Vibration of Structures and Machines

Giancarlo Genta

Vibration of Structures and Machines

Practical Aspects

Second Edition

With J 73 illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Giancarlo Genta Dipartimento di Meccanica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino, Italy

Cover illustration: See Figure 4-34 for details.

Library of Congress Cataloging-in-Publication Data Genta, G. (Giancarlo)

Vibration of structures and machines: practical aspects / Giancarlo Genta. - 2nd ed.

p. cm. Includes bibliographical references and index. ISBN-13 978-1-4684-0238-4 e-ISBN-13 978-1-4684-0236-0 DOl 10.1007/978-1-4684-0236-0

1. Vibration-Mathematical models. 2. Structural dynamics. 3. Machinery- Vibration. 4. Rotors- Vibration. I. Title. TA355.G44 1995 621.8' ll-dc20 94-35403

Printed on acid-free paper.

© 1995,1993 Springer-Verlag New York, Inc.

Softcover reprint of the hardcover 2nd edition 1995

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf­ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Hal Henglein; manufacturing supervised by Jacqui Ashri. Camera-ready copy supplied by the author.

987654321

To Franca and Alessandro

Preface to the second edition

This second edition follows the previous one after about two years. Apart from correcting a number of printing errors, few additions have been introduced.

In the chapter on the finite element method, a short summary on finite elements in time (four-dimensional finite elements) has been introduced. In the chapter on rotordynamics, two new graphical representations are used; namely, the roots locus plot for free whirling and a tri-dimensional plot for the forced response for which the designation of "orbital tubes" has been proposed. The former has been borrowed from controlled systems dynamics, where it is widely used, while the latter is made practical by the availability of powerful graphical tools for postprocessing numerical or experimental results.

As a last point, a more detailed description of the behaviour of magnetic bearings has been introduced in Chapter 6.

Torino, September 1994 G. Genta

Preface

The present book originates from the need felt by the author to give a systematic form to the contents of the lectures he gives to mechanical and aeronautical engineering students of the Technical University (Politecnico) of Torino, within the frames of the courses of Principles and Methodologies of Mechanical Design and Construction of Aircraft Engines. Its main aim is that of summarizing the funda­mentals of mechanics of vibrations to give the needed theoretical background to the engineer who has to deal with vibration analysis and to show a number of design applications of the theory. As the emphasis is laid mostly on the practical aspects, the theoretical aspects are not dealt with in detail, particularly in areas in which a long and complex study would be needed.

It is structured in 6 chapters. The basic concepts of linear dynamics of discrete systems are summarized in chapter 1. Following the lines described above, some specialized topics, such as random vibrations, are just touched upon, more to remind the reader that they exist and to stimulate him to undertake a deeper study of these aspects rather than to supply detailed information.

The dynamics of continuous systems is the subject of chapter 2. As the analysis of the dynamic behaviour of continuous systems is now mostly performed using discretization techniques, the stress is laid mostly on these, particularly the finite element method, with the aim of supplying the users of commercial computer codes with the theoretical background needed to built adequate mathematical models and to evaluate critically the results obtained from the computer.

The behaviour of nonlinear systems is studied in chapter 3, with the aim of stressing the aspects of this subject that can be of interest to engineers more than to theoretical mechanicists. The recent advances in all fields of technology result often in an increased nonlinearity of machines and structural elements and design engineers must increasingly face nonlinear problems: This chapter is meant to be of help in this instance.

Chapters 4 and 5 are devoted to the study of the dynamics of rotating and recip­rocating machines. They are meant as specific applications of the more general topics studied before, and they intend to be more applicative than the previous ones. However, methods and mathematical models that have not yet entered everyday design practice and are still regarded as research topics are dealt with.

The last chapter constitutes an introduction to the dynamics of controlled structural systems, which are increasingly entering design practice and will unquestionably be used more often in the future.

x Preface

The subjects here studied are usually considered different fields of applied mechanics or mechanical design. Specialists in rotor dynamics, torsional vibration, modal analysis, nonlinear mechanics and controlled systems often speak different languages, and it is difficult for students to be aware of the unifying ideas that are at the base of all these different specialized fields. Particularly confusing can be the inconsistency of the symbols used in the different fields. In order to use a consistent symbol system throughout the whole book some deviation from the common practice is unavoidable.

The author believes that it is possible to explain all the various aspects related to mechanical vibrations (actually not only mechanical) using a unified approach. The present book is an effort in this direction.

S.I. units are used in the whole book, with few exceptions. The first exception is the measure of angles, for which in some cases the old unit degree is preferred to the S.1. unit radian, particularly where phase angles are concerned. Frequencies and angular velocities should be measured in rad/s. Sometimes the older units (Hz for frequencies and revolutions per minute [rpm]) are used, when the author feels that this makes things more intuitive or where normal engineering practice suggests it. In all formulae, at any rate, consistent units must be used. In very few cases is this rule not followed, but the reader is expressly warned in the text.

For frequencies no distinction is generally made between frequency in Hz and circular frequency in rad/s. While being aware of the subtle differences existing between the two quantities, or better, between the two different ways of seeing the same quantity, which are subtended by the use of two different names, the author chose to regard the two concepts as equivalent. A single symbol (A.) is then used for the two and the symbol f is never used for a frequency in Hz. The period is then always equal to T = 21r1A. as consistent units (in this case, rad/s) must be used in all formulae. A similar rule holds for angular velocities, which are always referred to with the symbol (0. No different symbol is used for angular velocities in rpm, which in some texts are referred to with symbol n. The use of symbol A. instead of (0 for frequencies is due to the need for avoiding confusion between frequencies and angular velocities. In rotor dynamics the speed at which the whirling motion takes place is regarded as a whirl frequency and not a whirl angular velocity (even if the expression whirl speed is used sometimes in opposition to spin speed), and symbols are used accordingly. It can be said that the concept of angular velocity is used only for the rotation of material objects, while the rotational speed of a vector in the Argand plane or of the deformed shape df a rotor (which does not involve actual rotation of a material object) is considered a frequency.

The author is grateful to the colleagues and the students of the Mechanics Department of the Politecnico of Torino for their suggestions, criticism and general exchange of ideas and, in particular, to the postgraduate students working in the dynamics field at the Department, for reading the whole manuscript and checking most equations. Particular thanks are due to my wife, Franca, both for her encour­agement and for having done the tedious work of revising the manuscript.

Torino, October 1992 G. Genta

Contents

Preface to the second edition Preface Symbols Introduction

1. Discrete linear systems

1.1 Systems with a single degree of freedom 1.2 Systems with many degrees of freedom 1.3 State space 1.4 Free behaviour 1.5 Uncoupling of the equations of motion 1.6 Excitation due to the motion of the constraints 1.7 Forced oscillations with harmonic excitation 1.8 System with structural damping 1.9 Systems with frequency dependent parameters 1.10 Co-ordinate transformation based on Ritz vectors 1.11 Structural modifications 1.12 Parameter identification 1.13 Laplace transforms, block diagrams and transfer functions 1.14 Response to non-harmonic excitation 1.15 Short account on random vibrations 1.16 Concluding examples 1.17 Exercises

2. Continuous linear systems

2.1 General considerations 2.2 Beams and bars 2.3 Flexural vibration of rectangular plates 2.4 Propagation of elastic waves in taut strings and pipes 2.5 The assumed modes methods 2.6 Lumped parameters methods 2.7 The finite element method 2.8 Reduction of the number of the degrees of freedom 2.9 Exercises

3. Nonlinear systems

3.1 Linear versus nonlinear systems 3.2 Equation of motion 3.3 Free oscillations of the undamped system. 3.4 Forced oscillations of the undamped system. 3.5 Free oscillations of the damped systems

vii ix

xiii xvii

1 4 6 8

14 21 22 34 38 40 41 43 45 48 53 58 66

69 73 91 94 98

101 108 127 135

139 140 144 158 166

xii Contents

3.6 Forced oscillations of the damped system 3.7 Parametrically excited systems 3.8 An outline on chaotic vibrations 3.9 Exercises

4. Dynamics of rotating machinery

4.1 Rotors and structures 4.2 Vibration of rotors: the Campbell diagram 4.3 Forced vibrations of rotors: critical speeds 4.4 Fields of instability 4.5 The linear "Jeffcott rotor" 4.6 Model with 4 degrees of freedom: gyroscopic effect 4.7 Dynamic study of rotors with many degrees of freedom 4.8 Non isotropic systems 4.9 Introduction to nonlinear rotordynamics 4.10 Rotors on hydrodynamic bearings (oil whirl and oil whip) 4.11 Flexural vibration dampers 4.12 Signature of rotating machinery 4.13 Rotor balancing 4.14 Exercises

5. Dynamic problems of reciprocating machines

5.1 Specific problems of reciprocating machines 5.2 "Equivalent system" for the study oftorsional vibrations 5.3 Computation of the natural frequencies 5.4 Forced vibrations 5.5 Torsional instability of crank mechanisms 5.6 Dampers for torsional vibrations 5.7 Experimental measurements of torsional vibrations 5.8 Axial vibrations of crankshafts 5.9 Short outline on balancing of reciprocating machines 5.10 Exercises

6. Short outline on controlled and active systems

6.1 General considerations 6.2 Control systems 6.3 Controlled discrete linear systems 6.4 Modal approach to structural control 6.5 Dynamic study of rotors on magnetic bearings 6.6 Exercises

Appendix. Solution methods

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

Bibliography

Index

176 197 203 207

209 211 213 215 218 238 252 266 288 304 320 322 325 335

337 338 348 350 370 373 381 382 383 386

389 391 395 412 422 438

439 440 443 454 456

465

471

Symbols

C viscous damping coefficient, clearance. c· complex viscous damping coefficient. [c] damping matrix of the element (FEM). Ccr critical value of c. Ceq equivalent damping coefficient. d distance (between axis of cylinder and center of crank). e base of natural logarithms. if} vector of the external forces on the element (FEM). !. amplitude of the force F(t).

g acceleration of gravity. h thickness of oil film, relaxation factor. h(t) response to a unit-impulse function. g (t) response to a unit-step function.

imaginary unit (i = ...[1). k stiffness, gain. k" complex stiffness (k" =k' +ik"). [k] stiffness matrix of the element (FEM). 1 length, length of the connecting rod. 10 length in a reference condition. m mass, number of modes taken into account, number of outputs. [m] mass matrix of the element (FEM). n number of degrees of freedom. p pressure. qj ith generalized coordinate. {q} vector of the (complex) coordinates. {q;} ith eigenvector. qj(x,y,z) ith eigenfunction. r radius, number of inputs. {r} Ritz vector, vector of the complex coordinates (rotating frame), vector of the

command inputs. s Laplace variable.

time, thickness. u displacement. {u} vector of the inputs. u(t) unit-step function. v velocity. v, velocity of sound. xyz (fixed) reference frame. {x} vector of the coordinates.

xiv Symbols

Xo amplitude of x(t). xm maximum value of periodic law x(t).

{y} vector of the outputs. z complex coordinate (z = x + iy). {z} state vector. A area of the cross section. [A] dynamic matrix (state space approach). [B] input gain matrix. [C] damping matrix. [C] output gain matrix. [C] modal damping matrix. [D] dynamic matrix. E Young's modulus. F force. :r Rayleigh dissipation function. {F} vector of the excitation. Fi ith modal force. G shear modulus, balance-quality grade. G (5) transfer function. H()..) frequency response. [H] controllability matrix. / area moment of inertia. [I] identity matrix. J mass moment of inertia. L work. [K] stiffness matrix. [K"] imaginary part of the stiffness matrix. Ki ith modal stiffness. M moment. [M] mass matrix. Mi ith modal mass. [N] matrix of the shape functions. o load factor (Ocvirk number). [0] observability matrix. Q quality factor. Qi ith generalized force. [R] rotation matrix. R radius. S Sommerfeld number. {S} state vector. S(A) power spectral density. [S] Jacobian matrix. {R} vector ofthe modal-participation factors. 'T kinetic energy. T period of the free oscillations. [T] transfer matrix, matrix linking the forces to the inputs. 'l1 potential energy. V velocity, volume. a slenderness of a beam, phase of static unbalance, nondimensional parameter. ~ attitude angle, phase of couple unbalance, nondimensional parameter. [~] compliance matrix.

y ~

~t) ~L fu E

Symbols

shear strain. logarithmic decrement, phase in phase-angle diagrams. unit-impulse function (Dirac delta). virtual work. virtual displacement. strain, eccentricity.

xv

~ damping factor (~= c!ecr); nondimensional coordinate (~= zll), complex coordi­

nate (~ = ~ + ill). II {ll} 9 A A' [A2] An Ap Il. V

~ll~ P 0

Oy

't

~ X 'If CO

COcr

ell [ell] [ell"] n ~( 9t(

) )

loss factor. modal coordinates. angular coordinate. frequency, complex frequency, whirl speed. whirl speed in the rotating frame. eigenvalue matrix. natural frequency of the undamped system. frequency of the resonance peak in damped systems. coefficient of the nonlinear term of stiffness, viscosity. Poisson's ratio, complex frequency. rotating reference frame. density. decay rate, stress. yield stress. shear stress. angular displacement, complex coordinate (~= ~y - i~x).

shear factor, angular error for couple unbalance. relative damping. angular velocity (spin speed). critical speed. phase angle. eigenvector matrix. eigenvector matrix reduced to m modes. angular velocity. imaginary part. real part. complex conjugate Gi is the conjugate of a).

SUBSCRIPTS

d deviatoric. m mean. n nonrotating. r rotating. I imaginary part. R real part.

Introduction

Vibration is one of the most common aspects of life. Many natural phenomena, as well as man-made devices, involve periodic motion of some sort. Our own bodies include many organs that perform periodic motion, with a wide spectrum of fre­quencies, from the relatively slow motion of the lungs or the 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 and indeed one of the most impressive and catastrophic natural phenomena is the earthquake, a manifestation of vibration. In man-made devices vibration is often less impressive but can be a symptom of malfunctioning and frequently a signal of danger. When travelling by vehicle, particularly driving or flying, any increase of the vibration level makes us feel uncomfortable and alerts us. Vibration is also what causes sound, from the most unpleasant noise to the most delightful music.

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

Vibration produced by natural phenomena and, increasingly, by man-made devices is also a particular type of pollution, which can be felt as noise if the frequencies that characterize the phenomenon lie within the audible range (spanning from about 18 Hz to 20 kHz) or otherwise directly as vibration. This type of pollution can cause severe discomfort. The discomfort due to noise depends much on the intensity of the noise and on its frequency, but many other features are of great importance. The sound of a bell and the noise from some machine can have the same intensity and frequency but can create very different sensations. Although even the psychological disposition of the subject can be important in assessing how much discomfort a certain sound creates, some standards must be assessed in order to evaluate the acceptability of noise sources.

Generally speaking, there is growing awareness of the problem and designers are asked, sometimes forced, to reduce to lower and lower levels the noise 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 dis­comfort is not the amplitude of the vibration but the peak value of the acceleration. The level of acceleration that causes discomfort depends on the frequency and on

xviii Introduction

the time of exposition, but other factors such as the position of the human body and the part that is in contact with the source are important as well. Also, for this case, some standards have been stated. The maximum r.m.s. (root mean square) values of acceleration that cause reduced proficiency when applied for a stated time in a vertical direction to a sitting subject are plotted as a function of frequency in Figure 1. The figure, which is reported from ISO 2631-1978 standard, deals with a field from 1 to 80 Hz and with daily exposure times from 1 minute to 24 hours.

The exposure limits can be obtained by multiplying the values reported in Figure 1 by 2, while the reduced comfort boundary is obtained by dividing the 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 man is more affected by vibration lies between 4 and 8 Hz. Frequencies lower than 1 Hz produce sensations that can be assimilated to motion sickness. They depend on many parameters other than acceleration and are variable from individual to individual. At over 80 Hz, the effect of vibration is too dependent on the part of the body involved and on the skin conditions as local vibrations become the governing factor to give general guidelines. An attempt to classify the effects of vibration with different frequencies on man is shown in Table 1. Note that there are resonance fields at which some parts of the body vibrate with particularly large amplitudes.

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

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

6 r .m .•.

10 8

[m/s2] 4

1 8

0.1 88 2 468

10 Frequency [Hz] 100

Introduction

~ ~~~~~----------------~------~ ~ ~~~~~~~~~~~~~+-____ ~L-_

i!~ri~:S~i~~C \ i "'" Ai,,:~~b!~II" I ~~~r~i~~ ... n 0.1 1 10 102 10' 10' 106 10

Frequency [Hz]

xix

Table I. Effects of vibration and noise (intended as airborne vibration) on man as functions of frequency (R.E.D. Bishop, Vibration, Cambridge University Press, Cambridge, 1979).

In English, as in many other languages, there are two terms used to designate oscillatory motion: oscillation and vibration. The two terms are used with almost the same meaning; however, if a difference can be found, the fIrst is more often used to emphasize the kinematic aspects of the phenomenon (i.e., the time history of the motion in itself), while the second implies dynamic considerations (i.e., consider­ations on the relationships between the motion and the causes from which it orig­inates). Actually, not all oscillatory motions can be considered vibrations: For a vibration to take place, it is necessary that a continuous exchange of energy between two different forms occurs. In mechanical systems, the particular forms of energy that are involved with the phenomenon are kinetic energy and potential (elastic or gravitational) energy. Oscillations in electrical circuits are due to exchange of energy between the electrical and the magnetic fIelds.

Many periodic motions taking place at low frequencies are, consequently, oscil­lations but not vibrations, such as the motion of the lungs. It is not, however, the slowness of the motion that is important but the lack of dynamic effects. To be subject to vibration, a system must be able to store energy in two different forms and allow energy to be transferred from one to the other. The simplest mechanical oscillators are the pendulum and the spring-mass system. The corresponding simplest electrical oscillator is the capacitor-inductor system. Their behaviour can be studied using the same linear second-order differential equation with constant coefficients, even if in the case of the pendulum the application of a simple linear model requires the assumption that the amplitude of the oscillation is small.

For centuries the pendulum, and later, the spring-mass system (later still the capacitor-inductor system), has been more than a model. It constituted a paradigm through which the oscillatory behaviour of actual systems has been interpreted. All oscillatory phenomena in real life are more complex than that, at least for the presence of dissipative mechanisms owing to which at each vibration cycle, i.e., each time the

xx Introduction

energy is transformed forward and backward between the two main energy forms, some of the energy of the system is dissipated, usually being transformed into heat. This causes the vibration amplitude to decay in time until the system comes to rest, unless some form of excitation sustains the motion by providing the required energy.

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

Although the very concept of periodic motion was well known, ancient natural philosophy failed to understand vibratory phenomena, with the exception of the study of sound and music. This is not surprising, as vibration could neither be predicted theoretically, owing to the lack of the concept of inertia, nor observed experimentally, as the wooden or stone structures were not prone to vibrate, and, above all, ancient machines were heavily damped owing to the very high friction. The beginnings of the theoretical study of vibrating systems is traced back to observations made by Galileo Galilei in 1583 regarding the motions of one of the lamps hanging from the ceiling of the cathedral of Pisa.1t is said that he timed the period of oscillations using the beat of his heart as a time standard to conclude that the period of the oscillations is independent from the amplitude.

Be this true or not, he described in detail the motion of the pendulum in his Dialo go sopra i due massimi sistemi del mondo, published in 1638, and stated clearly that its oscillations are isochronous. It is not surprising that the beginning of the studies of vibratory mechanics occurred at 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 the theoretical research in this field. While Galileo seems to have believed that the oscillations of a pendulum have a fixed period even if the amplitude is large (he quotes a displacement from the vertical as high as 50°), certainly Huygens knew that this is true only in linear systems and introduced around 1656 a modified pendulum whose oscillations would have been truly isochronous even at large amplitudes. He published his results in his Horologium Oscillatorium in 1673.

The great development of theoretical mechanics in the eighteenth and nineteenth 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 to theoretical sciences, with the possibility of performing very complex numerical experiments, revealed completely new phenomena and disclosed unexpected per­spectives. The study of chaotic motion in general and of chaotic vibrations of non­linear systems in particular will hopefully clarify some phenomena that were up to now beyond the possibility of scientific study and shed new light on known aspects of mechanics of vibration.

Mechanics of vibration is not just a field for theoretical study. Design engineers had to deal with vibration for a long time, but recently the current tendencies of technology have made the dynamic analysis of machines and structures more and more important.

Introduction xxi

The load conditions the designer has to take into account in the structural analysis of any member of a machine or a structure can be conventionally considered as static, quasi-static or dynamic. A load condition belongs to the first category if it is cOnstant and is applied to the structure for all its life or for a considerable part of it. A typical example is the self-weight of a building. The task of the structural analyst is usually limited to determining whether the stresses caused are within the allowable limits of the material, taking into account all possible environmental effects (creep, corrosion, etc.). Sometimes the analyst ensures that the deformations of the structure are con­sistent with the regular working of the machine. Also, loads that are repeatedly exerted on the structure, but that are applied and removed slowly and stay at a constant value for a long enough time, are assimilated to static loads. An example of these static load conditions is the pressure loading on the structure of the pressurized fuselage of an airliner and the thermal loading of many pressure vessels. In this case, the designer also has to take into account the fatigue phenomena that can be caused by the repeated application of the load. As the number of stress cycles is usually low, low-cycle fatigue usually is encountered.

Quasi-static load conditions are those conditions that, although due to dynamic phenomena, share with static loads the characteristics of being applied slowly and of remaining for comparatively long times at more or less constant values. Examples are the centrifugal loading of rotors and the loads on the structures of space vehicles due to inertia forces during launch or re-entry. Also, in this case, fatigue phenomena can be of prominent importance in the structural analysis.

Dynamic load conditions are those in which the loads are rapidly varying and cause strong dynamic effects. The distinction is due mainly to the speed at which loads vary in time. As it is necessary to state in some way a time scale to assess whether a certain load is applied slowly, it is possible to say that a load condition is static or quasi-static if the characteristic times of its variation are far longer than the longest period of the free vibrations of the structure.

The same load can be considered static if applied to a structure whose first natural frequency is high, and dynamic if applied to a structure that vibrates at low frequency. Dynamic loads can cause the structure to vibrate and sometimes can produce a resonant response. Causes of dynamic loading can be the motion of what supports the structure (as in the case of seismic loading of buildings or of the stressing of the structure of ships due to wave motion), the motion of the structure (as in the case of ground vehicles moving on uneven roads), or the interaction of the two motions (as in the case of aircraft flying in gusty air). Other sources of dynamic loads are unbalanced rotating machinery and aero- or gas-dynamic phenomena in jet and rocket engines.

The task the structural analyst must perform in these cases is much more demanding. To check that the structure can withstand the dynamic loading for the required time and that the amplitude of the vibration does not affect the ability of the machine to perform its tasks, he must acquire a knowledge of the dynamic behaviour, which is often quite detailed. The natural frequencies of the structure and the corresponding mode shapes must first be obtained, and then its motion under the

xxii Introduction

action of the dynamic loads and the resulting stresses in the material must be com­puted. Fatigue must generally be taken into account, and very often the methods based on fracture mechanics must be applied.

Fatigue is not necessarily due to vibration, as it can be defined more generally as the possibility that a structural member fails under repeated loading at stress levels lower than those that could cause failure if applied only once. However, the most common way in which this repeated loading takes place is linked with vibration. If a part of a machine or structure vibrates, particularly if the frequency of the vibration is high, it can be called upon to withstand a high number of stress cycles in a com­paratively short time, and this is usually the mechanism triggering fatigue damage.

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

Progress causes machines to be lighter, faster, and, generally speaking, more sophisticated. All these trends make the tasks of the structural analyst more complex and demanding. Increasing the speed of machines is often a goal in itself, as in the transportation field. This is sometimes useful in increasing production and lowering costs (as in machine tools), or causing more power to be produced, transmitted, or converted (as in energy-related devices). Faster machines however are likely to be the cause of more intense vibrations and also, quite often, are prone to suffer damages due to vibrations. Speed is just one of the aspects. Machines tend to be lighter, and materials with higher strength are developed incessantly. Better design procedures allow the exploitation of these characteristics with higher stress levels, and all these efforts often result in less stiff structures, which are more prone to vibrate. All these aspects compel designers to deal in more detail with the dynamic behaviour of machines.

Dynamic problems, which in the past were accounted for by simple overdesign of the relevant elements, must now be studied in detail, and dynamic design is increasingly the most important part of the design of many machines. It is not a case that most of the methods used nowadays in dynamic structural analysis were first developed for nuclear or aerospace applications, where safety and lightness are of the utmost importance. These methods are spreading around in other fields of industry, and the number of engineers working in the design area, particularly those involved in dynamic analysis, is growing. A good technical background in this field, at least enough to understand the existence and the importance of these problems, is increasingly important for persons not directly involved in structural analysis, such as production engineers, managers, and users of machinery.

It is now almost commonplace to state that about one-half of the engineers working in mechanical industries, and particularly in the motor-vehicle industry, are employed in tasks directly related with design. A detailed analysis of the tasks in which engineers are engaged in an industrial group working in the field of energy systems is reported in Figure 2a. While more and more engineers are engaged in design, the relative economic weight of design activities on total production costs is rapidly increasing. An increase of 300% in the period from 1950 to 1990 has been recorded.

Introduction xxiii

Management 14% a) Order management 9% Design (conceptual) 14%

Design (analysis) 25% Research 12%

Production 12% Quality control

Building yard 4% Sales

Purchases 5%

0 5~ 10~ 15~ 20~ 25~ 30~ 100~

CL> u

Dynamic analysis c: ro ....,

80% ... 0 p..

.9 u 60% Thermal analysis ·s 0 c: 40~ 0 u Stress analysis CL> Drafting CL>

.~ 20% ...., ~ b)

CL> P:::

0 1950 1960 1970 1980 Years 2000

Figure 2. (a) Tasks in which engineers are employed in an Italian industrial group working in the field of energy systems; (b) relative economic weight of the various activities linked with structural design (P.G. Avanzini, La Jormazione universitaria net campo delle grandi costruzioni meccaniche, Giomata di studio sUll'insegnamento della costruzione delle mac­chine, Pisa, 31/3/1989).

Within design activities the relative importance of structural analysis, mainly dynamic analysis, is increasing, while that of activities generally indicated as drafting is greatly reduced (Figure 2b).

Economic reasons advocate the use of predictive methods for the study of the dynamic behaviour of machines from the earliest stages of design, without having to wait until prototypes are built and experimental data are available. The cost of design changes increases very rapidly during the progress of the development of a machine, from the very low cost of changes introduced very early in the design stage to the dreadful costs (also in terms ofloss of image) that occur when a product already on the market has to be recalled to the factory to be modified. On the other side, the effectiveness of the changes decreases while new constraints due to the progress of the design process are stated. This situation is summarized in the graph of Figure 3. Since many design changes can be necessary as a result of dynamic structural

xxiv Introduction

analysis, it must be started as early as possible in the design process, at least in the form of first-approximation studies. The analysis must then be refined and detailed when the machine takes a more definite form.

The quantitative prediction, and not only the qualitative understanding, of the dynamic behaviour of structures is then increasingly important. To understand and, even more, to predict quantitatively the behaviour of any system, it is necessary to resort to models that can be analyzed using mathematical tools. Such analysis work is unavoidable, even if in some of its aspects it can seem that the physical nature of the problem is lost within the mathematical intricacy of the analytical work. After the analysis has been performed it is necessary to extract results and to interpret them in order to obtain a synthetic picture of the relevant phenomena. The analytical work is a necessity to ensure a correct interpretation of the relevant phenomena, but if it is not followed by a synthesis, it remains only a sterile mathematical exercise. The tasks designers are facing in modern technology force them to understand increas­ingly complex analytical techniques. They must, however, retain the physical insight and engineering common sense without which no sound synthesis can be performed.

If the technological advancement forces the designer to perform increasingly complex tasks, it also provides the instruments for the fulfilment of the new duties with powerful means of theoretical and experimental analysis.

The availability of more and more powerful computers has deeply changed the methods, the mathematical means, and even the language of structural analysis, while extending the capability of mathematical study to problems that previously could be tackled only through experiments. However, the basic concepts and theories of structural dynamics have not changed: its roots are very deep and strong and can doubtless sustain the new rapid growth. Moreover, only the recent increase of the computational power made possible a deeper utilization of the body of knowledge that accumulated in the last two centuries and often remained unexploited owing to the impossibility of handling the exceedingly complex computations. The numerical solution of problems that, until a few years ago, required an experimental approach can only be attempted by applying the aforementioned methods of theoretical mechanics.

At the same time, together with computational instruments, test machines and techniques also had striking progress. Designers can now base their choices on large quantities of experimental data obtained on machines similar to those he must study, which are often not only more plentiful, but also more detailed and less linked with the ability and experience of the experimenter than those that were available in the past. Tests on prototypes or on physical models of the machine (even if today numerical experimentation is increasingly substituting physical experimentation) not only yield a large amount of information on the actual behaviour of machines, but also allow validation of theoretical and computational techniques.

Modern instrumentation is increasingly used to allow a more or less continuous monitoring of machines in operating conditions. This allows designers to collect many data on how machines work in their actual service conditions and to reduce safety margins without endangering, but actually increasing, safety.

Figure 3. Cost and effectiveness of design changes as function of the stage at which the changes are intro-duced.

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Introduction

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As already said, designers can nowadays rely on very powerful computational instruments that are widely used in structural analysis. Their use is not, however, free of dangers. Often a sort of disease, called number crunching syndrome, has been identified as affecting those who deal with computational mechanics. Oden and Bathe· defined it as "blatant overconfidence, indeed the arrogance, of many working in the field [of computational mechanics] ... that is becoming a disease of epidemic proportions in the computational mechanics community. Acute symptoms are the naive viewpoint that because gargantuan computers are now available, one can code all the complicated equations of physics, grind out some numbers, and thereby describe every physical phenomena of interest to mankind".

Methods and instruments that give the user a feeling of omnipotence, since they supply numerical results on problems that can be of astounding complexity, without allowing him to control the various stages of the computation, are clearly potentially dangerous. They give the user a feeling of confidence and objectivity, since the computer cannot at any rate be wrong or have its own subjective bias. The finite element method, perhaps the most powerful computational method used for many tasks, among which the solution of problems of structural dynamics is one of the most important, is, without doubt, the most dangerous from this viewpoint.

At the beginning computers entered into the field of structural analysis in a quiet and reserved way. From the beginning of the fifties computers were used to perform automatically those computational procedures that required long and tedious work, for which electromechanical calculators were widely employed. As the computations required for the solution of many problems (like the evaluation of the critical speeds of complex rotors or the torsional vibration analysis of crankshafts) were very long, the use of automatic computing machines was an obvious progress of great practical

* Oden T.1., Bathe K.1.,A Commentary on Computational Mechanics, Applied Mechanics Review, 31, p.1053, 1978.

xxvi Introduction

impact. At the end of the fifties computations that nobody could even think to perfonn without using computers became routine work. Programs of increasing complexity were often prepared by specialists, and analysts started to concentrate their attention on the preparation of data and on the interpretation of results more than on how the computation was perfonned. In the sixties the situation evolved further and the first commercial finite element codes appeared on the market. Soon they had some sort of preprocessors and postprocessors to help the user handle the large amount of data and results.

In the seventies general-purpose codes, able to tackle a wide variety of different problems, were commonly used. These codes, which are often prepared by specialists who have little knowledge of the specific problems for which the code can be used, are generally considered by the users tools to use without bothering too much about how they work and about the assumptions on which work is based. More and more often the designer who must use these commercial codes tends to accept noncritically any result that comes out of the computer. Moreover, these codes allow a specialist in a single field to design a complex system without seeking the cooperation of other specialists in the relevant matters in the belief that the code can act as a most reliable and unbiased consultant. The user must, on the contrary, know very well what the code can do and the assumptions that are at its foundations. He must have a good physical perception of the meaning of the data he introduces and the results he gets in order to be able to give a critical evaluation.

There are two main possible sources of errors in the results obtained from a code. First there can be errors (bugs, in the jargon of computer users) in the code itself. This can also happen in well-known commercial codes, particularly when the problem in study requires the use of parts of the code that are seldom used or insufficiently tested. The user can try to solve problems the programmer has never imagined his code could be asked to tackle and, thus, can follow, obviously without having the least suspicion of doing so, paths that have never been imagined possible and consequently have never been tested.

More often it is the modelling of the physical problem that is to blame for poor results. The user must always be aware that even the most sophisticated code always deals with a simplified model of the real world, and that it is a part of his task to ascertain that the model retains the relevant features of the actual problem.

Generally speaking, a model is acceptable only in the measure it yields predictions that are close to the actual behaviour of the physical system. Other than this, only its internal consistency can be unquestionable, but internal consistency alone has little interest for the applications of a model.

The availability of programs that automatically prepare data (preprocessors) can make things worse. Together with the advantages of reducing the workrequired from the user and avoiding the errors linked with the manual preparation and introduction of a large mass of data, there is the drawback of giving a fallacious confidence. The mathematical model prepared by the machine is neither better nor more objective than a handmade one, and it is always the operator who must use his engineering knowledge and common sense in order to reach a satisfactory model. The use of general-purpose codes requires from the designer a knowledge of the physical fea­tures of the actual systems and of the modelling methods not much less than that

Introduction xxvii

required for the preparation of the code. The designer must also be familiar with the older simplified methods through which he can quickly obtain approximated results or at least an order of magnitude allowing him to keep a process over which he has little influence under close control.

The use of sophisticated computational methods must not decrease the skill of building very simple models that retain the basic feature of the actual system with a minimum of complexity. Some very ingenious analysts are able to create models, often with only one, or very few, degrees of freedom, which can simulate the actual behaviour of a much complicated physical system. The need of this skill is actually increasing, and such models often constitute a base for a physical insight that cannot be reached using complex numerical procedures. The latter are then mandatory for the collection of quantitative information, whose interpretation is made easier by the insight already gained.

Concern about vibration and dynamic analysis is not restricted to designers. No matter how good the dynamic design of a machine is, if it is not properly maintained, the level of vibration it produces can increase to a point at which it becomes dangerous or causes discomfort. The balance conditions of a rotor, for instance, can change in time and periodic rebalancing can be required. Maintenance engineers must be aware of vibration-related problems to the same extent as design engineers. The analysis of the vibrations produced by a machine can be a very powerful tool for the engineer who has to maintain a machine in working condition. It has the same importance that the study of the symptoms of disease has for medical doctors.

In the past the experimental study of the vibration characteristics of a machine was a matter of experience and was more an art than a science: Some maintenance engineers could immediately recognize problems developed by machines and sometimes foretell future problems just by pressing an ear against the back of a screwdriver whose blade was in contact with carefully chosen parts of the outside of the machine. The study of the motion of water in a transparent bag put on the machine or of a white powder distributed on a dark vibrating panel gave other important indications. Nowadays modern instrumentation, particularly electronic computer-controlled instruments, gives a scientific basis to this aspect of the mechanics of machines.

The ultimate goal of "preventive maintenance" is that of continuously obtaining a complete picture of the working conditions of a machine in such a way as to plan the required maintenance operations in advance, without having to wait for mal­functions to actually take place. In some, more advanced, fields of technology, such as aerospace or nuclear engineering, this approach is already entering everyday practice. In other fields these are more indications for future developments than current reality. Unfortunately, the subject of vibration analysis is a complex one and the use of modern instrumentation requires a theoretical background that is beyond the knowledge of many maintenance or practical engineers.