Formulas for Dynamics, Acoustics and Vibration E-Book

Robert D. Blevins

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Series in Acoustics Noise and Vibration
• Sprache: Englisch

Seitenzahl: 449

Veröffentlichungsjahr: 2016

Beschreibung

With Over 60 tables, most with graphic illustration, and over 1000 formulas, Formulas for Dynamics, Acoustics, and Vibration will provide an invaluable time-saving source of concise solutions for mechanical, civil, nuclear, petrochemical and aerospace engineers and designers. Marine engineers and service engineers will also find it useful for diagnosing their machines that can slosh, rattle, whistle, vibrate, and crack under dynamic loads.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

Chapter 1: Definitions, Units, and Geometric Properties

1.1 Definitions

1.2 Symbols

1.3 Units

1.4 Motion on the Surface of the Earth

1.5 Geometric Properties of Plane Areas

1.6 Geometric Properties of Rigid Bodies

1.7 Geometric Properties Defined by Vectors

References

Chapter 2: Dynamics of Particles and Bodies

2.1 Kinematics and Coordinate Transformations

2.2 Newton's Law of Particle Dynamics

2.3 Rigid Body Rotation

References

Chapter 3: Natural Frequency of Spring–Mass Systems, Pendulums, Strings, and Membranes

3.1 Harmonic Motion

3.2 Spring Constants

3.3 Natural Frequencies of Spring–Mass Systems

3.4 Modeling Discrete Systems with Springs and Masses

3.5 Pendulum Natural Frequencies

3.6 Tensioned Strings, Cables, and Chain Natural Frequencies

3.7 Membrane Natural Frequencies

References

Chapter 4: Natural Frequency of Beams

4.1 Beam Bending Theory

4.2 Natural Frequencies and Mode Shapes of Single-Span and Multiple-Span Beams

4.3 Axially Loaded Beam Natural Frequency

4.4 Beams with Masses, Tapered Beams, Beams with Spring Supports, and Shear Beams

4.5 Torsional and Longitudinal Beam Natural Frequencies

4.6 Wave Propagation in Beams

4.7 Curved Beams, Rings, and Frames

References

Chapter 5: Natural Frequency of Plates and Shells

5.1 Plate Flexure Theory

5.2 Plate Natural Frequencies and Mode Shapes

5.3 Cylindrical Shells

5.4 Spherical and Conical Shells

References

Chapter 6: Acoustics and Fluids

6.1 Sound Waves and Decibels

6.2 Sound Propagation in Large Spaces

6.3 Acoustic Waves in Ducts and Rooms

6.4 Acoustic Natural Frequencies and Mode Shapes

6.5 Free Surface Waves and Liquid Sloshing

6.6 Ships and Floating Systems

6.7 Added Mass of Structure in Fluids

References

Further Reading

Chapter 7: Forced Vibration

7.1 Steady-State Forced Vibration

7.2 Transient Vibration

7.3 Vibration Isolation

7.4 Random Vibration Response to Spectral Loads

7.5 Approximate Response Solution

References

Chapter 8: Properties of Solids, Liquids, and Gases

8.1 Solids

8.2 Liquids

8.3 Gases

References

Appendix A: Approximate Methods for Natural Frequency

A.1 Relationship between Fundamental Natural Frequency and Static Deflection

A.2 Rayleigh Technique

A.3 Dunkerley and Southwell Methods

A.4 Rayleigh–Ritz and Schmidt Approximations

A.5 Galerkin Procedure for Continuous Structures

References

Appendix B: Numerical Integration of Newton's Second Law

References

Appendix C: Standard Octaves and Sound Pressure

C.1 Time History and Overall Sound Pressure

C.2 Peaks and Crest

C.3 Spectra and Spectral Density

C.4 Logarithmic Frequency Scales and Musical Tunings

C.5 Human Perception of Sound (Psychological Acoustics)

References

Appendix D: Integrals Containing Mode Shapes of Single-Span Beams

Reference

Appendix E: Finite Element Programs

Professional/Commercial Programs

Open Source /Low-Cost Programs

Index

End User License Agreement

Pages

xi

xii

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

37

38

39

31

32

33

34

35

36

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

439

440

441

442

443

444

445

446

447

448

Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 1: Definitions, Units, and Geometric Properties

Figure 1.1 Newton's second law in consistent units,

Figure 1.2 Motion of the surface of rotating earth and Coriolis deflection of moving particles relative to the earth. Latitude is zero at the equator.

Figure 1.3 A plane section with centroid (

C

) and rotated and translated coordinate systems and a solid body with a translated coordinate system and a rotated vector

r

Figure 1.4 Geometric sections for Examples 1.3 and 1.4

Chapter 2: Dynamics of Particles and Bodies

Figure 2.1 Three cases of rotation

Figure 2.2 Three cases of mass impact against an uncompressed spring

Figure 2.3 Car with velocity strikes a stationary car. They slide together distance

d

before stopping.

Figure 2.4 Irregular object with perimeter

P

and initial velocity strikes and penetrates a stationary plate. is its exit velocity

Figure 2.5 Circular, elliptical, parabolic, and hyperbolic orbits are conic sections

Figure 2.6 Elliptical orbit nomenclature

Figure 2.7 Inertial coordinate system

x–y–z

is instantaneously fixed in the rotating body on the axis of rotation

Figure 2.8 Static balancing a rotor by resting the shaft on knife-edges. Two-plane imbalance generates a transverse moment on the bearings. After Ref. [18]

Figure 2.9 Freely swinging pendulum angular velocity versus angle

Figure 2.10 Spinning top (a) and constrained rotor precession (b)

Figure 2.11 A spinning bicycle wheel is supported by a point at one end of its axle. The wheel precesses about this point. See case 7 of Table 2.7.

Chapter 3: Natural Frequency of Spring–Mass Systems, Pendulums, Strings, and Membranes

Figure 3.1 Sinusoidal motion during free vibration of a spring–mass system. Displacement and velocity are out of phase. The period

Figure 3.2 The three-axis plot of acceleration, velocity, and peak-to-peak displacement amplitudes for harmonic motion is used to specify a vibration environment envelope of a vibrating base

Figure 3.3 Mass with a linear extension spring

Figure 3.4 Cantilever plate spring and a box on a two-dimensional spring suspension

Figure 3.5 Examples of a spring–mass system and a torsional pendulum

Figure 3.6 Three models for a spring in a spring–mass system

Figure 3.7 Bellows modeling

Figure 3.8 Spring–mass discretized longitudinal beam natural frequencies as a function of number of elements

Figure 3.9 Linearization of the gravity-induced torque on the pendulum for small deflections

Figure 3.10 Tensioned cables

Figure 3.11 Flat and cylindrical tensioned membranes

Figure 3.12 Out-of-plane modal deformation patterns of flat membranes

Chapter 4: Natural Frequency of Beams

Figure 4.1 Coordinates for stress and bending deformation of a beam. is the axial beam bending stress and

R

is the radius of curvature

Figure 4.2 Cross section of an asymmetric sandwich beam and beam on an elastic foundation, (a) sandwich beam and (b) beam on elastic foundation

Figure 4.3 Mode shapes of single-span beams (Table 4.2)

Figure 4.4 Fundamental mode shape of a pinned–pinned beam as a function of position of intermediate support (case 1 of Table 4.4)

Figure 4.5 Fundamental natural frequency of an

N

-span beam with ends clamped, variable length outer spans, and intermediate pinned supports spaced at length

L

[14]

Figure 4.6 Example of a clamped–clamped tube

Figure 4.7 Axially loaded beams with pinned ends. force/length, (a) uniform axial load, (b) traction reacted at one end, and (c) traction reacted at both ends

Figure 4.8 End conditions for beams with traction load ([6]; Table 4.7)

Figure 4.9 Effect of axial loads on natural frequency parameter (Eqs. 4.11, 4.21) of clamped–clamped beam (a) and pinned–pinned beam (b) with axial tractions reacted at both ends 21

Figure 4.10 Mode shape of beams with spring-supported ends

Figure 4.11 First-mode deflection of cantilever flexure and shear beams

Figure 4.12 Natural frequencies of transverse modes of a pinned–pinned round steel rod compared to flexural and shear beam theories

Figure 4.13 An aluminum honeycomb beam suspended on strings

Figure 4.14 Torsional deformation of a shaft with rigid end disks (case 8 of Table 4.15)

Figure 4.15 In-plane modes of a complete ring

Figure 4.16 In-plane natural frequency of a circular arc with pinned end with circumferential constraint. Cases 1 and 2 of Table 4.17 in comparison with Henrych [83] and Tufekci [84]

Figure 4.17 Effect of arc rise and axial restraint on transverse in-plane mode of slightly curved beams with pinned and clamped ends (Eq. 4.47)

Figure 4.18 Pin-ended frame and circular arc

Chapter 5: Natural Frequency of Plates and Shells

Figure 5.1 First six modes of simply supported rectangular and circular plates. Natural frequencies increase left to right in each row

Figure 5.2 A rectangular plate with a riveted Z stiffener. Attaching the stiffener to the surrounding structure increases the stiffener's ability to resist plate deformation

Figure 5.3 Coordinates for a circular cylinder, nodal patterns, and deformation of a simply supported cylinder without axial constraint. After Ref. 110 in part

Figure 5.4 Variation in strain energy of a cylindrical shell with increasing number of waves 119

Figure 5.5 Deformed shapes for simply supported shell without axial constraint

Figure 5.6 Natural frequencies of vibration of a cylindrical shell with ends simply supported without axial constraint using the Donnell shell theory and computed using Equation 5.39

Figure 5.7 Section of a cylindrical panel. Simply supported edge conditions without circumferential constraint prevents radial displacement but does not restrict circumferential motion and rotation

Figure 5.8 Shallow spherically curved shell and a flat rectangular plate with the same planform

Chapter 6: Acoustics and Fluids

Figure 6.1 Speed of sound in air, water, steam, and ammonia as a function of temperature. [4]

Figure 6.2 Reduction in speed of sound for propagation through an array of rigid rods [8, 9]

Figure 6.3 A fluid element for derivation of the one-dimensional wave equation

Figure 6.4 Wavelength, traveling, and standing waves

Figure 6.5 Measurement of sound pressure level

Figure 6.6 Sound absorption in atmospheric air at 1 atm and 19

Figure 6.7 Ray and room acoustics

Figure 6.8 Waves in ducts and transfer matrix model for a duct acoustical component

Figure 6.9 Acoustic pressure on the closed end of a duct as a function of piston velocity amplitude and circular frequency (case 2 of Table 6.4)

Figure 6.10 Expansion muffler and Helmholtz resonator transmission loss. -sectional area of chamber; -sectional area of inlet and exit pipes. Curves between and 0.5 repeat between and 1 and its multiples

Figure 6.11 Lined ducts. The circular duct has a 360° liner, whereas the rectangular duct shown has only one side lined

Figure 6.12 Insertion loss of a lined duct in air as a function of the mean airflow velocity. The liner duct has 20% opening and and is tuned to 1600 Hz

Figure 6.13 Systems with compressible fluid in variable volume cavities

Figure 6.14 Polytropic constant as a function of the size of an air bubble in water at [65]

Figure 6.15 Fundamental sloshing mode in a rectangular basin (case 1 of Table 6.7). The

x–y

plane is parallel to the mean fluid level and perpendicular to the paper

Figure 6.16 Response of a rectangular harbor with various size openings to sea waves at frequency

f

for . The response is measured at

x

101

Figure 6.17 Stability of a ship in roll. The moment increases roll of an unstable ship

Chapter 7: Forced Vibration

Figure 7.1 Forced vibration of a spring–mass damped system and phase of response relative to the force

Figure 7.2 Dynamic amplification factor and phase

Figure 7.3 (a) and (b) Spring–mass systems on vibrating base

Figure 7.13 Four cases of isolation. Cases a and d minimize force on the base from motions of a vibrating mass, Cases b and c isolate mass from vibrating base

Figure 7.4 (a–c) Three cases of transverse excitation of a beam

Figure 7.5 Dynamic response of a pinned–pinned tube on a vibrating support as a function of support vibration frequency

f

Figure 7.7 Superposition of loads to create a finite rectangular pulse

Figure 7.6 Response to a suddenly applied and held force (Figure 7.3). period

Figure 7.8 Transient response to a rectangular pulse lasting two natural periods

Figure 7.9 Response spectra (Maximum response) of a damped single-degree-of-freedom system to a duration limited step force

Figure 7.10 Maximum displacement of a damped single-degree-of-freedom oscillator to sawtooth force

Figure 7.11 A portal frame with pine legs is loaded by a ground shock wave

Figure 7.12 Sawtooth shock from Mil-Std M-810C

Figure 7.14 Transmissibility and isolation as a function of frequency and damping

Figure 7.15 Vibration isolation suspensions. Upper row suspensions isolate vibration sensitive component (

M

) from support vibration. Bottom row isolates the floor from machinery vibration

Figure 7.17 Two-stage vibration isolation. The upper mass has substantially higher isolation at high frequencies than the lower mass for forcing frequency above the natural frequencies

Figure 7.16 Two-stage vibration isolation of a rotating machine and its model

Figure 7.18 Tuned mass damper is tuned to the natural frequency of

M

1

. The tuned mass has 8% damping and 5% of the mass of the large mass . It reduces resonant response by a factor of 8

Figure 7.19 Base excitation section in

g

as a function of frequency from MIL-STD-810C

Appendix A: Approximate Methods for Natural Frequency

Figure 1.1 Elastic systems

List of Tables

Chapter 1: Definitions, Units, and Geometric Properties

Table 1.6 Properties of homogeneous solids

Table 1.1 Nomenclature

Table 1.2 Consistent sets of engineering units

Table 1.3 Decimal unit multipliers

Table 1.4 Conversion factors

Table 1.5 Properties of plane sections

Table 1.7 Properties of elements described by points and vectors

Chapter 2: Dynamics of Particles and Bodies

Table 2.1 Kinematic motion

Table 2.2 Coordinate transformations

Table 2.3 Particle dynamics

Table 2.4 Rockets and orbits

Table 2.5 Rigid body rotation theory

Table 2.6 Rotation about single axis

Table 2.7 Multiaxis rotation

Chapter 3: Natural Frequency of Spring–Mass Systems, Pendulums, Strings, and Membranes

Table 3.1 Harmonic motion

Table 3.2 Spring stiffness

Table 3.3 Natural frequency of spring–mass systems

Table 3.5 Natural frequency of pendulum systems

Table 3.6 Natural frequencies of strings, cables, and chains

Table 3.7 Natural frequencies of membranes

Chapter 4: Natural Frequency of Beams

Table 4.1 Beam bending theory

Table 4.2 Natural frequencies of single-span beams

Table 4.8 Natural frequencies of beams with concentrated masses

Table 4.3 Beam mode shapes and their derivatives (Ref. [9])

Table 4.4 Natural frequencies of two-span beams

Table 4.5 Natural frequencies of multi-span beams with pinned intermediate supports

Table 4.6 Buckling load and mode for single span beams

Table 4.7 Values of

λ

i

for beams with axial traction

Table 4.9 Natural frequencies of tapered and stepped beams

Table 4.10 Natural frequencies of beams with spring supports

Table 4.11 Section shear coefficients

Table 4.12 Natural frequencies of shear beams

Table 4.17 Natural frequencies of arcs, bends, and frames

Table 4.13 Natural frequencies of longitudinal vibration, continued

Table 4.14 Torsion constants of beam cross sections

Table 4.15 Natural frequencies of shafts in torsion

Table 4.16 Natural frequencies of circular rings

Chapter 5: Natural Frequency of Plates and Shells

Table 5.1 Plate bending stress and strain theory

Table 5.2 Natural frequencies of round, annular, and elliptical plates

Table 5.5 Natural frequencies of grillages, stiffened, and orthotropic plates

Table 5.3 Natural frequency of rectangular plates

Table 5.4 Natural frequencies of parallelogram, triangular, and other plates

Table 5.6 Cylindrical shell stress and strain theory

Table 5.7 Natural frequencies of cylindrical shells

Table 5.8 Natural frequencies of spherical shells

Table 5.9 Natural frequencies of conical shells

Chapter 6: Acoustics and Fluids

Table 6.1 Speed of sound

Table 6.2 Acoustic wave equation

Table 6.3 Acoustic wave propagation

Table 6.4 Duct and room acoustics

Table 6.5 Acoustic natural frequencies

Table 6.6 Cumulative modal density

Table 6.7 Natural frequencies of surface waves in basins

Table 6.7 Natural frequencies of waves in U-tubes, continued

Table 6.8 Ship motions

Table 6.10 Added mass of bodies

Table 6.9 Added mass of sections

Chapter 7: Forced Vibration

Table 7.1 Harmonically forced vibration of spring-mass systems

Table 7.2 Harmonic forced vibration of continuous systems

Table 7.3 Transient vibration of a spring-mass system

Table 7.4 Transient vibration of continuous structures

Table 7.6 Equivalent static load and approximate response

Table 7.5 Response to random broad band excitation

Chapter 8: Properties of Solids, Liquids, and Gases

Table 8.1 Modulus of elasticity and Poisson's ratio of ferrous and nickel alloys

Table 8.3 Modulus of elasticity and density of various metals

Table 8.4 Modulus and density of wood

a

Table 8.5 Density and modulus of elasticity of plastics, glass, fibers, and laminates

Table 8.6 Properties of freshwater

Table 8.8 Properties of various liquids

a

Table 8.9 Properties of various gases

Table 8.10 Properties of air – metric units, 1 atm pressure

Appendix A: Approximate Methods for Natural Frequency

Table A.1 Approximate methods for computing natural frequencies

Appendix B: Numerical Integration of Newton's Second Law

Table 2.1 Numerical integration of Newton's second law

Appendix C: Standard Octaves and Sound Pressure

Table C.1 Decibel scales of sound

Table C.2 Standard one and one-third octave bands

Formulas for Dynamics, Acoustics and Vibration

Robert D. Blevins

This edition first published 2016

© 2016 John Wiley & Sons, Ltd

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought

Library of Congress Cataloging-in-Publication Data

Blevins, Robert D.

Formulas for dynamics, acoustics and vibration / Robert D. Blevins.

pages cm

Includes bibliographical references and index.

ISBN 978-1-119-03811-5 (cloth)

1. Engineering mathematics–Formulae. 2. Dynamics–Mathematics. 3. Vibration–Mathematical

models. I. Title.

TA332.B59 2015

620.001′51– dc23

2015015479

A catalogue record for this book is available from the British Library.

Preface

This book is an illustrated compilation of formulas for solving dynamics, acoustics, and vibration problems in engineering. Over 1000 formulas for solution of practical problems are presented in 60 tables for quick reference by engineers, designers, and students.

The majority of the formulas are exact solutions for dynamics, acoustics, and vibrations. Their origin lies in the development of calculus and its application to physics by Sir Isaac Newton and Gottfried Wilhelm Leibnitz in the 17th century. Sir Isaac Newton (1642–1727) formulated the inverse square law of gravitation and proved all three of Kepler's laws of planetary motion. Leonhard Euler (1707–1783) derived the equations of rigid body dynamics and wave propagation. In 1732, Daniel Bernoulli described the normal modes of hanging chains in terms of Bessel functions. Gustav Kirchhoff developed the theory of plate vibrations in 1850. The modern age of exact classical dynamic solutions began with publication of Lord Rayleigh's The Theory of Sound (first edition 1877), followed by A. E. H. Love's A Treatise on the Mathematical Theory of Elasticity (first edition 1892–1893), Horace Lamb's The Dynamical Theory of Sound (first edition 1910), and Stephen Timoshenko's Vibration Problems in Engineering (first edition 1928). These books are foundations of vibration analysis and they are still in print.

Formulas in this book span the technical literature from the second edition of The Theory of Sound (1894) to technical journals of the 21st century. Most were generated by modern contributors including Arthur W. Leissa, H. Max Irvine, Phil McIver, Werner Soedel, Daniel J. Gorman, and W. Kitpornchai. Systems too large and complex for exact solution by formulas are analyzed approximately with numerical methods (Appendices A and B) that are formulated, interpreted, and checked with exact classical formulas and reasoning.

As structures become lighter and more flexible, they become more prone to vibration and dynamic failure. Vibration failures include wind-induced vibration of the Tacoma Narrows suspension bridge in 1940 and the steam flow-induced radiation leaks from vibrating tubes that shut down the San Onofre Nuclear Generating Station in 2012. Dynamic and vibration analysis differs from static analysis by including time and inertia. Dynamic analysis starts with identification of a flexible system in a dynamic environment. The next step is calculation of the system mass, stiffness, natural frequency, mode shape, and transmissibility. The third step is calculation of dynamic response to the time-dependent loads. Then design changes can be made to increase reliability and reduce noise and vibration.

Methods to reduce unwanted noise and vibration fatigue failures include the following: (1) increase stiffness, (2) increase damping to reduce amplitude, (3) reduce excitation with load paths to ground, (4) detune natural frequencies from excitation frequencies, (5) reduce stress concentrations and installation preload, (6) use fatigue- and wear-resistant materials, and (7) inspect and repair to limit propagating damage. Increasing stiffness solves many vibration problems, but the size and weight of the stiffened structure may not meet design goals. Design optimization of light, flexible systems requires dynamic analysis and test.

Formulas and data for dynamic analysis are presented in the tables. The text discusses examples, explanation, and some derivations. Chapter 1 provides definitions, symbols, units, and geometric properties. Chapter 2 provides dynamics of point masses and rigid bodies. Chapters 3 through 5 provide natural frequencies and mode shapes for elastic beams, plates, shells, and spring–mass systems. Chapter 6 provides fluid and acoustic solutions to the wave equation and added mass. Chapter 7 presents formulas for the response of elastic structures to sinusoidal, transient, and random loads. Chapter 8 presents properties of structural solids, liquids, and gases that support the formulas provided in Chapters 1 through 7. The property data is given in both SI (metric) and US customary (ft-lb) units. Appendices A and B present approximate and numerical methods for natural frequency and time history analysis.

The formulas are ready to apply with pencils, spreadsheets, and digital devices. There are 35 worked-out examples. Supporting content and software is available at www.aviansoft.com. The table formats were developed by Raymond J. Roark. Many results from my previous formulas book are included and updated with knowledge gleaned over the intervening 30 years. Many contributed to this book. The author would like to thank György Szász, Dr. Hwa-Wan Kwan, Mark Holcomb, Ryan Bolin, Larry Julyk, Alex Rasche, and Keith Riley for reviewing chapters. Beocky Yalof skillfully edited the original manuscript. This book is dedicated to the individuals who developed the solutions herein.

Chapter 1Definitions, Units, and Geometric Properties

1.1 Definitions

Acceleration

The rate of change of velocity. The second derivative of displacement with respect to time.

Added mass

The mass of fluid entrained by a vibrating structure immersed in fluid. The natural frequency of vibration of a structure surrounded by fluid is lower than that of the structure vibrating in a vacuum owing to the added mass of fluid (

Tables 6.9

and

6.10

).

Amplitude

The maximum excursion from the equilibrium position during a vibration cycle.

Antinode

Point of maximum vibration amplitude during free vibration in a single mode. See node.

Attenuation, acoustic

Difference in sound or vibration between two points along the path of energy propagation. Also see damping, insertion loss.

Bandwidth

The range of frequencies through which vibration energy is transferred.

Beam

Slender structure whose cross section and deflection vary along a single axis. Beams support tension, compression, and bending loads. Shear deformations are negligible compared to bending deformations in slender beams.

Boundary condition

Time-independent constraints that represent idealized structural interfaces, such as zero force, displacement, velocity, rotation, or pressure.

Broad band

A process consisting of a large number of component frequencies, none of which is dominant, distributed over a broad frequency band, usually more than one octave. Also see narrow band and tone.

Bulk modulus of elasticity