Vibrations of shells and plates, third edition

DK1310_FM 7/8/04 10:14 AM Page A Vibrations of Shells and Plates DK1310_FM 7/8/04 10:14 AM Page B MECHANICAL ENGINEERING A Series of Textbooks and Reference Books Founding Editor L L Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Spring Designer’s Handbook, Harold Carlson Computer-Aided Graphics and Design, Daniel L Ryan Lubrication Fundamentals, J George Wills Solar Engineering for Domestic Buildings, William A Himmelman Applied Engineering Mechanics: Statics and Dynamics, G Boothroyd and C Poli Centrifugal Pump Clinic, Igor J Karassik Computer-Aided Kinetics for Machine Design, Daniel L Ryan Plastics Products Design Handbook, Part A: Materials and Components; Part B: Processes and Design for Processes, edited by Edward Miller Turbomachinery: Basic Theory and Applications, Earl Logan, Jr Vibrations of Shells and Plates, Werner Soedel Flat and Corrugated 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Drives: Selection and Application, Frederick W Heilich III and Eugene E Shube 25 Finite Element Methods: An Introduction, Ronald L Huston and Chris E Passerello 26 Mechanical Fastening of Plastics: An Engineering Handbook, Brayton Lincoln, Kenneth J Gomes, and James F Braden 27 Lubrication in Practice: Second Edition, edited by W S Robertson 28 Principles of Automated Drafting, Daniel L Ryan 29 Practical Seal Design, edited by Leonard J Martini 30 Engineering Documentation for CAD/CAM Applications, Charles S Knox 31 Design Dimensioning with Computer Graphics Applications, Jerome C Lange 32 Mechanism Analysis: Simplified Graphical and Analytical Techniques, Lyndon O Barton 33 CAD/CAM Systems: Justification, Implementation, Productivity Measurement, Edward J Preston, George W Crawford, and Mark E Coticchia 34 Steam Plant Calculations Manual, V Ganapathy 35 Design Assurance for Engineers and Managers, John A Burgess 36 Heat Transfer Fluids and Systems for Process and Energy Applications, Jasbir Singh 37 Potential Flows: Computer Graphic Solutions, Robert H Kirchhoff 38 Computer-Aided Graphics and Design: Second Edition, Daniel L Ryan 39 Electronically Controlled Proportional Valves: Selection and Application, Michael J Tonyan, edited by Tobi Goldoftas 40 Pressure Gauge Handbook, AMETEK, U.S Gauge Division, edited by Philip W Harland 41 Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R P Donovan 42 Design of Mechanical Joints, Alexander Blake 43 CAD/CAM Dictionary, Edward J Preston, George W Crawford, and Mark E Coticchia 44 Machinery Adhesives for Locking, Retaining, and Sealing, Girard S Haviland 45 Couplings and Joints: Design, Selection, and Application, Jon R Mancuso 46 Shaft Alignment Handbook, John Piotrowski 47 BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer, V Ganapathy 48 Solving Mechanical Design Problems with Computer Graphics, Jerome C Lange 49 Plastics Gearing: Selection and 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High Vacuum Technology: A Practical Guide, Marsbed H Hablanian 72 Pressure Sensors: Selection and Application, Duane Tandeske 73 Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter 74 Thermal Fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski 75 Classical and Modern Mechanisms for Engineers and Inventors, Preben W Jensen 76 Handbook of Electronic Package Design, edited by Michael Pecht 77 Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by Marc A Meyers, Lawrence E Murr, and Karl P Staudhammer 78 Industrial Refrigeration: Principles, Design and Applications, P C Koelet 79 Applied Combustion, Eugene L Keating 80 Engine Oils and Automotive Lubrication, edited by Wilfried J Bartz 81 Mechanism Analysis: Simplified and Graphical Techniques, Second Edition, Revised and Expanded, Lyndon O Barton 82 Fundamental Fluid Mechanics for the Practicing Engineer, James W Murdock DK1310_FM 7/8/04 10:14 AM Page E 83 Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Second Edition, Revised and Expanded, P K Mallick 84 Numerical Methods for Engineering Applications, Edward R Champion, Jr 85 Turbomachinery: Basic Theory and Applications, Second Edition, Revised and Expanded, Earl Logan, Jr 86 Vibrations of Shells and Plates: Second Edition, Revised and Expanded, Werner Soedel 87 Steam Plant Calculations Manual: Second Edition, Revised and Expanded, V Ganapathy 88 Industrial Noise Control: Fundamentals and Applications, Second Edition, Revised and Expanded, Lewis H Bell and Douglas H Bell 89 Finite Elements: Their Design and Performance, Richard H MacNeal 90 Mechanical Properties of Polymers and Composites: Second Edition, Revised and Expanded, Lawrence E Nielsen and Robert F Landel 91 Mechanical Wear Prediction and Prevention, Raymond G Bayer 92 Mechanical Power Transmission Components, edited by David W South and Jon R Mancuso 93 Handbook of Turbomachinery, edited by Earl Logan, Jr 94 Engineering Documentation Control Practices and Procedures, Ray E Monahan 95 Refractory Linings Thermomechanical Design and Applications, Charles A Schacht 96 Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D Meadows 97 An Introduction to the Design and Behavior of Bolted Joints: Third Edition, Revised and Expanded, John H Bickford 98 Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski 99 Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa 100 Friction Science and Technology, Peter J Blau 101 Introduction to Plastics and Composites: Mechanical Properties and Engineering Applications, Edward Miller 102 Practical Fracture Mechanics in Design, Alexander Blake 103 Pump Characteristics and Applications, Michael W Volk 104 Optical Principles and Technology for Engineers, James E Stewart 105 Optimizing the Shape of Mechanical Elements and Structures, A A Seireg and Jorge Rodriguez 106 Kinematics and Dynamics of Machinery, Vladimír Stejskal and Michael Valásek 107 Shaft Seals for Dynamic Applications, Les Horve 108 Reliability-Based Mechanical Design, edited by Thomas A Cruse 109 Mechanical Fastening, Joining, and Assembly, James A Speck 110 Turbomachinery Fluid Dynamics and Heat Transfer, edited by Chunill Hah 111 High-Vacuum Technology: A Practical Guide, Second Edition, Revised and Expanded, Marsbed H Hablanian 112 Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D Meadows DK1310_FM 7/8/04 10:14 AM Page F 113 Handbook of Materials Selection for Engineering Applications, edited by G T Murray 114 Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka 115 Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M Lepi 116 Applied Computational Fluid Dynamics, edited by Vijay K Garg 117 Fluid Sealing Technology, Heinz K Muller and Bernard S Nau 118 Friction and Lubrication in Mechanical Design, A A 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Ispas, and Dan Boboc DK1310_FM 7/8/04 10:14 AM Page G 145 Implementing Concurrent Engineering in Small Companies, Susan Carlson Skalak 146 Practical Guide to the Packaging of Electronics: Thermal and Mechanical Design and Analysis, Ali Jamnia 147 Bearing Design in Machinery: Engineering Tribology and Lubrication, Avraham Harnoy 148 Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing, R E Little 149 Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V Ganapathy 150 The CAD Guidebook: A Basic Manual for Understanding and Improving Computer-Aided Design, Stephen J Schoonmaker 151 Industrial Noise Control and Acoustics, Randall F Barron 152 Mechanical Properties of Engineered Materials, Wolé Soboyejo 153 Reliability Verification, Testing, and Analysis in Engineering Design, Gary S Wasserman 154 Fundamental Mechanics of Fluids: Third Edition, I G Currie 155 Intermediate Heat Transfer, Kau-Fui Vincent Wong 156 HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W Stanford III 157 Gear Noise and Vibration: Second Edition, Revised and Expanded, J Derek Smith 158 Handbook of Turbomachinery: Second Edition, Revised and Expanded, edited by Earl Logan, Jr., and Ramendra Roy 159 Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A Antaki 160 Turbomachinery: Design and Theory, Rama S R Gorla and Aijaz Ahmed Khan 161 Target Costing: Market-Driven Product Design, M Bradford Clifton, Henry M B Bird, Robert E Albano, and Wesley P Townsend 162 Fluidized Bed Combustion, Simeon N Oka 163 Theory of Dimensioning: An Introduction to Parameterizing Geometric Models, Vijay Srinivasan 164 Handbook of Mechanical Alloy Design, edited by George E Totten, Lin Xie, and Kiyoshi Funatani 165 Structural Analysis of Polymeric Composite Materials, Mark E Tuttle 166 Modeling and Simulation for Material Selection and Mechanical Design, edited by George E Totten, Lin Xie, and Kiyoshi Funatani 167 Handbook of Pneumatic Conveying Engineering, David Mills, Mark G Jones, and Vijay K Agarwal 168 Clutches and Brakes: Design and Selection, Second Edition, William C Orthwein 169 Fundamentals of Fluid Film Lubrication: Second Edition, Bernard J Hamrock, Steven R Schmid, and Bo O Jacobson 170 Handbook of Lead-Free Solder Technology for Microelectronic Assemblies, edited by Karl J Puttlitz and Kathleen A Stalter 171 Vehicle Stability, Dean Karnopp 172 Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G Bayer DK1310_FM 7/8/04 173 174 175 176 177 178 179 180 181 182 183 184 10:14 AM Page H Liquid Pipeline Hydraulics, E Shashi Menon Solid Fuels Combustion and Gasification, Marcio L de Souza-Santos Mechanical Tolerance Stackup and Analysis, Bryan R Fischer Engineering Design for Wear, Raymond G Bayer Vibrations of Shells and Plates: Third Edition, Revised and Expanded, Werner Soedel Refractories Handbook, edited by Charles A Schacht Practical Engineering Failure Analysis, Hani M Tawancy, Anwar UlHamid, and Nureddin M Abbas Mechanical Alloying and Milling, C Suryanarayana Mechanical Vibration: Analysis, Uncertainties, and Control, Second Edition, Revised and Expanded, Haym Benaroya Design of Automatic Machinery, Stephen J Derby Practical Fracture Mechanics in Design: Second Edition, Revised and Expanded, Arun Shukla Practical Guide to Designed Experiments, Paul D Funkenbusch Additional Volumes in Preparation Mechanical Engineering Software Spring Design with an IBM PC, Al Dietrich Mechanical Design Failure Analysis: With Failure Analysis System Software for the IBM PC, David G Ullman DK1310_FM 7/8/04 10:14 AM Page i Vibrations of Shells and Plates Third Edition, Revised and Expanded Werner Soedel Dept of Mechanical Engineering Purdue University West Lafayette, Indiana Marcel Dekker, Inc New York Index Moment distributed, 281 line, 281 loading, 281, 282 point, 281 resultants, 46 Moment Green’s function shells, 295–298 plates, 299 Momentum, 130, 279 change of, 216, 230, 231 distributed, 216 Momentum-impulse, 230 Mote, C D., 316, 317, 318 Motion in-phase, 364 liquid dominated, 511 out-of-phase, 364 Multiring stiffener, 360 Mushtari, K M., 155 Naghdi, P M., 44, 150, 336 Napolean, Narrow strip load, 227 Natural frequencies, 75 barrel shell, 173 beam, 77 sandwich, 413 shear deformation, 325 cylindrical shell, 98, 157, 159, 162, 187, 406 bending approximation, 152 Donnell-Mushtari-Vlasov, 157, 159, 187, 406 Galerkin’s method, 187 initial stress, 320 panel, 131, 162 orthotropic, 402, 406 rotatory inertia, 333, 335 shear deformation, 333, 335 Southwell’s principle, 196–198 spring loaded, 365 547 Yu’s approximation, 158 curtain, 410 discrete systems, 533 Dunkerley’s principle, 199–201 dynamic absorber, 348 multi-degree of freedom, 350 elastic foundation, general, 447 plates, 448 ring, 449 shell, 447, 451 finite elements, 529 finite differences, 519 Galerkin’s method, 184, 186 general approach, 75 inextensional ring, 168 liquid on plate, 510 membrane, 315 net, 407, 408, 410 orthotropic plate, 401 shell, 406 panel, 131, 163, 175 curved, 476 dynamic absorber, 347 interior support, 344 mass attached, 339 panel attached, 352 similitude, 476 spring attached, 342 stiffened, 356, 359 plate, 285 continuous, 378 beam functions, 182, 183 circular, 105, 128 doubly curved, 174 Dunkerley’s method, 199 elastic foundation, 460 Green’s function, 260, 272, 275 finite difference, 519 in-plane, 124, 128, 260 liquid on, 510 548 mass on, 200 mass void, 369 orthotropic, 401 rectangular, 86, 89, 92, 110, 124, 272, 274, 283, 285, 329 rotatory inertia, 329 sandwich, 413 shear deformation, 329 strain energy, 203, 205 three connected, 374 Rayleigh-Ritz method, 191 receptance equation, 339, 362, 367, 372 ring, 82, 123, 168, 191, 450, 463 rotating beam, 425 inextensional ring, 428 ring, 428 string, 419 saw blade, 315 sensitivity to curvature, 63 similitude, 469–479 approximate, 477–479 general, 469, 471 plate, 472 spherical panels, 474 shallow shell, 163, 174 fan blade, 154 spherical cap, 165 shell on elastic foundation, 447 Southwell’s principle, 198 spherical shell, 149 Timoshenko beam, 327 toroidal shell, 170 system subtracted, 368 variational method, 182 Natural modes, 75 antinodes, 300 axisymmetric, 151 base, 113 beam, 77 breathing, 149 Index conical shell, 196 cylindrical shell, 95, 98 damping influence, 117 experimental, 117 liquid on plate, 510 membrane, 315 nonpreferential orientation, 233 orthogonality, 106, 109, 234 orthogonalization, 113 panel, 131 with mass, 340 plate, 333 circular, 105, 128 in-plane, 126 rectangular, 86, 89, 92, 126 square, 89, 92, 110 Rayleigh-Ritz, 192–196 rigid body, 149, 229 ring, 82, 123, 450 rotating, beam, 425 ring, 429 string, 419 saw blade, 317 selective excitation, 217–220 series expansion, 133–143 spherical cap, 166 spherical shell, 149 stiffening ring, 356 superposition, 109 Timoshenko beam, 327 variational method, 179, 181 Yu’s approximation, 159 Navier, C L., Negative curvature, 165 Nets, 408 Neutral surface, 411, 412 Newton, I., 1, 2, 39 Nodal circle, 105 displacement, 521 force vector, 523 Index lines, 119 Node line interpretation, 119 Noncrossing node lines, 119 Nondimensional form, 471 Nonuniform thickness, 46 Normal coordinates, 506 shear strains, 322 strains, 15, 21 stress, 15 velocity, 502 Novozhilov, V V., 44, 45, 51, 156 Nowacki, W., 51, 156, 207 Numerical differentiation, 63 Off-resonance response, 215 Orthogonal coordinates, Orthogonality, 106, 210, 211 condition, 109, 210, 211, 511, 536 integral, 109, 210, 211 matrix equation, 533 Orthogonalization process, 113 Orthogonal modes, 2, 533 Orthotropic cylindrical shell, 402 lamina, 393 material, 392 net, 406 plate, 400 strip, 393, 395 Ovaling motion, 422 Overdamped, 537 Pagani, M., 3, 315 Panel, 162, 221, 354, 476 point support, 343 with mass, 339 with spring, 338, 342 Parallel fibers, 392 Particle velocity, 500 Pendulum, 549 Periodic forcing response, 250 formulation, 248 shells, 248 plates, 248, 388 Phase angle, 215 Piston slap, 216 Plane stress, 392 Plate, 3, 4, 69 attached mass, 200 boundary conditions, 71, 88 circular, 71, 102 continuous, 377 elastic foundation, 448 element, 524 finite differences, 516 forcing function, 214 elliptical, 71 equation, 3, 69 hysteretically damped, 386 in-plane vibration, 124, 128 liquid on, 506 moment forcing, 286 rectangular, 86, 89, 182, 200, 217, 243, 251, 253, 386 rotatory inertia, 331 shear deformation, 331 similitude, 472 square, 89 strain-displacement, 69, 70 strain energy, 203 thermal effects, 443 thickness effect, 473 three connected, 376 transverse vibration, 86, 102 Plunkett, R., 388 Ply, 391 Point force, 220, 231, 236, 267, 276 impact, 227, 231 mass impact, 277 moment, 285 550 support, 343, 462 Poisson, S D., 3, effect, 444 ratio, 474 Polar coordinates, 71 Potential energy, 417, 423 function, 501, 510 Power series solutions, 133–142 rod, 134 beam, 135 Prescott’s equation, 139 Powder, D P., 227 Prasad, M G., 344 Principal curvature, directions, 399 Prescott, J., 68 Prescott’s equation, 68, 139 Pretension, 406 Rails, 451 Radius of curvature, 9, 47, 61 Rapoport, I M., 506 Rayleigh, J W S., 4, 7, 145, 151, 191, 506 frequency, 194 quotient, 191, 192, 193, 194 method, 191, 195, 384 Rayleigh-Ritz method, 178, 191, 193, 194, 201, 317 Robound, 231 Receptance complex, 354 continuous plate, 377 cross, 346 graphical representation, 341 harmonic response, 344, 346 line, 356 magnitude, 355 method, 337–379 moment coupling, 377 Index multiple connections, 360, 374 phase, 355 point, 363, 462 ring, 357 three systems, 370, 372, 373, 374 two systems, 338, 344, 346, 347, 352, 360, 366 Rectangular box, 337 contour, 476 membrane, 110 plate, 86, 89, 92, 110, 124, 272, 274, 283, 285, 329 strip, 392 Reduced systems, 214, 260 Reference surface, 8, 27, 46, 47 temperature, 439 Reinforcing material, 392 Response, 215 harmonic, 215 impulse, 216 magnitude, 215 phase lag, 215 step, 216 Reissner, E., 7, 44 Repeated roots, 534 Residual stress, 302 strain energy, 302 Resonant modes, 118 Response curve, 215 magnitude, 215 phase, 215 Restoring mechanism, 301 Reynolds, D D., 505 Rigid body mode, 454 motion, 85 rotation, 264 translation, 229 Index Rigid wheel, 453 Ring, 4, 68, 82, 122, 454 closed, 123 elastic foundation, 449 element, 530 floating, 82 Green’s function, 261, 263 impact, 277 orthogonal modes, 235 point contact, 462 segment, 189 stiffening, 356 strain energy, 204 thermal effect, 443 Ritz, W., 193, 195 Rod, 2, 41, 67, 205, 214, 443 Rolling tire, 270 Ross, D., 383, 389 Rotating beam, 422 circular cylindrical shell, 431 disk, 436 machinery, 415 point moment, plate, 285 cylindrical shell, 287 ring, 425 shells of revolution, 433 string, 415, 423 structures, 415–436 Rotational speed, 415 Rotatory inertia, 31, 322 Saigal, S., 301 Saint-Venant, Sandwich shells, 410 plates, 412 beams, 412 Sakharov, I E., 356 Sanders, J L., 44 551 Sauveur, J., Saw blade, 301, 315 Scaling approximate, 478 law, 470 Schmidt orthogonalization, 113 Schwartz, L., 221 Separation of variables, 120 Shaker, 386 Shallow panel, 474 shell, 165 Shear center, 424 deformation, 19, 322 force, 27 modulus elastic foundation, 446 fiber, 394 matrix, 394 resultant, 323 strain, 13, 20, 21 stress, 15 velocity, 495 vibration, 328 Shell barrel-shaped, 170 closed, 149 conical, 54 composite material, 400, 402 cylindrical, 10, 56, 152, 161, 202, 236, 240, 287, 402 elastic foundation, 447 element, 529–532 imperfection, 220 noncircular, 60 parabolic, 61 revolution, 233, 235, 435 thick, 43, 149 toroidal, 48, 59, 168 shallow, 153 simplified equations, 145–176 spherical, 57, 146, 165, 228, 301 552 stiffening, 356 rule, 359 Similtude, 469–479 approximate, 478 exact, 448, 451, 469, 472, 473, 474 forced response, 476 shell, 470, 471 Simple oscillator, 215 Skew-symmetric, 220 load, 220 Soedel, D T., 369 Soedel, F P., 142 Soedel, S M., 422, 506 Soedel, W., 43, 110, 113, 142, 166, 196, 201, 227, 255, 256, 270, 281, 301, 324, 328, 337, 342, 344, 358, 360, 369, 378, 406, 409, 410, 430, 451, 473, 476, 477, 497, 506, 530 Sokolnikoff, I S 492 Speed of sound, 472, 499 Smokestack, 161 Southwell, R V 196, 315, 317 principle, 196, 197 Spherical cap, 165 Spring Connections, 350, 373 Spin speed, 318, 418 Static deflection, 1, 243 equilibrium, 243, 244, 245 initial condition, 243 sag, 1, 243 Stationary motion, 430 Steady state, 215 Steel, 99 Step function, 216 response, 216, 274 Stiffener, 356, 359 Stiffening rule, 359 Stiffness Index matrix, 524, 527 principle direction, 395 superposition, 196 Strain, arch, 65 barrel, 171 bending, 23 conical shell, 55 cylindrical shell, 57 displacement, 15, 44, 302, 324, 439, 482 energy, 191, 201–205, 306, 427, 440 membrane, 23 parabolic shell, 62 plates, 69, 124 shells of revolution, 53 spherical shell, 58 String, 1, 3, 415 Stringer, 360, 406 Strip element, 530 Stress functions, 402 resultants, 313, 397, 438 Stress-strain, 13, 439, 482 Structural combinations, 337–378 Structurally axisymmetric, 220 Stutts, D S., 430 Subcritical damping, 212 solution, 212 Superposition modes, 109, 110 Supporting matrix, 392 Suspension system, 346 Symmetry, 220, 242, 455, 461, 464 Tangential motion, 467 Tauchert, T R., 438 Taylor, B., Taylor series, 16, 17 Temperature distribution, 439 Index field, 438 model, 444 Tensile test, 381 Tension, 319, 407 Tensioning, 318 Textiles, 406 Thermal coefficient of expansion, 439 dynamic stress, 439 effects, 438–444 expansion, 438 shock, 438 static stress, 444 strain, 439 strain energy, 440 stresses, 318, 438, 439 Time, separation of, 120 Timoshenko, S., 43, 51, 64, 207, 264, 325, 326 beam equation, 325 Tires, 301, 346, 353, 394, 454 Todhunter, L., Toroidal shell, 48, 59 Torsional circular cylindrical shell, 72 pendulum, vibration, Torsion bar, 72 Transfer function, 272 Transverse deflection, 84 impulse, 262 motion, response, 218 shear deflection, 22 forces, 27 vibration, 328 Traveling load, 264, 267 resonance, 267, 270 pressure wave, 264 Travel solution, 2, 264, 267 553 Triad, 431 Tsai, S W 394 Turbine blades, 301 Turbulent damping, 380 Twisting angle, 282, 283 Uncoupling, 122, 398 Underdamped, 212 Ungar, E E., 383, 389 Uniform pressure, 218 Unit impulse, 257 Variational displacements, 30, 37 integral method, 179, 201, 317 symbol, 39 Vector space, 208 Velocity potential, 507 vector, 415, 426 Vibroelasticity, 487 Virtual displacement, 31, 40 work, 7, 40, 282, 527 Viscoelasticity, 487 Viscous damping, 208 damping coefficient, 208 damping factor, 208 Vlasov, V Z., 45, 51, 155 Volterra, E., 193 Volume fraction, 394 Wah, T., 519 Wash machine basket, 45 Watson, G N., 504 Wave equation acoustic, 498 liquid, 511 one-dimensional, 2, 495 three-dimensional, 496, 498 torsional, ... Second Edition, Revised and Expanded, Earl Logan, Jr 86 Vibrations of Shells and Plates: Second Edition, Revised and Expanded, Werner Soedel 87 Steam Plant Calculations Manual: Second Edition, ... my grandchildren: Amber, Nicholas, Ashleigh, April, Jackson, Broderic, Thomas, Carter, Kathleen and the yet unborn Preface to the Third Edition The third edition of Vibrations of Shells and Plates... 157 Gear Noise and Vibration: Second Edition, Revised and Expanded, J Derek Smith 158 Handbook of Turbomachinery: Second Edition, Revised and Expanded, edited by Earl Logan, Jr., and Ramendra Roy