COPYRIGHTED MATERIAL. Index

Transcription

1 Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method, 139, 670 Axial force in beams, 352 Axisymmetric modes of shells, 601 B Bar in axial vibration, 69, 234 Basic equations of elasticity, 700 Beam in transverse vibration, 71, 317 under axial force, 352 bending torsional vibration, 380 boundary conditions, 323 continuous, 359 on elastic foundation, 364 Euler Bernoulli theory, 317 flexural waves, 628 free vibration, 185, 197 infinite length, 385 under moving loads, 350 Rayleigh s theory, 369 Timoshenko theory, 371 transform method, 385 Beat, 23 Beat frequency, 23 Bessel functions, 445, 491 Bessel s differential equation, 445, 491 Bishop s theory, 260 Boundary conditions bar in axial vibration, 71, 239 beam in transverse vibration, 73, 323 circular plate, 489 extremization of functional, 100 forced, 101 free, 101 geometric, 101 kinematic, 101 Kirchhoff, 469 membrane, 421 natural, 101 plate, 79, 465 shaft, 281 shell, 579, 596 skew edge of a plate, 469, 508 string, 209,211 thick plate, 508 C Calculus of a single variable, 85 Calculus of variations, 86 Cauchy-residue theorem, 193 Characteristic equation, 46, 219 Characteristic vector, 46 Circular frequency, 22 Circular membrane, 444 forced vibration, 448 free vibration, 444 mode shapes, 447 Circular plate, 485 equation of motion, 485 forced vibration, 495 free vibration, 490, 533 mode shapes, 494, 516 variable thickness, 531 Circular rings, 393 classification of vibration, 397 equations of motion, 393 extensional vibration, 407 in-plane flexural vibrations, 398 torsional vibration, 406 twist bending vibration, 402 Classical plate theory, 457 Collocation method, 144, 680 boundary method, 680 interior method, 680 mixed method, 680 Comparison function, 163 Complementary energy, 105 Complex frequency response, 41 Complex number representation, 20 Compressional waves, 623 COPYRIGHTED MATERIAL 713

2 714 Index Concept of vibration, 1 Conical shell, 544, 560, 591 Constitutive relations, 703 Constraints, 96 Continuous beams, 359 Continuous systems general, 132 history, 8 literature, 29 notable contributions, 12 Convolution integral, 42 Convolution theorem, 190 Coupled bending torsional vibration, 380 Critical damping constant, 35 Curved beams, 393 thick, 414 thin, 408 Cylindrical shell, 543, 559, 582 axisymmetric modes, 601 Donnell Mushtari Vlasov theory, 584 equations of motion, 583 natural frequencies, 584, 598 rotary inertia and shear deformation, 592 D D Alembert force, 69 D Alembert s principle, 69 D Alembert s solution, 210, 608 Damped harmonic response, 40 Damped vibration, 15 Damping matrix, 43 Damping ratio, 35, 171 Definitions, 21 Degree of freedom, 11 Derivation of equations, 85, 123, 125 Dilatational waves, 631 Dirac delta function, 152, 680 Discrete systems, 11, 13, 33 Dispersive medium, 629 Distortional waves, 632 Distributed systems, 14 Donnell Mushtari Vlasov theory of shells, 584 natural frequencies, 584 Duhamel s integral, 42 E Eigenfunctions orthogonality, 133, 161, 165, 246 properties, 160 Eigenvalue problem, 45, 163 formulation, 130 positive definite, 164 self-adjoint, 163 semidefinite, 164 Eigenvalues, 46 properties, 160 Eigenvector, 46, 55 Elastic element, 1 Elastic foundation, 364, 521 Elasticity, 700 basic equations, 700 constitutive relations, 703 equations of motion, 704 Lamé s elastic constants, 703 rotations, 702 strain displacement relations, 700 stress, 700 stress strain relations, 703 Elastic wave propagation, 607 in infinite elastic medium, 631 traveling-wave solution, 608 Equation(s) of motion, 704 beam in transverse vibration, 71, 371 circular cylindrical shell, 583 circular plate, 485, 515 circular ring, 393, 398, 402 conical shell, 591 coupled bending torsional vibration of beams, 380 curved beam, 408 integral, 127 longitudinal vibration of bars, 69, 235, 236, 258, 260 membrane, 420, 444 plate with in-plane loads, 523 plate in transverse vibration, 73, 457 shell, 575 spherical shell, 591 thick beam, 371 thick shell, 595 thin beam, 317 torsional vibration of shafts, 271 transverse vibration of string, 205 Equilibrium approach, 68 bar in axial vibration, 69, 234 beam in transverse vibration, 71 D Alembert s principle, 69 membrane, 420 Newton s second law, 68 plate with in-plane loads, 523 plate in transverse vibration,

3 Index 715 shaft vibration, 271 string vibration, 205 thick plate, 499 thin ring, 393 Equivoluminal waves, 634 Euler Lagrange equation, 89, , 96, 109 Exceptional part, 123 Excitations, 17 Expansion theorem, 48, 151, 161, 166 Extensional vibration of circular rings, 407 Extremization of functional, 86, 96 F Finite-dimensional systems, 13 First approximation shell theory, 556 Flexural rigidity of plate, 78, 458 Flexural waves, 628 Forced response, 169 Forced vibration beam, 198, 344 beam on elastic foundation, 366 circular plate, 495 definition, 16 finite string, 183 longitudinal vibration of bar, 254, 264 membrane, 438, 448 under moving load, 367 multidegree-of-freedom system, 52, 53, 54 plate, 479 shaft in torsional vibration, 292 single-degree-of-freedom system, 36 string, 227 Foundation modulus, 364, 521 Fourier integral, 26 Fourier series, 24, 175 Fourier transform pair, 27 sine and cosine, 178 Fourier transforms, 175, 707 membrane vibration, 441 string vibration, 213 Free vibration beam, 185, 325 beam on elastic foundation, 364 circular plate, 490 definition, 16 finite string, 181, 194, 217 infinite beam, 385 longitudinal vibration of bar, 236, 249 membrane, 426, 444 Mindlin plate, 511 multidegree-of-freedom system, 47, 52 plate with in-plane loads, 528 plate with variable thickness, 533 rectangular plates, 471 shaft in torsional vibration, 276, 288 single-degree-of-freedom system, 33 string of infinite length, 210 transverse vibration of beams, 341 Frequency, 1 Frequency equation, 46 plate, 480 string, 219 Frequency ratio, 39 Frequency spectrum, 25 Functional, 86, 91, 93, 95 Fundamental frequency, 24, 46 G Galerkin equations, 674 Galerkin method, 143, 673 Gauss points, 146 General force, 41 Generalized coordinates, 48, 109 General viscous damping, 54, 679 Green s function, 126, 131, 132 Group velocity, 629 H Hamilton s principle, 107 applications, 115 conservative systems, 109 continuous systems, 111 discrete systems, 108 generalized, 111 longitudinal vibration of bars, 235 nonconservative systems, 110 shell, 575 system of masses, 110 Timoshenko beam, 371 torsional vibration of a shaft, 115 transverse vibration of a thin beam, 116, 317 use of generalized coordinates, 109 Harmonic analysis, 24 Harmonic force, 36 Harmonic functions, 18 Harmonic motion definition, 1 representation, 18 Harmonics, 25 Harmonic waves, 610 Heaviside, 174

4 716 Index Homogeneous equations, 153 solution, 153 I Importance of vibration, 4 Impulse response function, 126 Inertial element, 1 Infinite beam, 385 Infinite-dimensional systems, 14 Infinite elastic medium, 631 Initial conditions, 48 beam, 341 membrane, 421 shaft, 289 string, 209, 613 Initial excitation longitudinal vibration of bars, 249 In-plane flexural vibrations of rings, 398 In-plane loads on a plate, 523 Integral equation, 123 approach, 123 assumed modes method, 139 classification, 124 collocation method, 144 first kind, 124 Fredholm type, 124 Galerkin method, 143 homogeneous, 124 iterative method, 134 linear, 124 nonlinear, 123 normal, 125 numerical integration method, 146 Rayleigh Ritz method, 139 second kind, 124 singular, 125 solution, 133 third kind, 124 Volterra type, 124 Integral transform, 174 Integral transform methods, 174 Integrodifferential equation, 123 Inverse transforms, 42, 193 Irrotational waves, 634 Isoperimetric problem, 96 Iterative method, 134 K Kernel, 123, 174 Kinetic energies of structural elements, 652 Kirchhoff boundary condition, 469 Kirchhoff s hypothesis, 556 L Lagrange equations, 109, 140, 672 Lagrangian, 107, 109 Lamé parameters, 543, 703 Lamé s elastic constants, 543, 703 Laplace equation, 303 Laplace transforms, 41, 188, 707 convolution theorem, 190 partial fraction method, 191 properties, 189 shifting property, 189 string vibration, 215 Least squares method, 686 Left eigenvector, 55 Linear frequency, 22 Linear vibration, 16 Literature on vibration, 29 Longitudinal vibration of bars, 234 Bishop s theory, 260 boundary conditions, 236, 238, 239 equation of motion, 235, 236 forced vibration, 254 free vibration, 236, 237 initial excitation, 249 mode shapes, 247, 259, 262 natural frequencies, 236, 259, 262 orthogonality of eigenfunctions, 246 Rayleigh theory, 258 support motion, 257 wave solution, 237 Longitudinal waves, 634 Love s approximations, 556 Lumped-parameter systems, 13 M Magnification factor, 36 Mass element, 1 Mass matrix, 43 Membrane analogy, 308 Membranes, 420 circular, 444 forced vibration, 438, 448 Fourier transform approach, 441 free transverse vibration, 130 irregular shape, 452 mode shapes, 430 partial circular, 453 rectangular, 426 Method of undetermined coefficients, 134

5 Index 717 Mindlin theory of plates, 499 circular plate, 515 free vibration, 511 Modal analysis approach, 151, 167 forced vibration, 292 forced vibration of bars, 264 free vibration, 289 membranes, 438, 448 multidegree-of-freedom system, 52, 54 in state space, 54 torsional vibration of shafts, 289, 292 Modal coordinates, 48 Modal vector, 46 Mode shapes beam, 326 circular plate, 494 cylindrical shell, 601 longitudinal vibration of bar, 259, 262 membrane, 430, 447 rectangular plate, 480 string, 220 Moving load on a beam, 350, 367 Multidegree-of-freedom system, 43 eigenvalue problem, 45 equations of motion, 43 expansion theorem, 48 forced vibration analysis, 52, 53, 54 free vibration analysis, 47 generalized coordinates, 48 modal analysis, 47, 54 modal coordinates, 48 modal matrix, 47 orthogonality of modal vectors, 46 N Natural frequency beams, 326 circular plate, 520 circular rings, 398 cylindrical shell, 584, 598 definition, 23 Donnell Mushtari Vlasov theory of shells, 584 longitudinal vibration of bars, 259, 262 Love s theory of shells, 587 rotating beam, 359 shaft in torsional vibration, 277 single-degree-of-freedom system, 33 thick beams, 377 thick rings, 401 n-degree-of-freedom system, 44 Newton s second law of motion, 68 longitudinal vibration of bars, 234 Nodal lines, 430 Nodes, 612 Noncircular shafts, 295 torsional rigidity, 303 Nonhomogeneous equation, 167 Nonlinear vibration, 16 Nonperiodic function, 26 Nonperiodic motion, 1 Normalization, 46 Normal modes, 46 orthogonality, 339 Nucleus, 123 Numerical integration method, 146 O Orthogonality of eigenfunctions, 162, 165 in integral formulation, 133 longitudinal vibration of bars, 246 torsional vibration of shafts, 286 transverse vibration of beams, 339 Orthogonality of modal vectors, 46 P Partial fraction method, 191 Period, 22 Periodic functions, 24 Periodic motion, 1 Phase angle, 22, 41 Phase difference, 23 Phase velocity, 612 Plate in transverse vibration, 73, 457 additional contributions, 80 boundary conditions, 79 circular plate, 485 equations of motion, 78 flexural rigidity, 78, 458 forced vibration, 495 free vibration, 471, 490 initial conditions, 79 with in-plane loads, 523 Mindlin theory, 499 mode shapes, 480, 494 moment displacement relations, 78 on elastic foundation, 521 rotary inertia and shear deformation, 499 state of stress, 75 strain displacement relations, 76 variable thickness, 529

6 718 Index Potential energy, 104 Prandtl s membrane analogy, 308 Prandtl stress function, 303 Primary (P) waves, 634 Principle of minimum complementary energy, 105 Principle of minimum potential energy, 104 Principle of stationary Reissner energy, 106 Properties eigenfunctions, 160 eigenvalues, 160 Proportional damping, 53, 678 P waves, 623 R Random vibration, 1 Rayleigh Ritz method, 139, 661 Rayleigh s method, 650 Rayleigh s principle, 650 Rayleigh s quotient, 648 Rayleigh theory, 258, 369 Rayleigh waves, 635 Recent contributions approximate analytical methods, 693 circular rings and curved beams, 416 elastic wave propagation, 643 integral equation approach, 147 integral transform methods, 201 longitudinal vibration of bars, 267 membrane vibration, 453 modal analysis approach, 171 multidegree-of-freedom systems, 60 vibration of plates, 535 torsional vibration of shafts, 313 transverse vibration of beams, 387 variational approach, 119 vibration of shells, 603 vibration of strings, 231 Rectangular plate, 471 boundary conditions, 465 on elastic foundation, 521 equation of motion, 457 forced vibration, 479 free vibration, 471 frequency equations, 480 with in-plane loads, 523 mode shapes, 480 rotary inertia and shear deformation, 499 subjected to in-plane loads, 523 variable thickness, 529 Reference kinetic energy, 662 Reflection of waves, 617, 619, 622 Regular part, 123 Reissner energy, 106 Residual, 144, 673 Resonance, 38 Right eigenvector, 55 Ritz coefficients, 661 Rotary inertia, beams, 369 circular rings, 399, 403 plates, 499 shells, 592 Rotational waves, 634 Rotations, 702 S Saint-Venant s theory, 295 Scotch yoke mechanism, 19 Self-adjoint eigenvalue problem, 163 Separation of variables, 153 bar vibration, 237 beam vibration, 325 membrane vibration, 426, 444 rectangular plate, 471 string vibration, 217 Shafts torsional properties, 310 in torsional vibration, 271 Shear correction factor, 372 Shear deformation beams, 371 circular rings, 399, 403 curved beams, 414 plates, 499 shells, 592 Shear waves, 623, 625 Shell coordinates, 541 Shells, 541 boundary conditions, 579 conical shell, 544, 560, 568, 591 cylindrical shell, 543, 559, 568, 582 Donnell Mushtari Vlasov theory, 584 first quadratic form of surface, 543 force and moment resultants, 563 kinetic energy, 573 Kirchhoff s hypothesis, 556 Lamé parameters, 543 Love s approximations, 556 rotary inertia and shear deformation, 592 spherical shell, 546, 561, 570, 591 strain displacement relations, 552

7 Index 719 strain energy, 571 stress strain relations, 562 theory of surfaces, 541 Single-degree-of-freedom system, 33 critically damped, 36 damped harmonic response, 40 forced vibration, 36, 41 free vibration, 33 under general force, 41 under harmonic force, 36 overdamped, 36 underdamped, 35 Skew plate, 540 Solid mechanics. 104 Spectral diagram, 25 Spherical shell, 546, 561, 591 Spring element, 1 Standing wave, 612 State space, 54 State vector, 54 Static deflection, 39 Stiffness matrix, 43 Strain displacement relations, 700 Strain energies of structural elements, 652 Strain energy, 104 Stress, 700 Stress strain relations, 703 String boundary conditions, 211 finite length, 194 forced vibration, 183 free vibration, 181 harmonic waves, 611 infinite, 210 transverse vibration, 205 traveling wave solution, 210 wave motion, 611 Sturm Liouville problem, 154 classification, 155 periodic, 155 regular, 155 singular, 155 Subdomain method, 684 Support motion longitudinal vibration of bars, 257 Surface waves, 635 S waves, 625 Synchronous motion, 23 T Terminology, 21 Theory of surfaces, 541 Three-dimensional vibration of circular ring, 393 Timoshenko Gere theory, 300 Torsional properties of shafts, 310 Torsional rigidity, 303 Torsional vibration of circular rings, 406 Torsional vibration of shafts, 115, 271 elementary theory, 271 forced vibration, 292 free vibration, 276, 289 noncircular shafts, 295, 299 Timoshenko Gere theory, 300 Transformation of relations, 486 Transform method in beams, 385 Transform pair, 175 Transient motion, 1, 4 Transients, 42 Transmission of waves, 619 Transverse vibration of plates, 457 boundary conditions, 465, 489 circular plates, 485 on elastic foundation, 521 equation of motion, 457 forced vibration, 479 free vibration, 471, 511 frequency equations, 480 Mindlin theory, 499 mode shapes, 475, 480 rotary inertia and shear deformation, 499 with variable thickness, 529 Transverse vibration of strings, 205 Transverse vibration of thin beams, 71, 116, 185, 317 under axial force, 352 coupled bending torsional vibration, 380 on elastic foundation, 364 equation of motion, 317 Euler Bernoulli theory, 317 flexural waves, 628 forced vibration, 344 frequencies and mode shapes, 326 infinite length, 385 with in-plane loads, 523 on many supports, 359 under moving load, 350, 367 orthogonality of normal modes, 339 Rayleigh s theory, 369 response due to initial conditions, 341 rotating, 357 Timoshenko theory, 371 transformation methods, 385

8 720 Index Traveling wave solution, 210, 608 U Uncoupled equations, 48 Undamped system, 47, 52 Undamped vibration, 15 Underdamped system, 35 V Variational approach, 85 membrane, 423 plate, 458 shaft, 272 string, 235 thick plate, 505 Variational methods, 85 in solid mechanics, 104 Variation operator, 89 Vibration analysis, 16 beams, 317, 369, 371 beams on elastic foundation, 364 circular cylindrical shell, 582 circular rings, 393, 406 concept, 1 continuous beam, 359 curved beams, 393, 408 developments, 5 forced, 52, 53 free, 47 history, 8 importance, 4 membranes, 420 multidegree-of-freedom system, 43 origins, 5 plates, 457, 485, 499 problems, 15 rotating beam, 357 shafts, 271 shells, 541 single-degree-of-freedom system, 33 string, 205 thick beams, 371 Viscous damping coefficient, 170 Viscously damped system, 34, 54, 169 W Warping function, 296 Wave equation D Alembert s solution, 608 membrane, 421 one-dimensional, 607 string, 207 traveling-wave solution, 608 two-dimensional, 610 Wavelength, 612 Wave number, 612 Wave packet, 629 Wave propagation, 607 in infinite elastic medium, 631 traveling wave, 608 Waves compressional, 623 dilatational, 631 distortional, 631 equivoluminal, 634 flexural, 628 harmonic, 611 irrotational, 634 longitudinal, 634 primary, 634 P, 623, 634 Rayleigh, 635 rotational, 634 shear, 623, 625 standing, 612 surface, 635 S, 625 traveling, 210, 608 Wave solution compressional waves, 623 dilatational waves, 631 distortional waves, 632 flexural waves, 628 graphical interpretation, 614 group velocity, 629 interface of two materials, 619 membrane, 425 P waves, 623 Rayleigh waves, 635 reflection of waves, 617, 622 shear waves, 623, 625 string, 210, 611 surface waves, 635 S waves, 625 transmission of waves, 619 wave packet, 629 Weighted residual methods, 673