Table Of ContentPreface
This textbook represents the Mechanical Vibrations lecture course given to
students in the fourth year at the Department of Engineering Sciences (now
F.I.L.S.), English Stream, University Politehnica of Bucharest, since 1993.
It grew in time from a course taught in Romanian since 1972 to students in
the Production Engineering Department, followed by a special course given
between 1985 and 1990 to postgraduate students at the Strength of Materials Chair.
Mechanical Vibrations, as a stand alone subject, was first introduced in the
curricula of mechanical engineering departments in 1974. To sustain it, we
published with Professor Gh. Buzdugan the book Vibration of Mechanical Systems
in 1975, at Editura Academiei, followed by two editions of Mechanical Vibrations,
in 1979 and 1982, at Editura didactică şi pedagogică. In 1984 we published
Vibration Measurement at Martinus Nijhoff Publ., Dordrecht, which was the
English updated version of a book published in 1979 at Editura Academiei.
As seen from the Table of Contents, this book is application oriented and
limited to what can be taught in an one-semester (28 hours) lecture course. It also
contains material to support the tutorial that includes the use of finite element
computer programs and basic laboratory experiments. The course syllabus changed
in time due to the growing use of computers. We wrote simple finite element
programs to assist students in solving problems as homework. The course aims to:
(a) increase the knowledge of vibration phenomena; (b) further the understanding
of the dynamic behaviour of structures and systems; and (c) provide the necessary
physical basis for analytical and computational approaches to the development of
engineering solutions to vibration problems.
As a course taught for non-native speakers, it has been considered useful to
reproduce as language patterns some sentences from English texts.
Computational methods for large eigenvalue problems, model reduction,
estimation of system parameters based on the analysis of frequency response data,
transient responses, modal testing and vibration testing are treated in the second
volume. No reference is made to the dynamics of rotor-bearing systems and the
vibration of discs, impellers and blades which are studied in the Dynamics of
Machinery lecture course.
April 2006 Mircea Radeş
Prefaţă
Lucrarea reprezintă cursul de Vibraţii mecanice predat studenţilor anului
IV al Facultăţii de Inginerie în Limbi Străine, Filiera Engleză, la Universitatea
Politehnica Bucureşti, începând cu anul 1993. Conţinutul cursului s-a lărgit în timp,
pornind de la un curs predat din 1972 studenţilor de la facultatea T. C. M. (în
prezent I.M.S.T.), urmat de un curs postuniversitar organizat între 1985 şi 1990 în
cadrul Catedrei de Rezistenţa materialelor.
Vibraţiile mecanice au fost introduse în planul de învăţământ al facultăţilor
cu profil mecanic ca un curs de sine stătător în 1974. Pentru a susţine cursul, am
publicat, sub conducerea profesorului Gh. Buzdugan, monografia Vibraţiile
sistemelor mecanice la Editura Academiei în 1975, urmată de două ediţii ale
manualului Vibraţii mecanice la Editura didactică şi pedagogică în 1979 şi 1982. În
1984 am publicat Vibration Measurement la Martinus Nijhoff Publ., Dordrecht,
reprezentând versiunea revizuită în limba engleză a monografiei ce a apărut în
1979 la Editura Academiei.
După cum reiese din Tabla de materii, cursul este orientat spre aplicaţii
inginereşti, fiind limitat la ceea ce se poate preda în 28 ore. Materialul prezentat
conţine exerciţii rezolvate care susţin seminarul, în cadrul căruia se utilizează
programe cu elemente finite elaborate de autor şi se prezintă lucrări demonstrative
de laborator, fiind utile şi la rezolvarea temelor de casă. Cursul are un loc bine
definit în planul de învăţământ, urmărind a) descrierea fenomenelor vibratorii
întâlnite în practica inginerească; b) modelarea sistemelor vibratoare şi analiza
acestora cu metoda elementelor finite; şi c) înarmarea studenţilor cu baza fizică
necesară în modelarea analitică şi numerică a structurilor în vibraţie şi a maşinilor,
pentru elaborarea soluţiilor inginereşti ale problemelor de vibraţii.
Fiind un curs predat unor studenţi a căror limbă maternă nu este limba
engleză, au fost reproduse expresii şi fraze din cărţi scrise de vorbitori nativi ai
acestei limbi.
În volumul al doilea se vor prezenta metode de calcul pentru probleme de
valori proprii de ordin mare, reducerea ordinului modelelor, răspunsul tranzitoriu,
estimarea pametrilor sistemelor vibratoare pe baza analizei funcţiilor răspunsului în
frecvenţă, analiza modală experimentală şi încercările la vibraţii. Nu se tratează
dinamica sistemelor rotor-lagăre şi vibraţiile discurilor şi paletelor, acestea fiind
studiate în cadrul cursului de Dinamica maşinilor.
Aprilie 2006 Mircea Radeş
Contents
Preface 1
Prefaţă 2
Contents 3
1. Modelling Vibrating Systems 5
1.1 Vibrations vs. Oscillations 5
1.2 Discrete vs. Continuous Systems 6
1.3 Simple Vibrating Systems 7
1.4 Vibratory Motions 8
1.5 Damping 10
2. Simple Linear Systems 11
2.1 Undamped Free Vibrations 11
2.2 Undamped Forced Vibrations 22
2.3 Damped Free Vibrations 35
2.4 Damped Forced Vibrations 42
Exercices 73
3. Simple Non-Linear Systems 79
3.1 Non-Linear Harmonic Response 79
3.2 Cubic Stiffness 81
3.3 Combined Coulomb and Structural Damping 92
3.4 Quadratic Damping 97
3.5 Effect of Pre-Loading 103
4 MECHANICAL VIBRATIONS
4. Two-Degree-of-Freedom Systems 105
4.1 Coupled Translation 106
4.2 Torsional Systems 119
4.3 Flexural Systems 130
4.4 Coupled Translation and Rotation 145
4.5 Coupled Pendulums 151
4.6 Damped Systems 156
Exercices 179
5. Several Degrees of Freedom 183
5.1 Lumped Mass Systems 184
5.2 Plane Trusses 210
5.3 Plane Frames 220
5.4 Grillages 234
5.5 Frequency Response Functions 241
Exercices 247
6. Continuous Systems 259
6.1 Lateral Vibrations of Thin Beams 259
6.2 Longitudinal Vibration of Rods 275
6.3 Torsional Vibration of Rods 278
6.4 Timoshenko Beams 280
References 281
Index 289
1.
MODELLING VIBRATING SYSTEMS
Vibrations are dynamic phenomena encountered in everyday life, from the
heart beating and walking, trees shaking in gusty winds or boats floating on rough
waters, vibration of musical instruments and loudspeaker cones, to bouncing of
cars on corrugated roads, swaying of buildings due to wind or earthquakes,
vibrations of conveyers and road drills.
It is customary to term ‘vibrations’ only the undesired repetitive motions,
giving rise to noise or potentially damaging stress levels. The effect of vibrations
on humans, buildings and machines are of main concern. Modelling vibration
phenomena implies describing the structure and parameters of the vibrating body,
the excitation function and the response levels.
This introductory chapter focuses on definitions and classifications, to give
an overview of the main notions used in vibration analysis.
1.1 Vibrations vs. Oscillations
The Oxford Dictionary gives “vibration, n. Vibrating, oscillation; (phys)
rapid motion to and fro, esp. of the parts of a fluid or an elastic solid whose
equilibrium is disturbed”. It comes out that all matter, gaseous, liquid or solid is
capable of executing vibrations and, in fact, so are the elementary particles of
which the matter is composed.
Generally, oscillations are variations of a state parameter about the value
corresponding to a stable equilibrium position (or trajectory). Vibrations are
oscillations due to an elastic restoring force. To save confusion, a flexible beam or
string vibrates while a pendulum oscillates.
For practical engineering purposes it is usual to allocate the term
‘vibration’ predominantly to unwanted periodic motions. In music, the opposite is
the case, since all musical instruments use periodic vibrations to make sound. We
might say that vibration in engineering is more akin to noise in acoustics: an
6 MECHANICAL VIBRATIONS
annoying, but to a degree, inescapable by-product of the machine, either in terms of
external sound or damage within itself. Apart from harmful vibrations, there are
installations whose operation is based on vibratory motions, namely: concrete
tampers, pile driving vibrators, soil compaction machines, vibrating screens,
fatigue testing machines, etc.
All bodies possessing mass and elasticity are capable of vibration. A
vibrating system has both kinetic energy, stored in the mass by virtue of its
velocity, and potential energy, stored in the elastic element as strain energy. A
major feature of vibrations is the cyclic transformation of potential energy into
kinetic and back again. In a conservative system, when there is no dissipation of
energy, the total energy is constant. At the point of maximum displacement
amplitude, the instantaneous velocity is zero, the system has only potential energy.
At the static equilibrium position, the strain energy is zero and the system has only
kinetic energy. The maximum kinetic energy must equal the maximum potential
energy. Equating the two energies it is possible to obtain the natural frequency of
vibration. This is the basis of Rayleigh’s method.
Vibrating systems are subject to damping because energy is removed by
dissipation or radiation. Damping is responsible for the decay of free vibrations, for
the phase shift between excitation and response, and provides an explanation for
the fact that the forced response of a vibratory system does not grow without limit.
1.2 Discrete vs. Continuous Systems
The number of independent coordinates needed to specify completely the
configuration of a vibrating system at any instant gives the number of degrees of
freedom of the system.
It follows that, in order to describe the motion of every particle of a
system, the number of degrees of freedom has to be infinite. However, for practical
purposes, it is useful to use systems of approximate dynamical similarity to the
actual system, which have a small number of degrees of freedom.
The criteria used to determine how many degrees of freedom to ascribe to
any system under analysis are practical in nature. For instance, some of the
possible system motions may be so small that they are not of practical interest.
Some or most of the motions of particles in the system may be practically similar,
allowing such particles to be lumped into a single rigid body. The frequency range
of the excitation forces may be so narrow that only one, or at most a few, of the
natural frequencies of the system can give rise to resonances. Groups of particles
experiencing similar motions may be considered single bodies, thereby reducing
the number of degrees of freedom necessary to consider. All these practical
considerations lead to the concept of lumped masses which are rigid bodies
1. MODELLING VIBRATING SYSTEMS 7
connected by massless flexible members. The motions predicted by using such
approximate lumped-parameter or discrete systems are often close enough to the
actual vibrations to satisfy all practical demands and to provide useful design data
and allowable vibration limits.
In some systems, a second approximation can be made, by taking into
account the mass of the elastic members. This is necessary only when the flexible
members have distributed masses which are comparable in magnitude with the
masses of system components modelled as rigid bodies.
Finally, there are many systems of practical interest which have such
simple shapes that they can be considered as systems possessing an infinite number
of degrees of freedom. Such distributed-parameter or continuous systems may be
modelled as strings, beams, plates, membranes, shells and combinations of these.
In most engineering applications, geometrically complex structures are
replaced by discretized mathematical models. A successful discretization approach
is the finite element method. The infinite degree of freedom system is replaced by a
finite system exhibiting the same behaviour. The actual structure is divided
(hypothetically) into well-defined sub-domains (finite elements) which are so small
that the shape of the displacement field can be approximated without too much
error, leaving only the amplitude to be found. All individual elements are then
assembled together in such a way that their displacements are mating each other at
the element nodes or at certain points at their interfaces, the internal stresses are in
equilibrium with the applied loads reduced at nodes, and the prescribed boundary
conditions are satisfied. Modelling errors include inappropriate element types,
incorrect shape functions, improper supports and poor mesh.
1.3 Simple Vibrating Systems
A surprisingly large number of practical vibration problems which arise in
the machines and structures designed by engineers can be treated with a
sufficiently high degree of accuracy by imagining the actual system to consist of a
single rigid body, whose motion can be described by a single coordinate.
In reality, the simplest imaginable system consists of the body whose
motion is of interest and the fixed surrounding medium, relative to which the
motion is measured. The problem of treating such a simplified system is fourfold.
The first part consists in deciding what part of the system is the rigid body and
what part are the flexible members. The second part consists in calculating the
values of the dynamic parameters of the rigid body and flexible parts. The third
part consists in writing the equations of motion of the equivalent system, Finally,
the fourth part consists in solving the equations for the prescribed conditions of
8 MECHANICAL VIBRATIONS
free or forced vibrations. Alternatively, methods using the kinetic and potential
energies may be used in the place of the last two stages.
The first two parts require judgement and experience which come with
practice, that is, with the repeated process of assuming equivalent systems,
predicting their motions and checking the predictions against actual measurements
on the real systems. Model verification and validation may require updating of
system parameters or even of the model structure. The adequacy of the solution
depends largerly on the skill with which the basic simplifying assumptions are
made. A basic choice is between linear and non-linear models. Damping estimation
is another source of error, because damping cannot be calculated like the mass and
stiffness properties. The last two steps consist in applying procedures worked out
by mathematicians. The real engineering work lies in the first two stages, while the
last two stages may be considered as mere applications of recipies.
One degree of freedom systems are considered in Chapters 2 and 3.
Discrete systems are treated in Chapters 4 and 5. Chapter 6 is devoted to straight
beams and bars.
1.4 Vibratory Motions
According to the cause producing or sustaining the vibratory motion, one
can distinguish: free vibrations, produced by an impact or an initial displacement;
forced vibrations, produced by external forces or kinematic excitation; parametric
vibrations, due to the change, produced by an external cause, of a system
parameter; self-excited vibrations, produced by a mechanism inherent in the
system, by conversion of an energy obtained from a uniform energy source
associated with the system oscillatory excitation.
If the system is distorted from the equilibrium configuration and then
released, it will vibrate with free vibrations. If any part of the system is struck by a
blow, the system will vibrate freely. Musical instruments like drums are struck and
strings are plucked. Free vibrations exist when the forces acting on the system arise
solely from motion of the system itself. The frequencies of the free vibrations are
fixed functions of the mass, stiffness, and damping properties of the system itself.
They are called natural frequencies. For any particular system they have definite
constant values. When all particles of a body vibrate in a synchronous harmonic
motion, the deflected shape is a natural mode shape.
Vibrations which take place under the excitation of external forces are
forced vibrations. External forces in any system are forces which have their
reactions acting on bodies which are not parts of the system isolated for study. The
forcing function can be harmonic, complex periodic, impulse, transient, or random.
1. MODELLING VIBRATING SYSTEMS 9
When a system is excited by a periodic external force which has one
frequency equal to or nearly equal to a natural frequency of the system, the ensuing
vibratory motion becomes relatively large even for small amplitudes of the
disturbing force. The system then is in a state of resonance. An example is the
swing pushed at the right intervals. Other examples include vibrations of geared
systems at the tooth-meshing frequency, torsional vibrations of multi-cylinder
engine shafts at the firing frequency, vibrations of rolling element bearings at the
ball passing frequencies, etc.
There is an effect arising from the damping which causes the resonance
frequency to differ slightly from the natural frequency by an amount which
increases with the damping. Fortunately the distinction in practice is very small and
can be neglected in most engineering structures, unless very high damping is
provided on purpose.
Resonance relates to the condition where either a maximum motion is
produced by a force of constant magnitude, or a minimum force is required to
maintain a prescribed motion level. A resonance is defined by a frequency, a
response level and a bandwidth of the frequency response curve. Avoidance of
large resonant vibration levels can be accomplished by: a) changing the excitation
frequency; b) making stiffness and/or mass modifications to change the natural
frequencies; c) increasing or adding damping; and d) adding a dynamic vibration
absorber.
When the driving frequency is an integer multiple of the natural frequency
of the associated linear system, non-linear single-degree-of-freedom systems
described by Mathieu equations exhibit parametric instabilities, referred to as
parametric resonances.
The principal parametric resonance occurs when the excitation frequency
is twice the natural frequency. Parametric resonances of fractional order also exist.
Multi-degree-of-freedom systems can experience parametric resonance if the
driving frequency and two or more natural frequencies satisfy a linear relation with
integer coefficients.
Parametric resonance is a state of vibration in which energy flows into the
system from an external source at resonance, increasing the amplitude of the
system’s response. This energy is dependent upon both the natural frequency of the
system and the frequency of the parameter variation.
During resonant vibrations and self-excited vibrations, the system vibrates
at its own natural frequency. But while the former are forced vibrations, whose
frequency is equal to a whole-number ratio multiple of the external driving
frequency, the latter is independent of the frequency of any external stimulus.
In a self-excited vibration, the alternating force that sustains the motion is
created or controlled by the motion itself. When the motion stops, the alternating
force disappears. Well-known examples include the vibrations of a violin string
