Preface ........................................................ xi
1 Introduction ................................................. 1
1.1 What Is a Continuous System? ............................ 1
1.2 A Comparison of Frequencies Obtained from Continuous
and Discrete Models ..................................... 5
1.3 A Preview of the Subsequent Chapters .................... 7
2 Transverse Vibrations of Strings ............................ 11
2.1 Differential Equation of Motion ........................ 12
2.2 Free Vibrations; Classical Solution .................... 15
2.3 Initial Conditions ..................................... 19
2.4 Consideration of Transverse Gravity .................... 22
2.5 Free Vibrations; Traveling Wave Solution ............... 23
2.6 Other End Conditions ................................... 26
2.7 Discontinuous Strings .................................. 30
2.8 Damped Free Vibrations ................................. 35
2.9 Forced Vibrations; Eigenfunction Superposition
Method ................................................. 38
2.10 Forced Vibrations; Closed Form Exact Solutions ......... 48
2.11 Energy Functionals for a String ........................ 57
2.12 Rayleigh Method ........................................ 59
2.13 Ritz Method ............................................ 61
2.14 Large Amplitude Vibrations ............................. 66
2.15 Some Concluding Remarks ................................ 69
References .................................................. 71
Problems .................................................... 71
3 Longitudinal and Torsional Vibrations of Bars ............... 77
3.1 Equation of Motion for Longitudinal Vibrations ......... 78
3.2 Equation of Motion for Torsional Vibrations ............ 80
3.3 Free Vibration of Bars ................................. 83
3.4 Other Solutions by Analogy ............................. 86
3.5 Free Vibrations of Bars with Variable Cross-Section .... 86
3.6 Forced Vibrations of Bars; Material Damping ............ 91
3.7 Energy Functionals and Rayleigh and Ritz Methods ....... 96
References .................................................. 98
Problems .................................................... 99
4 Beam Vibrations ............................................ 103
4.1 Equations of Motion for Transverse Vibrations ......... 104
4.2 Solution of the Differential Equation for Free
Vibrations ............................................ 107
4.3 Classical Boundary Conditions - Frequencies and Mode
Shares ................................................ 108
4.4 Other Boundary Conditions - Added Masses or Springs ... 116
4.5 Orthogonality of the Eigenfunctions ................... 120
4.6 Initial Conditions .................................... 123
4.7 Continuous and Discontinuous Beams .................... 127
4.8 Forced Vibrations ..................................... 130
4.9 Energy Functionals - Rayleigh Method .................. 135
4.10 Ritz Method ........................................... 141
4.11 Effects of Axial Forces ............................... 144
4.12 Shear Deformation and Rotary Inertia .................. 151
4.13 Curved Beams—Equations of Motion ...................... 167
4.14 Curved Beams—Vibration Analysis ....................... 170
References ................................................. 174
Problems ................................................... 175
5 Membrane Vibrations ........................................ 181
5.1 Equation of Motion for Transverse Vibrations .......... 182
5.2 Free Vibrations of Rectangular Membranes .............. 186
5.3 Circular Membranes .................................... 193
5.4 Annular and Sectorial Membranes ....................... 196
5.5 Initial Conditions .................................... 200
5.6 Forced Vibrations ..................................... 204
5.7 Energy Functionals; Rayleigh and Ritz Methods ......... 208
References ................................................. 217
Problems ................................................... 218
6 Plate Vjbrations ........................................... 221
6.1 Equation of Motion for Transverse Vibrations .......... 222
6.2 Free Vibrations of Rectangular Plates; Exact
Solutions ............................................. 229
6.3 Circular Plates ....................................... 235
6.4 Annular and Sectorial Plates .......................... 240
6.5 Energy Functionals; Rayleigh and Ritz Methods ......... 242
6.6 Approximate Solutions for Rectangular Plates .......... 247
6.7 Other Free Vibration Problems for Plates According
to Classical Plate Theory ............................. 253
6.8 Complicating Effects in Plate Vibrations .............. 260
References ................................................. 265
Problems ................................................... 267
7 Shell Vibrations ........................................... 271
7.1 Introduction .......................................... 272
7.2 Equations of Motion for Shallow Shells ................ 275
7.3 Free Vibrations of Shallow Shells ..................... 280
7.4 Equations of Motion for Circular Cylindrical Shells ... 293
7.5 Solutions for Deep or Closed Circular Cylindrical
Shells ................................................ 296
References ................................................. 307
Problems ................................................... 308
8 Vibrations of Three-Dimensional Bodies ..................... 311
8.1 Equations of Motion in Rectangular Coordinates ........ 312
8.2 Exact Solutions in Rectangular Coordinates ............ 316
8.3 Approximate Solutions for Rectangular
Parallelepipeds ....................................... 318
8.4 Exact Solutions in Cylindrical Coordinates ............ 328
8.5 Approximate Solutions for Solid Cylinders ............. 333
8.6 Approximate Solutions for Hollow Cylinders ............ 346
8.7 Other Three-Dimensional Bodies ........................ 352
References ................................................. 359
Problems ................................................... 361
9 Vibrations of Composite Continuous Systems ................. 363
9.1 Differential Equation of a Laminated Body in
Rectangular Coordinates ............................... 365
9.2 Laminated Beams ....................................... 374
9.3 Laminated Thick Beams ................................. 380
9.4 Beams with Tubular Cross-Sections ..................... 385
9.5 Laminated Thin Curved Beams ........................... 388
9.6 Laminated Thick Curved Beams .......................... 393
9.7 Laminated Thin Plates ................................. 401
9.8 Thick Plates .......................................... 415
9.9 Laminated Shallow Shells .............................. 424
9.10 Laminated Thick Shallow Shells ........................ 444
9.11 Laminated Cylindrical Shells .......................... 450
9.12 Vibrations of Other Laminated Shells .................. 459
References ................................................. 462
Problems ................................................... 463
A Summary of One Degree-of-Freedom Vibrations (with Viscous
Damping) ................................................... 467
В Bessel Functions: Some Useful Information .................. 471
С Hyperbolic Functions: Some Useful Relations ................ 477
Index ......................................................... 479
| 
