Giancarlo Genta

Vibration Dynamics and Control

Spri ringer

Contents

Series Preface vii

Preface ix

Symbols xxi

Introduction 1

I Dynamics of Linear, Time Invariant, Systems 23

1 Conservative Discrete Vibrating Systems 25 1.1 Oscillator with a single degree of freedom 25 1.2 Systems with many degrees of freedom 28 1.3 Coefficients of influence and compliance matrix 32 1.4 Lagrange equations 32 1.5 Configuration space 40 1.6 State space 41 1.7 Exercises 46

2 Equations in the Time, Frequency, and Laplace Domains 49 2.1 Equations in the time domain 49 2.2 Equations in the frequency domain 50 2.3 Equations in the Laplace domain 54 2.4 Exercises 57

xiv Contents

3 Damped Discrete Vibrating Systems 59 3.1 Linear viscous damping 59 3.2 State-space approach 65 3.3 Rayleigh dissipation function 67 3.4 Structural or hysteretic damping 72 3.5 Non-viscous damping 78 3.6 Structural damping as nonviscous damping 84 3.7 Systems with frequency-dependent parameters 90 3.8 Exercises 92

4 Free Vibration of Conservative Systems 93 4.1 Systems with a single degree of freedom 93 4.2 Systems with many degrees of freedom 97 4.3 Properties of the eigenvectors 99 4.4 Uncoupling of the equations of motion 101 4.5 Modal participation factors 106 4.6 Structural modification I l l 4.7 Exercises 113

5 Free Vibration of Damped Systems 115 5.1 Systems with a single degree of freedom-

viscous damping 115 5.2 Systems with a single degree of freedom-hysteretic

damping 122 5.3 Systems with a single degree of freedom -

nonviscous damping 123 5.4 Systems with many degrees of freedom 125 5.5 Uncoupling the equations of motion: space of the

configurations 126 5.6 Uncoupling the equations of motion: state space 131 5.7 Exercises 133

6 Forced Response in the Frequency Domain: Conservative Systems 135 6.1 System with a single degree of freedom 135 6.2 System with many degrees of freedom 142 6.3 Modal computation of the response 144 6.4 Coordinate transformation based on Ritz vectors 148 6.5 Response to periodic excitation 151 6.6 Exercises 151

7 Forced Response in the Frequency Domain: Damped Systems 153 7.1 System with a single degree of freedom: steady-state

response 153

Contents xv

7.2 System with a single degree of freedom: nonstationary response 160

7.3 System with structural damping 161 7.4 System with many degrees of freedom 163 7.5 Modal computation of the response 165 7.6 Multi-degrees of freedom systems with hysteretic

damping 168 7.7 Response to periodic excitation 169 7.8 The dynamic vibration absorber 170 7.9 Parameter identification 175 7.10 Exercises 177

8 Response to Nonperiodic Excitation 179 8.1 Impulse excitation 179 8.2 Step excitation 181 8.3 Duhamel's integral 184 8.4 Solution using the transition matrix 187 8.5 Solution using Laplace transforms 187 8.6 Numerical integration of the equations of motion 189 8.7 Exercises 191

9 Short Account of Random Vibrations 193 9.1 General considerations 193 9.2 Random forcing functions 194 9.3 White noise 197 9.4 Probability distribution 198 9.5 Response of linear systems 199 9.6 Exercises 211

10 Reduction of the Number of Degrees of Freedom 213 10.1 General considerations 213 10.2 Static reduction of conservative models 214 10.3 Guyan reduction 216 10.4 Damped systems 218 10.5 Dynamic reduction 219 10.6 Modal reduction 219 10.7 Component-mode synthesis 220 10.8 Exercises 224

11 Controlled Linear Systems 225 11.1 General considerations 225 11.2 Control systems 227 11.3 Controllability and observability 231 11.4 Open-loop control 232 11.5 Closed-loop control 234 11.6 Basic control laws 237

xvi Contents

11.7 Delayed control 246 11.8 Control laws with frequency-dependent gains 248 11.9 Robustness of the controller 253 11.10 State feedback and state observers 253 11.11 Control design 257 11.12 Modal approach to structural control 259 11.13 Exercises 261

12 Vibration of Beams 263 12.1 Beams and bars 263 12.2 Axial behavior of straight bars 265 12.3 Torsional vibrations of straight beams 279 12.4 Flexural vibrations of straight beams: The Euler-Bernoulli

beam 280 12.5 Bending in the yz-pl&ne 305 12.6 Coupling between flexural and torsional vibrations

of straight beams 306 12.7 The prismatic homogeneous Timoshenko beam 310 12.8 Interaction between axial forces and flexural vibrations

of straight beams 314 12.9 Exercises 318

13 General Continuous Linear Systems 321 13.1 Elastic continuums 321 13.2 Flexural vibration of rectangular plates 324 13.3 Vibration of membranes 329 13.4 Propagation of elastic waves in taut strings 332 13.5 Propagation of sound waves in pipes 335 13.6 Linear continuous systems with structural damping . . . . 337 13.7 Exercises 338

14 Discretization of Continuous Systems 341 14.1 Overview of discretization techniques 341 14.2 The assumed-modes methods 343 14.3 Lumped-parameters methods 349 14.4 Transfer-matrices methods 350 14.5 Holtzer's method for torsional vibrations of shafts 353 14.6 Myklestadt's method for flexural vibrations of beams . . . 357 14.7 Exercises 360

15 The Finite Element Method 363 15.1 Element characterization 363 15.2 Timoshenko beam element 367 15.3 Mass and spring elements 376 15.4 Plate element: Kirchoff formulation 377 15.5 Plate element: Mindlin formulation 380

Contents xvii

15.6 Brick elements 382 15.7 Isoparametric elements 384 15.8 Some considerations on the consistent mass matrix . . . . 387 15.9 Assembling the structure 388 15.10 Constraining the structure 390 15.11 Dynamic stiffness matrix 393 15.12 Damping matrices 395 15.13 Finite elements in time 396 15.14 Exercises 399

16 Dynamics of Multibody Systems 401 16.1 General considerations 401 16.2 Lagrange equations in terms

of pseudo-coordinates 407 16.3 Motion of a rigid body 411 16.4 Exercises 420

17 Vibrating Systems in a Moving Reference Frame 421 17.1 General considerations 421 17.2 Vibrating system on a rigid carrier 426 17.3 Lumped-parameters discretization 430 17.4 Modal discretization 436 17.5 Planar systems 441 17.6 Beam attached to a rigid body: planar dynamics 442 17.7 The rotating beam 445 17.8 Exercises 446

II Dynamics of Nonlinear and t ime Variant Systems 449

18 Free Motion of Conservative Nonlinear Systems 451 18.1 Linear versus nonlinear systems 451 18.2 Equation of motion 452 18.3 Free oscillations 456 18.4 Direct integration of the equations of motion 458 18.5 Harmonic balance 463 18.6 Ritz averaging technique 466 18.7 Iterative techniques 469 18.8 Perturbation techniques 471 18.9 Solution in the state plane 475 18.10 Exercises 478

19 Forced Response of Conservative Nonlinear Systems 481 19.1 Approximate evaluation of the response to a harmonic

forcing function 481 19.2 Undamped Duffing's equation 483

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19.3 Conclusions 496 19.4 Exercises 498

20 Free Motion of Damped Nonlinear Systems 501 20.1 Nonlinear damping 501 20.2 Motion about an equilibrium position (in the small) . . . 503 20.3 Direct integration of the equation of motion 504 20.4 Equivalent damping 508 20.5 Solution in the state plane 509 20.6 Stability in the small 512 20.7 The Van der Pol oscillator 514 20.8 Exercises 516

21 Forced Response of Damped Nonlinear Systems 519 21.1 Reduction of the size of the problem 519 21.2 First approximation of the response to a harmonic forcing

function 521 21.3 Duffing's equation with viscous damping 525 21.4 Duffing's equation with structural damping 528 21.5 Backbone and limit envelope 529 21.6 Multiple Duffing equations 534 21.7 Approximated sub- and super-harmonic response 539 21.8 Van der Pol method: stability of the steady-state

solution 541 21.9 Strongly nonlinear systems 545 21.10 Poincare mapping 546 21.11 Chaotic vibrations 550 21.12 Exercises 555

22 Time Variant and Autoparametric Systems 557 22.1 Linear time-variant systems 557 22.2 Hill's equation 559 22.3 Pendulum on a moving support: Mathieu equation . . . . 560 22.4 The elastic pendulum 566 22.5 Autoparametric systems 569 22.6 Exercises 575

III Dynamics of Rotating and Reciprocating Machinery 577

23 Elementary Rot or dynamics: The Jeffcott Rotor 579 23.1 Elementary rotordynamics 579 23.2 Vibrations of rotors: the Campbell diagram 581

Contents xix

23.3 Forced vibrations of rotors: critical speeds 584 23.4 Fields of instability 587 23.5 The undamped linear Jeffcott rotor 590 23.6 Jeffcott rotor with viscous damping 596 23.7 Jeffcott rotor with structural damping 606 23.8 Equations of motion in real coordinates 608 23.9 Stability in the supercritical field 608 23.10 Acceleration through the critical speed 610 23.11 Exercises 614

24 Dynamics of Multi-Degrees-of-Freedom Rotors 615 24.1 Model with 4 degrees of freedom: gyroscopic effect . . . . 615 24.2 Rotors with many degrees of freedom 630 24.3 Real versus complex coordinates 634 24.4 Fixed versus rotating coordinates 636 24.5 State-space equations 637 24.6 Static solution 637 24.7 Critical-speed computation 638 24.8 Unbalance response 639 24.9 Campbell diagram and roots locus 640 24.10 Acceleration of a torsionally stiff rotor 641 24.11 Exercises 645

25 Nonisotropic Rotating Machines 647 25.1 Jeffcott rotor on nonisotropic supports 648 25.2 Nonisotropic Jeffcott rotor 652 25.3 Secondary critical speeds due to rotor weight 656 25.4 Equation of motion for an anisotropic machine 657 25.5 Exercises 670

26 Nonlinear Rotors 671 26.1 General considerations 671 26.2 Nonlinear Jeffcott rotor: equation of motion 672 26.3 Unbalance response 673 26.4 Free circular whirling 676 26.5 Stability of the equilibrium position 678 26.6 Exercises 688

27 Dynamic Problems of Rotating Machines 689 27.1 Rotors on hydrodynamic bearings 689 27.2 Dynamic study of rotors on magnetic bearings 706 27.3 Flexural vibration dampers 723 27.4 Signature of rotating machinery 728 27.5 Exercises 732

xx Contents

28 Rotor Balancing 733 28.1 General considerations 733 28.2 Rigid rotors 734 28.3 Flexible rotors 737 28.4 Exercises 744

29 Torsional Vibration of Crankshafts 745 29.1 Specific problems of reciprocating machines 745 29.2 Equivalent system for the study of torsional vibrations . . 746 29.3 Computation of the natural frequencies 758 29.4 Forced vibrations 762 29.5 Torsional instability of crank mechanisms 784 29.6 Exercises 787

30 Vibration Control in Reciprocating Machines 789 30.1 Dissipative dampers 789 30.2 Damped vibration absorbers 793 30.3 Rotating-pendulum vibration absorbers 796 30.4 Experimental measurement of torsional vibrations 799 30.5 Axial vibration of crankshafts 800 30.6 Short outline on balancing of reciprocating machines . . . 801 30.7 Exercises 805

A Solution Methods 807 A.l General considerations 807 A.2 Solution of linear sets of equations 808 A.3 Computation of eigenfrequencies 812 A.4 Solution of nonlinear sets of equations 822 A.5 Numerical integration in time of the equation of motion . 824

В Laplace Transform Pairs 831

Index 849