Mechanical Wave Vibrations An elegant and accessible exploration of the fundamentals of the analysis and control of vibration in structures from a wave standpoint
In Mechanical Wave Vibrations: Analysis and Control, Professor Chunhui Mei delivers an expert discussion of the wave analysis approach (as opposed to the modal-based approach) to mechanical vibrations in structures. The book begins with deriving the equations of motion using the Newtonian approach based on various sign conventions before comprehensively covering the wave vibration analysis approach. It concludes by exploring passive and active feedback control of mechanical vibration waves in structures.
The author discusses vibration analysis and control strategies from a wave standpoint and examines the applications of the presented wave vibration techniques to structures of various complexity. Readers will find in the book:
- A thorough introduction to mechanical wave vibration analysis, including the governing equations of various types of vibrations
- Comprehensive explorations of waves in simple rods and beams, including advanced vibration theories
- Practical discussions of coupled waves in composite and curved beams
- Extensive coverage of wave mode conversions in built-up planar and spatial frames and networks
- Complete treatments of passive and active feedback wave vibration control
- MATLAB® scripts both in the book and in a companion solutions manual for instructors
Mechanical Wave Vibrations: Analysis and Control is written as a textbook for both under-graduate and graduate students studying mechanical, aerospace, automotive, and civil engineering. It will also benefit researchers and educators working in the areas of vibrations and waves.
Table of Contents:
Preface xi
Acknowledgement xiii
About the Companion Website xv
1 Sign Conventions and Equations of Motion Derivations 1
1.1 Derivation of the Bending Equations of Motion by Various Sign Conventions 1
1.1.1 According to Euler–Bernoulli Bending Vibration Theory 2
1.1.2 According to Timoshenko Bending Vibration Theory 7
1.2 Derivation of the Elementary Longitudinal Equation of Motion by Various Sign Conventions 10
1.3 Derivation of the Elementary Torsional Equation of Motion by Various Sign Conventions 12
2 Longitudinal Waves in Beams 15
2.1 The Governing Equation and the Propagation Relationships 15
2.2 Wave Reflection at Classical and Non-Classical Boundaries 16
2.3 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 20
2.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 24
2.5 Numerical Examples and Experimental Studies 27
2.6 MATLAB Scripts 32
3 Bending Waves in Beams 39
3.1 The Governing Equation and the Propagation Relationships 39
3.2 Wave Reflection at Classical and Non-Classical Boundaries 40
3.3 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 46
3.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 50
3.5 Numerical Examples and Experimental Studies 55
3.6 MATLAB Scripts 59
4 Waves in Beams on a Winkler Elastic Foundation 69
4.1 Longitudinal Waves in Beams 69
4.1.1 The Governing Equation and the Propagation Relationships 69
4.1.2 Wave Reflection at Boundaries 70
4.1.3 Free Wave Vibration Analysis 71
4.1.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 72
4.1.5 Numerical Examples 76
4.2 Bending Waves in Beams 79
4.2.1 The Governing Equation and the Propagation Relationships 79
4.2.2 Wave Reflection at Classical Boundaries 82
4.2.3 Free Wave Vibration Analysis 84
4.2.4 Force Generated Waves and Forced Wave Vibration Analysis 84
4.2.5 Numerical Examples 89
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5 Coupled Waves in Composite Beams 97
5.1 The Governing Equations and the Propagation Relationships 97
5.2 Wave Reflection at Classical and Non-Classical Boundaries 100
5.3 Wave Reflection and Transmission at a Point Attachment 102
5.4 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 104
5.5 Force Generated Waves and Forced Vibration Analysis of Finite Beams 105
5.6 Numerical Examples 108
5.7 MATLAB Script 114
6 Coupled Waves in Curved Beams 119
6.1 The Governing Equations and the Propagation Relationships 119
6.2 Wave Reflection at Classical and Non-Classical Boundaries 121
6.3 Free Vibration Analysis in a Finite Curved Beam – Natural Frequencies and Modeshapes 127
6.4 Force Generated Waves and Forced Vibration Analysis of Finite Curved Beams 128
6.5 Numerical Examples 134
6.6 MATLAB Scripts 143
7 Flexural/Bending Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simply-supported 151
7.1 The Governing Equations of Motion 151
7.2 Closed-form Solutions 152
7.3 Wave Reflection, Propagation, and Wave Vibration Analysis Along the Simply-supported X Direction 154
7.4 Wave Reflection, Propagation, and Wave Vibration Analysis Along the y Direction 156
7.4.1 Wave Reflection at a Classical Boundary along the y Direction 157
7.4.2 Wave Propagation and Wave Vibration Analysis along the y Direction 159
7.5 Numerical Examples 159
8 In-Plane Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simply-supported 189
8.1 The Governing Equations of Motion 189
8.2 Closed-form Solutions 190
8.3 Wave Reflection, Propagation, and Wave Vibration Analysis along the Simply-supported X Direction 192
8.3.1 Wave Reflection at a Simply-supported Boundary Along the X Direction 192
8.3.2 Wave Propagation and Wave Vibration Analysis Along the X Direction 195
8.4 Wave Reflection, Propagation, and Wave Vibration Analysis along the y Direction 197
8.4.1 Wave Reflection at a Classical Boundary along the y Direction 198
8.4.2 Wave Propagation and Wave Vibration Analysis along the y Direction 201
8.5
Special Situation of k 0 = 0: Wave Reflection, Propagation, and Wave Vibration Analysis along the y Direction 201
8.5.1 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type I Simple Supports Along the X Direction When K 0 = 0 202
8.5.2 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type II Simple Supports Along the X Direction When K 0 = 0 203
8.5.3 Wave Propagation and Wave Vibration Analysis along the y Direction When k 0 = 0 205
8.6 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k 0 ≠ 0 207
8.6.1 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k 0 ≠ 0, k 1 ≠ 0, and k 2 ≠ 0 207
8.6.2 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along the y Direction When k 0 = 0, and either k 1 = 0 or k 2 = 0 209
8.7 Numerical Examples 212
8.7.1 Example 1: Two Pairs of the Same Type of Simple Supports Along the X and Y Directions 212
8.7.2 Example 2: One Pair of the Same Type Simple Supports Along the X Direction, Various Combinations of Classical Boundaries on Opposite Edges along the y Direction 217
8.7.3 Example 3: One Pair of Mixed Type Simple Supports Along the X Direction, Various Combinations of Classical Boundaries on Opposite Edges along the y Direction 223
9 Bending Waves in Beams Based on Advanced Vibration Theories 227
9.1 The Governing Equations and the Propagation Relationships 227
9.1.1 Rayleigh Bending Vibration Theory 227
9.1.2 Shear Bending Vibration Theory 228
9.1.3 Timoshenko Bending Vibration Theory 230
9.2 Wave Reflection at Classical and Non-Classical Boundaries 232
9.2.1 Rayleigh Bending Vibration Theory 232
9.2.2 Shear and Timoshenko Bending Vibration Theories 238
9.3 Waves Generated by Externally Applied Point Force and Moment on the Span 244
9.3.1 Rayleigh Bending Vibration Theory 245
9.3.2 Shear and Timoshenko Bending Vibration Theories 246
9.4 Waves Generated by Externally Applied Point Force and Moment at a Free End 247
9.4.1 Rayleigh Bending Vibration Theory 248
9.4.2 Shear and Timoshenko Bending Vibration Theories 249
9.5 Free and Forced Vibration Analysis 250
9.5.1 Free Vibration Analysis 250
9.5.2 Forced Vibration Analysis 250
9.6 Numerical Examples and Experimental Studies 252
9.7 MATLAB Scripts 257
10 Longitudinal Waves in Beams Based on Various Vibration Theories 263
10.1 The Governing Equations and the Propagation Relationships 263
10.1.1 Love Longitudinal Vibration Theory 263
10.1.2 Mindlin–Herrmann Longitudinal Vibration Theory 264
10.1.3 Three-mode Longitudinal Vibration Theory 265
10.2 Wave Reflection at Classical Boundaries 267
10.2.1 Love Longitudinal Vibration Theory 267
10.2.2 Mindlin–Herrmann Longitudinal Vibration Theory 268
10.2.3 Three-mode Longitudinal Vibration Theory 269
10.3 Waves Generated by External Excitations on the Span 271
10.3.1 Love Longitudinal Vibration Theory 271
10.3.2 Mindlin–Herrmann Longitudinal Vibration Theory 272
10.3.3 Three-mode Longitudinal Vibration Theory 273
10.4 Waves Generated by External Excitations at a Free End 275
10.4.1 Love Longitudinal Vibration Theory 275
10.4.2 Mindlin–Herrmann Longitudinal Vibration Theory 276
10.4.3 Three-mode Longitudinal Vibration Theory 276
10.5 Free and Forced Vibration Analysis 277
10.5.1 Free Vibration Analysis 278
10.5.2 Forced Vibration Analysis 278
10.6 Numerical Examples and Experimental Studies 281
11 Bending and Longitudinal Waves in Built-up Planar Frames 287
11.1 The Governing Equations and the Propagation Relationships 287
11.2 Wave Reflection at Classical Boundaries 289
11.3 Force Generated Waves 291
11.4 Free and Forced Vibration Analysis of a Multi-story Multi-bay Planar Frame 292
11.5 Reflection and Transmission of Waves in a Multi-story Multi-bay Planar Frame 304
11.5.1 Wave Reflection and Transmission at an L-shaped Joint 304
11.5.2 Wave Reflection and Transmission at a T-shaped Joint 308
11.5.3 Wave Reflection and Transmission at a Cross Joint 315
12 Bending, Longitudinal, and Torsional Waves in Built-up Space Frames 329
12.1 The Governing Equations and the Propagation Relationships 329
12.2 Wave Reflection at Classical Boundaries 333
12.3 Force Generated Waves 336
12.4 Free and Forced Vibration Analysis of a Multi-story Space Frame 338
12.5 Reflection and Transmission of Waves in a Multi-story Space Frame 341
12.5.1 Wave Reflection and Transmission at a Y-shaped Spatial Joint 343
12.5.2 Wave Reflection and Transmission at a K-shaped Spatial Joint 353
13 Passive Wave Vibration Control 369
13.1 Change in Cross Section or Material 369
13.1.1 Wave Reflection and Transmission at a Step Change by Euler–Bernoulli Bending Vibration Theory 371
13.1.2 Wave Reflection and Transmission at a Step Change by Timoshenko Bending Vibration Theory 372
13.2 Point Attachment 373
13.2.1 Wave Reflection and Transmission at a Point Attachment by Euler–Bernoulli Bending Vibration Theory 374
13.2.2 Wave Reflection and Transmission at a Point Attachment by Timoshenko Bending Vibration Theory 375
13.3 Beam with a Single Degree of Freedom Attachment 376
13.4 Beam with a Two Degrees of Freedom Attachment 378
13.5 Vibration Analysis of a Beam with Intermediate Discontinuities 380
13.6 Numerical Examples 381
13.7 MATLAB Scripts 390
14 Active Wave Vibration Control 401
14.1 Wave Control of Longitudinal Vibrations 401
14.1.1 Feedback Longitudinal Wave Control on the Span 401
14.1.2 Feedback Longitudinal Wave Control at a Free Boundary 405
14.2 Wave Control of Bending Vibrations 407
14.2.1 Feedback Bending Wave Control on the Span 407
14.2.2 Feedback Bending Wave Control at a Free Boundary 410
14.3 Numerical Examples 413
14.4 MATLAB Scripts 416
Index 421
About the Author :
Chunhui Mei is a Professor in the Department of Mechanical Engineering at the University of Michigan-Dearborn. She has over twenty years’ research and teaching experience on vibrations, controls, and instrumentation and measurement systems. She served as an Associate Editor for ASME Journal of Vibration and Acoustics.
