Vibrations of Shells and Plates

by Werner Soedel

Overview

Soedel (mechanical engineering, Purdue U.) goes beyond the usual examination of beam and rod vibration as he describes the vibration behavior of shells and plates for engineering graduate students and practicing engineers. He begins with the historical development of vibration analysis of continuous structural elements. Among the topics addressed in the chapters that follow: force vibration of shells by modal expansion, the dynamic influence (Green's) function, moment loading, combinations of structures, hysteresis damping, shells made of composite material, rotating structures, thermal effects, elastic foundations, similitude, interactions with liquids and gases, and discreting approaches. Material that has been updated or is new to this edition includes such topics as elastic foundations, simplified equations, and the first formulation of a moment loading dynamic for shells and plates. Annotation ©2004 Book News, Inc., Portland, OR

Product Details

ISBN-13:	9780203026304
Publisher:	CRC Press
Publication date:	08/11/2004
Series:	Mechanical Engineering
Sold by:	Barnes & Noble
Format:	eBook
Pages:	592
File size:	26 MB
Note:	This product may take a few minutes to download.

Table of Contents

1	Historical Development of Vibration Analysis of Continuous Structural Elements	1
	References	4
2	Deep Shell Equations	7
2.1	Shell Coordinates and Infinitesimal Distances in Shell Layers	8
2.2	Stress-Strain Relationships	13
2.3	Strain-Displacement Relationships	15
2.4	Love Simplifications	22
2.5	Membrane Forces and Bending Moments	24
2.6	Energy Expressions	28
2.7	Love's Equations by Way of Hamilton's Principle	30
2.8	Boundary Conditions	35
2.9	Hamilton's Principle	39
2.10	Other Deep Shell Theories	43
2.11	Shells of Nonuniform Thickness References	46
2.12	Radii of Curvature	47
	References	50
3	Equations of Motion for Commonly Occurring Geometries	51
3.1	Shells of Revolution	51
3.2	Circular Conical Shell	54
3.3	Circular Cylindrical Shell	56
3.4	Spherical Shell	57
3.5	Other Geometries	59
	References	63
4	Nonshell Structures	64
4.1	Arch	64
4.2	Beam and Rod	67
4.3	Circular Ring	68
4.4	Plate	69
4.5	Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar	72
	References	74
5	Natural Frequencies and Modes	75
5.1	General Approach	75
5.2	Transversely Vibrating Beams	77
5.3	Circular Ring	82
5.4	Rectangular Plates that are Simply supported Along Two Opposing Edges	86
5.5	Circular Cylindrical Shell Simply Supported	93
5.6	Circular Plates Vibrating Transversely	102
5.7	Example: Plate Clamped at Boundary	103
5.8	Orthogonality Property of Natural Modes	106
5.9	Superposition Modes	109
5.10	Orthogonal Modes from Nonorthogonal Superposition Modes	113
5.11	Distortion of Experimental Modes Because of Damping	117
5.12	Separating Time Formally	120
5.13	Uncoupling of Equations of Motion	122
5.14	In-Plane Vibrations of Rectangular Plates	124
5.15	In-Plane Vibration of Circular Plates	128
5.16	Deep Circular Cylindrical Panel Simply Supported at All Edges	131
5.17	Natural Mode Solutions by Power Series	133
5.18	On Regularities Concerning Nodelines	142
	References	143
6	Simplified Shell Equations	145
6.1	Membrane Approximation	145
6.2	Axisymmetric Eigenvalues of a Spherical Shell	146
6.3	Bending Approximation	151
6.4	Circular Cylindrical Shell	152
6.5	Zero In-Plane Deflection Approximation	153
6.6	Example: Curved Fan Blade	154
6.7	Donnell-Mushtari-Vlasov Equations	154
6.8	Natural Frequencies and Modes	157
6.9	Circular Cylindrical Shell	157
6.10	Circular Duct Clamped at Both Ends	159
6.11	Vibrations of a Freestanding Smokestack	161
6.12	Special Cases of the Simply Supported Closed Shell and Curved Panel	162
6.13	Barrel-Shaped Shell	163
6.14	Spherical Cap	165
6.15	Inextensional Approximation: Ring	167
6.16	Toroidal Shell	168
6.17	The Barrel-Shaped Shell Using Modified Love Equations	170
6.18	Doubly Curved Rectangular Plate	174
	References	176
7	Approximate Solution Techniques	178
7.1	Approximate Solutions by Way of the Variational Integral	179
7.2	Use of Beam Functions	181
7.3	Galerkin's Method Applied to Shell Equations	184
7.4	Rayleigh-Ritz Method	191
7.5	Southwell's Principle	196
7.6	Dunkerley's Principle	199
7.7	Strain Energy Expressions	201
	References	206
8	Forced Vibrations of Shells by Modal Expansion	207
8.1	Model Participation Factor	207
8.2	Initial Conditions	210
8.3	Solution of the Modal Participation Factor Equation	211
8.4	Reduced Systems	214
8.5	Steady-State Harmonic Response	215
8.6	Step and Impulse Response	216
8.7	Influence of Load Distribution	217
8.8	Point Loads	220
8.9	Line Loads	225
8.10	Point Impact	227
8.11	Impulsive Forces and Point Forces Described by Dirac Delta Functions	230
8.12	Definitions and Integration Property of the Dirac Delta Function	232
8.13	Selection of Mode Phase Angles for Shells of Revolution	233
8.14	Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered	236
8.15	Initial Velocity Excitation of a Simply Supported Cylindrical Shell	240
8.16	Static Deflections	243
8.17	Rectangular Plate Response to Initial Displacement Caused by Static Sag	243
8.18	The Concept of Modal Mass, Stiffness Damping and Forcing	246
8.19	Steady State Response of Shells to Periodic Forcing	248
8.20	Plate Response to a Periodic Square Wave Forcing	251
8.21	Beating Response to Steady state Harmonic Forcing	253
	References	255
9	Dynamic Influence (Green's) Function	256
9.1	Formulation of the Influence Function	257
9.2	Solution to General Forcing Using the Dynamic Influence Function	259
9.3	Reduced Systems	260
9.4	Dynamic Influence Function for the Simply Supported Shell	261
9.5	Dynamic Influence Function for the Closed Circular Ring	263
9.6	Traveling Point Load on Simply Supported Cylindrical Shell	264
9.7	Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction	267
9.8	Steady-State Harmonic Green's Function	271
9.9	Rectangular Plate Examples	272
9.10	Floating Ring Impacted by a Point Mass	277
	References	279
10	Moment Loading	281
10.1	Formulation of Shell Equations That Include Moment Loading	282
10.2	Modal Expansion Solution	284
10.3	Rotating Point Moment on a Plate	285
10.4	Rotating Point Moment on a Shell	287
10.5	Rectangular Plate Excited by a Line Moment	289
10.6	Response of a Ring on an Elastic Foundation to a Harmonic Point Moment	291
10.7	Moment Green's Function	295
	References	300
11	Vibration of Shells and Membranes Under the Influence of Initial Stresses	301
11.1	Strain-Displacement Relationships	302
11.2	Equations of Motion	305
11.3	Pure Membranes	309
11.4	Example: The Circular Membrane	311
11.5	Spinning Saw Blade	315
11.6	Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses	318
	References	320
12	Shell Equations with Shear Deformation and Rotatory Inertia	322
12.1	Equations of Motion	322
12.2	Beams with Shear Deflection and Rotatory Inertia	325
12.3	Plates with Transverse Shear Deflection and Rotatory Inertia	329
12.4	Circular Cylindrical Shells with Transverse Shear Deflection and Rotatory Inertia	333
	References	336
13	Combinations of Structures	337
13.1	Receptance Method	338
13.2	Mass Attached to Cylindrical Panel	339
13.3	Spring Attached to Shallow Cylindrical Panel	342
13.4	Harmonic Response of a System in Terms of Its Component Receptances	344
13.5	Dynamic Absorber	347
13.6	Harmonic Force Applied Though a Spring	350
13.7	Steady-State Response to Harmonic Displacement Excitation	353
13.8	Complex Receptances	354
13.9	Stiffening of Shells	356
13.10	Two Systems Joined by Two or More Displacement	360
13.11	Suspension of an Instrument Package in a Shell	362
13.12	Subtracting Structural Subsystems	365
13.13	Three and More Systems Connected	370
13.14	Examples of Three Systems Connected to Each Other	374
	References	378
14	Hysteresis Damping	380
14.1	Equivalent Viscous Damping Coefficient	381
14.2	Hysteresis Damping	381
14.3	Direct Utilization of Hysteresis Model in Analysis	384
14.4	Hysteretically Damped Plate Excited by Shaker	386
14.5	Steady State Response to Periodic Forcing	388
	References	390
15	Shells Made of Composite Material	391
15.1	Nature of Composites	391
15.2	Lamina-Constitutive Relationship	392
15.3	Laminated Composite	397
15.4	Equation of Motion	399
15.5	Orthotropic Plate	400
15.6	Circular Cylindrical Shell	402
15.7	Orthotropic Nets or Textiles Under Tension	406
15.8	Hanging Net or Curtain	408
15.9	Shells Made of Homogeneous and Isotropic Lamina	410
15.10	Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina	412
	References	414
16	Rotating Structures	415
16.1	String Parallel to Axis of Rotation	415
16.2	Beam Parallel to Axis of Rotation	422
16.3	Rotating Ring	425
16.4	Rotating Ring Using Inextensional Approximation	428
16.5	Cylindrical Shell Rotating with Constant Spin About Its Axis	431
16.6	General Rotations of Elastic Systems	432
16.7	Shells of Revolution with Constant Spin About their Axes of Revolution	433
16.8	Spinning Disk	436
	References	436
17	Thermal Effects	438
17.1	Stress Resultants	438
17.2	Equations of Motion	440
17.3	Plate	443
17.4	Arch, Ring, Beam, and Rod	443
17.5	Limitations	444
	References	445
18	Elastic Foundations	446
18.1	Equations of Motion for Shells on Elastic Foundations	447
18.2	Natural Frequencies and Modes	447
18.3	Plates on Elastic Foundations	448
18.4	Ring on Elastic Foundation	449
18.5	Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation	451
18.6	Forces Transmitted into the Base of the Elastic Foundation	451
18.7	Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel	453
18.8	Response of a Shell on an Elastic Foundation to Base Excitation	458
18.9	Plate Examples of Base Excitation and Force Transmission	460
18.10	Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point	462
18.11	Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement	464
	References	468
19	Similitude	469
19.1	General Similitude	469
19.2	Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells	471
19.3	Plates	472
19.4	Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature)	474
19.5	Forced Response	476
19.6	Approximate Scaling of Shells Controlled by Membrane Stiffness	477
19.7	Approximate Scaling of Shells Controlled by Bending Stiffness	478
	References	479
20	Interactions with Liquids and Gases	480
20.1	Fundamental Form in Three-Dimensional Curvilinear Coordinates	480
20.2	Stress-Strain-Displacement Relationships	482
20.3	Energy Expressions	486
20.4	Equations of Motion of Vibroelasticity with Shear	487
20.5	Example: Cylindrical Coordinates	492
20.6	Example: Cartesian Coordinates	493
20.7	One-Dimensional Wave Equations for Solids	495
20.8	Three-Dimensional Wave Equations for Solids	496
20.9	Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics)	498
20.10	Interface Boundary Conditions	502
20.11	Example: Acoustic Radiation	502
20.12	Incompressible Liquids	505
20.13	Example: Liquid on Plate	506
20.14	Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases	511
	References	513
21	Discretizing Approaches	515
21.1	Finite Differences	515
21.2	Finite Elements	520
21.3	Free and Forced Vibration Solutions	533
	References	538
	Index	539

Editorial Reviews

A revised, updated, and expanded reference-text that provides a unified presentation of shell vibrations, with plates, rings, and other structural elements treated as special cases--starting with the derivation of the simplest set of deep shell equations and continuing on to anisotropic shells and the effect of initial stresses. The significant amount of new material reflects the latest developments in the field and meets the need of graduate students and practicing engineers to become acquainted with additional topics such as traveling modes in rotating shells, thermal effects, and fluid loading. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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