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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.