Mechanical Engineering SeriesFrederick F. LingEditor-in-Chief

Mechanical Engineering Series

G. Genta, Vibration Dynamics and Control

R. Firoozian, Servomotors and Industrial Control Theory

G. Genta and L. Morello, The Automotive Chassis, Volumes 1 & 2

F. A. Leckie and D. J. Dal Bello, Strength and Stiffness of Engineering Systems

Wodek Gawronski, Modeling and Control of Antennas and Telescopes

Makoto Ohsaki and Kiyohiro Ikeda, Stability and Optimization of Structures: GeneralizedSensitivity Analysis

A.C. Fischer-Cripps, Introduction to Contact Mechanics, 2nd ed.

W. Cheng and I. Finnie, Residual Stress Measurement and the Slitting Method

J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory Methodsand Algorithms, 3rd ed.

J. Angeles, Fundamentals of Robotic Mechanical Systems:Theory, Methods, and Algorithms, 2nd ed.

P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory

J.M. Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis

I.J. Busch-Vishniac, Electromechanical Sensors and Actuators

J. Chakrabarty, Applied Plasticity

K.K. Choi and N.H. Kim, Structural Sensitivity Analysis an Optimization 1: Linear Systems

K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 2: NonlinearSystems and Applications

G. Chryssolouris, Laser Machining: Theory and Practice

V.N. Constantinescu, Laminar Viscous Flow

G.A. Costello, Theory of Wire Rope, 2nd ed.

K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems

M.S. Darlow, Balancing of High-Speed Machinery

W. R. DeVries, Analysis of Material Removal Processes

J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability

J.F. Doyle, Wave Propagation in Structures: Spectral Analysis Using FastDiscrete Fourier Transforms, 2nd Edition

P.A. Engel, Structural Analysis of Printed Circuit Board Systems

A.C. Fischer-Cripps, Introduction to Contact Mechanics

A.C. Fischer-Cripps, Nanoindentation, 2nd ed.

J. Garcıa de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems:The Real-Time Challenge

W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures

W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach

G. Genta, Dynamics of Rotating Systems

D. Gross and T. Seelig, Fracture Mechanics with Introduction to Micro-mechanics

K.C. Gupta, Mechanics and Control of Robots

(continued after index)

Giancarlo Genta

Vibration Dynamicsand Control

123

Giancarlo GentaPolitecnico di TorinoTorino, Italygiancarlo.genta@polito.it

ISBN: 978-0-387-79579-9 e-ISBN: 978-0-387-79580-5DOI: 10.1007/978-0-387-79580-5

Library of Congress Control Number: 2008934073

c© Springer Science+Business Media, LLC 2009All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether or notthey are subject to proprietary rights.

Printed on acid-free paper

springer.com

Mechanical Engineering Series

Frederick F. LingEditor-in-Chief

The Mechanical Engineering Series features graduate texts and researchmonographs to address the need for information in contemporary mechan-ical engineering, including areas of concentration of applied mechanics,biomechanics, computational mechanics, dynamical systems and control,energetics, mechanics of materials, processing, production systems, ther-mal science, and tribology.

Advisory Board/Series Editors

Applied Mechanics F.A. LeckieUniversity of California,Santa Barbara

D. GrossTechnical University of Darmstadt

Biomechanics V.C. MowColumbia University

Computational Mechanics H.T. YangUniversity of California,Santa Barbara

Dynamic Systems and Control/ D. BryantMechatronics University of Texas at Austin

Energetics J.R. WeltyUniversity of Oregon, Eugene

Mechanics of Materials I. FinnieUniversity of California, Berkeley

Processing K.K. WangCornell University

Production Systems G.-A. KlutkeTexas A&M University

Thermal Science A.E. BerglesRensselaer Polytechnic Institute

Tribology W.O. WinerGeorgia Institute of Technology

Series Preface

Mechanical engineering, and engineering discipline born of the needs of theindustrial revolution, is once again asked to do its substantial share inthe call for industrial renewal. The general call is urgent as we face pro-found issues of productivity and competitiveness that require engineeringsolutions, among others. The Mechanical Engineering Series is a series fea-turing graduate texts and research monographs intended to address theneed for information in contemporary areas of mechanical engineering.

The series is conceived as a comprehensive one that covers a broadrange of concentrations important to mechanical engineering graduate ed-ucation and research. We are fortunate to have a distinguished roster ofseries editors, each an expert in one of the areas of concentration. Thenames of the series editors are listed on page vi of this volume. The areasof concentration are applied mechanics, biomechanics, computational me-chanics, dynamic systems and control, energetics, mechanics of materials,processing, thermal science, and tribology.

Preface

After 15 years since the publication of Vibration of Structures and Machinesand three subsequent editions a deep reorganization and updating of thematerial was felt necessary. This new book on the subject of Vibrationdynamics and control is organized in a larger number of shorter chapters,hoping that this can be helpful to the reader. New material has been addedand many points have been updated. A larger number of examples and ofexercises have been included.

Since the first edition, these books originate from the need felt by theauthor to give a systematic form to the contents of the lectures he givesto mechanical, aeronautical, and then mechatronic engineering students ofthe Technical University (Politecnico) of Torino, within the frames of thecourses of Principles and Methodologies of Mechanical Design, Construc-tion of Aircraft Engines, and Dynamic Design of Machines. Their main aimis to summarize the fundamentals of mechanics of vibrations to give theneeded theoretical background to the engineer who has to deal with vibra-tion analysis and to show a number of design applications of the theory.Because the emphasis is mostly on the practical aspects, the theoreticalaspects are not dealt with in detail, particularly in areas in which a longand complex study would be needed.

The book is structured in 30 Chapters, subdivided into three Parts.The first part deals with the dynamics of linear, time invariant, systems.

The basic concepts of linear dynamics of discrete systems are summarizedin the first 10 Chapters. Following the lines just described, some specializedtopics, such as random vibrations, are just touched on, more to remind thereader that they exist and to stimulate him to undertake a deeper study ofthese aspects than to supply detailed information.

x Preface

Chapter 11 constitutes an introduction to the dynamics of controlledstructural systems, which are increasingly entering design practice and willunquestionably be used more often in the future.

The dynamics of continuous systems is the subject of the following twochapters. As the analysis of the dynamic behavior of continuous systemsis now mostly performed using discretization techniques, only the basicconcepts are dealt with. Discretization techniques are described in a gen-eral way in Chapter 14, while Chapter 15 deals more in detail with thefinite element method, with the aim of supplying the users of commercialcomputer codes with the theoretical background needed to build adequatemathematical models and critically evaluate the results obtained from thecomputer.

The following two Chapters are devoted to the study of multibody mod-eling and of the vibration dynamics of systems in motion with respect toan inertial reference frame. These subjects are seldom included in bookson vibration, but the increasing use of multibody codes and the inclusionin them of flexible bodies modeled through the finite element method welljustifies the presence of these two chapters.

Part II (including Chapters from 18 to 22 is devoted to the study of non-linear and non time-invariant systems. The subject is dealt with stressingthe aspects of these subjects that are of interest to engineers more than totheoretical mechanicists. The recent advances in all fields of technology of-ten result in an increased nonlinearity of machines and structural elementsand design engineers must increasingly face nonlinear problems: This partis meant to be of help in this instance.

Part III deals with more applied aspects of vibration mechanics. Chaptersfrom 23 to 28 are devoted to the study of the dynamics of rotating ma-chines, while Chapters 29 and 30 deal with reciprocating machines. Theyare meant as specific applications of the more general topics studied be-fore and intend to be more application oriented than the previous ones.However, methods and mathematical models that have not yet entered ev-eryday design practice and are still regarded as research topics are dealtwith herein.

Two appendices related to solution methods and Laplace transform arethen added.

The subjects studied in this book (particularly in the last part) are usu-ally considered different fields of applied mechanics or mechanical design.Specialists in rotor dynamics, torsional vibration, modal analysis, nonlinearmechanics, and controlled systems often speak different languages, and it isdifficult for students to be aware of the unifying ideas that are at the baseof all these different specialized fields. The inconsistency of the symbolsused in the different fields can be particularly confusing. In order to use aconsistent symbol system throughout the book, some deviation from thecommon practice is unavoidable.

Preface xi

The author believes that it is possible to explain all the aspects relatedto mechanical vibrations (actually not only mechanical) using a unifiedapproach. The current book is an effort in this direction.

S.I. units are used in the whole book, with few exceptions. The firstexception is the measure of angles, for which in some cases the old unitdegree is preferred to the S.I. unit radian, particularly where phase anglesare concerned. Frequencies and angular velocities should be measured inrad/s. Sometimes the older units (Hz for frequencies and revolutions perminute [rpm]) are used, when the author feels that this makes things moreintuitive or where normal engineering practice suggests it. In most formulas,at any rate, consistent units are used. In very few cases this rule is notfollowed, but the reader is explicitly warned in the text.

For frequencies, no distinction is generally made between frequency in Hzand circular frequency in rad/s. Although the author is aware of the subtledifferences between the two quantities (or better, between the two differentways of seeing the same quantity), which are subtended by the use of twodifferent names, he chose to regard the two concepts as equivalent. A singlesymbol (ω) is used for both, and the symbol f is never used for a frequencyin Hz. The period is then always equal to T = 2π/ω because consistentunits (in this case, rad/s) must be used in all formulas. A similar rule holdsfor angular velocities, which are always referred to with the symbol Ω.1 Nodifferent symbol is used for angular velocities in rpm, which in some textsare referred to by n.

In rotor dynamics, the speed at which the whirling motion takes placeis regarded as a whirl frequency and not a whirl angular velocity (even ifthe expression whirl speed is often used in opposition to spin speed), andsymbols are used accordingly. It can be said that the concept of angularvelocity is used only for the rotation of material objects, and the rotationalspeed of a vector in the complex plane or of the deformed shape of a rotor(which does not involve actual rotation of a material object) is considereda frequency.

The author is grateful to colleagues and students in the MechanicsDepartment of the Politecnico di Torino for their suggestions, criticism,and general exchange of ideas and, in particular, to the postgraduate stu-dents working in the dynamics field at the department for reading the wholemanuscript and checking most of the equations. Particular thanks are dueto my wife, Franca, both for her encouragement and for doing the tediouswork of revising the manuscript.

G. GentaTorino, July 2008

1In Vibration of Structures and Machines λ was used instead of ω for frequenciesto avoid using Ω for angular velocities. In the present text a more standard notation isadopted.

Contents

Series Preface vii

Preface ix

Symbols xxi

Introduction 1

I Dynamics of Linear, Time Invariant, Systems 23

1 Conservative Discrete Vibrating Systems 251.1 Oscillator with a single degree of freedom . . . . . . . . . 251.2 Systems with many degrees of freedom . . . . . . . . . . . 281.3 Coefficients of influence and compliance matrix . . . . . . 321.4 Lagrange equations . . . . . . . . . . . . . . . . . . . . . . 321.5 Configuration space . . . . . . . . . . . . . . . . . . . . . . 401.6 State space . . . . . . . . . . . . . . . . . . . . . . . . . . 411.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 Equations in the Time, Frequency, and Laplace Domains 492.1 Equations in the time domain . . . . . . . . . . . . . . . . 492.2 Equations in the frequency domain . . . . . . . . . . . . . 502.3 Equations in the Laplace domain . . . . . . . . . . . . . . 542.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xiv Contents

3 Damped Discrete Vibrating Systems 593.1 Linear viscous damping . . . . . . . . . . . . . . . . . . . 593.2 State-space approach . . . . . . . . . . . . . . . . . . . . . 653.3 Rayleigh dissipation function . . . . . . . . . . . . . . . . 673.4 Structural or hysteretic damping . . . . . . . . . . . . . . 723.5 Non-viscous damping . . . . . . . . . . . . . . . . . . . . . 783.6 Structural damping as nonviscous damping . . . . . . . . 843.7 Systems with frequency-dependent parameters . . . . . . . 903.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Free Vibration of Conservative Systems 934.1 Systems with a single degree of freedom . . . . . . . . . . 934.2 Systems with many degrees of freedom . . . . . . . . . . . 974.3 Properties of the eigenvectors . . . . . . . . . . . . . . . . 994.4 Uncoupling of the equations of motion . . . . . . . . . . . 1014.5 Modal participation factors . . . . . . . . . . . . . . . . . 1064.6 Structural modification . . . . . . . . . . . . . . . . . . . . 1114.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Free Vibration of Damped Systems 1155.1 Systems with a single degree of freedom–

viscous damping . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Systems with a single degree of freedom–hysteretic

damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3 Systems with a single degree of freedom –

nonviscous damping . . . . . . . . . . . . . . . . . . . . . 1235.4 Systems with many degrees of freedom . . . . . . . . . . . 1255.5 Uncoupling the equations of motion: space of the

configurations . . . . . . . . . . . . . . . . . . . . . . . . . 1265.6 Uncoupling the equations of motion: state space . . . . . . 1315.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Forced Response in the Frequency Domain:Conservative Systems 1356.1 System with a single degree of freedom . . . . . . . . . . . 1356.2 System with many degrees of freedom . . . . . . . . . . . 1426.3 Modal computation of the response . . . . . . . . . . . . . 1446.4 Coordinate transformation based on Ritz vectors . . . . . 1486.5 Response to periodic excitation . . . . . . . . . . . . . . . 1516.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Forced Response in the Frequency Domain: DampedSystems 1537.1 System with a single degree of freedom: steady-state

response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Contents xv

7.2 System with a single degree of freedom: nonstationaryresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 System with structural damping . . . . . . . . . . . . . . . 1617.4 System with many degrees of freedom . . . . . . . . . . . 1637.5 Modal computation of the response . . . . . . . . . . . . . 1657.6 Multi-degrees of freedom systems with hysteretic

damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.7 Response to periodic excitation . . . . . . . . . . . . . . . 1697.8 The dynamic vibration absorber . . . . . . . . . . . . . . . 1707.9 Parameter identification . . . . . . . . . . . . . . . . . . . 1757.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8 Response to Nonperiodic Excitation 1798.1 Impulse excitation . . . . . . . . . . . . . . . . . . . . . . 1798.2 Step excitation . . . . . . . . . . . . . . . . . . . . . . . . 1818.3 Duhamel’s integral . . . . . . . . . . . . . . . . . . . . . . 1848.4 Solution using the transition matrix . . . . . . . . . . . . 1878.5 Solution using Laplace transforms . . . . . . . . . . . . . . 1878.6 Numerical integration of the equations of motion . . . . . 1898.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

9 Short Account of Random Vibrations 1939.1 General considerations . . . . . . . . . . . . . . . . . . . . 1939.2 Random forcing functions . . . . . . . . . . . . . . . . . . 1949.3 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.4 Probability distribution . . . . . . . . . . . . . . . . . . . 1989.5 Response of linear systems . . . . . . . . . . . . . . . . . . 1999.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10 Reduction of the Number of Degrees of Freedom 21310.1 General considerations . . . . . . . . . . . . . . . . . . . . 21310.2 Static reduction of conservative models . . . . . . . . . . . 21410.3 Guyan reduction . . . . . . . . . . . . . . . . . . . . . . . 21610.4 Damped systems . . . . . . . . . . . . . . . . . . . . . . . 21810.5 Dynamic reduction . . . . . . . . . . . . . . . . . . . . . . 21910.6 Modal reduction . . . . . . . . . . . . . . . . . . . . . . . 21910.7 Component-mode synthesis . . . . . . . . . . . . . . . . . 22010.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

11 Controlled Linear Systems 22511.1 General considerations . . . . . . . . . . . . . . . . . . . . 22511.2 Control systems . . . . . . . . . . . . . . . . . . . . . . . . 22711.3 Controllability and observability . . . . . . . . . . . . . . 23111.4 Open-loop control . . . . . . . . . . . . . . . . . . . . . . . 23211.5 Closed-loop control . . . . . . . . . . . . . . . . . . . . . . 23411.6 Basic control laws . . . . . . . . . . . . . . . . . . . . . . . 237

xvi Contents

11.7 Delayed control . . . . . . . . . . . . . . . . . . . . . . . . 24611.8 Control laws with frequency-dependent gains . . . . . . . 24811.9 Robustness of the controller . . . . . . . . . . . . . . . . . 25311.10 State feedback and state observers . . . . . . . . . . . . . 25311.11 Control design . . . . . . . . . . . . . . . . . . . . . . . . . 25711.12 Modal approach to structural control . . . . . . . . . . . . 25911.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

12 Vibration of Beams 26312.1 Beams and bars . . . . . . . . . . . . . . . . . . . . . . . . 26312.2 Axial behavior of straight bars . . . . . . . . . . . . . . . 26512.3 Torsional vibrations of straight beams . . . . . . . . . . . 27912.4 Flexural vibrations of straight beams: The Euler–Bernoulli

beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28012.5 Bending in the yz-plane . . . . . . . . . . . . . . . . . . . 30512.6 Coupling between flexural and torsional vibrations

of straight beams . . . . . . . . . . . . . . . . . . . . . . . 30612.7 The prismatic homogeneous Timoshenko beam . . . . . . 31012.8 Interaction between axial forces and flexural vibrations

of straight beams . . . . . . . . . . . . . . . . . . . . . . . 31412.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

13 General Continuous Linear Systems 32113.1 Elastic continuums . . . . . . . . . . . . . . . . . . . . . . 32113.2 Flexural vibration of rectangular plates . . . . . . . . . . . 32413.3 Vibration of membranes . . . . . . . . . . . . . . . . . . . 32913.4 Propagation of elastic waves in taut strings . . . . . . . . 33213.5 Propagation of sound waves in pipes . . . . . . . . . . . . 33513.6 Linear continuous systems with structural damping . . . . 33713.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

14 Discretization of Continuous Systems 34114.1 Overview of discretization techniques . . . . . . . . . . . . 34114.2 The assumed-modes methods . . . . . . . . . . . . . . . . 34314.3 Lumped-parameters methods . . . . . . . . . . . . . . . . 34914.4 Transfer-matrices methods . . . . . . . . . . . . . . . . . . 35014.5 Holtzer’s method for torsional vibrations of shafts . . . . . 35314.6 Myklestadt’s method for flexural vibrations of beams . . . 35714.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

15 The Finite Element Method 36315.1 Element characterization . . . . . . . . . . . . . . . . . . . 36315.2 Timoshenko beam element . . . . . . . . . . . . . . . . . . 36715.3 Mass and spring elements . . . . . . . . . . . . . . . . . . 37615.4 Plate element: Kirchoff formulation . . . . . . . . . . . . . 37715.5 Plate element: Mindlin formulation . . . . . . . . . . . . . 380

Contents xvii

15.6 Brick elements . . . . . . . . . . . . . . . . . . . . . . . . 38215.7 Isoparametric elements . . . . . . . . . . . . . . . . . . . . 38415.8 Some considerations on the consistent mass matrix . . . . 38715.9 Assembling the structure . . . . . . . . . . . . . . . . . . . 38815.10 Constraining the structure . . . . . . . . . . . . . . . . . . 39015.11 Dynamic stiffness matrix . . . . . . . . . . . . . . . . . . . 39315.12 Damping matrices . . . . . . . . . . . . . . . . . . . . . . 39515.13 Finite elements in time . . . . . . . . . . . . . . . . . . . . 39615.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

16 Dynamics of Multibody Systems 40116.1 General considerations . . . . . . . . . . . . . . . . . . . . 40116.2 Lagrange equations in terms

of pseudo-coordinates . . . . . . . . . . . . . . . . . . . . . 40716.3 Motion of a rigid body . . . . . . . . . . . . . . . . . . . . 41116.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

17 Vibrating Systems in a Moving Reference Frame 42117.1 General considerations . . . . . . . . . . . . . . . . . . . . 42117.2 Vibrating system on a rigid carrier . . . . . . . . . . . . . 42617.3 Lumped-parameters discretization . . . . . . . . . . . . . 43017.4 Modal discretization . . . . . . . . . . . . . . . . . . . . . 43617.5 Planar systems . . . . . . . . . . . . . . . . . . . . . . . . 44117.6 Beam attached to a rigid body: planar dynamics . . . . . 44217.7 The rotating beam . . . . . . . . . . . . . . . . . . . . . . 44517.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

II Dynamics of Nonlinear and time VariantSystems 449

18 Free Motion of Conservative Nonlinear Systems 45118.1 Linear versus nonlinear systems . . . . . . . . . . . . . . . 45118.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . 45218.3 Free oscillations . . . . . . . . . . . . . . . . . . . . . . . . 45618.4 Direct integration of the equations of motion . . . . . . . 45818.5 Harmonic balance . . . . . . . . . . . . . . . . . . . . . . . 46318.6 Ritz averaging technique . . . . . . . . . . . . . . . . . . . 46618.7 Iterative techniques . . . . . . . . . . . . . . . . . . . . . . 46918.8 Perturbation techniques . . . . . . . . . . . . . . . . . . . 47118.9 Solution in the state plane . . . . . . . . . . . . . . . . . . 47518.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

19 Forced Response of Conservative Nonlinear Systems 48119.1 Approximate evaluation of the response to a harmonic

forcing function . . . . . . . . . . . . . . . . . . . . . . . . 48119.2 Undamped Duffing’s equation . . . . . . . . . . . . . . . . 483

xviii Contents

19.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 49619.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

20 Free Motion of Damped Nonlinear Systems 50120.1 Nonlinear damping . . . . . . . . . . . . . . . . . . . . . . 50120.2 Motion about an equilibrium position (in the small) . . . 50320.3 Direct integration of the equation of motion . . . . . . . . 50420.4 Equivalent damping . . . . . . . . . . . . . . . . . . . . . 50820.5 Solution in the state plane . . . . . . . . . . . . . . . . . . 50920.6 Stability in the small . . . . . . . . . . . . . . . . . . . . . 51220.7 The Van der Pol oscillator . . . . . . . . . . . . . . . . . . 51420.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

21 Forced Response of Damped Nonlinear Systems 51921.1 Reduction of the size of the problem . . . . . . . . . . . . 51921.2 First approximation of the response to a harmonic forcing

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52121.3 Duffing’s equation with viscous damping . . . . . . . . . . 52521.4 Duffing’s equation with structural damping . . . . . . . . 52821.5 Backbone and limit envelope . . . . . . . . . . . . . . . . 52921.6 Multiple Duffing equations . . . . . . . . . . . . . . . . . . 53421.7 Approximated sub- and super-harmonic response . . . . . 53921.8 Van der Pol method: stability of the steady-state

solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54121.9 Strongly nonlinear systems . . . . . . . . . . . . . . . . . . 54521.10 Poincare mapping . . . . . . . . . . . . . . . . . . . . . . . 54621.11 Chaotic vibrations . . . . . . . . . . . . . . . . . . . . . . 55021.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

22 Time Variant and Autoparametric Systems 55722.1 Linear time-variant systems . . . . . . . . . . . . . . . . . 55722.2 Hill’s equation . . . . . . . . . . . . . . . . . . . . . . . . . 55922.3 Pendulum on a moving support: Mathieu equation . . . . 56022.4 The elastic pendulum . . . . . . . . . . . . . . . . . . . . . 56622.5 Autoparametric systems . . . . . . . . . . . . . . . . . . . 56922.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

III Dynamics of Rotating and ReciprocatingMachinery 577

23 Elementary Rotordynamics:The Jeffcott Rotor 57923.1 Elementary rotordynamics . . . . . . . . . . . . . . . . . . 57923.2 Vibrations of rotors: the Campbell diagram . . . . . . . . 581

Contents xix

23.3 Forced vibrations of rotors: critical speeds . . . . . . . . . 58423.4 Fields of instability . . . . . . . . . . . . . . . . . . . . . . 58723.5 The undamped linear Jeffcott rotor . . . . . . . . . . . . . 59023.6 Jeffcott rotor with viscous damping . . . . . . . . . . . . . 59623.7 Jeffcott rotor with structural damping . . . . . . . . . . . 60623.8 Equations of motion in real coordinates . . . . . . . . . . 60823.9 Stability in the supercritical field . . . . . . . . . . . . . . 60823.10 Acceleration through the critical speed . . . . . . . . . . . 61023.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

24 Dynamics of Multi-Degrees-of-Freedom Rotors 61524.1 Model with 4 degrees of freedom: gyroscopic effect . . . . 61524.2 Rotors with many degrees of freedom . . . . . . . . . . . . 63024.3 Real versus complex coordinates . . . . . . . . . . . . . . 63424.4 Fixed versus rotating coordinates . . . . . . . . . . . . . . 63624.5 State-space equations . . . . . . . . . . . . . . . . . . . . . 63724.6 Static solution . . . . . . . . . . . . . . . . . . . . . . . . . 63724.7 Critical-speed computation . . . . . . . . . . . . . . . . . 63824.8 Unbalance response . . . . . . . . . . . . . . . . . . . . . . 63924.9 Campbell diagram and roots locus . . . . . . . . . . . . . 64024.10 Acceleration of a torsionally stiff rotor . . . . . . . . . . . 64124.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

25 Nonisotropic Rotating Machines 64725.1 Jeffcott rotor on nonisotropic supports . . . . . . . . . . . 64825.2 Nonisotropic Jeffcott rotor . . . . . . . . . . . . . . . . . . 65225.3 Secondary critical speeds due to rotor weight . . . . . . . 65625.4 Equation of motion for an anisotropic machine . . . . . . 65725.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

26 Nonlinear Rotors 67126.1 General considerations . . . . . . . . . . . . . . . . . . . . 67126.2 Nonlinear Jeffcott rotor: equation of motion . . . . . . . . 67226.3 Unbalance response . . . . . . . . . . . . . . . . . . . . . . 67326.4 Free circular whirling . . . . . . . . . . . . . . . . . . . . . 67626.5 Stability of the equilibrium position . . . . . . . . . . . . . 67826.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

27 Dynamic Problems of Rotating Machines 68927.1 Rotors on hydrodynamic bearings . . . . . . . . . . . . . . 68927.2 Dynamic study of rotors on magnetic bearings . . . . . . . 70627.3 Flexural vibration dampers . . . . . . . . . . . . . . . . . 72327.4 Signature of rotating machinery . . . . . . . . . . . . . . . 72827.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

xx Contents

28 Rotor Balancing 73328.1 General considerations . . . . . . . . . . . . . . . . . . . . 73328.2 Rigid rotors . . . . . . . . . . . . . . . . . . . . . . . . . . 73428.3 Flexible rotors . . . . . . . . . . . . . . . . . . . . . . . . . 73728.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

29 Torsional Vibration of Crankshafts 74529.1 Specific problems of reciprocating machines . . . . . . . . 74529.2 Equivalent system for the study of torsional vibrations . . 74629.3 Computation of the natural frequencies . . . . . . . . . . . 75829.4 Forced vibrations . . . . . . . . . . . . . . . . . . . . . . . 76229.5 Torsional instability of crank mechanisms . . . . . . . . . 78429.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

30 Vibration Control in Reciprocating Machines 78930.1 Dissipative dampers . . . . . . . . . . . . . . . . . . . . . 78930.2 Damped vibration absorbers . . . . . . . . . . . . . . . . . 79330.3 Rotating-pendulum vibration absorbers . . . . . . . . . . 79630.4 Experimental measurement of torsional vibrations . . . . . 79930.5 Axial vibration of crankshafts . . . . . . . . . . . . . . . . 80030.6 Short outline on balancing of reciprocating machines . . . 80130.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

A Solution Methods 807A.1 General considerations . . . . . . . . . . . . . . . . . . . . 807A.2 Solution of linear sets of equations . . . . . . . . . . . . . 808A.3 Computation of eigenfrequencies . . . . . . . . . . . . . . 812A.4 Solution of nonlinear sets of equations . . . . . . . . . . . 822A.5 Numerical integration in time of the equation of motion . 824

B Laplace Transform Pairs 831

Index 849

Symbols

c viscous damping coefficient, clearancec∗ complex viscous damping coefficientccr critical value of cceq equivalent damping coefficientd distance (between axis of cylinder and center of crank)e base of natural logarithmsf force vector

f i ith modal forcef0 amplitude of the force F (t)g acceleration of gravityg(t) response to a unit-step inputh thickness of oil film, relaxation factorh(t) response to a unit-impulse functioni imaginary unit (i =

√−1)

k stiffness, gaink∗ complex stiffness (k∗ = k′ + ik′′)l length, length of the connecting rodl0 length at rest, length in a reference conditionm mass, number of modes taken into account, number of outputsn number of degrees of freedomp pressureqi ith generalized coordinateq vector of the (complex) coordinatesqi ith eigenvectorqi(xyz) ith eigenfunctionr radius, number of inputs

xxii Symbols

r Ritz vector, vector of the complex coordinates (rotating frame),vector of the command inputs, vector of modal participation factors

s Laplace variables state vector (transfer-matrices method)t time, thicknessu displacementu vector of the inputs, displacement vector (FEM)u(t) unit-step functionv velocityv vector containing the derivatives of the generalized coordinatesvs velocity of soundw vector of the generalized velocitiesx vector of the coordinatesx0 amplitude of x(t)xm maximum value of periodic law x(t)xyz (fixed) reference framey vector of the outputsz complex coordinate (z = x + iy)z state vectorA area of the cross-sectionA matrix linking vectors w and vA dynamic matrix (state-space approach)B matrix linking vectors w and vB input gain matrixC damping matrixC output gain matrix

C modal-damping matrixD dynamic matrix (configuration space approach)D direct link matrixE Young’s modulusE stiffness matrix of the material (FEM)F forceF Rayleigh dissipation functionG shear modulus, balance-quality gradeG(s) transfer functionG gyroscopic matrixG(s) matrix of the transfer functionsH(ω) frequency responseH controllability matrixH(ω) dynamic compliance matrixI area moment of inertiaI identity matrixJ mass moment of inertiaL workL(f) Laplace transform of function fK stiffness matrix.Kdyn dynamic stiffness matrixKg geometric stiffness matrix

Symbols xxiii

K′′ imaginary part of the stiffness matrix

Ki ith modal stiffnessM momentM mass matrix

M i ith modal massMg geometric stiffness matrixN matrix of the shape functionsO load factor (Ocvirk number)O observability matrixQ quality factorQi ith generalized forceR radiusR rotation matrixS Sommerfeld numberS Jacobian matrixS(λ) power spectral densityT period of the free oscillationsT kinetic energyT transfer matrix, matrix linking the forces to the inputsU potential energyV velocity, volumew powerα slenderness of a beam, phase of static unbalance, nondimensional

parameterβ attitude angle, phase of couple unbalance, nondimensional parameterγ shear strainδ logarithmic decrement, phase in phase-angle diagramsδ(t) unit-impulse function (Dirac delta)δL virtual workδx virtual displacementε strain, eccentricityε strain vectorζ damping factor (ζ = c/ccr); nondimensional coordinate (ζ = z/l),

complex coordinate (ζ = ξ + iη)η loss factorηη modal coordinatesθ angular coordinate, pitch angleμ coefficient of the nonlinear term of stiffness, viscosityν Poisson’s ratio, complex frequencyξηζ rotating reference frameρ densityσ decay rate, stressσ stress vectorσy yield stressτ shear stressφ angular displacement, roll angle, complex coordinate (φ = φy − iφx)χ Torsional stiffness, shear factor, angular error for couple unbalance.ψ specific damping capacity, yaw angleω frequency, complex frequency, whirl speed

xxiv Symbols

ω′ whirl speed in the rotating frame[ω2] eigenvalue matrixωn natural frequency of the undamped systemωp frequency of the resonance peak in damped systemsB compliance matrixΓ torsional damping coefficientΦ phase angleΦ eigenvector matrixΦ∗ eigenvector matrix reduced to m modesΩ angular velocity (spin speed)Ω angular velocity vectorΩcr critical speed� imaginary part� real part

complex conjugate (a is the conjugate of a)

˜ Laplace transform (f is the Laplace transform of f).

SUBSCRIPTS

d deviatoricm meann nonrotatingr rotatingI imaginary partR real part.

Introduction

Vibration

Vibration is one of the most common aspects of life. Many natural phe-nomena, as well as man-made devices, involve periodic motion of some sort.Our own bodies include many organs that perform periodic motion, with awide spectrum of frequencies, from the relatively slow motion of the lungsor heart to the high-frequency vibration of the eardrums. When we shiver,hear, or speak, even when we snore, we directly experience vibration.

Vibration is often associated with dreadful events; indeed one of themost impressive and catastrophic natural phenomena is the earthquake,a manifestation of vibration. In man-made devices vibration is often lessimpressive, but it can be a symptom of malfunctioning and is often a signalof danger. When traveling by vehicle, particularly driving or flying, anyincrease of the vibration level makes us feel uncomfortable. Vibration is alsowhat causes sound, from the most unpleasant noise to the most delightfulmusic.

Vibration can be put to work for many useful purposes: Vibrating sieves,mixers, and tools are the most obvious examples. Vibrating machines alsofind applications in medicine, curing human diseases. Another useful aspectof vibration is that it conveys a quantity of useful information about themachine producing it.

Vibration produced by natural phenomena and, increasingly, by man-made devices is also a particular type of pollution, which can be heard asnoise if the frequencies that characterize the phenomenon lie within theaudible range, spanning from about 18 Hz to 20 kHz, or felt directly as

2 Introduction

vibration. This type of pollution can cause severe discomfort. The discom-fort due to noise depends on the intensity of the noise and its frequency, butmany other features are also of great importance. The sound of a bell andthe noise from some machine may have the same intensity and frequencybut create very different sensations. Although even the psychological dispo-sition of the subject can be important in assessing how much discomfort acertain sound creates, some standards must be assessed in order to evaluatethe acceptability of noise sources.

Generally speaking, there is growing awareness of the problem and de-signers are asked, sometimes forced by standards and laws, to reduce thenoise produced by all sorts of machinery.

When vibration is transmitted to the human body by a solid surface,different effects are likely to be felt. Generally speaking, what causes dis-comfort is not the amplitude of the vibration but the peak value (or better,the root mean square value) of the acceleration. The level of accelerationthat causes discomfort depends on the frequency and the time of exposure,but other factors like the position of the human body and the part thatis in contact with the source are also important. Also, for this case, somestandards have been stated. The maximum r.m.s. (root mean square) val-ues of acceleration that cause reduced proficiency when applied for a statedtime in a vertical direction to a sitting subject are plotted as a function offrequency in Fig. 1. The figure, taken from the ISO 2631-1978 standard,deals with a field from 1 to 80 Hz and with daily exposure times from 1minto 24 h.

The exposure limits can be obtained by multiplying the values reportedin Fig. 1 by 2, while the reduced comfort boundary is obtained by dividingthe same values by 3.15 (i.e., by decreasing the r.m.s. value by 10 dB).From the plot, it is clear that the frequency field in which humans aremore affected by vibration lies between 4 and 8 Hz.

Frequencies lower than 1 Hz produce sensations similar to motion sick-ness. They depend on many parameters other than acceleration and arevariable from individual to individual.

At frequencies greater than 80 Hz, the effect of vibration is also depen-dent on the part of the body involved and on the skin conditions and it isimpossible to give general guidelines.

An attempt to classify the effects of vibration with different frequencieson the human body is shown in Fig. 2. Note that there are resonance fieldsat which some parts of the body vibrate with particularly large amplitudes.

As an example, the thorax–abdomen system has a resonant frequencyof about 3–6 Hz, although all resonant frequency values are dependent onindividual characteristics. The head–neck–shoulder system has a resonantfrequency of about 20–30 Hz, and many other organs have more or lesspronounced resonances at other frequencies (e.g., the eyeball at 60–90 Hz,the lower jaw-skull system at 100–220 Hz).

In English, as in many other languages, there are two terms used todesignate oscillatory motion: oscillation and vibration. The two terms are

Introduction 3

FIGURE 1. Vertical vibration exposure criteria curves defining the ‘fatigue-decreased proficiency boundary’ (ISO 2631-1978 standard).

used almost interchangeably; however, if a difference can be found, oscil-lation is more often used to emphasize the kinematic aspects of the phe-nomenon (i.e., the time history of the motion in itself), while vibrationimplies dynamic considerations (i.e., considerations on the relationshipsbetween the motion and the causes from which it originates).

FIGURE 2. Effects of vibration and noise (intended as airborne vibration) onthe human body as functions of frequency (R.E.D. Bishop, Vibration, CambridgeUniv. Press, Cambridge, 1979).

4 Introduction

Actually, not all oscillatory motions can be considered vibrations: For avibration to take place, it is necessary that a continuous exchange of energybetween two different forms occurs. In mechanical systems, the particularforms of energy that are involved are kinetic energy and potential (elastic orgravitational) energy. Oscillations in electrical circuits are due to exchangeof energy between the electrical and magnetic fields.

Many periodic motions taking place at low frequencies are thus oscil-lations but not vibrations, including the motion of the lungs. It is not,however, the slowness of the motion that is important but the lack of dy-namic effects.

Theoretical studies

The simplest mechanical oscillators are the pendulum and the spring–masssystem. The corresponding simplest electrical oscillator is thecapacitor–inductor system. Their behavior can be studied using the samelinear second-order differential equation with constant coefficients, even ifin the case of the pendulum the application of a simple linear model requiresthe assumption that the amplitude of the oscillation is small.

For centuries, the pendulum, and later the spring–mass system (later stillthe capacitor–inductor system), has been more than a model. It constituteda paradigm through which the oscillatory behavior of actual systems hasbeen interpreted. All oscillatory phenomena in real life are more complexthan that, at least for the presence of dissipative mechanisms causing someof the energy of the system is dissipated, usually being transformed intoheat, at each vibration cycle, i.e., each time the energy is transformed for-ward and backward between the two main energy forms. This causes thevibration amplitude to decay in time until the system comes to rest, un-less some form of excitation sustains the motion by providing the requiredenergy.

The basic model can easily accommodate this fact, by simply addingsome form of energy dissipator to the basic oscillator. The spring–mass–damper and the damped-pendulum models constitute a paradigm formechanical oscillators, while the inductor–capacitor–resistor system is thebasic damped electrical oscillator.

Although the very concept of periodic motion was well known, ancientnatural philosophy failed to understand vibratory phenomena, with the ex-ception of the study of sound and music. This is not surprising, as vibrationcould neither be predicted theoretically, owing to the lack of the conceptof inertia, nor observed experimentally, as the wooden or stone structureswere not prone to vibrate, and, above all, ancient machines were heavilydamped owing to high friction.

The beginnings of the theoretical study of vibrating systems are tracedback to observations made by Galileo Galilei in 1583 regarding the motions

Introduction 5

of one of the lamps hanging from the ceiling of the cathedral of Pisa. It issaid that he timed the period of oscillations using the beat of his heart as atime standard to conclude that the period of the oscillations is independentfrom the amplitude.

Whether or not this is true, he described in detail the motion of thependulum in his Dialogo sopra i due massimi sistemi del mondo, publishedin 1638, and stated clearly that its oscillations are isochronous. It is notsurprising that the beginning of the studies of vibratory mechanics occurredat the same time as the formulation of the law of inertia.

The idea that a mechanical oscillator could be used to measure time,due to the property of moving with a fixed period, clearly stimulated thetheoretical research in this field. While Galileo seems to have believed thatthe oscillations of a pendulum have a fixed period even if the amplitudeis large (he quotes a displacement from the vertical as high as 50◦), cer-tainly Huygens knew that this is true only in linear systems and around1656 introduced a modified pendulum whose oscillations would have beentruly isochronous even at large amplitudes. He published his results in hisHorologium Oscillatorium in 1673.

The great development of theoretical mechanics in the eighteenth andnineteenth centuries gave the theory of vibration very deep and solid roots.When it seemed that theoretical mechanics could not offer anything new,the introduction of computers, with the possibility of performing very com-plex numerical experiments, revealed completely new phenomena and dis-closed unexpected perspectives.

The study of chaotic motion in general and of chaotic vibrations of non-linear systems in particular will hopefully clarify some phenomena thathave been beyond the possibility of scientific study and shed new light onknown aspects of mechanics of vibration.

Vibration analysis in design

Mechanics of vibration is not just a field for theoretical study. Design en-gineers had to deal with vibration for a long time, but recently the currenttendencies of technology have made the dynamic analysis of machines andstructures more important.

The load conditions the designer has to take into account in the struc-tural analysis of any member of a machine or a structure can be conven-tionally considered as static, quasi-static, or dynamic. A load conditionbelongs to the first category if it is constant and is applied to the struc-ture for all or most of its life. A typical example is the self-weight of abuilding. The task of the structural analyst is usually limited to deter-mining whether the stresses static loads produce are within the allowablelimits of the material, taking into account all possible environmental ef-

6 Introduction

fects (creep, corrosion, etc.). Sometimes the analyst must check that thedeformations of the structure are consistent with the regular working of themachine.

Also, loads that are repeatedly exerted on the structure, but that areapplied and removed slowly and stay at a constant value for a long enoughtime, are assimilated to static loads. An example of these static load condi-tions is the pressure loading on the structure of the pressurized fuselage ofan airliner and the thermal loading of many pressure vessels. In this case,the designer also has to take into account the fatigue phenomena that canbe caused by repeated application of the load. Because the number of stresscycles is usually low, low-cycle fatigue is encountered.

Quasi-static load conditions are those conditions that, although due todynamic phenomena, share with static loads the characteristics of beingapplied slowly and remaining for comparatively long periods at more orless constant values. Examples are the centrifugal loading of rotors and theloads on the structures of space vehicles due to inertia forces during launchor re-entry. Also, in this case, fatigue phenomena can be very important inthe structural analysis.

Dynamic load conditions are those in which the loads are rapidly varyingand cause strong dynamic effects. The distinction is due mainly to the speedat which loads vary in time. Because it is necessary to state in some way atime scale to assess whether a certain load is applied slowly, it is possible tosay that a load condition is static or quasi-static if the characteristic timesof its variation are far longer than the longest period of the free vibrationsof the structure.

A given load can thus be considered static if it is applied to a struc-ture whose first natural frequency is high or dynamic if it is applied to astructure that vibrates at low frequency.

Dynamic loads may cause the structure to vibrate and can sometimesproduce a resonant response. Causes of dynamic loading can be the mo-tion of what supports the structure (as in the case of seismic loading ofbuildings or the stressing of the structure of ships due to wave motion),the motion of the structure (as in the case of ground vehicles moving onuneven roads), or the interaction of the two motions (as in the case of air-craft flying in gusty air). Other sources of dynamic loads are unbalancedrotating or reciprocating machinery and aero- or gas-dynamic phenomenain jet and rocket engines.

The task the structural analyst must perform in these cases is much moredemanding. To check that the structure can withstand the dynamic loadingfor the required time and that the amplitude of the vibration does not affectthe ability of the machine to perform its tasks, the analyst must acquire aknowledge of its dynamic behavior that is often quite detailed. The naturalfrequencies of the structure and the corresponding mode shapes must firstbe obtained, and then its motion under the action of the dynamic loadsand the resulting stresses in the material must be computed. Fatigue must

Introduction 7

generally be taken into account, and often the methods based on fracturemechanics must be applied.

Fatigue is not necessarily due to vibration; it can be defined more gener-ally as the possibility that a structural member fails under repeated load-ing at stress levels lower than those that could cause failure if appliedonly once. However, the most common way in which this repeated load-ing takes place is linked with vibration. If a part of a machine or struc-ture vibrates, particularly if the frequency of the vibration is high, it canbe called on to withstand a high number of stress cycles in a compar-atively short time, and this is usually the mechanism triggering fatiguedamage.

Another source of difficulty is the fact that, while static loads are usuallydefined in deterministic terms, often only a statistical knowledge of dynamicloads can be reached.

Progress causes machines to be lighter, faster, and, generally speaking,more sophisticated. All these trends make the tasks of the structural an-alyst more complex and demanding. Increasing the speed of machines isoften a goal in itself, like in the transportation field. This is sometimesuseful in increasing production and lowering costs (as in machine tools)or causing more power to be produced, transmitted, or converted (as inenergy-related devices). Faster machines, however, are likely to be the causeof more intense vibrations, and, often, they are prone to suffer damages dueto vibrations.

Speed is just one of the aspects. Machines tend to be lighter, and ma-terials with higher strength are constantly being developed. Better designprocedures allow the exploitation of these characteristics with higher stresslevels, and all these efforts often result in less stiff structures, which aremore prone to vibrate. All these aspects compel designers to deal in moredetail with the dynamic behavior of machines.

Dynamic problems, which in the past were accounted for by simpleoverdesign of the relevant elements, must now be studied in detail, anddynamic design is increasingly the most important part of the design ofmany machines.

Most of the methods used nowadays in dynamic structural analysis werefirst developed for nuclear or aerospace applications, where safety and,in the latter case, lightness are of utmost importance. These methods arespreading to other fields of industry, and the number of engineers working inthe design area, particularly those involved in dynamic analysis, is growing.A good technical background in this field, at least enough to understandthe existence and importance of these problems, is increasingly importantfor persons not directly involved in structural analysis, such as productionengineers, managers, and users of machinery.

It is now almost commonplace to state that about half of the engineersworking in mechanical industries, and particularly in the motor-vehicle in-dustry, are employed in tasks directly related to design. A detailed analysis

8 Introduction

of the tasks in which engineers are engaged in an industrial group workingin the field of energy systems is reported in Fig. 3a. An increasing numberof engineers are engaged in design and the relative economic weight of de-sign activities on total production costs is rapidly increasing. An increaseof 300% in the period from 1950 to 1990 has been recorded.

Within design activities, the relative importance of structural analysis,mainly dynamic analysis, is increasing, while that of activities generallyindicated as drafting is greatly reduced (Fig. 3b).

Economic reasons advocate the use of predictive methods for the studyof the dynamic behavior of machines from the earliest stages of design,

FIGURE 3. (a) Tasks in which engineers are employed in an Italian industrialgroup working in the field of energy systems; (b) relative economic weight of thevarious activities linked with structural design (P.G. Avanzini, La formazioneuniversitaria nel campo delle grandi costruzioni meccaniche, Giornata di studiosull’insegnamento della costruzione delle macchine, Pisa, March 31, 1989).

Introduction 9

without having to wait until prototypes are built and experimental data areavailable. The cost of design changes increases rapidly during the progress ofthe development of a machine, from the very low cost of changes introducedvery early in the design stage to the dreadful costs (mostly in terms of lossof image) that occur when a product already on the market has to berecalled to the factory to be modified. On the other hand, the effectivenessof the changes decreases while new constraints due to the progress of thedesign process are stated.

This situation is summarized in the plot of Fig. 4. Because many designchanges can be necessary as a result of dynamic structural analysis, it mustbe started as early as possible in the design process, at least in the form offirst-approximation studies. The analysis must then be refined and detailedwhen the machine takes a more definite form.

The quantitative prediction, and not only the qualitative understanding,of the dynamic behavior of structures is then increasingly important. Tounderstand and, even more, to predict quantitatively the behavior of anysystem, it is necessary to resort to models that can be analyzed usingmathematical tools. Such analysis work is unavoidable, even if in some ofits aspects it can seem that the physical nature of the problem is lost withinthe mathematical intricacy of the analytical work.

After the analysis has been performed it is necessary to extract resultsand interpret them to obtain a synthetic picture of the relevant phenomena.The analytical work is necessary to ensure a correct interpretation of therelevant phenomena, but if it is not followed by a synthesis, it remains onlya sterile mathematical exercise. The tasks designers are facing in modern

FIGURE 4. Cost and effectiveness of design changes as a function of the stage atwhich the changes are introduced.

10 Introduction

technology force them to understand increasingly complex analytical tech-niques. They must, however, retain the physical insight and engineeringcommon sense without which no sound synthesis can be performed.

Mathematical modeling

The computational predictions of the characteristics and the performance ofa physical system are based on the construction of a mathematical model,1

constructed from a number of equations, whose behavior is similar to thatof the physical system it replaces. In the case of discrete dynamic models,such as those used to predict the dynamic behavior of discrete mechanicalsystems, the model usually is made by a number of ordinary differentialequations2 (ODE).

The complexity of the model depends on many factors that are the firstchoice the analyst has to make. The model must be complex enough toallow a realistic simulation of the system’s characteristics of interest, butno more. The more complex the model, the more data it requires, and themore complicated are the solution and the interpretation of results. Todayit is possible to built very complex models, but overly complex models yieldresults from which it is difficult to extract useful insights into the behaviorof the system.

Before building the model, the analyst must be certain about what hewants to obtain from it. If the goal is a good physical understanding of theunderlying phenomena, without the need for numerically precise results,simple models are best. Skilled analysts were able to simulate even complexphenomena with precision using models with a single degree of freedom.If, on the contrary, the aim is precise quantitative results, even at theprice of more difficult interpretation, the use of complex models becomesunavoidable.

Finally, it is important to take into account the data available at the stagereached by the project: Early in the definition phase, when most data arenot yet available, it is useless to use complex models, into which more or lessarbitrary estimates of the numerical values must be introduced. Simplified,or synthetic, models are the most suitable for a preliminary analysis. As thedesign is gradually defined, new features may be introduced into the model,reaching comprehensive and complex models for the final simulations.

1Simulations are not always based on a mathematical models in a strict sense. In thecase of analog computers, the model was an electric circuit whose behavior simulated thatof the physical system. Simulation on digital computers is based on actual mathematicalmodels.

2A dynamic model, or a dynamic system, is a model expressed by one or more dif-ferential equations containing derivatives with respect to time.

Introduction 11

Such complex models, useful for simulating many characteristics of themachine, may be considered as true virtual prototypes. Virtual reality tech-niques allow these models to yield a large quantity of information, not onlyon performance and the dynamic behavior of the machine, but also on thespace taken by the various components, the adequacy of details, and theiresthetic qualities, that is comparable to what was once obtainable onlyfrom physical prototypes.

The models of a given machine thus evolve initially toward a greatercomplexity, from synthetic models to virtual prototypes, to return later tosimpler models.

Models are useful not only to the designer but also to the test engineer ininterpreting the results of testing and performing all adjustments. Simpli-fied models allow the test engineer to understand the effect of adjustmentsand reduce the number of tests required, provided they are simple enoughto give an immediate idea of the effect of the relevant parameters. Herethe final goal is to adjust the virtual prototype on the computer, transfer-ring the results to the physical machine and hoping that at the end of thisprocess only a few physical validation tests are required.

Simplified models that can be integrated in real time on relatively low-power hardware are also useful in control systems. A mathematical model ofthe controlled system (plant, in control jargon) may constitute an observer(always in the sense of the term used in control theory) and be a part ofthe control architecture.

The analyst has the duty not only of building, implementing, and usingthe models correctly but also of updating and maintaining them. The needto build a mathematical model of some complexity is often felt at a certainstage of the design process, but the model is then used much less thanneeded, and above all is not updated with subsequent design changes, withthe result that it becomes useless or must be updated when the need for itarises again.

There are usually two different approaches to mathematical modeling:models made by equations describing the physics of the relevant phenom-ena, − these may be defined as analytical models − and empirical models,often called black box models.

In analytical models the equations approximating the behavior of thevarious parts of the system, along with the required approximations andsimplifications, are written. Even if no real-world spring behaves exactlylike the linear spring, producing a force proportional to the relative dis-placement of its ends through a constant called stiffness, and even if nodevice dissipating energy is a true linear damper, the dynamics of a mass–spring–damper system (see Chapter 1) can be described, often to a verygood approximation, by the usual ordinary differential equation (ODE)

mx + cx + kx = f(t) .

12 Introduction

The behavior of some systems, on the other hand, is so complex thatwriting equations to describe it starting from the physical and geometricalcharacteristics of their structure is forbiddingly difficult. Their behavior isstudied experimentally and then a mathematical expression able to describeit is sought, identifying the various parameters from the experimental data.While each of the parameters m, c, and k included in the equation ofmotion of the mass–spring–damper system refers to one of the parts of thesystem and has a true physical meaning, the many coefficients appearingin empirical models usually have no direct physical meaning and refer tothe system as a whole.

Among the many ways to build black box models, that based on neu-ral networks must be mentioned.3 Such networks can simulate complexand highly nonlinear systems, adapting their parameters (the weights ofthe network) to produce an output with a relationship to the input thatsimulates the input–output relationship of the actual system.

Actually, the difference between analytical and black box models is notas clear-cut as it may seem. The complexity of the system is often suchthat it is difficult to write equations precisely describing the behavior ofits parts, while the values of the parameters cannot always be known withthe required precision. In such cases the model is built by writing equa-tions approximating the general pattern of the response of the system,with the parameters identified to make the response of the model as closeas possible to that of the actual system. In this case, the identified pa-rameters lose a good deal of their physical meaning related to the variousparts of the system they are conceptually linked to and become globalparameters.

In this book primarily analytical models will be described and an attemptwill be made to link the various parameters to the components of thesystem.

Once the model has been built and the equations of motion written,there is no difficulty in studying the response to any input, assuming theinitial conditions are stated. A general approach is to numerically inte-grate the ordinary differential equation constituting the model, using oneof the many available numerical integration algorithms. In this way, thetime history of the generalized coordinates (or of the state variables) isobtained from any given time history of the inputs (or of the forcingfunctions)

This approach, usually referred to as simulation or numerical experi-mentation, is equivalent to physical experimentation, where the system issubjected to given conditions and its response measured.

3Strictly speaking, neural networks are not sets of equations and thus do not belongto the mathematical models here described. However, at present neural networks areusually simulated on digital computers, in which case their model is made of a set ofequations.

Introduction 13

This method is broadly applicable, because it

• may be used on models of any type and complexity

• allows the response to any type of input to be computed

Its limitations are also clear:

• it does not allow the general behavior of the system to be known, butonly its response to given experimental conditions,

• it may require long computation time (and thus high costs) if themodel is complex, or has characteristics that make numerical inte-gration difficult, and

• it allows the effects of changes of the values of the parameters to bepredicted only at the cost of a number of different simulations.

If the model can be reduced to a set of linear differential equations withconstant coefficients, it is possible to obtain a general solution of the equa-tions of motion. The free behavior of the system can be studied indepen-dently from its forced behavior, and it is possible to use mathematicalinstruments such as Fourier or Laplace transforms to obtain solutions inthe frequency domain or in the Laplace domain. These solutions are oftenmuch more expedient than solutions in the time domain that are in generalthe only type of solution available for nonlinear systems.

The possibility of obtaining general results makes it convenient to startthe study by writing a linear model through suitable linearization tech-niques. Only after a good insight of the behavior of the linearized modelsis obtained will the study of the nonlinear model be undertaken. Whendealing with nonlinear systems it is also expedient to begin with simplifiedmethods, based on techniques like harmonic balance, or to look for seriessolutions before starting to integrate the equations numerically.

Computational vibration analysis

If technological advances force the designer to perform increasingly complextasks, it also provides the instruments for the fulfillment of the new dutieswith powerful means of theoretical and experimental analysis.

The availability of computers of increasing power has deeply changedthe methods, the mathematical means, and even the language of struc-tural analysis, while extending the mathematical study to problems thatpreviously could be tackled only through experiments. However, the ba-sic concepts and theories of structural dynamics have not changed: Itsroots are very deep and strong and can doubtless sustain the new rapid

14 Introduction

growth. Moreover, only the recent increase of computational power enableda deeper utilization of the body of knowledge that accumulated in the lasttwo centuries and often remained unexploited owing to the impossibility ofhandling the exceedingly complex computations. The numerical solution ofproblems that, until a few years ago, required an experimental approach canonly be attempted by applying the aforementioned methods of theoreticalmechanics.

At the same time, together with computational instruments, there was astriking progress in test machines and techniques. Designers can now basetheir choices on large quantities of experimental data obtained on machinessimilar to those being studied, which are often not only more plentiful butalso more detailed and less linked with the ability and experience of theexperimenter than those that were available in the past. Tests on prototypesor on physical models of the machine (even if numerical experimentationis increasingly replacing physical experimentation) not only yield a largeamount of information on the actual behavior of machines but also allowvalidation of theoretical and computational techniques.

Modern instruments are increasingly used to monitor more or less con-tinuously machines in operating conditions. This allows designers to collecta great deal of data on how machines work in their actual service conditionsand to reduce safety margins without endangering, but actually increasing,safety.

As already said, designers can now rely on very powerful computationalinstruments that are widely used in structural analysis. Their use is not,however, free of dangers. A sort of disease, called number crunching syn-drome, has been identified as affecting those who deal with computationalmechanics. Oden and Bathe4 defined it as ‘blatant overconfidence, indeedthe arrogance, of many working in the field [of computational mechanics] ...that is becoming a disease of epidemic proportions in the computational me-chanics community. Acute symptoms are the naive viewpoint that becausegargantuan computers are now available, one can code all the complicatedequations of physics, grind out some numbers, and thereby describe everyphysical phenomena of interest to mankind’.

Methods and instruments that give the user a feeling of omnipotence, be-cause they supply numerical results on problems that can be of astoundingcomplexity, without allowing the user to control the various stages of thecomputation, are clearly potentially dangerous. They give the user a feel-ing of confidence and objectivity, because the computer cannot be wrongor have its own subjective bias.

The finite element method, perhaps the most powerful computationalmethod used for many tasks, among which the solution of problems of

4Oden T.J., Bathe K.J., A Commentary on Computational Mechanics, Applied Me-chanics Review, 31, p. 1053, 1978.

Introduction 15

structural dynamics is one of the most important is, without doubt, themost dangerous from this viewpoint.

In the beginning, computers entered the field of structural analysis in aquiet and reserved way. From the beginning of the 1950s computers wereused to automatically perform those computational procedures that re-quired long and tedious work, for which electromechanical calculators werewidely used. Because the computations required for the solution of manyproblems (like the evaluation of the critical speeds of complex rotors orthe torsional vibration analysis of crankshafts) were very long, the use ofautomatic computing machines was an obvious improvement.

At the end of the 1950s computations that nobody could even think ofperforming without using computers became routine work. Programs of in-creasing complexity were often prepared by specialists, and analysts startedto concentrate their attention on the preparation of data and the interpre-tation of results more than on how the computation was performed. In the1960s the situation evolved further, and the first commercial finite elementcodes appeared on the market. Soon they had some sort of preprocessorsand postprocessors to help the user handle the large amounts of data andresults.

In the 1970s general-purpose codes that can tackle a wide variety ofdifferent problems were commonly used. These codes, which are often pre-pared by specialists who have little knowledge of the specific problems forwhich the code can be used, are generally considered by the users to betools to use without bothering to find out how they work and the assump-tions on which the work is based. Often the designer who must use thesecommercial codes tends to accept noncritically any result that comes outof the computer.

Moreover, these codes allow a specialist in a single field to design acomplex system without seeking the cooperation of other specialists in therelevant matters in the belief that the code can act as a most reliable andunbiased consultant.

On the contrary, the user must know well what the code can do and theassumptions at its foundation. He must have a good physical perception ofthe meaning of the data being introduced and the results obtained in orderto be able to give a critical evaluation.

There are two main possible sources of errors in the results obtainedfrom a code. First there can be errors (bugs, in the jargon of computerusers) in the code itself. This may even happen in well-known commercialcodes, particularly if the problem being studied requires the use of partsof the code that are seldom used or insufficiently tested. The user may tryto solve problems the programmer never imagined the code could be askedto tackle and may thus follow (without having the least suspicion of doingso) paths that were never imagined and thus never tested.

More often, it is the modeling of the physical problem that is to blame forpoor results. The user must always be aware that even the most

16 Introduction

sophisticated code always deals with a simplified model of the real world,and it is a part of the user’s task to ascertain that the model retains therelevant features of the actual problem.

Generally speaking, a model is acceptable only if it yields predictionsclose to the actual behavior of the physical system. Other than this, onlyits internal consistency can be unquestionable, but internal consistencyalone has little interest for the applications of a model.

The availability of programs that automatically prepare data (preproces-sors) can make things worse. Together with the advantages of reducing thework required from the user and avoiding the errors linked with the manualpreparation and introduction of a large amount of data, there is the draw-back of giving a false confidence. The mathematical model prepared by themachine is neither better nor more objective than a handmade one, and itis always the operator who must use engineering knowledge and commonsense to reach a satisfactory model.

The use of general-purpose codes requires the designer to have a knowl-edge of the physical features of the actual systems and of the modelingmethods not much less than that required to prepare the code. The de-signer must also be familiar with the older simplified methods throughwhich approximate, or at least order-of-magnitude, results can be quicklyobtained, allowing the designer to keep a close control over a process inwhich he has little influence.

The use of sophisticated computational methods must not decrease theskill of building very simple models that retain the basic feature of the ac-tual system with a minimum of complexity. Some very ingenious analystscan create models, often with only one, or very few, degrees of freedom,which can simulate the actual behavior of a complicated physical system.The need for this skill is actually increasing, and such models often con-stitute a base for a physical insight that cannot be reached using complexnumerical procedures. The latter are then mandatory for the collection ofquantitative information, whose interpretation is made easier by the insightalready gained.

Concern about vibration and dynamic analysis is not restricted to de-signers. No matter how good the dynamic design of a machine is, if it isnot properly maintained, the level of vibration it produces can increaseto a point at which it becomes dangerous or causes discomfort. The bal-ance conditions of a rotor, for example, may change in time, and periodicrebalancing may be required.

Maintenance engineers must be aware of vibration-related problems tothe same extent as design engineers. The analysis of the vibration producedby a machine can be a very powerful tool for the engineer who has tomaintain a machine in working condition. It has the same importance thatthe study of the symptoms of disease has for medical doctors.

In the past, the experimental study of the vibration characteristics of amachine was a matter of experience and was more an art than a

Introduction 17

science: Some maintenance engineers could immediately recognize prob-lems developed by machines and sometimes even foretell future problemsjust by pressing an ear against the back of a screwdriver whose blade isin contact with carefully chosen parts of the outside of the machine. Thestudy of the motion of water in a transparent bag put on the machineor of a white powder distributed on a dark vibrating panel could giveother important indications. Modern instrumentation, particularly elec-tronic computer-controlled instruments, gives a scientific basis to this as-pect of the mechanics of machines.

The ultimate goal of preventive maintenance is that of continuously ob-taining a complete picture of the working conditions of a machine in sucha way as to plan the required maintenance operations in advance, withouthaving to wait for malfunctions to actually take place.

In some more advanced fields of technology, such as aerospace or nu-clear engineering, this approach has already entered everyday practice. Inother fields, these are more indications for future developments than currentreality.

Unfortunately, the subject of vibration analysis is complex and the useof modern instrumentation requires a theoretical background beyond theknowledge of many maintenance or practical engineers.

Active vibration control

The revolution in all fields of technology, and increasingly in everyday life,due to the introduction of computers, microprocessors, and other electronicdevices did not only change the way machines and structures are designed,built, and monitored but also had an increasingly important impact onhow they work and will deeply change the very idea of machines. A typicalexample is the expression intelligent machines, which until a few decadesago would have been considered an intrinsically contradictory statement,but now is commonly accepted.

The recent developments in the fields of electronics, information, andcontrol systems made it possible to tackle dynamic problems of structuresin a new and often more effective way. While the traditional approachfor reducing dynamic stressing has always been that of changing (usuallyincreasing) the stiffness of the structure or adding damping, now controlsystems that can either adapt the behavior of the structure to the changingdynamic requirements or fight vibrations directly by applying adequatedynamic forces to the structure are increasingly common. This trend iswidespread in all fields of structural mechanics, with civil, mechanical, andaeronautical engineering applications. For example, structural control hasbeen successfully attempted in tall buildings and bridges, machine tools,aircraft, bearing systems for rotating machinery, robots, space structures,

18 Introduction

and ground vehicles. In the latter case, the term active suspensions haseven become popular among the general public.

The advantages of this approach over the conventional one are clearand can be easily evidenced by the example of a large lightweight struc-ture designed to be deployed in space in a microgravity environment. Theabsence or the low value of static forces allows the design of very lightstructures, and lightness is a fundamental prerequisite for any structurethat has to be brought into orbit. This leads to very low natural frequen-cies and corresponding vibration modes that can easily be excited and arevery lightly damped. Any attempt to maintain the dynamic stresses anddisplacements within reasonable limits with conventional techniques, i.e.,by stiffening the structure and adding damping, would lead to large in-creases in the mass and, hence, the cost of the structure. The applicationof suitable control devices can achieve the same goals in a far lighter andcheaper way.

A structure provided with actuators that can adapt its geometric shapeor modify its mechanical characteristics to stabilize a number of workingparameters (e.g., displacements, stresses, and temperatures) is said to bean adaptive structure. An adaptive structure can be better defined as astructure with actuators allowing controlled alterations of the system statesand characteristics.

If there are sensors, the structure can be defined a sensory structure.The two things need not go together, as in the case of a structure providedwith embedded optical fibers that supply information about the structuralintegrity of selected components or in the case of a machine with a built-indiagnostic system. If, however, the structure is both adaptive and sensory,it is a controlled structure.

Active structures are a subset of controlled structures in which there isan external source of power, aimed at supplying the control energy andmodulated by the control system using the information supplied by thesensors. Another typical characteristic is that the integration between thestructure and the control system is so strong that the distinction betweenstructural functionality and control functionality is blurred and no separateoptimization of the parts is possible.

Intelligent structures can be tentatively differentiated from active struc-tures by the presence of a highly distributed control system that takes careof most of the functions. Most biological structures fall in this category;a good example of the operation of an intelligent structure is the way thewing of a bird regulates the aerodynamic forces needed to fly. Not only isthe shape constantly adapted, but the dynamic behavior of the structure isalso controlled. Although a central control system coordinates all this, mostof the control action is committed to peripheral subsystems, distributed onthe whole structure.

A tentative classification of adaptive and sensory structures is shown inFig. 5.

Introduction 19

FIGURE 5. Tentative classification of adaptive and sensory structures. A: adap-tive structures; B: sensory structures; C: controlled structures; D: active struc-tures; and E: intelligent structures.

In all types of controlled structures the control system may need to per-form different tasks, with widely different requirements. For example, it canbe used to change some critical parameter to adapt the characteristics ofthe system to the working conditions, like a device that varies the stiffnessof the supports of a rotor with the aim of changing its critical speed duringstart-up to allow a shift from subcritical to supercritical conditions withouthaving to actually pass a critical speed. In this case, there is no need tohave a very complicated control system, and even a manual control canbe used, if a slow start-up is predicted. Other examples requiring a slowcontrol system are the suspension systems for ground vehicles that are ableto maintain the vehicle body in a prescribed attitude even when variationsof static or quasi-static forces (e.g., centrifugal forces in road bends) occur.

In the case where the control system has to supply forces to controlvibrations, its response has to be faster. If only a few modes of a large andpossibly soft structure are to be controlled, as in the case of tall buildings,bridges, and some space structures, the requirements for the control systemmay not be severe, but they become tougher when the characteristic timeof the phenomena to be kept under control gets shorter since the relevantfrequencies are high.

In other cases, when the structural elements are movable and a controlsystem is already present to control the rigid-body motions, the control ofthe dynamic behavior of the system can be achieved by suitably modulatingthe inputs to the devices that operate the machine. This is the case ofrobot arms or deployable space structures in which the dynamic behavioris strongly affected by the way the actuators perform their task of drivingthe structural elements to the required positions.

20 Introduction

It is easy to predict that the application of structural control, particularlyusing active control systems, will become more popular in the future. Theadvances in performance and cost reduction of control systems are goingto make it cost-effective, but a key factor for its success will be the incor-poration of complex microprocessor-based control systems into machinesof different kinds. They, although basically introduced for reasons differentfrom structural control, can also take care of the latter in an effective andeconomical way. The advances in the field of neural networks may also openpromising perspectives in the field of structural control.

However, if the control system must perform the vibration control ofa structure, a malfunctioning of the first can cause a structural failure.The reliability required is that typical of control systems performing vitalfunctions, like in fly-by-wire systems, and this requirement can have heavyeffects on costs, on both the component and the system level, and can slowdown the application of structural control in low-cost, mass-productionapplications.

As already stated, the trend is toward an increasing integration betweenthe structural and control functions, and this leads to the need for a unifiedapproach at the design and analysis stages. The control subsystem mustno longer be seen as something added to an already existing structuralsubsystem that has been designed independently.

There is a trend toward a unified approach to many aspects of struc-tural dynamics and control, from both the theoretical viewpoint and itspractical applications. A further interdisciplinary effort must also includethose aspects that are more strictly linked with the electrical and electroniccomponents that are increasingly found in all kinds of machinery.

This interdisciplinary approach to the design of complex machines isincreasingly referred to as mechatronics.

Although there are many definitions of what mechatronics is, it can besafely stated that it deals with the integration of mechanics, electronics,and control science to design products that reach their specifications mainlythrough a deep integration of their structural and control subsystems. Atentative graphical definition is shown in Fig. 6, which must be regardedto as an approximation.5 First, the sets defining the various componenttechnologies are not crisply defined; they are fuzzy sets. Second, it is ques-tionable whether computer technology is to be so much stressed, as in thisway analogic devices seem to be ruled out.

But what is actually lacking in Fig. 6 are the economic aspects, whichmust enter such an interdisciplinary approach from the onset of any prac-tical application. The very need for an integrated approach allowing a truesimultaneous engineering of the various components of any machine comes

5S. Ashley, “Getting a hold on mechatronics”, Mechanical Engineering, 119 (5), May1997.

Introduction 21

FIGURE 6. Tentative definition of mechatronics.

from economic consideration, even before thinking of the performance orother technical aspects. No wonder that among the first applications ofmechatronics were consumer goods like cameras and accessories for per-sonal computers.

It is the integration of a sound mechanical design, which includes staticand dynamic analysis and simulation, with electronic and control designwhich allows construction of machines that offer better performance withincreased safety levels at potentially lower costs.