4.4  Covariance of X,Y ....................................... 71
2.1.4.5  Correlation Coefficient ............................... 71
2.1.5  Linear Independence ..................................................72
2.1.5.1  Relationship between Random
Variables Xand Y ........................................72
2.1.5.2  Expected Value of Sum of Random
Variables Xand Y ........................................73
2.1.6  CDF and PDFs of Random Variables .........................73
2.1.6.1  Discrete Variables ....................................... 74
2.1.6.2  Continuous Variables .................................. 76
2.2  Sums of Random Variables .....................................................80
2.2.1  Discrete Variables ......................................................80
2.2.2  Continuous Variables ................................................. 81
2.2.2.1  Sums of Normally Distributed PDF ...........82
2.2.2.2  Sums of nNormally Distributed Variable ... 83
2.3  Other Functions of Random Variables ....................................84
2.3.1  Distributions of Multiplication of Xand Y .................84
2.3.2  Distributions of Sample Variance, Chi-Square (χ
2
) ...85
2.3.2.1  Sample Variance .........................................85
2.3.2.2  Chi-Square Distribution..............................86
2.3.2.3  CDF of Chi-Square, n= 1 ...........................86
2.3.2.4  PDF of Chi-Square, n = 1 ...........................86
2.3.2.5  Mean ...........................................................86
2.3.2.6  Variance ......................................................86
2.3.2.7  PDF of Chi-Square, n> 1 ...........................86
2.3.2.8  Reproductive ...............................................87
2.3.2.9  Approximation ............................................87
2.3.2.10  Mean of Y ...................................................87
2.3.2.11  Variance of Y ..............................................87
2.3.2.12  Square Root of Chi-Square (χ
2
) ..................88
2.3.2.13  Gamma Distribution and Chi-Square
Distribution .................................................88
2.3.2.14  Relation between Chi-Square χn
2
and
Sample Variance SX
2
....................................89
2.3.3  Distributions of Ratios of Random Variables ............90
2.3.3.1  Distribution of Variable Ratios ...................90
2.3.3.2  Student’s Distribution .................................90
2.3.3.3  FDistribution.............................................. 91
2.4  Design Considerations .............................................................92
2.4.1  Further Discussion of Probability-Based Design .......92
2.4.2  Combination of Loads ................................................95
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2.5  Central Limit Theorems and Applications ..............................97
2.5.1  Central Limit Theorems .............................................98
2.5.1.1  Lyapunov Central Limit Theorem  .............98
2.5.1.2  Lindeberg–Levy Central Limit Theorem ...99
2.5.1.3  De Moirve–Laplace Central Limit
Theorem ....................................................100
2.5.2  Distribution of Product of Positive Random
Variables ................................................................... 102
2.5.3  Distribution of Extreme Values ................................ 103
2.5.3.1  CDF and PDF of Distribution of
Extreme Values ......................................... 103
2.5.4  Special Distributions ................................................ 104
2.5.4.1  CDF and PDF of Extreme Value of
Rayleigh Distributions .............................. 104
2.5.4.2  Extreme Value Type I Distribution ........... 104
2.5.4.3  Distribution of Minimum Values .............. 107
2.5.4.4  Extreme Value Type II Distribution ......... 108
2.5.4.5  Extreme Value Type III Distribution ........ 109
Section ii  Random Process
Chapter 3  Random Processes in the Time Domain .......................................... 115
3.1  Definitions and Basic Concepts ............................................. 115
3.1.1  State Spaces and Index Sets ..................................... 115
3.1.1.1  Definition of Random Process .................. 115
3.1.1.2  Classification of Random Process ............ 116
3.1.1.3  Distribution Function of Random Process ... 117
3.1.1.4  Independent Random Process................... 121
3.1.2  Ensembles and Ensemble Averages.......................... 123
3.1.2.1  Concept of Ensembles............................... 123
3.1.2.2  Statistical Expectations and Moments ......124
3.1.3  Stationary Process and Ergodic Process .................. 129
3.1.3.1  Stationary Process .................................... 129
3.1.3.2  Ergodic Process ........................................ 133
3.1.4  Examples of Random Process .................................. 134
3.1.4.1  Gaussian Process ...................................... 135
3.1.4.2  Poisson Process ......................................... 136
3.1.4.3  Harmonic Process ..................................... 142
3.2  Correlation Analysis .............................................................. 144
3.2.1  Cross-Correlation ..................................................... 144
3.2.1.1  Cross-Correlation Function ...................... 145
3.2.1.2  Cross-Covariance Function ...................... 146
3.2.2  Autocorrelation ......................................................... 147
3.2.2.1  Physical Meaning of Correlation .............. 147
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3.2.2.2  Characteristics of Autocorrelation
Function .................................................... 148
3.2.2.3  Examples of Autocorrelation Function ..... 152
3.2.3  Derivatives of Stationary Process ............................ 154
3.2.3.1  Stochastic Convergence ............................ 154
3.2.3.2  Mean-Square Limit ................................... 155
3.2.3.3  Mean-Square Continuity .......................... 157
3.2.3.4  Mean-Square Derivatives of Random
Process ...................................................... 159
3.2.3.5  Derivatives of Autocorrelation Functions .... 159
3.2.3.6  Derivatives of Stationary Process ............. 161
3.2.3.7  Derivatives of Gaussian Process ............... 162
Chapter 4  Random Processes in the Frequency Domain .................................. 165
4.1  Spectral Density Function ..................................................... 165
4.1.1  Definitions of Spectral Density Functions ............... 165
4.1.1.1  Mean-Square Integrable of Random
Process ...................................................... 165
4.1.1.2  Stationary Process: A Review .................. 169
4.1.1.3  Autospectral Density Functions ................ 170
4.1.1.4  Spectral Distribution Function Ψ(ω) ......... 175
4.1.1.5  Properties of Auto-PSD Functions ........... 176
4.1.2  Relationship with Fourier Transform ....................... 179
4.1.2.1  Fourier Transformation Random Process .... 179
4.1.2.2  Energy Equation ....................................... 179
4.1.2.3  Power Density Functions .......................... 182
4.1.3  White Noise and Band-Pass Filtered Spectra .......... 184
4.1.3.1  White Noise .............................................. 184
4.1.3.2  Low-Pass Noise......................................... 186
4.1.3.3  Band-Pass Noise ....................................... 187
4.1.3.4  Narrow-Band Noise .................................. 188
4.2  Spectral Analysis ................................................................... 188
4.2.1  Definition .................................................................. 188
4.2.1.1  Cross-Power Spectral Density Function ... 188
4.2.1.2  Estimation of Cross-PSD Function ........... 190
4.2.2  Transfer Function ..................................................... 191
4.2.2.1  Random Process through Linear
Systems ................................................... 191
4.2.2.2  Estimation of Transfer Functions ............. 197
4.2.2.3  Stationary Input ........................................ 199
4.2.3  Coherence Analysis .................................................. 199
4.2.3.1  Coherence Function ..................................200
4.2.3.2  Attenuation and Delay .............................. 201
4.2.3.3  Sum of Two Random Processes ................ 201
4.2.4  Derivatives of Stationary Process ............................202
x Contents
4.3  Practical Issues of PSD Functions .........................................203
4.3.1  One-Sided PSD.........................................................203
4.3.1.1  Angular Frequency versus Frequency ......203
4.3.1.2  Two-Sided Spectrum versus SingleSided Spectrum .........................................204
4.3.1.3  Discrete Fourier Transform ......................204
4.3.2  Signal-to-Noise Ratios .............................................205
4.3.2.1  Definition ..................................................205
4.3.2.2  Engineering Significances ........................207
4.4  Spectral Presentation of Random Process .............................208
4.4.1  General Random Process .........................................208
4.4.2  Stationary Process .................................................... 210
4.4.2.1  Dynamic Process in the Frequency
Domain ..................................................... 210
4.4.2.2  Relationship between the Time and the
Frequency Domains .................................. 210
4.4.2.3  Spectral Distribution and Representation .... 213
4.4.2.4  Analogy of Spectral Distribution
Function to CDF ....................................... 216
4.4.2.5  Finite Temporal and Spectral Domains .... 217
Chapter 5  Statistical Properties of Random Process ........................................ 221
5.1  Level Crossings .....................................................................222
5.1.1  Background ..............................................................222
5.1.1.1  Number of Level Crossings ......................222
5.1.1.2  Correlations between Level Crossings......224
5.1.2  Derivation of Expected Rate ....................................224
5.1.2.1  Stationary Crossing...................................224
5.1.2.2  Up-Crossing ..............................................225
5.1.2.3  Limiting Behavior .....................................225
5.1.3  Specializations .........................................................227
5.1.3.1  Level Up-Crossing, Gaussian Process ......227
5.1.3.2  Zero Up-Crossing .....................................228
5.1.3.3  Peak Frequency .........................................229
5.1.3.4  Bandwidth and Irregularity ......................230
5.1.4  Random Decrement Methods ................................... 232
5.1.4.1  Random Decrement (Level Up-Crossing) ... 232
5.1.4.2  Lag Superposition (Zero Up-Crossing) .... 235
5.1.4.3  Lag Superposition (Peak Reaching) ......... 237
5.1.5  Level Crossing in Clusters ........................................238
5.1.5.1  Rice’s Narrow-Band Envelopes ................238
5.2  Extrema .................................................................................243
5.2.1  Distribution of Peak Values ......................................243
5.2.1.1  Simplified Approach .................................243
5.2.1.2  General Approach .....................................245
xi Contents
5.2.2  Engineering Approximations ...................................247
5.2.2.1  Background ...............................................247
5.2.2.2  Probability Distributions of Height,
Peak, and Valley .......................................249
5.3  Accumulative Damages ......................................................... 252
5.3.1  Linear Damage Rule: The Deterministic
Approach ............................................................ 253
5.3.1.1  S–N Curves ............................................... 253
5.3.1.2  Miner’s Rule ............................................. 253
5.3.2  Markov Process ........................................................254
5.3.2.1  General Concept ....................................... 255
5.3.2.2  Discrete Markov Chain ............................. 255
5.3.3  Fatigue ......................................................................263
5.3.3.1  High-Cycle Fatigue ...................................263
5.3.3.2  Low-Cycle Fatigue ....................................266
5.3.4  Cascading Effect ...................................................... 270
5.3.4.1  General Background ................................. 270
5.3.4.2  Representation of Random Process .......... 271
5.3.4.3  Occurrence Instance of Maximum
Load ..................................................... 273
Section iii  Vibrations
Chapter 6  Single-Degree-of-Freedom Vibration Systems ................................ 279
6.1  Concept of Vibration .............................................................280
6.1.1  Basic Parameters ......................................................280
6.1.1.1  Undamped Vibration Systems .................. 281
6.1.1.2  Damped SDOF System ............................. 291
6.1.2  Free Decay Response ...............................................297
6.1.2.1  Amplitude dand Phase ϕ .........................297
6.2  Periodically Forced Vibration ............................................... 301
6.2.1  Harmonic Excitation ................................................ 301
6.2.1.1  Equation of Motion ................................... 301
6.2.1.2  Harmonically Forced Response ................302
6.2.1.3  Dynamic Magnification ............................307
6.2.1.4  Transient Response under Zero Initial
Conditions .................................................308
6.2.2  Base–Excitation and Force Transmissibility ............ 313
6.2.2.1  Model of Base Excitation .......................... 313
6.2.2.2  Force Transmissibility .............................. 317
6.2.3  Periodic Excitations .................................................. 319
6.2.3.1  General Response ..................................... 319
6.2.3.2  The nth Steady-State Response ................320
6.2.3.3  Transient Response ................................... 321
xii Contents
6.3  Response of SDOF System to Arbitrary Forces .................... 321
6.3.1  Impulse Responses ................................................... 321
6.3.1.1  Unit Impulse Response Function .............. 322
6.3.2  Arbitrary Loading and Convolution ......................... 323
6.3.2.1  Convolution ............................................... 323
6.3.2.2  Transient Response under Harmonic
Excitation f
0
sin(ωt) ................................... 325
6.3.3  Impulse Response Function and Transfer Function ... 327
6.3.4  Frequency Response and Transfer Functions ........... 329
6.3.5  Borel’s Theorem and Its Applications ...................... 330
6.3.5.1  Borel’s Theorem ........................................ 330
Chapter 7  Response of SDOF Linear Systems to Random Excitations ............ 335
7.1  Stationary Excitations ............................................................ 335
7.1.1  Model of SDOF System ........................................... 335
7.1.1.1  Equation of Motion ................................... 335
7.1.1.2  Zero Initial Conditions ............................. 335
7.1.1.3  Solution in Terms of Convolution ............. 336
7.1.1.4  Nature of Forcing Function ...................... 336
7.1.1.5  Response ................................................... 336
7.1.2  Mean of Response Process ....................................... 336
7.1.3  Autocorrelation of Response Process ....................... 337
7.1.3.1  Autocorrelation ......................................... 337
7.1.3.2  Mean Square ............................................. 338
7.1.4  Spectral Density of Response Process ..................... 341
7.1.4.1  Auto-Power Spectral Density Function .... 341
7.1.4.2  Variance .................................................... 342
7.1.5  Distributions of Response Process ........................... 342
7.2  White Noise Process .............................................................. 342
7.2.1  Definition .................................................................. 342
7.2.2  Response to White Noise ......................................... 343
7.2.2.1  Auto-PSD Function ................................... 343
7.2.2.2  Variance .................................................... 343
7.2.3  White Noise Approximation .................................... 345
7.3  Engineering Examples ...........................................................346
7.3.1  Comparison of Excitations .......................................346
7.3.1.1  Harmonic Excitation .................................346
7.3.1.2  Impulse Excitation ....................................348
7.3.1.3  Random Excitation ................................... 351
7.3.1.4  Other Excitations ...................................... 353
7.3.2  Response Spectra...................................................... 355
7.3.2.1  Response Spectrum .................................. 355
7.3.2.2  Design Spectra .......................................... 356
7.3.3  Criteria of Design Values ......................................... 357
7.3.3.1  Pseudo Spectrum ...................................... 357
xiii Contents
7.3.3.2  Correlation of Acceleration and
Displacement ............................................ 358
7.4  Coherence Analyses .............................................................. 359
7.4.1  Estimation of Transfer Function ............................... 359
7.4.2  Coherence Function .................................................. 363
7.4.3  Improvement of Coherence Functions .....................365
7.5  Time Series Analysis .............................................................365
7.5.1  Time Series ...............................................................366
7.5.1.1  General Description ..................................366
7.5.1.2  Useful Models of Time Series ..................366
7.5.2  Characters of ARMA Models .................................. 367
7.5.2.1  Moving-Average Process MA(q) .............. 367
7.5.2.2  Autoregressive Process AR(p) .................. 370
7.5.2.3  ARMA(p, q) .............................................. 372
7.5.3   Analyses of Time Series in the Frequency Domain .... 376
7.5.3.1  Z-Transform .............................................. 376
7.5.3.2  Sampling of Signals .................................. 377
7.5.3.3  Transfer Function of Discrete Time
System ....................................................... 378
7.5.3.4  PSD Functions .......................................... 379
7.5.4  Time Series of SDOF Systems .................................380
7.5.4.1  Difference Equations ................................380
7.5.4.2  ARMA Models ......................................... 382
7.5.4.3  Transfer Functions .................................... 383
7.5.4.4  Stability of Systems .................................. 385
Chapter 8  Random Vibration of MDOF Linear Systems ................................. 391
8.1  Modeling ................................................................................ 391
8.1.1  Background .............................................................. 391
8.1.1.1  Basic Assumptions .................................... 391
8.1.1.2  Fundamental Approaches .........................392
8.1.2  Equation of Motion ..................................................392
8.1.2.1  Physical Model ..........................................392
8.1.2.2  Stiffness Matrix ........................................ 395
8.1.2.3  Mass and Damping Matrices ....................396
8.1.3  Impulse Response and Transfer Functions ............... 398
8.1.3.1  Scalar Impulse Response Function and
Transfer Function ...................................... 398
8.1.3.2  Impulse Response Matrix and Transfer
Function Matrix ........................................399
8.1.3.3  Construction of Transfer Functions ..........399
8.1.3.4  Principal Axes of Structures .....................400
8.2  Direct Model for Determining Responses.............................400
8.2.1  Expression of Response ............................................400
8.2.2  Mean Values ............................................................. 401
xiv Contents
8.2.2.1  Single Coordinate ..................................... 401
8.2.2.2  Multiple Coordinates ................................402
8.2.3  Correlation Functions ...............................................404
8.2.4  Spectral Density Function of Response ...................405
8.2.4.1  Fourier Transforms of f(t) and x(t) ............405
8.2.4.2  Power Spectral Density Function .............405
8.2.4.3  Mean Square Response .............................407
8.2.4.4  Variance ....................................................407
8.2.4.5  Covariance ................................................407
8.2.5  Single Response Variable: Spectral Cases ...............407
8.2.5.1  Single Input ...............................................407
8.2.5.2  Uncorrelated Input ....................................408
8.3  Normal Mode Method ...........................................................408
8.3.1  Proportional Damping ..............................................408
8.3.1.1  Essence of Caughey Criterion ..................408
8.3.1.2  Monic System ...........................................409
8.3.2  Eigen-Problems ........................................................ 410
8.3.2.1  Undamped System .................................... 410
8.3.2.2  Underdamped Systems ............................. 411
8.3.3  Orthogonal Conditions ............................................. 411
8.3.3.1  Weighted Orthogonality ........................... 412
8.3.3.2  Modal Analysis ......................................... 413
8.3.4  Modal Superposition ................................................ 416
8.3.5  Forced Response and Modal Truncation .................. 418
8.3.5.1  Forced Response ....................................... 418
8.3.5.2  Rayleigh Quotient ..................................... 419
8.3.5.3  Ground Excitation and Modal
Participation Factor ...................................420
8.3.5.4  Modal Superposition, Forced Vibration ...420
8.3.5.5  Modal Truncation ..................................... 422
8.3.6  Response to Random Excitations ............................. 423
8.3.6.1  Modal and Physical Response .................. 423
8.3.6.2  Mean .........................................................424
8.3.6.3  Covariance ................................................424
8.3.6.4  Probability Density Function for xi
(t) .......426
8.4  Nonproportionally Damped Systems, Complex Modes ........ 428
8.4.1  Nonproportional Damping ....................................... 428
8.4.1.1  Mathematical Background ........................ 428
8.4.1.2  The Reality of Engineering ...................... 429
8.4.2  State Variable and State Equation ............................ 429
8.4.3  Eigen-Problem of Nonproportionally Damped
System ...................................................................... 430
8.4.3.1  State Matrix and Eigen-Decomposition ... 430
8.4.3.2  Eigenvectors and Mode Shapes ................ 432
8.4.3.3  Modal Energy Transfer Ratio ................... 435
xv Contents
8.4.4  Response to Random Excitations ............................. 437
8.4.4.1  Modal and Physical Response .................. 438
8.4.4.2  Mean ......................................................... 439
8.4.4.3  Covariance ................................................440
8.4.4.4  Brief Summary .........................................446
8.5  Modal Combination ...............................................................446
8.5.1  Real Valued Mode Shape .........................................446
8.5.1.1  Approximation of Real Valued Mode
Shape .........................................................446
8.5.1.2  Linear Dependency and Representation ...447
8.5.2  Numerical Characteristics ........................................449
8.5.2.1  Variance ....................................................449
8.5.2.2  Root Mean Square ....................................449
8.5.3  Combined Quadratic Combination .......................... 451
Section iV  Applications and Further Discussions
Chapter 9  Inverse Problems .............................................................................. 459
9.1  Introduction to Inverse Problems .......................................... 459
9.1.1  Concept of Inverse Engineering ............................... 459
9.1.1.1  Key Issues ................................................. 459
9.1.1.2  Error ..........................................................460
9.1.1.3  Applications .............................................. 461
9.1.2  Issues of Inverse Problems ....................................... 461
9.1.2.1  Modeling ................................................... 461
9.1.2.2  Identification, Linear System ....................463
9.1.2.3  Identification, General System ..................463
9.1.2.4  Simulations ...............................................464
9.1.2.5  Practical Considerations ...........................464
9.1.3  The First Inverse Problem of Dynamic Systems ......465
9.1.3.1  General Description ..................................465
9.1.3.2  Impulse Response .....................................466
9.1.3.3  Sinusoidal Response .................................466
9.1.3.4  Random Response ....................................466
9.1.3.5  Modal Model ............................................468
9.1.4  The Second Inverse Problem of Dynamic Systems ... 471
9.1.4.1  General Background ................................. 471
9.1.4.2  White Noise .............................................. 472
9.1.4.3  Practical Issues ......................................... 472
9.2  System Parameter Identification ............................................ 472
9.2.1  Parameter Estimation, Random Set ......................... 473
9.2.1.1  Maximum Likelihood ............................... 473
9.2.1.2  Bias and Consistency ................................ 479
xvi Contents
9.2.2  Confidence Intervals................................................. 481
9.2.2.1  Estimation and Sampling Distributions .... 481
9.2.3  Parameter Estimation, Random Process ..................482
9.2.3.1  General Estimation ...................................482
9.2.3.2  Stationary and Ergodic Process ................487
9.2.3.3  Nonstationary Process ..............................489
9.2.4  Least Squares Approximation and Curve Fitting ....492
9.2.4.1  Concept of Least Squares .........................492
9.2.4.2  Curve Fitting ............................................. 493
9.2.4.3  Realization of Least Squares Method .......494
9.3  Vibration Testing ................................................................... 495
9.3.1  Test Setup ................................................................. 495
9.3.1.1  Mathematical Model ................................. 495
9.3.1.2  Numerical Model ......................................496
9.3.1.3  Experimental Model .................................496
9.3.2  Equipment of Actuation and Measurement ..............497
9.3.2.1  Actuation ...................................................497
9.3.2.2  Measurement ............................................. 501
9.3.3  Signal and Signal Processing ...................................505
9.3.3.1  Data-Acquisition System ..........................505
9.3.3.2  Single Processing and Window Functions...507
9.3.4  Nyquist Circle ........................................................... 511
9.3.4.1  Circle and Nyquist Plot ............................. 511
9.3.4.2  Circle Fit ................................................... 513
9.3.4.3  Natural Frequency and Damping Ratio .... 515
Chapter 10  Failures of Systems .......................................................................... 519
10.1  3σCriterion ........................................................................... 519
10.1.1  Basic Design Criteria ............................................... 519
10.1.2  3σCriterion .............................................................. 520
10.1.3  General Statement .................................................... 520
10.1.3.1  General Relationship between Sand R ..... 520
10.1.3.2  System Failure, Further Discussion .......... 522
10.2  First Passage Failure .............................................................. 522
10.2.1  Introduction .............................................................. 523
10.2.2  Basic Formulation .................................................... 523
10.2.2.1  General Formulation ................................. 523
10.2.2.2  Special Cases ............................................524
10.2.3  Largest among Independent Peaks ........................... 525
10.2.3.1  Exact Distribution ..................................... 525
10.2.3.2  Extreme Value Distribution ...................... 527
10.2.3.3  Design Value Based on Return Period ..... 529
10.3  Fatigue ................................................................................... 529
10.3.1  Physical Process of Fatigue ...................................... 529
xvii Contents
10.3.2  Strength Models ....................................................... 530
10.3.2.1  High-Cycle Fatigue ................................... 530
10.3.2.2  Miner’s Rule, More Detailed Discussion ... 531
10.3.3  Fatigue Damages ...................................................... 532
10.3.3.1  Narrowband Random Stress ..................... 532
10.3.3.2  Wideband Random Stress ......................... 538
10.3.4  Damages due to Type D Low Cycle ......................... 542
10.3.4.1  Fatigue Ductility Coefficient .................... 543
10.3.4.2  Variation of Stiffness ................................ 543
10.4  Considerations on Reliability Design .................................... 545
10.4.1  Further Discussion of Probability-Based Design ..... 545
10.4.1.1  Random Variable vs. Random Process ..... 545
10.4.1.2  Necessity of Distinguishing TimeInvariant and Time-Variable Loads ..........546
10.4.1.3  Time-Variable Load at a Given Time
Spot ........................................................... 547
10.4.1.4  Combination of Time-Variable Loads
in a Given Time Period ............................. 547
10.4.1.5  Additivity of Distribution Functions.........548
10.4.2  Failure Probability under MH Load .........................549
10.4.2.1  Failure Probability Computation ..............549
10.4.2.2  Time-Invariant and -Variable Loads .........549
10.4.2.3  Principles of Determining Load and
Load Combination ....................................549
10.4.2.4  Total and Partial Failure Probabilities ...... 550
10.4.2.5  Independent Events ................................... 550
10.4.2.6  Mutually Exclusive Failures,
the Uniqueness Probabilities..................... 552
10.4.3  General Formulations ............................................... 556
10.4.3.1  Total Failure Probability ........................... 556
10.4.3.2  Occurrence of Loads in a Given Time
Duration .................................................... 556
10.4.3.3  Brief Summary ......................................... 556
10.4.4  Probability of Conditions ......................................... 557
10.4.4.1  Condition for Occurrence of Partial
Failure Probabilities .................................. 558
10.4.4.2  Event of Single Type of Loads .................. 558
10.4.5  Brief Summary ......................................................... 571
Chapter 11  Nonlinear Vibrations and Statistical Linearization ......................... 575
11.1  Nonlinear Systems ................................................................. 575
11.1.1  Examples of Nonlinear Systems .............................. 575
11.1.1.1  Nonlinear System ..................................... 576
11.1.1.2  Memoryless Nonlinear System ................. 578
xviii Contents
11.1.2  General Nonlinear System, Volterra Model ............. 579
11.1.3  Structure Nonlinearity ............................................. 579
11.1.3.1  Deterministic Nonlinearity ....................... 579
11.1.3.2  Random Nonlinearity ............................... 585
11.2  Nonlinear Random Vibrations ..............................................594
11.2.1  General Concept of Nonlinear Vibration .................594
11.2.1.1  The Phase Plane ........................................ 595
11.2.1.2  Example of Nonlinear Vibration with
Closed-Form Solution ...............................596
11.2.1.3  System with Nonlinear Damping Only .... 601
11.2.1.4  System with Nonlinear Spring .................. 601
11.2.2  Markov Vector ..........................................................602
11.2.2.1  ItōDiffusion and Kolmogorov Equations ...602
11.2.2.2  Solutions of FPK Equation .......................603
11.2.3  Alternative Approaches ............................................605
11.2.3.1  Linearization .............................................605
11.2.3.2  Perturbation ..............................................606
11.2.3.3  Special Nonlinearization ..........................606
11.2.3.4  Statistical Averaging .................................606
11.2.3.5  Numerical Simulation ...............................606
11.3  Monte Carlo Simulations .......................................................607
11.3.1  Basics of Monte Carlo Method .................................607
11.3.1.1  Applications ..............................................609
11.3.2  Monte Carlo and Random Numbers ........................ 613
11.3.2.1  Generation of Random Numbers .............. 613
11.3.2.2  Transformation of Random Numbers ....... 614
11.3.2.3  Random Process ....................................... 617
11.3.3  Numerical Simulations ............................................. 617
11.3.3.1  Basic Issues ............................................... 617
11.3.3.2  Deterministic Systems with Random
Inputs ........................................................ 619
11.3.3.3  Random Systems ...................................... 619

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