Vibration Dynamics and Control

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1 Giancarlo Genta Vibration Dynamics and Control Spri ringer

Contents Series Preface Preface Symbols vii ix xxi Introduction 1 I Dynamics of Linear, Time Invariant, Systems 23 1 Conservative Discrete Vibrating Systems 25 1.

2 Contents Series Preface Preface Symbols vii ix xxi Introduction 1 I Dynamics of Linear, Time Invariant, Systems 23 1 Conservative Discrete Vibrating Systems Oscillator with a single degree of freedom Systems with many degrees of freedom Coefficients of influence and compliance matrix Lagrange equations Configuration space State space Exercises 46 2 Equations in the Time, Frequency, and Laplace Domains Equations in the time domain Equations in the frequency domain Equations in the Laplace domain 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.

3 xiv Contents 3 Damped Discrete Vibrating Systems Linear viscous damping State-space approach Rayleigh dissipation function Structural or hysteretic damping Non-viscous damping Structural damping as nonviscous damping Systems with frequency-dependent parameters Exercises 92 4 Free Vibration of Conservative Systems Systems with a single degree of freedom Systems with many degrees of freedom Properties of the eigenvectors Uncoupling of the equations of motion Modal participation factors Structural modification Ill 4.7 Exercises Free Vibration of Damped Systems Systems with a single degree of freedomviscous damping Systems with a single degree of freedom-hysteretic damping Systems with a single degree of freedom - nonviscous damping Systems with many degrees of freedom Uncoupling the equations of motion: space of the configurations Uncoupling the equations of motion: state space Exercises Forced Response in the Frequency Domain: Conservative Systems System with a single degree of freedom System with many degrees of freedom Modal computation of the response Coordinate transformation based on Ritz vectors Response to periodic excitation Exercises Forced Response in the Frequency Domain: Damped Systems 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.

4 Contents xv 7.2 System with a single degree of freedom: nonstationary response System with structural damping System with many degrees of freedom Modal computation of the response Multi-degrees of freedom systems with hysteretic damping Response to periodic excitation The dynamic vibration absorber Parameter identification Exercises Response to Nonperiodic Excitation Impulse excitation Step excitation Duhamel's integral Solution using the transition matrix Solution using Laplace transforms Numerical integration of the equations of motion Exercises Short Account of Random Vibrations General considerations Random forcing functions White noise Probability distribution Response of linear systems Exercises Reduction of the Number of Degrees of Freedom General considerations Static reduction of conservative models Guyan reduction Damped systems Dynamic reduction Modal reduction Component-mode synthesis Exercises Controlled Linear Systems General considerations Control systems Controllability and observability Open-loop control Closed-loop control 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.

5 xvi Contents 11.7 Delayed control Control laws with frequency-dependent gains Robustness of the controller State feedback and state observers Control design Modal approach to structural control Exercises Vibration of Beams Beams and bars Axial behavior of straight bars Torsional vibrations of straight beams Flexural vibrations of straight beams: The Euler-Bernoulli beam Bending in the yz-pl&ne Coupling between flexural and torsional vibrations of straight beams The prismatic homogeneous Timoshenko beam Interaction between axial forces and flexural vibrations of straight beams Exercises General Continuous Linear Systems Elastic continuums Flexural vibration of rectangular plates Vibration of membranes Propagation of elastic waves in taut strings Propagation of sound waves in pipes Linear continuous systems with structural damping Exercises Discretization of Continuous Systems Overview of discretization techniques The assumed-modes methods Lumped-parameters methods Transfer-matrices methods Holtzer's method for torsional vibrations of shafts Myklestadt's method for flexural vibrations of beams Exercises The Finite Element Method Element characterization Timoshenko beam element Mass and spring elements Plate element: Kirchoff formulation 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.

6 Contents xvii 15.6 Brick elements Isoparametric elements Some considerations on the consistent mass matrix Assembling the structure Constraining the structure Dynamic stiffness matrix Damping matrices Finite elements in time Exercises Dynamics of Multibody Systems General considerations Lagrange equations in terms of pseudo-coordinates Motion of a rigid body Exercises Vibrating Systems in a Moving Reference Frame General considerations Vibrating system on a rigid carrier Lumped-parameters discretization Modal discretization Planar systems Beam attached to a rigid body: planar dynamics The rotating beam Exercises 446 II Dynamics of Nonlinear and time Variant Systems Free Motion of Conservative Nonlinear Systems Linear versus nonlinear systems Equation of motion Free oscillations Direct integration of the equations of motion Harmonic balance Ritz averaging technique Iterative techniques Perturbation techniques Solution in the state plane Exercises Forced Response of Conservative Nonlinear Systems Approximate evaluation of the response to a harmonic forcing function Undamped Duffing's equation 483

xviii Contents 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.

7 xviii Contents 19.3 Conclusions Exercises Free Motion of Damped Nonlinear Systems Nonlinear damping Motion about an equilibrium position (in the small) Direct integration of the equation of motion Equivalent damping Solution in the state plane Stability in the small The Van der Pol oscillator Exercises Forced Response of Damped Nonlinear Systems Reduction of the size of the problem First approximation of the response to a harmonic forcing function Duffing's equation with viscous damping Duffing's equation with structural damping Backbone and limit envelope Multiple Duffing equations Approximated sub- and super-harmonic response Van der Pol method: stability of the steady-state solution Strongly nonlinear systems Poincare mapping Chaotic vibrations Exercises Time Variant and Autoparametric Systems Linear time-variant systems Hill's equation Pendulum on a moving support: Mathieu equation The elastic pendulum Autoparametric systems Exercises 575 III Dynamics of Rotating and Reciprocating Machinery Elementary Rot or dynamics: The Jeffcott Rotor Elementary rotordynamics 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.

8 Contents xix 23.3 Forced vibrations of rotors: critical speeds Fields of instability The undamped linear Jeffcott rotor Jeffcott rotor with viscous damping Jeffcott rotor with structural damping Equations of motion in real coordinates Stability in the supercritical field Acceleration through the critical speed Exercises Dynamics of Multi-Degrees-of-Freedom Rotors Model with 4 degrees of freedom: gyroscopic effect Rotors with many degrees of freedom Real versus complex coordinates Fixed versus rotating coordinates State-space equations Static solution Critical-speed computation Unbalance response Campbell diagram and roots locus Acceleration of a torsionally stiff rotor Exercises Nonisotropic Rotating Machines Jeffcott rotor on nonisotropic supports Nonisotropic Jeffcott rotor Secondary critical speeds due to rotor weight Equation of motion for an anisotropic machine Exercises Nonlinear Rotors General considerations Nonlinear Jeffcott rotor: equation of motion Unbalance response Free circular whirling Stability of the equilibrium position Exercises Dynamic Problems of Rotating Machines Rotors on hydrodynamic bearings Dynamic study of rotors on magnetic bearings Flexural vibration dampers Signature of rotating machinery 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.

9 xx Contents 28 Rotor Balancing General considerations Rigid rotors Flexible rotors Exercises Torsional Vibration of Crankshafts Specific problems of reciprocating machines Equivalent system for the study of torsional vibrations Computation of the natural frequencies Forced vibrations Torsional instability of crank mechanisms Exercises Vibration Control in Reciprocating Machines Dissipative dampers Damped vibration absorbers Rotating-pendulum vibration absorbers Experimental measurement of torsional vibrations Axial vibration of crankshafts Short outline on balancing of reciprocating machines 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

Table of Contents. Preface... 13

Table of Contents. Preface... 13 Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...