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Engineering Vibration Handbook

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Engineering Vibration Handbook

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Engineering Vibration Handbook

The effects of vibrations on the behavior of mechanical and structural systems are often of critical importance to their design, performance, and survival. For this reason the subject of mechanical vibrations is offered at both the advanced undergraduate level and graduate level at most engineering schools. I have taught vibrations to mechanical and aerospace engineering students, primarily seniors, for a number of years and have used a variety of textbooks in the process. As with many books of this type, the emphasis is often a matter of taste. Some texts emphasize mathematics, but generally fall short on physical interpretation and demonstrative examples, while others emphasize methodology and application but tend to oversimplify the mathematical development and fail to stress the fundamental principles. Moreover, both types fail to stress the underlying mechanics and physics to a satisfactory degree, if at all. For these reasons, there appeared to be a need for a textbook that couples thorough mathematical development and physical interpretation, and that emphasizes the mechanics and physics of the phenomena. The book would need to be readable for students with the background afforded by a typical university engineering curriculum, and would have to be self-contained to the extent that concepts are developed, advanced and abstracted using that background as a base. The present volume has been written to meet these goals and fill the apparent void.

Engineering Vibrations provides a systematic and unified presentation of the subject of mechanical and structural vibrations, emphasizing physical interpretation, fundamental principles and problem solving, coupled with rigorous mathematical development in a form that is readable to advanced undergraduate and graduate university students majoring in engineering and related fields. Abstract concepts are developed and advanced from principles familiar to the student, and the interaction of theory, numerous illustrative examples and discussion form the basic pedagogical approach. The text, which is extensively illustrated, gives the student a thorough understanding of the basic concepts of the subject, and enables him or her to apply these principles and techniques to any problem of interest. In addition, the pedagogy encourages the reader’s physical sense and intuition, as well as analytical skills. The text also provides the student with a solid background for further formal study and research, as well as for self study of specialized techniques and more advanced topics. Particular emphasis is placed on developing a connected string of ideas, concepts and techniques that are sequentially advanced and generalized throughout the text. In this way, the reader is provided with a thorough background in the vibration of single degree of freedom systems, discrete multi-degree of freedom systems, onedimensional continua, and the relations between each, with the subject viewed as a whole. Some distinctive features are as follows. The concept of mathematical modeling is introduced in the first chapter and the question of validity of such models is emphasized throughout. An extensive review of elementary dynamics is presented as part of the introductory chapter. A discussion and demonstration of the underlying physics accompany the introduction of the phenomenon of resonance. A distinctive approach incorporating generalized functions and elementary dynamics is used to develop the general impulse response. Structural damping is introduced and developed from first principle as a phenomenological theory, not as a heuristic empirical result as presented in many other texts. Continuity between basic vector operations including the scalar product and normalization in three-dimensions and their extensions to N-dimensional space is clearly established. General (linear) viscous damping, as well as Rayleigh (proportional) damping, of discrete multi-degree of freedom systems is discussed, and represented in state space. Correspondence between discrete and continuous systems is established and the concepts of linear differential operators are introduced. A thorough development of the mechanics of pertinent 1-D continua is presented, and the dynamics and vibrations of various structures are studied in depth. These include axial and torsional motion of rods and transverse motion of strings, transverse motion of Euler-Bernoulli Beams and beam-columns, beams on elastic foundations, Rayleigh Beams and Timoshenko Beams. Unlike in other texts, the Timoshenko Beam problem is stated and solved in matrix form. Operator notation is introduced throughout. In this way, all continua discussed are viewed from a unified perspective. Case studies provide a basis for comparison of the various beam theories with one another and demonstrate quantitatively the limitations of single degree of freedom approximations. Such studies are examined both as examples and as exercises for the student.

The background assumed is typical of that provided in engineering curricula at U.S. universities. The requisite background includes standard topics in differential and integral calculus, linear differential equations, linear algebra, boundary value problems and separation of variables as pertains to linear partial differential equations of two variables, sophomore level dynamics and mechanics of materials. MATLAB is used for root solving and related computations, but is not required. A certain degree of computational skill is, however, desirable.

The text can basically be partitioned into preliminary material and three major parts: single degree of freedom systems, discrete multi-degree of freedom systems, and one-dimensional continua. For each class of system the fundamental dynamics is discussed and free and forced vibrations under various conditions are studied. A breakdown of the eleven chapters that comprise the text is provided below. The first chapter provides introductory material and includes discussions of degrees of freedom, mathematical modeling and equivalent systems, a review of complex numbers and an extensive review of elementary dynamics. Chapters 2 through 4 are devoted to free and forced vibration of single degree of freedom systems. Chapter 2 examines free vibrations and includes undamped, viscously damped and Coulomb damped systems. An extensive discussion of the linear and nonlinear pendulum is also included. In Chapter 3 the response to harmonic loading is established and extended to various applications including support excitation, rotating imbalance and whirling of shafts. The mathematical model for structural damping is developed from first principle based on local representation of the body as comprised of linear hereditary material. The chapter closes with a general Fourier Series solution for systems subjected to general periodic loading and its application. The responses of systems to nonperiodic loading, including impulse, step and ramp loading and others, as well as general loading, are discussed in Chapter 4. The Dirac Delta Function and the Heaviside Step Function are first introduced as generalized functions. The relation and a discussion of impulsive and nonimpulsive forces follow. The general impulse response is then established based on application of these concepts with basic dynamics. The responses to other types of loading are discussed throughout the remainder of the chapter. Chapter 5, which is optional and does not affect continuity, covers Laplace transforms and their application as an alternate, less physical/nonphysical, approach to problems of vibration of single degree of freedom systems. The dynamics of multi-degree of freedom systems is studied in Chapter 6. The first part of the chapter addresses Newtonian mechanics and the derivation of the equations of motion of representative systems in this context. It has been my experience (and I know I’m not alone in this) that many students often have difficulty and can become preoccupied or despondent with setting up the equations of motion for a given system. As a result of this they often lose sight of, or never get to, the vibrations problem itself. To help overcome this difficulty, Lagrange’s equations are developed in the second part of Chapter 6, and a methodology and corresponding outline are established to derive the equations of motion for multi-degree of freedom systems. Once mastered, this approach provides the student a direct means of deriving the equations of motion of complex multi-degree of freedom systems. The instructor who chooses not to cover Lagrange’s equations may bypass these sections. The chapter closes with a fundamental discussion of the symmetry of the mass, stiffness and damping matrices with appropriate coordinates.

The free vibration problem for multi-degree of freedom systems with applications to various systems and conditions including semi-definite systems is presented in Chapter 7. The physical meanings of the modal vectors for undamped systems are emphasized and the properties of the modal vectors are discussed. The concepts of the scalar product, orthogonality and normalization of three-dimensional vectors are restated in matrix form and abstracted to N-dimensional space, where they are then discussed in the context of the modal vectors. The chapter closes with extensive discussions of the free vibration of discrete systems with viscous damping. The problem is examined in both N-dimensional space and in the corresponding state space. Analogies to the properties of the modal vectors for undamped systems are then abstracted to the complex eigenvectors for the problem of damped systems viewed in state space. Forced vibration of discrete multi-degree of freedom systems is studied in Chapter 8. A simple matrix inversion approach is first introduced for systems subjected to harmonic excitation. The introductory section concludes with a discussion of the simple vibration absorber. The concepts of coordinate transformations, principal coordinates and modal coordinates are next established. The bulk of the chapter is concerned with modal analysis of undamped and proportionally damped systems. The chapter concludes with these procedures abstracted to systems with general (linear) viscous damping in both N-dimensional space and in state space. The dynamics of one-dimensional continua is discussed in Chapter 9. Correlation between discrete and continuous systems is first established, and the concept of differential operators is introduced. The correspondence between vectors and functions is made evident as is that of matrix operators and differential operators. This enables the reader to identify the dynamics of continua as an abstraction of the dynamics of discrete systems. The scalar product and orthogonality in function space then follow directly. The kinematics of deforming media is then developed for both linear and geometrically nonlinear situations. The equations governing various onedimensional continua are established, along with corresponding possibilities for boundary conditions. It has been my experience that students have difficulty in stating all but the simplest boundary conditions when approaching vibrations problems. This discussion will enlighten the reader in this regard and aid in alleviating that problem. Second order systems that are studied include longitudinal and torsional motion of elastic rods and transverse motion of strings. Various beam theories are developed from a general, first principle, point of view with the limitations of each evident from the discussion. Euler-Bernoulli Beams and beam-columns, Rayleigh Beams and Timoshenko Beams are discussed in great detail, as is the dynamics of accelerating beam-columns. The various operators pertinent to each system are summarized in a table at the end of the chapter.

The general free vibration of one-dimensional continua is established in Chapter 10 and applied to the various continua discussed in Chapter 9. The operator notation introduced earlier permits the student to perceive the vibrations problem for continua as merely an extension of that discussed for discrete systems. Case studies are presented for various rods and beams, allowing for a direct quantitative evaluation of the one degree of freedom approximation assumed in the first five chapters. It further allows for direct comparison of the effectiveness and validity of the various beam theories. Properties of the modal functions, including the scalar product, normalization and orthogonality are established. The latter is then used in the evaluation of amplitudes and phase angles. Forced vibration of one-dimensional continua is discussed in Chapter 11. The justification for generalized Fourier Series representation of the response is established and modal analysis is applied to the structures of interest under various loading conditions.

The material covered in this text is suitable for a two-semester sequence or a one-semester course. The instructor can choose appropriate chapters and/or sections to suit the level, breadth and length of the particular course being taught. To close, I would like to thank Professor Haim Baruh, Professor Andrew Norris, Ms. Pamela Carabetta, Mr. Lucian Iorga and Ms. Meghan Suchorsky, all of Rutgers University, for reading various portions of the manuscript and offering helpful comments and valuable suggestions. I would also like to express my gratitude to Ms. Carabetta for preparing the index. I wish to thank Glen and Maria Hurd for their time, effort and patience in producing the many excellent drawings for this volume. Finally, I wish to thank all of those students, past and present, who encouraged me to write this book.

William J. Bottega

TOC:

1. PRELIMINARIES 1
1.1 Degrees of Freedom..................... 2
1.2 Equivalent Systems...................... 6
1.2.1 Extension/Contraction of Elastic Rods........... 6
1.2.2 Bending of Elastic Beams.................. 8
1.2.3 Torsion of Elastic Rods................. 16
1.2.4 Floating Bodies.................... 20
1.2.5 The Viscous Damper................. 22
1.2.6 Aero/Hydrodynamic Damping (Drag)............ 24
1.3 Springs Connected in Parallel and in Series........... 25
1.3.1 Springs in Parallel................... 26
1.3.2 Springs in Series...................26
1.4 A Brief Review of Complex Numbers............. 28
1.5 A Review of Elementary Dynamics................ 30
1.5.1 Kinematics of Particles................. 31
1.5.2 Kinetics of a Single Particle................ 38
1.5.3 Dynamics of Particle Systems................ 49
1.5.4 Kinematics of Rigid Bodies................ 56
1.5.5 (Planar) Kinetics of Rigid Bodies............. 60
1.6 Concluding Remarks..................... 66
Bibliography......................... 67
Problems......................... 67

2. FREE VIBRATION OF SINGLE DEGREE OF FREEDOM SYSTEMS 75
2.1 Free Vibration of Undamped Systems............. 75
2.1.1 Governing Equation and System Response........... 76
2.1.2 The Effect of Gravity................. 87
2.1.3 Work and Energy.................... 93
2.1.4 The Simple Pendulum................... 94
2.2 Free Vibration of Systems with Viscous Damping.......... 109
2.2.1 Equation of Motion and General System Response..... 109
2.2.2 Underdamped Systems................ 111
2.2.3 Logarithmic Decrement................ 115
2.2.4 Overdamped Systems.................. 119
2.2.5 Critically Damped Systems............... 121
2.3 Coulomb (Dry Friction) Damping...............127
2.3.1 Stick-Slip Condition................. 127
2.3.2 System Response...................129
2.4 Concluding Remarks................... 133
Bibliography....................... 135

3. FORCED VIBRATION OF SINGLE DEGREE OF FREEDOM
SYSTEMS – 1: PERIODIC EXCITATION 143
3.1 Standard Form of the Equation of Motion........... 143
3.2 Superposition...................... 144
3.3 Harmonic Forcing..................... 147
3.3.1 Formulation..................... 147
3.3.2 Steady State Response of Undamped Systems....... 149
3.3.3 Steady State Response of Systems with Viscous Damping... 162
3.3.4 Force Transmission and Vibration Isolation........179
3.4 Structural Damping..................... 184
3.4.1 Linear Hereditary Materials.............. 185
3.4.2 Steady State Response of Linear Hereditary Materials...... 186
3.4.3 Steady State Response of Single Degree of Freedom Systems... 189
3.5 Selected Applications..................... 192
3.5.1 Harmonic Motion of the Support............ 192
3.5.2 Unbalanced Motor................... 201
3.5.3 Synchronous Whirling of Rotating Shafts.......... 206
3.6 Response to General Periodic Loading............ 211
3.6.1 General Periodic Excitation.............. 211
3.6.2 Steady State Response................ 213
3.7 Concluding Remarks................... 219
Bibliography....................... 220
Problems.......................... 220

4. FORCED VIBRATION OF SINGLE DEGREE OF FREEDOM
SYSTEMS – 2: NONPERIODIC EXCITATION 229
4.1 Two Generalized Functions.................. 229
4.1.1 The Dirac Delta Function (Unit Impulse)......... 230
4.1.2 The Heaviside Step Function (Unit Step)......... 232
4.1.3 Relation Between the Unit Step and the Unit Impulse..... 233
4.2 Impulse Response..................... 234
4.2.1 Impulsive and Nonimpulsive Forces............ 234
4.2.2 Response to an Applied Impulse............. 235
4.3 Response to Arbitrary Excitation............... 239
4.4 Response to Step Loading................. 241
4.5 Response to Ramp Loading.................. 246
4.6 Transient Response by Superposition............... 248
4.6.1 The Rectangular Pulse................ 249
4.6.2 Linear Transition to Constant Load Level.......... 255
4.7 Shock Spectra....................... 257
4.8 Concluding Remarks................... 268
Bibliography....................... 269
Problems......................... 269

5. OPERATIONAL METHODS 273
5.1 The Laplace Transform.................. 273
5.1.1 Laplace Transforms of Basic Functions........... 274
5.1.2 Shifting Theorem.................. 276
5.1.3 Laplace Transforms of the Derivatives of Functions...... 277
5.1.4 Convolution.................... 278
5.2 Free Vibrations..................... 279
5.3 Forced Vibrations..................... 281
5.3.1 The Governing Equations................. 281
5.3.2 Steady State Response................ 282
5.3.3 Transient Response.................. 283
5.4 Concluding Remarks................... 285
Bibliography....................... 285
Problems......................... 285

6. DYNAMICS OF MULTI-DEGREE OF FREEDOM SYSTEMS 287
6.1 Newtonian Mechanics of Discrete Systems........... 288
6.1.1 Mass-Spring Systems................. 288
6.1.2 The Double Pendulum................ 296
6.1.3 Two-Dimensional Motion of a Rigid Frame......... 300
6.2 Lagrange’s Equations.................... 303
6.2.1 Virtual Work..................... 304
6.2.2 The Canonical Equations............... 306
6.2.3 Implementation................... 309
6.2.4 The Rayleigh Dissipation Function............ 321
6.3 Symmetry of the System Matrices.............. 324
6.3.1 The Stiffness Matrix................. 324
6.3.2 The Mass Matrix................... 327
6.3.3 The Damping Matrix.................. 328
6.4 Concluding Remarks................... 329
Bibliography....................... 330
Problems......................... 330

7. FREE VIBRATION OF MULTI-DEGREE OF FREEDOM SYSTEMS 341
7.1 The General Free Vibration Problem and Its Solution....... 341
7.2 Unrestrained Systems.................... 371
7.3 Properties of Modal Vectors................. 374
7.3.1 The Scalar Product................... 375
7.3.2 Orthogonality..................... 377
7.3.3 Normalization.................... 384
7.4 Systems with Viscous Damping................. 387
7.4.1 System Response.................. 387
7.4.2 State Space Representation............... 394
7.5 Evaluation of Amplitudes and Phase Angles.......... 400
7.5.1 Undamped Systems.................. 401
7.5.2 Systems with General Viscous Damping......... 403
7.6 Concluding Remarks................... 404
Bibliography....................... 405
Problems.......................... 405

8. FORCED VIBRATION OF MULTI-DEGREE OF FREEDOM
SYSTEMS 415
8.1 Introduction........................ 416
8.1.1 Steady State Response to Harmonic Excitation........ 416
8.1.2 The Simple Vibration Absorber.............. 418
8.2 Modal Coordinates.................... 422
8.2.1 Principal Coordinates.................. 422
8.2.2 Coordinate Transformations.............. 424
8.2.3 Modal Coordinates................... 427
8.3 General Motion in Terms of the Natural Modes........... 431
8.3.1 Linear Independence of the Set of Modal Vectors....... 431
8.3.2 Modal Expansion.................. 432
8.4 Decomposition of the Forced Vibration Problem......... 433
8.5 Solution of Forced Vibration Problems............ 440
8.6 Mode Isolation....................... 468
8.7 Rayleigh Damping.................... 474
8.8 Systems with General Viscous Damping............ 479
8.8.1 Steady State Response to Harmonic Excitation........ 480
8.8.2 Eigenvector Expansion................483
8.8.3 Decomposition of the Forced Vibration Problem...... 484
8.8.4 Solution of Forced Vibration Problems........... 487
8.9 Concluding Remarks................... 498
Bibliography....................... 500
Problems......................... 500

9. DYNAMICS OF ONE-DIMENSIONAL CONTINUA 511
9.1 Mathematical Description of 1-D Continua........... 511
9.1.1 Correspondence Between Discrete and Continuous Systems.. 512
9.1.2 The Scalar Product and Orthogonality.......... 517
9.2 Characterization of Local Deformation............ 520
9.2.1 Relative Extension of a Material Line Element........ 521
9.2.2 Distortion..................... 524
9.3 Longitudinal Motion of Elastic Rods............. 525
9.4 Torsional Motion of Elastic Rods............... 530
9.5 Transverse Motion of Strings and Cables............ 534
9.6 Transverse Motion of Elastic Beams............. 539
9.6.1 Kinematical and Constitutive Relations........... 539
9.6.2 Kinetics........................ 543
9.6.3 Euler-Bernoulli Beam Theory............. 544
9.6.4 Rayleigh Beam Theory.................. 549
9.6.5 Timoshenko Beam Theory............... 552
9.7 Geometrically Nonlinear Beam Theory............ 558
9.8 Translating 1-D Continua.................. 562
9.8.1 Kinematics of a Material Particle............ 562
9.8.2 Kinetics....................... 565
9.9 Concluding Remarks................... 569
Bibliography....................... 570
Problems......................... 571

10. FREE VIBRATION OF ONE-DIMENSIONAL CONTINUA 579
10.1 The General Free Vibration Problem............. 579
10.2 Free Vibration of Uniform Second Order Systems......... 581
10.2.1 The General Free Vibration Problem and Its Solution..... 581
10.2.2 Longitudinal Vibration of Elastic Rods......... 582
10.2.3 Torsional Vibration of Elastic Rods........... 591
10.2.4 Transverse Vibration of Strings and Cables........ 595
10.3 Free Vibration of Euler-Bernoulli Beams........... 599
10.4 Free Vibration of Euler-Bernoulli Beam-Columns......... 617
10.5 Free Vibration of Rayleigh Beams.............. 622
10.6 Free Vibration of Timoshenko Beams.............. 627
10.7 Normalization of the Modal Functions............. 634
10.8 Orthogonality of the Modal Functions.............. 636
10.8.1 Systems Whose Mass Operators Are Scalar Functions.... 637
10.8.2 Second Order Systems................. 639
10.8.3 Euler-Bernoulli Beams and Beam-Columns........ 646
10.8.4 Rayleigh Beams.................. 652
10.8.5 Timoshenko Beams.................. 656
10.9 Evaluation of Amplitudes and Phase Angles.......... 660
10.9.1 Systems Possessing a Single Scalar Mass Operator..... 660
10.9.2 Rayleigh Beams.................. 666
10.9.3 Timoshenko Beams.................. 669
10.10 Concluding Remarks................... 673
Bibliography...................... 675
Problems......................... 675

11. FORCED VIBRATION OF ONE-DIMENSIONAL CONTINUA 683
11.1 Modal Expansion..................... 684
11.1.1 Linear Independence of the Modal Functions........ 684
11.1.2 Generalized Fourier Series.............. 685
11.2 Decomposition of the Forced Vibration Problem......... 686
11.3 Solution of Forced Vibration Problems............ 690
11.3.1 Axially Loaded Elastic Rods............. 690
11.3.2 Torsion of Elastic Rods............... 692
11.3.3 Strings and Cables................. 694
11.3.4 Euler-Bernoulli Beams................ 697
11.3.5 Rayleigh Beams.................. 708
11.3.6 Timoshenko Beams.................. 711
11.4 Concluding Remarks................... 714
Bibliography....................... 714
Problems......................... 715
INDEX............................. 721