Preface ......................................................... v
Chapter 1: Three-Dimensional Theories ........................... 1
1.1 Nonlinear Electroelasticity for Strong Fields ............... 1
1.2 Linear Piezoelectricity for Infinitesimal Fields ............ 6
1.2.1 Linearization ......................................... 6
1.2.2 Polarized ceramics ................................... 12
1.2.3 Quartz and langasite ................................. 13
1.3 Linear Theory for Small Fields Superposed on a Finite
Bias ....................................................... 16
1.3.1 The reference state .................................. 16
1.3.2 The initial state .................................... 17
1.3.3 The present state .................................... 17
1.3.4 Equations for the incremental fields ................. 18
1.3.5 Small bias ........................................... 20
1.3.6 Frequency perturbation due to a small bias ........... 21
1.4 Cubic Theory for Weak Nonlinearity ......................... 22
Chapter 2: Thickness-Shear Modes of Plate Resonators ........... 25
2.1 Static Thickness-Shear Deformation ......................... 25
2.1.1 A plate under a mechanical load ...................... 25
2.1.2 A plate under an electrical load ..................... 28
2.2 Nonlinear Thickness-Shear Deformation ...................... 31
2.2.1 General analysis ..................................... 32
2.2.2 An example ........................................... 33
2.3 Effects of Initial Fields on Thickness-Shear Deformation ... 36
2.3.1 General analysis ..................................... 37
2.3.2 An example ........................................... 39
2.4 Linear Thickness-Shear Vibration ........................... 43
2.4.1 Governing equations .................................. 43
2.4.2 Free vibration ....................................... 45
2.4.3 Forced vibration ..................................... 47
2.4.4 An unelectroded plate ................................ 48
2.5 Effects of Electrode Inertia ............................... 49
2.5.1 General analysis ..................................... 49
2.5.2 Identical electrodes ................................. 51
2.6 Inertial Effects of Imperfectly Bounded Electrodes ......... 52
2.6.1 General analysis ..................................... 52
2.6.2 Identical electrodes ................................. 54
2.7 Effects of Electrode Inertia and Shear Stiffness ........... 60
2.7.1 General analysis ..................................... 60
2.7.2 Special cases ........................................ 62
2.7.3 Numerical results .................................... 64
2.8 Nonlinear Thickness-Shear Vibration ........................ 66
2.8.1 Governing equations .................................. 66
2.8.2 Free vibration ....................................... 67
2.8.3 Forced vibration ..................................... 68
2.9 Effects of Initial Fields on Thickness-Shear Vibration ..... 70
2.9.1 Governing equations .................................. 71
2.9.2 Open electrodes ...................................... 71
2.9.3 Shorted electrodes ................................... 72
Chapter 3: Slowly Varying Thickness-Shear Modes ................ 73
3.1 Exact Waves in a Plate .................................... 73
3.1.1 Eigenvalue problem ................................. 74
3.1.2 An example ......................................... 77
3.1.3 A special case: Thickness-twist waves in
a ceramic plate .................................... 78
3.1.4 A special case: Thickness-twist waves in a quartz
plate .............................................. 85
3.2 An Approximate Equation for Thickness-Shear Waves ......... 87
3.3 Thickness-Shear Vibration of Finite Plates ................ 91
3.3.1 Sinusoidal modes ................................... 92
3.3.2 Hyperbolic modes ................................... 93
3.4 Energy Trapping in Mesa Resonators ........................ 93
3.5 Contoured Resonators ...................................... 97
3.6 Energy Trapping due to Material Inhomogeneity ............. 98
3.6.1 General analysis ................................... 98
3.6.2 An example ........................................ 101
3.7 Energy Trapping by Electrode Mass ........................ 103
3.8 Effects of Non-uniform Electrodes ........................ 104
3.9 Effects of Electromechanical Coupling on Energy
Trapping ................................................. 106
3.9.1 Governing equations ............................... 107
3.9.2 Free vibration solution ........................... 108
3.9.3 Discussion ........................................ 109
3.10 Coupling to Flexure ...................................... 110
3.10.1 Governing equations ............................... 110
3.10.2 Waves in unbounded plates ......................... 111
3.10.3 Vibrations of finite plates ....................... 112
3.11 Coupling to Face-Shear and Flexure ....................... 114
3.11.1 Governing equations ............................... 115
3.11.2 Waves in unbounded plates ......................... 115
3.11.3 Vibrations of finite plates ....................... 117
3.12 Effects of Middle Surface Curvature ...................... 119
3.12.1 Governing equations ............................... 120
3.12.2 Thickness-shear approximation ..................... 122
3.12.3 Coupled thickness-shear and extension ............. 124
Chapter 4: Mass Sensors ....................................... 127
4.1 Inertial Effect of a Mass Layer by Perturbation .......... 127
4.1.1 Governing equations ............................... 127
4.1.2 Abstract notation ................................. 128
4.1.3 Perturbation ...................................... 129
4.2 Thickness-Shear Modes of a Plate ......................... 131
4.3 Anti-Plane Modes of a Wedge .............................. 132
4.4 Torsional Modes of a Conical Shell ....................... 134
4.5 Effects of Inertia and Stiffness of a Mass Layer by
Perturbation ............................................. 135
4.5.1 Governing equations ............................... 136
4.5.2 Perturbation ...................................... 138
4.6 Effects of Inertia and Stiffness of a Mass Layer by
Variation ................................................ 139
4.6.1 Variational formulation ........................... 140
4.6.2 Frequency shift ................................... 141
4.6.3 Discussion ........................................ 142
4.7 Radial Modes of a Ring ................................... 143
4.7.1 The unperturbed mode .............................. 143
4.7.2 Perturbation solution ............................. 144
4.7.3 Variation solution ................................ 144
4.7.4 Discussion ........................................ 145
4.8 Effects of Shear Deformability of a Mass Layer ........... 146
4.8.1 Governing equations ............................... 147
4.8.2 Variational formulation ........................... 150
4.8.3 Frequency shift ................................... 151
4.9 Thickness-Shear Modes of a Plate with Thick Mass
Layers ................................................... 153
4.10 An Ill-Posed Problem in Elasticity for Mass Sensors ...... 157
4.10.1 Formulation of the problem ........................ 157
4.10.2 Known solutions ................................... 158
4.10.3 An open problem ................................... 159
4.11 Thickness-Shear Modes of a Circular Cylinder ............. 160
4.11.1 Thickness-shear modes in a circular cylinder ...... 160
4.11.2 Mass sensitivity .................................. 161
4.12 Mass Sensitivity of Surface Waves ........................ 163
4.12.1 Governing equations ............................... 163
4.12.2 A half-space with an electroded surface ........... 165
4.12.3 A half-space with an unelectroded surface ......... 166
4.13 Thickness-Twist Waves in a Ceramic Plate ................. 167
4.13.1 Governing equations ............................... 168
4.13.2 Anti-symmetric waves .............................. 168
4.13.3 Symmetric waves ................................... 170
4.14 Bechmann's Number for Thickness-Twist Waves .............. 171
4.14.1 Thickness-twist waves in an unbounded,
unelectroded plate ................................ 171
4.14.2 Thickness-twist waves in an unbounded,
electroded plate .................................. 172
4.14.3 Bechmann's number ................................. 173
4.15 Thickness-Twist Waves in a Quartz Plate .................. 175
4.15.1 Symmetric mass layers ............................. 175
4.15.2 Asymmetric mass layers ............................ 176
Chapter 5: Fluid Sensors ...................................... 181
5.1 An Ill-Posed Problem in Elasticity for Fluid Sensors ...... 181
5.1.1 Formulation of the problem .......................... 181
5.1.2 Known solutions ..................................... 182
5.1.3 Boundary integral equation formulation .............. 185
5.1.4 Generalization to piezoelectricity .................. 186
5.2 Perturbation Analysis ..................................... 187
5.2.1 Governing equations ................................. 188
5.2.2 Perturbation analysis ............................... 189
5.3 Thickness-Shear Modes of a Plate .......................... 191
5.4 Torsional Modes of a Cylindrical Shell .................... 192
5.4.1 Governing equations ................................. 193
5.4.2 Interior problem .................................... 194
5.4.3 Exterior problem .................................... 195
5.5 Thickness-Shear Modes of a Circular Cylinder .............. 196
5.5.1 Governing equations ................................. 196
5.5.2 Interior problem .................................... 197
5.5.3 Exterior problem .................................... 198
5.5.4 Axial thickness-shear ............................... 199
5.6 Surface Wave Fluid Sensors ................................ 199
5.6.1 Governing equations ................................. 200
5.6.2 A half-space with an electroded surface ............. 201
5.6.3 A half-space with an unelectroded surface ........... 202
5.7 Thickness-Twist Waves in a Ceramic Plate .................. 204
5.7.1 Governing equations ................................. 204
5.7.2 Anti-symmetric waves ................................ 205
5.7.3 Symmetric waves ..................................... 206
Chapter 6: Gyroscopes — Frequency Effect ...................... 209
6.1 High Frequency Vibrations of a Small Rotating
Piezoelectric Body ........................................ 209
6.2 Propagation of Plane Waves ................................ 210
6.2.1 General solution .................................... 210
6.2.2 Polarized ceramics .................................. 215
6.2.3 Quartz .............................................. 217
6.3 Thickness Vibrations of Plates ............................ 221
6.3.1 General solution .................................... 222
6.3.2 Ceramic plates ...................................... 223
6.3.3 Quartz plates ....................................... 227
6.4 Propagating Waves in a Rotating Piezoelectric Plate ....... 228
6.4.1 General solution .................................... 228
6.4.2 Ceramic plates ...................................... 232
6.5 Surface Waves over a Rotating Piezoelectric Half-Space .... 236
6.5.1 General solution .................................... 236
6.5.2 Ceramic half-space .................................. 238
Chapter 7: Gyroscopes - Charge Effect ......................... 245
7.1 A Rectangular Beam ........................................ 245
7.1.1 Governing equations ................................. 247
7.1.2 Forced vibration solution ........................... 250
7.1.3 An example .......................................... 251
7.2 A Circular Tube ........................................... 258
7.2.1 Governing equations ................................. 260
7.2.2 Forced vibration solution ........................... 261
7.2.3 An example .......................................... 263
7.3 A Beam Bimorph ............................................ 264
7.3.1 Governing equations ................................. 265
7.3.2 Forced vibration solution ........................... 270
7.3.3 Free vibration solution ............................. 271
7.3.4 An example .......................................... 272
7.4 An Inhomogeneous Shell .................................... 274
7.4.1 Structure ........................................... 275
7.4.2 Governing equations ................................. 276
7.4.3 Forced vibration solution ........................... 279
7.4.4 Discussion .......................................... 282
7.5 A Ceramic Ring ............................................ 282
7.5.1 Governing equations ................................. 284
7.5.2 Free vibration solution ............................. 286
7.5.3 Forced vibration solution ........................... 287
7.6 A Concentrated Mass and Ceramic Rods ...................... 288
7.6.1 Governing equations ................................. 289
7.6.2 Forced vibration solution ........................... 290
7.7 A Ceramic Plate by Two-Dimensional Equations .............. 291
7.7.1 Driving ............................................. 292
7.7.2 Sensing ............................................. 294
7.7.3 Discussion .......................................... 295
7.8 A Ceramic Plate by Zero-Dimensional Equations ............. 296
7.8.1 Governing equations ................................. 297
7.8.2 Free vibration solution ............................. 298
7.8.3 Forced vibration solution ........................... 299
Chapter 8: Acceleration Sensitivity ........................... 301
8.1 Deformation of a Quartz Plate under Normal Acceleration ... 301
8.1.1 Classical flexure ................................... 301
8.1.2 Flexure induced in-plane extension and thickness
contraction ......................................... 303
8.1.3 One-dimensional deformation ......................... 305
8.1.4 Two-dimensional deformation ......................... 308
8.2 First-Order Acceleration Sensitivity ...................... 311
8.3 An Estimate of Second-Order Acceleration Sensitivity and
its Reduction ............................................. 313
8.3.1 An estimate ......................................... 314
8.3.2 Reduction of normal acceleration sensitivity ........ 315
8.4 Second-Order Perturbation Analysis ........................ 318
8.4.1 Relatively large biasing deformations ............... 318
8.4.2 Equations for incremental vibrations ................ 319
8.4.3 Second-order perturbation analysis .................. 321
8.5 Second-Order Normal Acceleration Sensitivity .............. 324
8.5.1 Second-order acceleration sensitivity ............... 325
8.5.2 One-dimensional deformation ......................... 326
8.5.3 Two-dimensional deformation ......................... 328
8.6 Effects of Middle Surface Curvature ....................... 329
8.6.1 Biasing deformation ................................. 329
8.6.2 Unperturbed modes ................................... 332
8.6.3 Frequency shift ..................................... 332
8.7 Vibration Sensitivity ..................................... 334
8.7.1 Governing equations ................................. 335
8.7.2 First-order solution ................................ 336
8.7.3 Second-order solution ............................... 337
8.7.4 An example .......................................... 338
Chapter 9: Pressure Sensors ................................... 341
9.1 A Rectangular Plate in a Circular Cylindrical Shell ....... 341
9.1.1 Analysis ............................................ 343
9.1.2 Numerical results ................................... 345
9.2 A Circular Plate in a Circular Cylindrical Shell .......... 347
9.2.1 Analysis ............................................ 347
9.2.2 Numerical results ................................... 350
9.3 A Rectangular Plate in a Shallow Shell .................... 352
9.3.1 Analysis ............................................ 352
9.3.2 Numerical results ................................... 354
9.4 A Bimorph ................................................. 355
9.4.1 Analysis ............................................ 355
9.4.2 Numerical results ................................... 358
9.5 Surface Wave Pressure Sensors Based on Extension .......... 361
9.6 Surface Wave Pressure Sensors Based on Flexure ............ 364
9.5 Biasing deformations ...................................... 364
9.6.2 Frequency shifts .................................... 368
Chapter 10: Temperature Sensors ............................... 371
10.1 Thermoelectroelasticity .................................. 371
10.2 Linear Theory ............................................ 374
10.3 Small Fields Superposed on a Thermal Bias ................ 376
10.4 Thickness-Shear Modes of a Free Plate .................... 379
10.5 Thickness-Shear Modes of a Constrained Plate ............. 382
10.5.1 Analysis .......................................... 382
10.5.2 An example ........................................ 385
Chapter 11: Piezoelectric Generators .......................... 387
11.1 Thickness-Stretch of a Ceramic Plate ..................... 387
11.1.1 Analysis .......................................... 388
11.1.2 An example ........................................ 392
11.2 A Circular Shell ......................................... 395
11.3 A Beam Bimorph ........................................... 399
11.3.1 Analysis .......................................... 400
11.3.2 An example ........................................ 406
11.4 A Spiral Bimorph ......................................... 407
11.5 Nonlinear Behavior near Resonance ........................ 409
11.5.1 Analysis .......................................... 410
11.5.2 Numerical results ................................. 413
Chapter 12: Piezoelectric Transformers ........................ 417
12.1 A Thickness-Stretch Mode Plate Transformer ............... 417
12.1.1 Governing equations ............................... 417
12.1.2 Analytical solution ............................... 419
12.1.3 Numerical results ................................. 421
12.2 Rosen Transformer ........................................ 425
12.2.1 One-dimensional model ............................. 425
12.2.2 Free vibration analysis ........................... 428
12.2.3 Forced vibration analysis ......................... 429
12.2.4 Numerical results ................................. 432
12.3 A Thickness-Shear Mode Transformer - Free Vibration ...... 434
12.3.1 Governing equations ............................... 435
12.3.2 Free vibration analysis ........................... 437
12.3.3 Ceramic transformers .............................. 439
12.4 A Thickness-Shear Mode Transformer - Forced Vibration .... 444
12.4.1 Governing equations ............................... 444
12.4.2 Forced vibration analysis ......................... 447
Chapter 13: Power Transmissiion through an Elastic Wall ....... 451
13.1 Formulation of the Problem ............................... 451
13.2 Theoretical Analysis ..................................... 454
13.3 Numerical Results ........................................ 457
Chapter 14: Acoustic Wave Amplifiers .......................... 463
14.1 Equations for Piezoelectric Semiconductors ............... 463
14.2 Equations for a Thin Film ................................ 464
14.3 Surface Waves ............................................ 466
14.3.1 Analytical solution ............................... 466
14.3.2 Discussion ........................................ 470
14.3.3 Numerical results ................................. 472
14.4 Interface Waves .......................................... 474
14.4.1 Analytical solution ............................... 474
14.4.2 Discussion ........................................ 477
14.5 Waves in a Plate ......................................... 478
14.5.1 Symmetric waves ................................... 479
14.5.2 Anti-symmetric waves .............................. 482
14.5.3 Discussion ........................................ 483
14.6 Gap Waves ................................................ 484
14.6.1 Analytical solution ............................... 484
14.6.2 Discussion ........................................ 487
References .................................................... 491
Appendix 1 Notation .......................................... 501
Appendix 2 Electroelastic Material Constants ................. 503
Index ......................................................... 517
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