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Table of Contents 1. INTRODUCTION TO MECHANICAL VIBRATION: APPLICATIONS AND VIBRATION TYPES ........................................................1
1.1. HISTORY OF VIBRATION ....................................................................................................................................1
1.2. WHY IS THE STUDY OF VIBRATION REQUIRED? ......................................................................................................2
1.3. APPLICATION OF MECHANICAL VIBRATION ..........................................................................................................2
1.4. VIBRATION TYPES ............................................................................................................................................3
2. MODELING OF DYNAMIC VIBRATING SYSTEMS: ELEMENTS OF VIBRATING SYSTEMS; SCHEMATIC AND MATHEMATICAL
MODELING; HIGHLIGHTS ON CONCEPTS OF ENGINEERING MECHANICS, DIFFERENTIAL EQUATIONS, AND LINEAR ALGEBRA OF
MATRICES MANIPULATION. .............................................................................................................................................5
2.1. MODELING VIBRATION OF DYNAMIC SYSTEMS .....................................................................................................5
2.2. ELEMENTS OF VIBRATING SYSTEM ......................................................................................................................4
2.3. GOVERNING EQUATIONS OF VIBRATING SYSTEMS: ................................................................................................6
2.4. MATRIX MANIPULATION: .................................................................................................................................8
2.5. SOLUTION METHODS FOR DIFFERENTIAL EQUATION (PDE & ODE):........................................................................9
3. FREE VIBRATIONS OF SINGLE DOF SYSTEMS ............................................................................................................ 10
3.1. INTRODUCTION ............................................................................................................................................ 10
3.2. UNDAMPED FREE SYSTEMS VIBRATION: ............................................................................................................ 11
3.3. DAMPED FREE SYSTEMS VIBRATION: ................................................................................................................ 15
3.4. FORCED VIBRATION ................................................................................................................................. 21
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1. INTRODUCTION TO MECHANICAL VIBRATION: APPLICATIONS AND VIBRATION TYPES
1.1. HISTORY OF VIBRATION
The origins of the theory of vibration can be traced back to the design and development of musical
instruments (good vibration). It is known that drums, flutes, and stringed instruments existed in China
and India for several millennia B.C. Also, ancient Egyptians and Greeks explored sound and vibration
from both practical and analytical points of view. For example, while Egyptians had known of a harp
since at least 3000 B.C., the Greek philosopher, mathematician, and musician Pythagoras (of the
Pythagoras theorem fame) who lived during 582 to 502 B.C., experimented on sounds generated by
blacksmiths and related them to music and physics. The Chinese developed a mechanical seismograph
(an instrument to detect and record earthquake vibrations) in the 2nd century A.D. The foundation of
the modern-day theory of vibration was probably laid by scientists and mathematicians such as Robert
Hooke (1635–1703) of the Hooke’s law fame, who experimented on the vibration of strings; Sir Isaac
Newton (1642–1727), who gave us calculus and the laws of motion for analyzing vibrations; Daniel
Bernoulli (1700–1782) and Leonard Euler (1707–1783), who studied beam vibrations (Bernoulli-Euler
beam) and also explored dynamics and fluid mechanics; Joseph Lagrange (1736–1813), who studied
vibration of strings and also explored the energy approach to formulating equations of dynamics;
Charles Coulomb (1736–1806), who studied torsional vibrations and friction; Joseph Fourier (1768–
1830), who developed the theory of frequency analysis of signals; and Simeon-Dennis Poisson (1781–
1840), who analyzed vibration of membranes and also analyzed elasticity (Poisson’s ratio). As a result of
the industrial revolution and associated developments of steam turbines and other rotating machinery,
an urgent need was felt for developments in the analysis, design, measurement, and control of
vibration. Motivation for many aspects of the existing techniques of vibration can be traced back to
related activities since the industrial revolution. Much credit should go to scientists and engineers of
more recent history, as well. Among the notable contributors are Rankin (1820–1872), who studied
critical speeds of shafts; Kirchhoff (1824–1887), who analyzed vibration of plates; Rayleigh (1842–1919),
who made contributions to the theory of sound and vibration and developed computational techniques
for determining natural vibrations; de Laval (1845–1913), who studied the balancing problem of rotating
disks; Poincaré (1854–1912), who analyzed nonlinear vibrations; and Stodola (1859–1943), who studied
vibrations of rotors, bearings, and continuous systems. Distinguished engineers who made significant
contributions to the published literature and also to the practice of vibration include Timoshenko, Den
Hartog, Clough, and Crandall
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1.2. WHY IS THE STUDY OF VIBRATION REQUIRED?
The study of vibration is concerned with the oscillatory motions of bodies and the forces associated with
them. Vibration is a repetitive, periodic, or oscillatory response of a mechanical system. The rate of the
vibration cycles is termed “frequency”. Repetitive motions that are somewhat clean and regular, and
that occur at relatively low frequencies, are commonly called oscillations, while any repetitive motion,
even at high frequencies, with low amplitudes, and having irregular and random behavior falls into the
general class of vibration. Nevertheless, the terms “vibration” and “oscillation” are often used
interchangeably. In general, they can be defined as oscillation is the displacement of bodies which is
repeated periodically, and hence vibration is the oscillation of either rigid or elastic bodies with small
displacement that may vary at different instant of times.
Vibrations can naturally occur in an engineering system and may be representative of its free and
natural dynamic behavior. Also, vibrations may be forced onto a system through some form of
excitation. The excitation forces may be either generated internally within the dynamic system, or
transmitted to the system through an external source. When the frequency of the forcing excitation
coincides with that of the natural motion, the system will respond more vigorously with increased
amplitude. This condition is known as resonance, and the associated frequency is called the resonant
frequency. There are “good vibrations,” which serve a useful purpose. Also, there are “bad vibrations,”
which can be unpleasant or harmful. For many engineering systems, operation at resonance would be
undesirable and could be destructive. Suppression or elimination of bad vibrations and generation of
desired forms and levels of good vibration are general goals of vibration engineering.
1.3. APPLICATION OF MECHANICAL VIBRATION
Applications of vibration are found in many branches of engineering such as aeronautical and aerospace,
civil, manufacturing, mechanical, and even electrical. Usually, an analytical or computer model is needed
to analyze the vibration in an engineering system. Models are also useful in the process of design and
development of an engineering system for good performance with respect to vibrations. Vibration
monitoring, testing, and experimentation are important as well in the design, implementation,
maintenance, and repair of engineering systems. All these are important topics of study in the field of
vibration engineering.
In particular, practical applications and design considerations related to modifying the vibrational
behavior of mechanical devices and structures will be studied. This knowledge will be useful in the
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practice of vibration regardless of the application area or the branch of engineering; for example, in the
analysis, design, construction, operation, and maintenance of complex structures such as the Space
Shuttle and the International Space Station.
In specific terms, the science and engineering of vibration involve two broad categories of applications
which are:
Elimination or suppression of undesirable vibrations and
Generation of the necessary forms and quantities of useful vibrations.
Undesirable and harmful types of vibration include structural motions generated due to earthquakes,
dynamic interactions between vehicles and bridges or guide ways, noise generated by construction
equipment, vibration transmitted from machinery to its supporting structures or environment, and
damage, malfunction, and failure due to dynamic loading, unacceptable motions, and fatigue caused by
vibration.
Therefore, rigorous analysis and design are needed, particularly with regard to vibration, in the
development of dynamic systems. Lowering the levels of vibration will result in reduced noise and
improved work environment, maintenance of a high performance level and production efficiency,
reduction in user/operator discomfort, and prolonging the useful life of industrial machinery.
Desirable types of vibration include those generated by musical instruments, devices used in physical
therapy and medical applications, vibrators used in industrial mixers, vibrating rollers used in road
construction, vibrating sieves used in flour milling plants, part feeders and sorters, and vibratory
material removers such as drills and polishers (finishers).
1.4. VIBRATION TYPES
Based on various conditions, vibration of systems can be classified as:
Free and Forced vibrations
There are two general classes of vibrations - free and forced. Natural or Free vibration takes place
when a system oscillates under the action of forces inherent in the system itself, and when external
impressed forces are absent. The system under free vibration will vibrate at one or more of its
natural frequencies, which are properties of the dynamic system established by its mass and
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stiffness distribution. Vibration that takes place under the excitation of external forces is called
forced vibration.
Undamped and Damped Vibrations
Damping is a means of reducing velocity through resistance to the motion of vibrating systems when
energy is dissipated, in particular by forcing an object through a liquid or gas, or along another body.
It can be represented by dashpot in schematic representations of vibrating systems model. Units of
damping are often given as Newtons per meters per second (N/m/s, which is also expressed as
Ns/m.). A basic understanding of this concept is essential for vibration analysis. Free vibration refers
to the vibration of a damped (as well as undamped) system of masses with motion entirely
influenced by their potential energy. Forced vibration occurs when motion is sustained or driven by
an applied periodic force in either damped or undamped systems. Therefore, both free and forced
vibrations of systems can be termed as damped when the systems are required to dissipate energy
otherwise undamped.
Degrees of freedom; SDOF and MDOF Vibrations
The minimum number of independent coordinates required to describe the motion of a system is
called degrees of freedom of the system, and hence systems with one independent coordinate used
to represent completely the systems are called a single degrees of freedom (SDOF) system and
those with more than one independent coordinate are termed as multi degrees of freedom (MDOF)
systems.
Thus, a free particle undergoing general motion in space will have three degrees of freedom that it is
going to be considered as multi degrees of freedom (MDOF) system, and a rigid body will have six
degrees of freedom, i.e., three components of position and three angles defining its orientation.
Furthermore, a continuous elastic body will require an infinite number of coordinates (three for
each point on the body) to describe its motion; hence, its degrees of freedom must be infinite.
However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be
considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a
surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing
the system to one having a few degrees of freedom.
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2. MODELING OF DYNAMIC VIBRATING SYSTEMS: ELEMENTS OF VIBRATING SYSTEMS; SCHEMATIC AND MATHEMATICAL
MODELING; HIGHLIGHTS ON CONCEPTS OF ENGINEERING MECHANICS, DIFFERENTIAL EQUATIONS, AND LINEAR
ALGEBRA OF MATRICES MANIPULATION.
2.1. MODELING VIBRATION OF DYNAMIC SYSTEMS
Dynamic analysis can be carried out most conveniently by adopting the following three-stage approach:
Stage I: Devise a mathematical or physical model of the system to be analyzed.
Stage II: From the model, write the equations of motion.
Stage III: Evaluate the system response to relevant specific excitation.
Stage I: The mathematical model
Although it may be possible to analyze the complete dynamic system being considered, this often leads
to a very complicated analysis, and the production of much unwanted information. A simplified
mathematical model of the system is therefore usually sought which will, when analyzed, produce the
desired information as economically as possible and with acceptable accuracy. The derivation of a
simple mathematical model to represent the dynamics of a real system is not easy, if the model is to
give useful and realistic information.
However, to model any real system a number of simplifying assumptions can often be made. For
example, a distributed mass may be considered as a lumped mass, or the effect of damping in the
system may be ignored particularly if only resonance frequencies are needed or the dynamic response
required at frequencies well away from a resonance, or a non-linear spring may be considered linear
over a limited range of extension, or certain elements and forces may be ignored completely if their
effect is likely to be small. Furthermore, the directions of motion of the mass elements are usually
restrained to those of immediate interest to the analyst.
Thus the model is usually a compromise between a simple representation which is easy to analyze but
may not be very accurate, and a complicated but more realistic model which is difficult to analyze but
gives more useful results. Consider for example, the analysis of the vibration of the front wheel of a
motor car. Fig. 2.1 shows a typical suspension system. As the car travels over a rough road surface, the
wheel moves up and down, following the contours of the road. This movement is transmitted to the
upper and lower arms, which pivot about their inner mountings, causing the coil spring to compress
and extend. The action of the spring isolates the body from the movement of the wheel, with the shock
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absorber or damper absorbing vibration and sudden shocks. The tie rod controls longitudinal movement
of the suspension unit.
Fig.2.1. A typical suspension system: Rover 800 front suspension (By courtesy of Rover Group).
Fig.2.2(a) is a very simple model of this same system, which considers translational motion in a vertical
direction only: this model is not going to give much useful information, although it is easy to
analyze. The more complicated model shown in Fig.2.2 (b) is capable of producing some meaningful
results at the cost of increased labor in the analysis, but the analysis is still confined to motion in a
vertical direction only. A more refined model, shown in Fig.2.2(c), shows the whole car considered, with
the analysis of translational and rotational motion of the car body being allowed.
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Fig.2.2 (a) Simplest model - motion only in
a vertical direction
Fig.2.2 (b) More complex model-motion
only in a vertical direction
Fig.2.2(c) A more refined model than those in Fig.2.2 (a) and (b)-Motion in a vertical direction, roll, and
pitch of the road contour can be analyzed.
If the modeling of the car body by a rigid mass is too crude to be an acceptable assumption, a finite
element analysis may prove useful. This technique would allow the body to be represented by a number
of mass elements.
Another example of modeling is that the vibration of a machine tool such as a lathe can be analyzed by
modeling the machine structure by the two degree of freedom system shown in Fig.2.3. In the simplest
analysis the bed can be considered to be a rigid body with mass and inertia, and the headstock and
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tailstock are each modeled by lumped masses. The bed is supported by springs at each end as shown.
Such a model would be useful for determining the lowest or fundamental natural frequency of vibration.
A refinement to this model, which may be essential in some designs of machine where the bed cannot
be considered rigid, is to consider the bed to be a flexible beam with lumped masses attached as before.
Fig.2.3. Machine tool vibration analysis model
However, none of these models include the effect of damping in the structure. Damping in most
structures is very low so that the difference between the undamped and the damped natural
frequencies is negligible. It is usually only necessary to include the effects of damping in the mode if the
response to a specific excitation is sought, particularly at frequencies in the region of a resonance.
It is inevitable to note that vibrating systems can be modeled as distributed mass system for better
results through tiresome and complex analysis. But, because of the approximate nature of most models,
whereby small effects are neglected and the environment is made independent of the system
motions, it is usually reasonable to assume constant parameters and linear relationships. This means
that the coefficients in the equations of motion are constant and the equations themselves are linear:
these are real aids to simplifying the analysis. Distributed masses can often be replaced by lumped mass
elements to give ordinary rather those partial differential equations of motion. Usually the numerical
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value of the parameters can, substantially, be obtained directly from the system being analyzed.
However, model system parameters are sometimes difficult to assess, and then an intuitive estimate is
required, engineering judgment being of the essence.
In general, vibrating system can be modeled using the following two independent approaches:
Discrete (Lamped Mass) System Element:
The vibrating systems modeling with the assumption that the inertial (mass), flexibility (spring), and
dissipative (damping) characteristics could be “lumped” as a finite number of “discrete” elements is
a sort of modeling. Such models are termed as lumped-parameter or discrete parameter systems. If
all the mass elements move in the same direction, one has a one-dimensional system (i.e.,
“rectilinear” motion), with each mass having a single degree of freedom. If the masses can move
independently of each other, the number of degrees of freedom of such a one-dimensional system
will be equal to the number of lumped masses, and will be finite. In a “planar” system, each lumped
mass will be able to move in two orthogonal directions and hence, will have two degrees of
freedom; and similarly in a “spatial” system, each mass will have three degrees of freedom. As long
as the number of lumped inertia elements is finite, then one has a lumped parameter (or, discrete)
system with a finite degree of freedom.
Continuous System element:
Continuous systems such as beams, rods, cables and strings can be modeled by discrete mass and
stiffness parameters and analyzed as multi degree of freedom systems, but such a model is not
sufficiently accurate for most purposes and practical applications of the real world vibration
problems. Furthermore, mass and elasticity cannot always be separated in models of real systems.
Thus mass and elasticity have to be considered as distributed parameters in continuous systems
vibration analysis. For the analysis of systems with distributed mass and elasticity it is necessary to
assume a homogeneous, isotropic material which follows Hooke’s law.
Therefore, beams, rods, cables and strings are treated as continuous system element. Generally, in
practical vibrating systems, inertial, elastic, and dissipative effects are found continuously
distributed in one, two, or three dimensions.
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Correspondingly, there are line structures, surface/planar structures, or spatial structures. They will
possess an infinite number of mass elements, continuously distributed in the structure, and
integrated with some connecting flexibility (elasticity) and energy dissipation.
In view of the connecting flexibility, each small element of mass will be able to move out of phase
(or somewhat independently) with the remaining mass elements. It follows that a continuous
system (or a distributed-parameter system) will have an infinite number of degrees of freedom and
will require an infinite number of coordinates to represent its motion. In other words, when
extending the concept of a finite-degree-of-freedom system, an infinite-dimensional vector is
needed to represent the general motion of a continuous system (practical and real system).
Equivalently, a one-dimensional continuous system (a line structure) will need one independent
spatial variable, in addition to time, to represent its response. In view of the need for two
independent variables in this case — one for time and the other for space — the representation of
system dynamics will require partial differential equations (PDEs) rather than ordinary differential
equations (ODEs). Furthermore, the system will depend on the boundary conditions as well as the
initial conditions.
Stage II: The equations of motion
Several methods are available for obtaining the equations of motion from the mathematical
model, the choice of method often depending on the particular model and personal preference. For
example, analysis of the free-body diagrams drawn for each body of the model usually produces the
equations of motion quickly: but it can be advantageous in some cases to use an energy method such as
conservation of energy and the Lagrange equation. From the equations of motion the characteristic or
frequency equation is obtained, yielding data on the natural frequencies, modes of vibration, general
response, and stability.
Stage III: Response to specific excitation
Although Stage II of the analysis gives much useful information on natural frequencies, response, and
stability, it does not give the actual system response to specific excitations. It is necessary to know the
actual response in order to determine such quantities as dynamic stress, noise, output position, or
steady-state error for a range of system inputs, either force or motion, including harmonic, step and
ramp. This is achieved by solving the equations of motion with the excitation function present.
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Remember:
A few examples have been given above to show how real systems can be modeled, and the principles of
their analysis. To be competent to analyze system models it is first necessary to study the analysis of
damped and undamped, free and forced vibration of single degree of freedom which will be discussed in
the subsequent chapters. This not only allows the analysis of a wide range of problems to be carried out,
but it is also essential background to the analysis of systems with more than one degree of freedom.
Systems with distributed mass, such as beams, will also be analyzed.
2.2. ELEMENTS OF VIBRATING SYSTEM
Spring elements:
A spring is an elastic object used to store mechanical energy specifically potential energy. Springs
are usually made out of hardened steel; larger ones are made from annealed steel and hardened
after fabrication, while small springs can be wound from pre-hardened stock for vibrating systems.
The rate of a spring (stiffness) is the change in the force it exerts, divided by the change in deflection
of the spring. That is, it is the gradient of the force versus deflection curve. An extension or
compression spring has units of force divided by distance, for example lbf/in or N/m. Torsion springs
have units of force multiplied by distance divided by angle, such as N·m/rad or ft·lbf/degree. The
inverse of spring rate (stiffness) is compliance, which is if a spring has a rate of 10 N/mm, it has a
compliance of 0.1 mm/N. The stiffness (or rate) of springs in parallel is additive, as is the compliance
of springs in series.
Depending on the design and required operating environment, any material can be used to
construct a spring; so long the material has the required combination of rigidity and elasticity:
technically, a wooden bow is a form of spring.
Mass and inertia elements:
In physics, mass commonly refers to any of three properties of matter, which have been shown
experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational
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mass. With respect to vibration, the inertial mass of an object in a vibrating system is concerned and
it is used to determine the acceleration of the object in the presence of an applied force. According
to Isaac Newton's second law of motion, if a body of mass m is subjected to a force F, its
acceleration, a, is given by F/m.
Damping elements:
Damping is the phenomenon by which mechanical energy is dissipated (usually converted into
internal thermal energy) in dynamic systems. Knowledge of the level of damping in a dynamic
system is important in utilization, analysis, and testing of the system. In characterizing damping in a
dynamic system, it is important, first, to understand the major mechanisms associated with
mechanical-energy dissipation in the system. Then, a suitable damping model should be chosen to
represent the associated energy dissipation.
In the modeling of systems, damping can be neglected if the mechanical energy that is dissipated
during the time duration of interest is small in comparison to the initial total mechanical energy of
excitation in the system. Even for highly damped systems, it is useful to perform an analysis with the
damping terms neglected, in order to study several crucial dynamic characteristics; for example,
modal characteristics (undamped natural frequencies and mode shapes).
Several types of damping are inherently present in a mechanical system. If the level of damping that
is available in this manner is not adequate for proper functioning of the system, external damping
devices can be added either during the original design or in a subsequent stage of design
modification of the system. Three primary mechanisms of damping (damping elements) are
important in the study of mechanical systems. They are internal damping (of material), structural
damping (at joints and interfaces), and fluid damping (through fluid-structure interactions).
Internal (material) damping results from mechanical-energy dissipation within the material due to
various microscopic and macroscopic processes.
Structural damping is caused by mechanical energy dissipation resulting from relative motions
between components in a mechanical structure that has common points of contact, joints, or
supports.
Fluid damping arises from the mechanical energy dissipation resulting from drag forces and
associated dynamic interactions when a mechanical system or its components move in a fluid.
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Note that dampers are schematically represented by dashpots for their technical names include
damper and dashpot as they are a subset of dashpots and thus are sometimes called "dashpots",
just as cars are a subset of vehicles and are sometimes called "vehicles".
2.3. GOVERNING EQUATIONS OF VIBRATING SYSTEMS:
The concepts developed in this sub chapter constitute an introductory review of concepts to be
used in vibration analysis and serve as an introduction for extending these concepts to more
complex systems in later chapters. In addition, basic ideas relating to measurement of SDOF
vibrations are introduced that will later be extended to multiple degrees-of- freedom systems and
distributed-parameter systems. This chapter is intended to be a review of vibration basics and an
introduction to a more formal and general analysis approaches for more complicated models in the
following chapters.
Therefore, the following basic concepts that can be applied in the forthcoming analysis of vibration
problems will be introduced.
Equilibrium Equations: Newton’s Equations of Motion for Force Balance
Here is to show the application of Newton’s Law in the analysis of vibration problems. For example,
simple harmonic motion, or oscillation, may be exhibited by structures that have elastic restoring
forces. Such systems can be modeled, in some situations, by a spring–mass schematic, as illustrated
in Figure 2.1. This constitutes the most basic vibration model of a structure and can be used
successfully to describe a surprising number of devices, machines, and structures. The methods
presented here for solving such a simple mathematical model may seem to be more sophisticated
than the problem requires. However, the purpose of the analysis is to lay the groundwork for the
analysis in the following chapters of more complex systems.
If x=x (t) denotes the displacement (m) of the mass m (kg) from its equilibrium position as a function
of time t(s), the equation of motion for this system as per the second Newton’s Law becomes [upon
summing forces in Figure 2.1(b)]
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Figure 2.1 (a) Spring–mass schematic, (b) free body diagram, and (c) free body diagram of the static
Spring-mass system.
(2.1)
Where k is the stiffness of the spring (N/m), xs is the static deflection (m) of the spring under gravity
load, g is the acceleration due to gravity (m/s2), and the over dots denote differentiation with
respect to time. From summing forces in the free body diagram for the static deflection of the spring
[Figure 2.1(c)], mg = kxs and the above equation of motion becomes
(2.2)
where k and m are to be determined from static experiments.
This last expression is the equation of motion of a single-degree-of-freedom system and is a linear,
second-order, ordinary differential equation with constant coefficients. In this example, the second
Newton’s Law has been shown how much powerful it is in modeling a single degree-of-freedom
vibration problem, and its application can be extended to be used in more complicated systems such
as in multiple degrees-of-freedom and continuous systems in the same manner shown in the above.
Geometric Equations: Strain-Displacement Equations:
The concepts of mechanics of materials are also vital in the analysis of vibration problems especially
in continuous systems vibration. The theory of strain rests solely on geometric parameters and
hence the topic geometric equations which in general relate strain and displacements of a system
under consideration. Therefore, it is mandatory to refresh the concepts of mechanics of materials.
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The strain-displacement relations are given by such an equation in polar coordinates as:
(2.3)
Constitutive Equations:
Constitutive equations are the formulations developed from the concepts of mechanics of materials
to relate stresses and strains. These equations are highly employed in the analysis of vibration
problems especially for continuous systems vibration analysis. They can be of the form
(2.4)
Therefore, it is needed to review the concepts of mechanics of materials.
2.4. MATRIX MANIPULATION:
Linear algebra, the algebra of sets, vectors, and matrices, is useful in the study of mechanical
vibration. In practical vibrating systems, interactions among various components are inevitable.
There are many response variables associated with many excitations. It is thus convenient to
consider all excitations (inputs) simultaneously as a single variable, and also all responses (outputs)
as a single variable. The use of linear algebra makes the analysis of such a system convenient. The
subject of linear algebra is complex and is based on a rigorous mathematical foundation. The basics
of vectors and matrices form the foundation of linear algebra to be applied in vibration analysis.
The nature of the free response of a single-degree-of-freedom system is determined by the roots of
the characteristic equation of the differential equation (2.2). In addition, the exact solution is
calculated using these roots. A similar situation exists for the multiple-degree-of-freedom systems
described in the previous example of section 2.3.
Motivated by the single-degree-of-freedom system, this section reminds one of the methods used
to examine the problem of characteristic roots for systems in matrix notation and extends the same
ideas discussed in section 2.4 to the multiple-degree-of-freedom systems. The mathematical tools
needed to extend the ideas of section 2.4 are those of linear algebra, which are to be introduced
later on the coming chapters as needed.
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In fact, if one attempts to follow the method of solving single-degree-of-freedom vibration problems
in solving multiple-degree-of-freedom systems, one is led immediately to a standard matrix problem
called the algebraic eigenvalue problem. This section reminds that the matrix eigenvalue problem
approach is one of the methods to apply to the multiple-degree-of-freedom vibration problems that
will be introduced in the coming chapters. The eigenvalues and eigenvectors can be used to
determine the time response to initial conditions by the process called modal analysis which is to be
introduced later. Therefore, one should review the linear algebra concept of eigenvalue problem
and the related matrix manipulations.
2.5. SOLUTION METHODS FOR DIFFERENTIAL EQUATION (PDE & ODE):
A vibrating system can be interpreted as a collection of mass particles. In the case of distributed
systems, the number of particles is infinite. The flexibility and damping effects can be introduced as
forces acting on these particles. It follows that Newton’s second law for a mass particle forms the basis
of describing vibratory motions. System of differential equations can be obtained directly by applying
Newton’s second law to each particle, and once the equations are obtained, they have to be solved for
their solutions.
It is convenient, however, to use the standard solution methods of differential equations for simpler
system, and Lagrange’s equations for the purpose of determining the solutions for the responses of the
vibration problems when the system is relatively complex.
Therefore, it is essential to review methods for solving differential equations governing the motion of
each particle and the likes.
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3. FREE VIBRATIONS OF SINGLE DOF SYSTEMS
3.1. INTRODUCTION
In this chapter the free vibration of a single-degree-of-freedom system will be analyzed and reviewed.
Analysis, measurement, and design of undamped and damped vibrations of a single-degree-of-freedom
system (often abbreviated SDOF) is discussed. The concepts developed in this chapter constitute an
introductory review of free vibrations and serve as an introduction for extending these concepts to
more complex systems in later chapters.
In addition, basic ideas relating to measurement and control of vibrations are introduced that will later
be extended to multiple-degree-of- freedom systems and distributed-parameter systems. This chapter is
intended to be a review of free vibration basics and an introduction to a more formal and general
analysis for more complicated models in the forthcoming chapters.
Vibration technology has grown and taken on a more interdisciplinary nature. This has been caused by
more demanding performance criteria and design specifications for all types of machines and structures.
Hence, in addition to the standard material usually found in introductory chapters of vibration and
structural dynamics texts, several topics from control theory and vibration measurement theory are
presented. This material is included not to train the reader in control methods (the interested student
should study control and system theory texts) but rather to point out some useful connections between
vibration and control as related disciplines. In addition, structural control has become an important
discipline requiring the coalescence of vibration and control topics.
Vibrations are oscillatory responses of dynamic systems. Natural (free) vibrations occur in these systems
due to the presence of two modes of energy storage. Specifically, when the stored energy is converted
from one form to the other, repeatedly back and forth, the resulting time response of the system is
oscillatory in nature. In a mechanical system, natural vibrations can occur because kinetic energy, which
is manifested as velocities of mass (inertia) elements, can be converted into potential energy (which has
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two basic types: elastic potential energy due to the deformation in spring-like elements, and
gravitational potential energy due to the elevation of mass elements against the Earth’s gravitational
pull) and back to kinetic energy, repetitively, during motion. Similarly, natural oscillations of electrical
signals occur in circuits due to the presence of electrostatic energy (of the electric charge storage in
capacitor-like elements) and electromagnetic energy (due to the magnetic fields in inductor-like
elements).
Fluid systems can also exhibit natural oscillatory responses as they possess two forms of energy. But
purely thermal systems do not produce natural oscillations because they, as far as anyone knows, have
only one type of energy.
Note, however, that an oscillatory forcing function is able to make a dynamic system respond with an
oscillatory motion (usually at the same frequency as the forcing excitation) even in the absence of two
forms of energy storage. Such motions are forced responses rather than natural or free responses.
Hence mechanical vibrations can occur as both free (natural) responses and forced responses in
numerous practical situations.
Therefore, in this introductory chapter, single-degree-of-freedom systems that require only one
coordinate (or one independent displacement variable) in their model are considered almost exclusively.
It provides an introduction to the response analysis of mechanical vibrating systems in the time domain
only (free vibration only).
3.2. UNDAMPED FREE SYSTEMS VIBRATION:
This section first shows that many types of oscillatory systems can be represented by the equation of an
undamped simple oscillator, in particular, mechanical systems are considered.
The conservation of energy is a straightforward approach for deriving the equations of motion for
undamped oscillatory systems (or conservative systems). The equations of motion for mechanical
systems can be derived using the free-body diagram approach with the direct application of Newton’s
12
second law. An alternative and rather convenient approach is the use of Lagrange equations, which will
be discussed later on the next chapters.
The natural (free) response of an undamped simple oscillator is a simple harmonic motion. This is a
periodic, sinusoidal motion, and this simple time response is discussed under here.
3.2.1. Conservation of Energy:
There is no energy dissipation in undamped systems, which contain energy storage elements only. In
other words, energy is conserved in these systems, which are known as conservative systems. For
mechanical systems, conservation of energy gives
KE + PE = Constant (3.1)
These systems tend to be oscillatory in their natural motion, as noted before.
Fig.3.1. An example of SDOF system
Figure 3.1 shows a translatory mechanical system (an undamped oscillator) that has just one degree of
freedom x. This can represent a simplified model of a rail car that is impacting against a snubber. The
conservation of energy (equation (3.1)) gives
constkxxm 22
2
1
2
1
(3.2)
Here, m is the mass and k is the spring stiffness, which are both constants of the system considered.
Differentiate equation (3.2) with respect to time t to obtain
0 xxkxxm (3.3)
Since 0x at all t, in general, one can cancel it out. Hence, by the method of conservation of energy,
and obtains the equation of motion
13
0 xm
kx
(3.4)
It can be noted that the general form of the equation of free (i.e., no excitation force) motion of linear
systems which are similar to that considered above (Fig.3.1) is in the form of equation (3.4). This is the
equation of an undamped SDOF oscillator.
For mechanical system of mass m and stiffness k, equation (3.4) can be rewritten as
02 xx n (3.5)
Where m
kn
To determine the time response x of this system, one can use the trial solution
)sin( tAx n (3.6)
in which A and are unknown constants, to be determined by the initial conditions (for x and x ); say,
0)0( xx and 0)0( vx
(3.7)
Substitution of the trial solution into equation (3.4) to obtain
0)sin()( 22 tAA nnn (3.8)
This equation is identically satisfied for all t. Hence, the general solution of equation (3.4) is indeed
equation (3.6), which is periodic and sinusoidal. This response is sketched in Fig.3.2.
Fig.3.2. Free response of undamped simple oscillator
14
To determine A and which are constants, the boundary conditions will be used and hence
sin0 Ax
and cos0 nAv
(3.9)
From this equation
0
01tanv
xn
(3.10)
and using the trigonometric identity 1cossin 22
2
202
0
n
vxA
(3.11)
3.2.2. Free-body diagram Approach:
This is a method which employs the direct application of Newton’s second law using the free body diagrams of
the system. The following procedure can be followed:
1. Identifying the external and effective forces acting on the mass of the system.
2. Drawing separately free body diagrams of external and effective forces acting on the mass.
3. Applying the Newton’s second law for the balance of external and effective forces.
This approach can be illustrated using the system shown in Figure 3.1. The external force on the system
considered is identified to be only the restoring force due to the spring with stiffness k, and hence the
free body diagram for the external force acting on the system of mass m is
x
kx m
Fig.3.3. External force free body diagram
and the effective force free body diagram is also shown below as
x
m xm
Fig.3.4. Effective force free body diagram
15
Using the free body diagrams shown in Figures 3.3 and 3.4 and applying Newton’s second law for force
balance such that
EffevtiveExternal FF (3.12)
This implies that
xmkx or 0 kxxm (3.13)
This equation is the same as that of equation (3.4) which in turn gives the similar result as the above
method of energy conservation.
3.3. DAMPED FREE SYSTEMS VIBRATION:
Now consider the free (natural) response of a single degree-of-freedom system in the presence of energy
dissipation (damping).
Assume viscous damping, and consider the oscillator shown in Figure 3.5. The free-body diagram of the system is
shown separately. The following notation is used in this material of this sub-section.
n Undamped natural frequency
d Damped natural frequency
r Resonant frequency
Frequency of excitation load.
The viscous damping coefficient is denoted by c to be introduced here, and applying Newton’s second law, from
the free-body diagram in Fig.3.5, one has the equation of motion as
xckxxm or 0 xckxxm (3.14)
Or 02 2 xxx nn (3.15)
Where m
cn 2 and is called the damping ratio. The formal definition and the rationale for this
terminology will be discussed later.
16
Fig.3.5. A damped simple oscillator and its free-body diagram.
Equation (3.15) is a free (or unforced, or homogeneous) vibration equation of motion when there is energy
dissipation. Its solution is the free (natural) response of the system and is also known as the homogeneous
solution. Note thatm
kn , which is the natural frequency when there is no damping.
Hence, km
c
2
1
Assuming an exponential solution for equation (3.15) as
tCex (3.16)
This is justified by the fact that linear systems have exponential or oscillatory (i.e., complex exponential) free
responses.
Substitution of equation (3.16) into (3.15) gives
0)2( 22 tnn Ce (3.17)
Note that tCe is not zero in general. It follows that when λ satisfies the equation
0)2( 22 nn (3.18)
Then, equation (3.16) is the solution of equation (3.17). Equation (3.18) is called the characteristic equation of
the system. It is shown that this equation depends on the natural dynamics of the system, not on forcing excitation
or initial conditions.
Solution of equation (3.18) gives the two roots:
nn 12 (3.19)
1 and
2
17
These are called eigenvalues of the system and, in line with the values of , there are three conditions to be
observed.
Condition I: When λ1 ≠ λ2,
The general solution is
tt eCeCx 21
21 (3.20)
The two unknown constants C1 and C2 are related to the integration constants, and can be determined by two
initial conditions, which should be known.
Condition II: When 21 , one has the case of repeated roots.
In this case, the general solution (3.20) does not hold because C1 and C2 would no longer be independent
constants, to be determined by two initial conditions. The repetition of the roots suggests that one term of the
homogenous solution should have the multiplier t (a result of the double-integration of zero). Then, the general
solution is
tt teCeCx 21
21 (3.21)
One can identify three categories of damping level and the nature of the response will depend on the particular
category of damping levels to be discussed below.
CASE 1: Underdamped Motion ( 1 )
In this case, it follows from equation (3.19) that the roots of the characteristic equation are
dnnn ii 21 1 and 2 (3.22)
where, the damped natural frequency is given by
nd 21 (3.23)
Note that 1 and 2 are complex conjugates. The response (3.20) in this case can be expressed as
)( 21titit ddn eCeCex (3.24)
The term within the square brackets of equation (3.24) has to be real because it represents the time response of a
real physical system. It follows that C1 and C2, as well, have to be complex conjugates.
Note: tite ddti d sincos
tite ddti d sincos
Thus, an alternative form of the general solution would be
]sincos[ 21 tAtAex ddtn (3.25)
Here, A1 and A2 are the two unknown constants. By equating the coefficients, it can be shown that
18
211 CCA
)( 212 CCiA (3.26)
Hence,
2112
1iAAC
2122
1iAAC (3.27)
which are complex conjugates, as it is required.
For determination of the constants A1 and A2, it is possible to use the initial conditions: x (0) = xo, v(0) = vo as before.
Then, 10 Ax and (3.28)
21 AAv dno or
d
on
d
o xvA
_2 (3.29)
Yet, another form of the solution would be
)sin( tAex dtn (3.30)
Here, A and are the unknown constants with
(3.31)
Note that the response 0x as t . This means that the system is asymptotically stable (exponentially
decaying harmonic oscillation) with a circular frequency of 21 nvd . This can be shown by the
following figure. Note that the circular frequency is the damped natural frequency of vibration.
19
Fig.3.7. The response of an underdamped system
CASE 2: Overdamped Motion ( 1 )
This case occurs if the parameters of the system are such that 1 so that the discriminant in Equation (3.19) is
positive and the roots are a pair of negative real numbers. The solution of Equation (3.15) then becomes
tt nn eCeCx )1(2
)1(1
22 (3.32)
Where C1 and C2 are again constants determined by v0 and x0 which are the initial conditions.
The overdamped response has the form given in Figure 3.8. An overdamped system does not oscillate, but rather
returns to its rest position exponentially as t and is said to be asymptotically stable.
Fig.3.8. Response of an overdamped system
CASE 3: Critically damped Motion ( 1 )
This case occurs if the parameters of the system are such that 1 so that the discriminant in Equation (3.19) is
zero and the roots are a pair of negative real repeated numbers. The solution of Equation (3.15) then becomes
oonot xtxvetx n )( (3.33)
20
Fig.3.9. Response of a critically damped system.
The critically damped response is shown in Figure 3.9 for various values of the initial conditions v0 and x0.
It should be noted that critically damped systems can be thought of in several ways. First, they represent systems
with the minimum value of damping rate that yields a nonoscillating system. Critical damping can also be
thought of as the case that separates nonoscillation from oscillation motion of dynamic systems.
In other words, critical damping represents the limit of periodic motion; hence the displaced body is restored to
equilibrium in the shortest possible time, and without oscillation or overshoot. Many devices, particularly
electrical instruments, are critically damped to take advantage of this property.
Example:
A light rigid rod of length L is pinned at one end O and has a body of mass m attached at the other end. A spring
and viscous damper connected in parallel are fastened to the rod at a distance a from the support. The system is
set up in a horizontal plane: a plan view is shown in Fig.3.10. Assuming that the damper is adjusted to provide
critical damping, obtain the motion of the rod as a function of time if it is rotated through a small angle ϴo,
and then released. Given that ϴo = 2o and the undamped natural frequency of the system is 3rad/s, calculate the
displacement 1s after release.
Fig.3.10. A light rigid rod system.
Solution:
Taking moments about the pivot O gives
22 kacaIo
21
Where 2mlIo , so the equation of motion is
0222 kacaml
Now the system is adjusted for critical damping, so that 1 . The solution to the equation is therefore of the
form tneBtA
Now, o when 0t and 0dt
dwhen 0t
Hence, Ao and tn
t nn eBtABe 0
So that noB
Therefore,
tno
net 1
Ifsec
3rad
n , oo 2 , and sec1t ,
oe 4.0312 3
3.4. FORCED VIBRATION
Many real systems are subjected to periodic excitation. This may be due to unbalanced rotating parts,
reciprocating components, or a shaking foundation. Sometimes large motions of the suspended body are desired
as in vibratory feeders and compactors, but usually we require very low vibration amplitudes over a large
range of exciting forces and frequencies. Some periodic forces are harmonic, but even if they are not, they can
be represented as a series of harmonic functions using Fourier analysis techniques. Because of this the
response of elastically supported bodies to harmonic exciting forces and motions must be studied.
Here, the response of a viscous damped system to a simple harmonic exciting force with constant amplitude is
considered.
In the system shown in Fig.3.11, the body of mass m is connected by a spring and viscous damper to a fixed
support, whilst an harmonic force of circular frequency v and amplitude F acts upon it, in the line of motion.
The equation of motion is
vtFkxxcxm sin
The solution to 0 kxxcxm , which has already been studied, is the complementary function; it
represents the initial vibration which quickly dies away. The sustained motion is given by the particular solution.
A solution )sin( vtXx can be assumed, because this represents simple harmonic motion at the frequency of
22
the exciting force with a displacement vector which lags the force vector by , that is, the motion occurs after the
application of the force.
Fig.3.11. Single degree of freedom model of a forced system with viscous damping
Assuming )sin( vtXx ,
)2
sin()cos(
vtXvvtXvx
and )sin()sin( 22 vtXvvtXvx
Then, the equation of motion is
vtFvtkXvtcXvvtmXv sin)sin()2
sin()sin(2
A vector diagram of these forces can now be drawn as
Fig.3.12. Force vector diagram
From the diagram
2222 )(][ cXvmXvkXF or
222 cvmvk
FX
and 2
tanmXvkX
cvX
Thus, the steady-state solution for the motion equation is
23
)sin(
222
vtcvmvk
Fx
Where
2
1tanmXvkX
cvX
The complete solution includes the transient motion given by the complementary function:
tAex ntn 21sin
Fig.3.13 shows the combined motion
Fig.3.13. Forced vibration, combined motion
Equation (2.15) can be written in a more convenient form if we put
m
kn and
k
FX s
Then,
222
21
1
nn
svv
X
X
and
2
1
1
2
tan
n
n
v
v
24
sX
Xis known as the dynamic magnification factor, because sX is the static deflection of the system under
a steady force F, and X is the dynamic amplitude. By considering different values of the frequency ration
v
, we
can plot sX
Xand as functions of frequency for various values of . Fig.3.14 shows the results.
Fig.3.14.a) Amplitude-frequency response for system of Fig.3.11
Fig.3.14.b) Phase-frequency response for system of Fig.3.11
BIBLIOGRAPHY:
1. Engineering Vibration Analysis with Application to Control Systems, C. F. Beards BSc, PhD, CEng, MRAeS,
MIOA, 1st Edition, Great Britain, 1995.