1

ON THE SINUSOIDAL VIBRATION OF AUTOMOBILES

BY

OGUNMILUYI, IFEOLUWA MICHEAL

MATRICULATION NUMBER: 2009/1842

A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS,

COLLEGE OF NATURAL SCIENCES,

FEDERAL UNIVERSITY OF AGRICULTURE, ABEOKUTA.

IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE

AWARD OF BACHELOR OF SCIENCES (B.Sc.) DEGREE IN

MATHEMATICS

JANUARY, 2014.

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CERTIFICATION

This is to certify that this project was carried out by Ogunmiluyi Ifeoluwa Micheal, with

matriculation number 20091842, in the Department of Mathematics, College of Natural Sciences, Federal

University of Agriculture, Abeokuta, Ogun State, Nigeria.

………………………... ..………………… Ogunmiluyi Ifeoluwa Date

Student

………………………… ...………………… Dr. I.O Abiala Date (Supervisor)

………………………… ….………………. Dr. B.I Olajuwon Date (Head of Department)

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DEDICATION

This project work is dedicated to ALMIGHTY God (the creator of heaven and the earth) for his love

and mercy, who in His infinite mercy brought me thus far.

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ACKNOWLEDGEMENTS

First and foremost, I would like to thank my greatest teacher of all, God. I know that I am here and

that I am able to write all of this for a reason. I will do my best in never forgetting what a great fortune I

have had in just being here, and that it comes with a lesson and a responsibility. I hope I am doing the work

you have planned me to do.

I would like to thank my supervisor, Dr. I.O. Abiala, for being like a father to me and giving

invaluable suggestions to improvement of my project and for his patience throughout my project and also,

those first few days of uncertainty that you pulled with me are ones that I will not never forget, thank you for

believing in me, even if it’s only was for a few moments.

I would like to thank my friends and colleagues that I have met in this home far away from home

called Abeokuta. Specially, David, Moyo, Seyi (my editor) and Peter, who, even though have reduced me to

a fifth wheel in our relationship, have blossomed into a partnership that will not be forgotten. Whatever

happens with you too, do know that, throughout these couples of years, our relationship has provided me

with an impressively beautiful site to see, as it is when five friends fall in love with each other. You guys

have been more than friends.

I would like to thank honourable senior, Jinadu Ayo, though our relationship was born in a very odd

way, but I would not have expected otherwise, as both of us are odd in our own beautifully weird world. For

guiding me and helping my shortcomings. I have become a better man because of the mirror you held up for

me. Thank you.

A special thanks goes to my sister and my siblings, Ibukun and Joshua and also my dearest cousin,

Oluwadarasimi. I cannot but appreciate the effort of DLCF FUNAAB family, departmental mates and my

castle mates. Thank you, I love you all.

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Finally, my parents: Pastor A.I Ogunmiluyi and Mrs R.T Ogunmiluyi. They gave me my name, they

gave me my life, and everything else in between. I pride myself in having words for everything, but they

truly shut me up when it comes down to describing how much I love them and appreciate the efforts they

have put into giving me the life I have now. They are the reason I did this; they are the reason I thrive to be

better. Their pride for me is my main goal in life. As I have always taught and hoped; when they lay in their

death bed they would think, “I am proud of my son.” Thank you, thank you, thank you.

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TABLE OF CONTENTS

PAGES

Certification ii

Declaration iii

Acknowledgement iv

Table of contents vi

Nomenclature viii

Abstract ix

CHAPTER ONE

1.1 Introduction 1

1.2 Historical development of Sinusoidal vibration 3

1.3 Definition of key terms 5

CHAPTER TWO

2.1 Literature review 7

CHAPTER THREE

3.1 Introduction 10

3.2 Problem formulation 11

3.3 Problem analysis 14

3.4 Application or economic importance of sinusoidal vibration 21

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CHAPTER FOUR

4.1 Introduction 23

4.2 Method of solution to the problem 24

4.3 Numerical solution 28

CHAPTER FIVE

5.1 Conclusion 44

5.2 Recommendation 44

5.3 References 46

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NOMENCLATURE

m Mass of the automobiles measured in kilogram.

c This is the damping constant with unit force/unit velocity.

f This is the natural cyclic frequency of vibration (1/T ).

T The natural period of frequency of the vibration.

k Spring constant or stiffness.

x The displacement of the automobile which is measured in meter (m).

xdtdx

&= This is defined as the rate of change of displacement (x) and it’s often called the velocity

which is measured in m/s.

φ This is called the phase angle and they are arbitrary constant to be determined by initial condition.

mk=ω This is the square of the stiffness ‘k’ per the mass ‘m’ and is called the natural frequency of

the circular vibration and is measured in rad/sec.

F This is the external applied force on the automobile which is measured in Newton per meters (N/m).

kstωδ = This the static displacement.

ccc=ζ This is the fraction of critical damping.

ωmcc 2= This is the critical damping coefficient.

dω This is the damped natural frequency.

g This is the gravitational acceleration, g = 32.16

W This is the weight of member or structure (F)

2

2

dtxd This is the acceleration of the automobile which is measured in m/s2.

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ABSTRACT

This project work attempt to express the problem of sinusoidal vibration of automobiles as the

general equation of motion which is a second order linear differential equation comprising of the inertial

force, damping force, stiffness force and the external force and also explained the level of the damping force

acting on the automobile such as under damping, critical damping and over damping. Also, this project

attempt to express solution to the problem using the step-by-step integration with the central difference

method and Excel. It was discovered that increase in the time step leads to increase in the response.

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CHAPTER ONE

1.1 INTRODUCTION

Sinusoidal in relation to dynamics can be defined as having a magnitude that varies as the sine of an

independent variable. There are few machines that will vibrate in pure sinusoidal fashion.

Vibration is the motion of a particle or a body or system of connected bodies released from a position

of equilibrium. Most vibrations has much disadvantages in machines and structures because they have the

tendency to produce increased stresses, energy losses(damping), caused added wear, induce fatigue and

create passenger discomfort in vehicles. Also in rotating parts like gear needs to be given a lot attention to

when balancing in order to prevent damage from vibrations.

It also occurs when a system is released from its state of being balance. The system tends to return to

this balanced position under the action of restoring forces (such as it is well known in simple pendulum).

The system keeps moving back and forth across its position of equilibrium. The combination of elements

with intention of accomplishing a goal is called a system. For example, an automobile is a system whose

elements are the wheels, suspension, car body, brake and so on.

Also, Vibration can be defined as “the cyclical change in the position of an object as it moves

alternately to one side and the other of some reference or datum position” (Macinante, 1984). Vibration of

rigid bodies can be rectilinear (or translational), rotational, or a combination of the two. Rectilinear vibration

refers to a point whose path of vibration is a straight line, and rotational vibration refers to a rigid body

whose vibration is angular about some reference line. Additionally, vibration of flexible bodies can be

described by flexural or other elastic vibrations such as longitudinal, tension and compression, and torsional

or twisting.

Many types of engines, compressors, pumps, and other machinery that run continuously generate a

form of periodic vibration. If a motion is periodic, its velocity and acceleration are also periodic.

The three terms used in describing vibration are amplitude, frequency, and type. Thus, a vibration is

said to be sinusoidal if it corresponds to a sinusoidal function of time.

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Sinusoidal vibration which is also known as simple harmonic motion is the simplest form of

vibration, in which a body moves around an equilibrium position in a periodic (changes with time) and

smooth way or it can be defined as a type of vibration that smoothly changes with time. The best known

example of sinusoidal vibration is the simple pendulum where a ball is attached to a spring and its displaced

from its equilibrium with time.

Now relating sinusoidal vibration to automobiles, the terms used that changes with time are

displacement (distance), velocity (speed) and acceleration. Sinusoidal motion often occurs in our day to day

activities. When riding a Mazda car, each compartment moves in a circular manner as it changes with time

and when tracking the height of the compartment the motion is clearly sinusoidal.

The motion of any vehicle depends upon all the forces and moments that act upon it. These forces

and moments are caused by interaction of the vehicle with the surrounding medium(s) such as air or water

(fluid static and dynamic forces), gravitational attraction (gravity forces), Earth’s surface (support, ground,

or landing gear forces), and also for ship or aircrafts (propulsion forces).

Another important parameter to discuss when describing vibration is damping. Structural damping

occurs as material layers slide over one another during vibration. It is important to remember that damping is

one of the most difficult phenomena to model in vibrating systems. In fact, in the twenty years from 1945 to

1965, 2000 papers were published in the area of damping technology. Damping is usually best estimated

experimentally. Although damping mechanisms in real systems are rarely viscous, the nice analytical

properties of vibrating systems with viscous damping are worth exploiting if possible. In fact, the concept of

equivalent viscous damping is in wide use within the noise & vibration engineering community. If a system

is initially displaced at a certain distance and then released, such as a pendulum, it will vibrate about a

certain datum line for a finite amount of time before coming to rest. The amplitude of the motion decays,

and the cause of this decay in motion, or dissipation of energy, is referred to as damping. It is present

naturally, and if a system is not being forced to vibrate by an external source, its motion will eventually

decay because of the intrinsic damping that is present. Damping can also be introduced into a system as a

means of controlling the vibrations.

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1.2 HISTORICAL DEVELOPMENT OF SINUSOIDAL VIBRATION

The critical aspects of our knowledge about vibrations, we owe to Galileo Galileo, who in 1610 gave

the concept of mass, and also Hooke joined with Marriott in 1660 propound the Hooke’s law and also Isaac

Newton in 1665 who gave the laws of motion and the various equations to each ones, with Leibnitz in 1684,

they gave the calculus and Newton who in 1687 declare the laws of motion acceptable and valid.

Far back in 1100-1165, Hibat Allah Abu’l-Barakat al-Baghdaadi discovers that force is proportional

to acceleration rather than speed, which is now a fundamental law in classical mechanics under which there

is sinusoidal vibration. Later, Newton also developed calculus which is necessary to perform the

mathematical calculations involved in classical mechanics. However, it was Gottfried Leibniz who

independently of Newton, developed a calculus with the notation of the derivative and integral which are

used to this day, but classical mechanics retains the Newton’s dot notation for time derivatives which is

applied in sinusoidal vibration which is also respective of time. With the help of Hooke’s law in 1660, the

restoring force was used and Taylor’s in 1713 when coming up with the dawn of vibration analysis also used

the Hooke’s law and deduced an expression for the resultant force. Therefore, the experimental work of

Isaac Newton in 1665 and Leibnitz in 1684 led to the general equation of motion of second-order non-

homogenous linear equation.

The foundations of vibration theory for continuous media were established between 1733 and 1735

by Daniel Bernoulli and Leonard Euler. These two mathematical scientists by 1734 finally achieved the

fourth-order equation using an infinite series approach. The solutions were given by Eigen value equations

for several kinds of end conditions, which are common knowledge today. In 1739, Euler had another

discovery of the generality of the exponential method for the solution of differential equations with

constants coefficients. This method is the basic method used by analysts today to solve problems involving

differential equations of linear systems. He also solves the ordinary differential equation for a forced

harmonic oscillator and notices the resonance phenomenon.

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Another method that is used in solving the second-order differential equation is the finite difference

which is a numerical method for approximating the solutions to differential equations using finite difference

equations to approximate derivatives. But the possible and likely sources of error in finite difference

methods are round-off error which is the loss of precision due to computer rounding of decimal quantities

and truncation error which is the difference between the exact solution of the finite difference equation and

the exact quantity assuming perfect arithmetic.

Looking back at the short history of finite difference method. The finite difference method (FDM)

was first developed by A. Thom in the 1920s under the title “the method of square” to solve nonlinear

hydrodynamic equations which is in his book, ‘A. Thom and C. J. Apelt, Field Computations in Engineering

and Physics. London: D. Van Nostrand, 1961’. After this, there was some history in the 1930s and further

development of the finite difference method. Although some ideas may be traced back further, we begin the

fundamental paper by Courant, Friedrichs and Lewy (1928) on the solutions of the problems of

mathematical physics by means of finite differences. A finite difference approximation was first defined for

the wave equation and the CFL stability condition was shown to be necessary for convergence. Error bounds

for difference approximations of elliptic problems were first derived by Gerschgorin (1930) whose work was

based on a discrete analogue of the maximum principle for Laplace’s equation. This approach was pursued

through the 1960s and various approximations of elliptic equations and associated boundary conditions were

analysed. The finite difference theory for general initial value problems and parabolic problems then had an

intense period of development during 1950s and 1960s, when the concept of stability was explored in the lax

equivalence theorem and the Kreiss matrix lemmas. Independently of the engineering applications, a number

of papers appeared in the mathematical literature in the mid-1960s which were concerned with the

construction and analysis of finite difference schemes by the Rayleigh-Ritz procedure with piecewise linear

approximating functions.

Beginning in the mid-1950s, efforts to solve continuum problems in elasticity using small, discrete

"elements" to describe the overall behaviour of simple elastic bars began to appear. Argyris (1954) and

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Turner, et al. (1956) were the first to publish use of such techniques for the aircraft industry. Actual coining

of the term "finite element" appeared in a paper by Clough (1960).

The early use of finite elements lay in the application of such techniques for structural-related

problems. However, others soon recognized the versatility of the method and its underlying rich

mathematical basis for application in non-structural areas. Sienkiewicz and Cheung (1965) were among the

first to apply the finite element method to field problems (e.g., heat conduction, irrotational fluid flow, etc.)

involving solution of Laplace and Poisson equations, Gangadharan, et al. 2008 applied finite element

method to model the vehicle/track system and used Power Spectral Density (PSD) of track irregularities as

input to the system.

The underlying mathematical basis of the finite element method first lies with the classical

Rayleigh-Ritz and variational calculus procedures introduced by Rayleigh (1877) and Ritz (1909). These

theories provided the reasons why the finite element method worked well for the class of problems in which

variational statements could be obtained (e.g., linear diffusion type problems).

In finite difference method, three forms are commonly considered, these are; the forward, backward

and central differences. So also, the Newton’s series which also consist of the terms of the Newton forward

difference equation named after Isaac Newton, in essence, it is the Newton interpolation formula which was

published in his Principia Mathematica in 1687, namely the discrete analogue of the continuum Taylor

expansion

1.3 DEFINITIONS OF KEY TERMS

Damping: Dissipation of oscillatory or vibratory energy, with motion or with time.

Critical damping ( )cc : This is that value of damping that provides most rapid response to a step function

without overshoot.

Damping ratio: This is a fraction of cc .

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Displacement: Specified change of position, or distance, usually measured from the mean position or

position of rest. Usually applies to uniaxial, less often to angular motion.

Harmonic: A sinusoidal quality having a frequency that is an integral multiple (x2, x3, etc) of a fundamental

(x1) frequency.

Phase: A periodic quality, the fractional part of a period between a reference time (such as when

displacement = zero) and a particular time of interest; or between two motions or electrical signals having

the same fundamental frequency.

Stiffness: The ratio of force (or torque) to deflection of a spring-like element.

Velocity ( )xv

& : Rate of change of displacement with time, usually along a specified axis; it may refer to

angular motion as well as to uniaxial motion.

Vibration: Mechanical oscillation or motion about a reference point of equilibrium.

Natural Frequency ( )ω : The frequency of an undamped system’s free vibration; also, the frequency of any of

the normal modes of vibration. Natural frequency drops when damping is present.

Free Vibration: Free vibration occurs without force, similar after a reed is plucked.

Datum: This is a fixed starting point of a scale or operation from which inferences can be drawn from.

Datum Line: This is a standard on comparison or point of reference.

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CHAPTER TWO

2.1 LITERATURE REVIEW

Vibrations occur in many spheres of our life. For example, any unbalance in machines with rotating

parts such as fans, ventilators, centrifugal separators, washing machines, lathes, centrifugal pumps, rotary

presses, and turbines, can cause vibrations. For these machines, vibrations are generally undesirable.

Buildings and structures can experience vibrations due to operating machinery; passing vehicular (consisting

of vehicles), air, and rail traffic; or natural phenomena such as earthquakes and winds. Pedestrian bridges

and floors in buildings also experience vibrations due to human movement on them. In structural systems,

the fluctuating stresses due to vibrations can result in fatigue failure. Vibrations are also undesirable when

performing measurements with precision instruments such as an electron microscope and when fabricating

micro-electro-mechanical system

The study of vibration started in the sixteen century and since then, it has become a subject of intense

research. The behaviour of the solution is studied in a sufficiently small neighbourhood of a given solution,

for example, in a neighbourhood of stationary point or a periodic solution. The summary of some literature

review pertaining to some research on sinusoidal vibration, basically on damping will be presented in this

section.

In spite of a large amount of research, understanding of damping mechanisms is quite primitive. A

major reason for this is that, by contrast with inertia and stiffness forces, it is not in general clear which state

variables are relevant to determine the damping forces. Moreover, it seems that in a realistic situation it is

often the structural joints which are more responsible for the energy dissipation than the (solid) material.

There have been detailed studies on the material damping like (Bert, 1973) and also on energy dissipation

mechanisms in the joints (Earls, 1966, Beards and Williams, 1977). But here difficulty lies in representing

all these tiny mechanisms in different parts of the structure in a unified manner. Even in many cases these

mechanisms turn out be locally non-linear, requiring an equivalent linearization technique for a global

analysis (Bandstra, 1983). A well-known method to get rid of all these problems is to use the so called

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viscous damping. This approach was first introduced by (Rayleigh, 1877) via his famous dissipation

function, a quadratic expression for the energy dissipation rate with a symmetric matrix of coefficients, the

damping matrix.

A further idealization, also pointed out by Rayleigh, is to assume the damping matrix to be a linear

combination of the mass and stiffness matrices. Since its introduction this model has been used extensively

and is now usually known as ‘Rayleigh damping’, proportional damping or classical damping. With such a

damping model, the modal analysis procedure, originally developed for undamped systems, can be used to

analyse damped systems in a very similar manner. (Rayleigh, 1877) has shown that undamped linear

systems are capable of so-called natural motions. This essentially implies that all the system coordinates

execute harmonic oscillation at a given frequency and form a certain displacement pattern. The oscillation

frequency and displacement pattern are called natural frequencies and normal modes, respectively. Thus,

any mathematical representation of the physical damping mechanisms in the equations of motion of a

vibrating system will have to be a generalization and approximation of the true physical situation.

As (Scanlan, 1970) has observed, any mathematical damping model is really only a crutch which

does not give a detailed explanation of the underlying physics. Free oscillation of an undamped single

degree of frequency (SDOF) like sinusoidal vibration system never dies out and the simplest approach to

introduce dissipation is to incorporate an ideal viscous dashpot in the model. The damping force is assumed

to be proportional to the instantaneous velocity and the coefficient of proportionality which is known as the

dashpot-constant or viscous damping constant. The loss factor, which is the energy dissipation per radian to

the peak potential energy in the cycle, is widely accepted as a basic measure of the damping.

This dependence of the loss factor on the driving frequency has been discussed by (Crandall, 1970)

where it has been pointed out that the frequency dependence, observed in practice, is usually not of this

form. In such cases one often resorts to an equivalent ideal dashpot. Theoretical objections to the

approximately constant value of damping over a range of frequency, as observed in aero elasticity problems,

have been raised by (Naylor, 1970). Dissipation of energy takes place in the process of air flow and

coulomb-friction dominates around the joints. This damping behaviour has been studied by many authors in

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some practical situations, for example by (Cremer and Hecki, 1973). (Earls, 1966) has obtained the energy

dissipation in a lap joint over a cycle under different clamping pressure. (Beards and Williams, 1977) have

noted that significant damping can be obtained by suitably choosing the fastening pressure at the interfacial

slip in joints.

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CHAPTER THREE

3.1 INTRODUCTION

In this chapter, we shall introduce and formulate the problem on the sinusoidal vibration of

automobiles using the concept of second-order linear differential equation with the aid of Newton’s second

law of motion which states that when an applied force acts on a mass, the rate of change of momentum is

equal to the applied force which is the product of the mass and the acceleration. We shall be working on the

force ‘ F ’ applied to an automobile, the displacement ‘ x ’, the velocity ‘ v ’, the acceleration ‘ a ’, the mass ‘

m ’ of the automobile, the damping constant ‘ c ’ and the stiffness ‘ k ’.

When undergoing this project, Hooke’s law which states that force in the spring is proportional to

displacement from its equilibrium position where ‘ k ’ is the spring stiffness and the stiffness ‘ k ’ is

measured in Newton’s per meter (N/m), will be introduced in order to generate the stiffness ‘ k ’. The typical

stiffness parameters for a car: k = 17000 N/m for spring and k = 180000 N/m for a tire. For a truck the

stiffness can be 10 times the magnitude for a car.

However, the purpose and focus of this chapter is to explain how the second-order linear differential

equation

( ) ( ) ( ) ( )tFtkxtxctxm =++ &&& (3.1)

was formulated where the ‘ m ’ is the mass, ‘ x ’ is the displacement, ‘ k ’ is the stiffness, ‘ c ’ is the damping

constants, ‘ t ’ is the time of the vibration or motion and ‘ F ’ is the external force applied to the automobile.

Also, in this chapter, the analysis of the problem will be explained and the economic importance of

the problem will also be examined.

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3.2 PROBLEM FORMULATION

Sinusoidal vibration is also known as a single degree of freedom (SDOF) where the shape of the

system can be represented in terms of a single dynamic coordinate x(t). From the diagram below;

Figure 1

Where m = the mass of the automobile, k = stiffness, ( )tx = displacement and c = the damping which

leads to loss of energy in the vibrating system.

We shall now formulate the problem which is the second-order linear differential equation.

Let us consider when the object moves in a positive displacement ( )tx with a positive velocity ( )tx&

under an external force ( )tF . The accelerations, velocities and displacements in a system produce forces

when multiplied respectively by mass, damping and stiffness. For mass and inertia, the acceleration between

mass ‘ m ’ and acceleration ‘ x&& ’ is given by Newton’s second law which states that when a force acts on a

mass, the rate of change of momentum is equal to the applied force which is the product of mass and the

acceleration.

Mathematically;

Fdtdxm

dtd

=

21

This can also be written/expressed as;

xmF &&=

(This is a second-order linear differential equation)

Mass ( )m

Force ( )F Acceleration ( )x&&

Initial force ( )xm &&

Stiffness of the system.

The stiffness can be determined by any of the standard methods of static structure analysis. Using the

Hooke’s law which states that force in the spring is proportional to displacement from its equilibrium

position as in figure (a). The slope dxdf of the straight line between these extremes, where �� and ��

represent small changes in f and x respectively, is the stiffness ‘ k ’. The potential energy at any value of ‘

x ’ is the shaded triangle, Fx21 or 2

21 kx in the figure (b), where kx− is the restoring force where k is the

stiffness and is measured in Newton’s per meter and ‘ m ’ is the mass of the automobile and is measured in

kilograms.

aE, F elastic limit

F slope dxdf Area= 2

21 kx

L x

(a) (b)

22

Damping in a system

Damping is different, in that it dissipates energy, which is lost from the system. This happens

because nature has built in our system a retarding property, which implicitly acts against motion from the

advent of the motion and brings it to a stop, this is known as the damping of a system. The damping may be

deliberately added to a system or structure to reduce unwanted oscillations. Examples are discrete units,

usually using fluids, such as vehicle suspension dampers and viscoelastic damping layers on panels. In

vehicle suspension dampers, such a device typically produces a damping force, F , in response to closure

velocity, x& , by forcing fluid through a hole or opening in the system. This is inherently a square-law rather

than a linear effect, but can be made approximately linear by the use of a special valve, which opens

progressively with increasing flow. The damper is then known as an automotive damper. Then the force and

velocity are related by:

xcFd &= (3.21)

Where dF is the external applied damping force and x& the velocity at the same point. The quantity c is

called the damping constant having the dimensions force/unit velocity. Equation (3.21) is called the damping

force.

Thus;

[ ]∑ =−+ ttimeanyatforceExternalforcerestoringforcedampingforceinertial

Therefore, from the second law of motion, the equation of motion for automobile system having a degree of

two (2) at any time t is express as;

02

2

=++ kxdtdxc

dtxdm (3.22)

Equation (3.22) is a homogenous and a second-order differential equation where the external forces

neglected. If an external force is considered, we have;

( )tFkxdtdxc

dtxdm =++2

2

(3.23)

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3.3 PROBLEM ANALYSIS

The purpose of this section is to analyse the problem that will be solved in the chapter three of this

project. We shall analyse the general equation which is the second-order linear differential equation under

two conditions, which deals with the presence of damping force.

CASE 1(without damping)

For a system oscillating without an external force and damping being applied, then equation (3.23)

becomes;

0=+ kxxm &&

From the equation, m�� = ma = F where ‘a’ is the acceleration of the system, and since the acceleration is

describe as the change of velocity with time. This implies that 2

2

dtxda = , it follows that;

Acceleration - restoring force = 0

This gives, 02

2

=+ kxdt

xdm (3.31)

The auxiliary equation is; 02 =+ kmλ

From equation (3.1), let x be the instantaneous displacement varying with time, then;

Instantaneous displacement ( ) ftxtx π2sin=

Differentiating this and the result is called instantaneous velocity which is varying with time, then;

Instantaneous velocity ( ) ftfxxv ππ 2cos2== &

Also differentiating the velocity to get the instantaneous acceleration;

Instantaneous acceleration ( ) ftxfxva ππ 2sin4 22−=== &&&

Since fπω 2= , then, the above equations changes to;

Instantaneous displacement, velocity and acceleration are txtx ωωω cos,sin ,and tx ωω sin2− respectively.

Since ( ) txtx ωsin= and ( ) txtx ωω sin2−=&&

This implies,

0)sin()sin( 2 =+− txktxm ωωω

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)sin()sin( 2 txktxm ωωω =

mk

txmtxk

==)sin()sin(2

ωωω

mk=2ω

mk=ω

From the auxiliary equation 02 =+ kmλ

mk−=2λ

mki±=λ

ωλ i±=

Since equation (3.31) is a linear, homogenous second-order ordinary differential equation, the

solution is of the form,

( ) tCetx λ=

Where C and λ are constant and t is the time.

ωλ i±=

( ) titi eCeCtx ωω −+= 21 (3.32)

To change the complex component of ( )tx to a real component, we make use of Euler’s formula,

θθθ sincos ±=±e

Equation (3.32) now becomes;

( ) )sin(cos)sin(cos 21 titCtitCtx ωωωω −++=

tCCitCC ωω sin)(cos)( 2121 −++= (3.33)

Since ( )tx is not yet a real-valued function because of the component i , then )( 21 CC + and )( 21 CCi − must

be real-valued. Using the complex conjugate pair;

21

iBAC −=

22iBAC +

= (3.34)

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Substituting equation (3.34) back into equation (3.33) to have;

tiBAiBAitiBAiBA ωω sin22

cos22

+

−−

+

−

++

tiBAiBAitBAiBA ωω sin2

cos2

−−−

+

−++

ð tiBitA ωω sin2

2cos2

2

+

ð tiBtA ωω sincos +

Recall that 12 −=i ;

( ) tBtAtx ωω sincos += (3.35)

Therefore, the general solution of the undamped vibration is tBtA ωω sincos +

Let φsin1CA = and φcos1CB = (3.36)

Hence, φφ 221

2221

2 cossin CBandCA ==

Therefore, φφ 221

221

22 cossin CCBA +=+

( )φφ 2221 cossin += C

Recall that; 1cossin 22 =+

ð 21

22 CBA =+

Hence, the amplitude; 221 BAC += (3.37)

Substituting equation (3.36) into (3.35), we have;

( ) tCtCtx ωφωφ sincoscossin 11 +=

( )ttC ωφωφ sincoscossin1 +=

Recall from trigonometry,

( ) ABBABA sincossincossin +=+

( ) ( )φω += tCtx sin1 (3.38)

Also, from equation (3.36),

26

11

cossinCBand

CA

== φφ

ð

1

1cossintan

CBCA

== φφφ

BC

CA 1

1

× = ���

× ���

BA

=∴ φtan

ð ( )BA1tan −=φ

Where φ is the phase angle.

mk=ω is the natural frequency of circular vibration and T is the natural period of frequency which is

ωπ2

=T and f is the natural cyclic frequency of vibration denoted as;

mk

Tf

ππω

21

21

===

Where mg=ω and the static displacement kstωδ = and since

st

gkgg

mδπωπ

ω21

21

===

From equation (3.35);

( ) tBtAtx ωω sincos +=

We consider two cases to have the general equation;

For case 1: Suppose that the mass is pulled down to the point 0x and then released at time 0=t i.e.

( ) 00 xx = and ( ) 00 =x& , then;

( ) tBtAtx ωω sincos +=

( ) tBtAtx ωωωω cossin +−=&

At ( ) ( ) 000,0 0 === tandxxx &

27

( ) 0sin0cos0 += Ax

( ) Ax =0

0xA =∴

( ) 0cos0sin0 ωω BAx +−=&

ωB+= 00

0=B

Therefore, the motion of the system for case 1 is governed by;

( ) txtx ωcos0=

For case 2: Suppose that we have an impulse impacts initial velocity 0v to the mass and also

( ) ( ) 000 vxandtx == & , then;

( ) 0sin0cos0 BAx +=

00 += A

0=A

( ) 0cos0sin0 ωω BAx +−=&

ωBv += 00

ω

0vB =

Therefore, the motion of the system for case 2 is governed by;

( ) tv

tx ωω

sin0=

Therefore, the amplitude;

221 BAC +=

2

020

+=

ωvx

28

CASE 2(with damping)

For a system oscillating with damping and no external force being applied, then equation (3.23) becomes;

0=++ kxxcxm &&& (3.39)

From the auxiliary equation;

02 =+++ kcm λλ

Since it is an homogenous equation, then the solution will be of the form;

( ) tt eCeCtx 2121

λλ += (3.3.10)

The form of the solution of equation (3.39) depends upon whether the damping coefficient is equal to,

greater than or less than the critical damping coefficient cC and where C is the damping coefficient.

mkmmkmCc 222 === ω

The ratiocC

C=τ is the fraction of critical damping.

Considering each cases,

Case 1: When ( )1== τcCC (critical damping)

In this case, it is called the critical damping and the solution has no oscillation, then the solution is of the

form;

( ) ( ) mct

eBtAtx 2−

+=

( )tx 1 critical damping

0 1 2 3 t

-1 free vibration of a system with 1=cC

C

29

Case 2: When ( )1<< τcCC (under-damping)

In this case, it is also called the less than critical damping or under-damping and the solution is of the form;

( ) ( )tBtAetx ddm

ct

ωω cossin2 +=−

Or )sin(2 φω +=−

tCe dm

ct

tCe dt ωτω cos−=

Where φ is the phase angle and dω is the damped natural frequency which is related to the undamped

natural frequency ω by;

21 τωω −=d

When ( )1<τ , the solution consist of two actors, the first on decreasing exponential and the second a sine

wave. The combined result is exponentially decreasing sine wave lying in the space between the exponential

curve on both sides of the phase angle axis in the figure below;

Free vibration of a system with 1<cC

C

The smaller the damping constant C , the flatter will be the exponential curve and the more cycles will it

take for the vibration to be eliminated.

Case 3: When ( )1>> τcCC (overdamping)

In this case, it is called the greater-than-critical damping or the over-damping and the solution is of the form;

( ) ( )ttmct

BeAeetx 112 22 −−−−+= τωτω

30

When ( )1>τ ,the motion is not oscillating but rather a creeping back to the original position, this is due to

the fact that when 1>τ , then C is large.

Free vibration of a system with 1>cC

C

3.4 APPLICATION OR ECONOMIC IMPORTANCE OF THE PROBLEM

The economic importance of sinusoidal vibration in relation to the problem is numerous. Some of them are;

1. It is the most important and central point of physics. Anything that oscillates produces motion that is

partly or almost sinusoidal.

2. Another important use of sinusoidal vibration is that it is an Eigen-function of linear systems. This

means that it is important for the analysis of filters such as reverberators (objects that is repeated several

times as it bounces off different surfaces), equalizers, certain (but not all) ``effects'', etc.

3. From the point of view of computer music research, is that the human ear is a kind of spectrum

analyser. That is, the cochlea of the inner ear physically splits sound into its (near) sinusoidal components.

This is accomplished by the basilar membrane in the inner ear: a sound wave injected at the oval window

(which is connected via the bones of the middle ear to the ear drum), travels along the basilar membrane

inside the coiled cochlea. The membrane starts out thick and stiff, and gradually becomes thinner and more

compliant toward its apex (the helicotrema). A stiff membrane has a high resonance frequency while a thin,

compliant membrane has a low resonance frequency (assuming comparable mass density, or at least less of a

31

difference in mass than in compliance). Thus, as the sound wave travels, each frequency in the sound

resonates at a particular place along the basilar membrane. The highest frequencies resonate right at the

entrance, while the lowest frequencies travel the farthest and resonate near the helicotrema. The membrane

resonance effectively ``shorts out'' the signal energy at that frequency, and it travels no further. Along the

basilar membrane there are hair cells which feel the resonant vibration and transmit an increased firing rate

along the auditory nerve to the brain.

Thus, the ear is very literally a Fourier analyser for sound, albeit nonlinear and using analysis

parameters that are difficult to match exactly. Nevertheless, by looking at spectra (which display the amount

of each sinusoidal frequency present in a sound), we are looking at a representation much more like what the

brain receives when we hear.

4. It also bring about the concept of phase (i.e starting a system in motion that changes with time),

which is used in some advanced diagnostic techniques and the basic concept used in rotor balancing. For

example, in balancing a rotor and understanding what is happening, one must definitely understand

sinusoidal vibration and phase.

5. One of the major applications of sinusoids in Science and Engineering is the study of harmonic

motion. The equations for harmonic motion can be used to describe a wide range of phenomena, from the

motion of an object on a spring, to the response of an electronic circuit.

6. It’s appropriate for fatigue testing of products that operate primarily at a known speed (frequency)

under in-service conditions.

7. It helps in detecting sensitivity of a device to a particular excitation frequency.

8. It also helps in detecting resonances, natural frequencies, modal damping, and mode shapes.

9. It’s appropriate for calibration of vibration sensors and control systems.

10. For sinusoidal waveforms, it is easy to convert between acceleration, velocity and displacement.

11. Any vibration waveform, no matter how complex, can be decomposed into sinusoidal components.

This fact is the base of frequency analysis, perhaps the most known tool for vibration diagnostics.

32

CHAPTER FOUR

4.1 INTRODUCTION

From the problem analysed in the previous chapter, the method to be used to solve the problem is the

step-by-step integration coupled with the central difference method which is among the forms of the finite

difference method and it will be solved numerically. The finite difference techniques are based upon the

approximations that permit replacing differential equations by finite difference equations. These finite

difference approximations are algebraic in form, and the solutions are related to grid points.

Thus, a finite difference solution basically involves three steps: these are;

1. Dividing the solution into grids of nodes.

2. Approximating the given differential equation by finite difference equivalence that relates the solutions to

grid points.

3. Solving the difference equations subject to the prescribed boundary conditions and/or initial conditions.

In finite difference method, three forms are commonly considered, these are; the forward, backward and

central differences

Similarly, solutions and examples to this model are critically considered. For instance, it is a simple

matter to choose kandcm, so that the equation

02

2

=++ kxdtdxc

dtxdm (4.1)

is a valid linear equation. However, if one needs to specify the nature of the equation above, the settling time

and the peak time, then there may be a choice of kandcm, that will satisfy the equation.

33

4.2 METHOD OF SOLUTION TO THE PROBLEM

The central difference method combine with the direct integration techniques which is otherwise

known as the step-by-step integration will be used to solve the problem.

For the forward difference ( )[ ]xfδ = ( ) ( )xfhxf −+ (4.21)

For the backward difference ( )[ ] ( ) ( )hxfxfxf −−=δ (4.22)

The central difference is the summation of (4.21) and (4.22), we have;

( )[ ] ( ) ( )[ ] ( ) ( )[ ]hxfxfxfhxfxf −−+−+=δ

( ) ( ) ( ) ( )hxfxfxfhxf −−+−+=

( )[ ] ( ) ( )hxfhxfxf −−+=δ

Taking the average of the central difference = ( ) ( )[ ]hxfhxf −−+21

Now using the central difference to solve;

( ) ( ) ( )tFkxtxctxm =++ &&& (4.23)

From ( )[ ] ( ) ( )[ ]hxfhxfxf −−+=21δ

For this system, thtxxf ∆=== ,,

=> ( )[ ] ( ) ( )[ ]ttxttxtx ∆−−∆+=21δ

( ) ( )txtx =

Taking the derivative of ( )[ ]txδ with respect to t∆

34

( ) ( ) ( )t

ttxttxtx∆

∆−−∆+=

2&

( ) ( ) ( ) ( )2

2t

ttxtxttxtx∆

∆−+−∆+=&&

Substituting this into (4.23), we have;

( ) ( ) ( ) ( ) ( ) ( )[ ] ( )tFtxkt

ttxttxct

ttxtxttxm =+

∆∆−−∆+

+

∆∆−+−∆+

22

2

Opening the bracket;

( ) ( ) ( ) ( ) ( ) ( ) ( )tFtkxt

ttcxt

ttcxt

ttmxt

tmxt

ttmx=+

∆∆−

−∆

∆++

∆∆−

+∆

−∆

∆+22

2222

Collecting like terms;

( ) ( ) ( ) ( ) ( ) ( ) ( )tFtkxt

tmxt

ttcxt

ttmxt

ttcxt

ttmx=+

∆−

∆∆−

−∆

∆−+

∆∆+

+∆

∆+222

222

( )tFxktmx

tc

tmx

tc

tm

ttttt =

−

∆−

∆−

∆+

∆+

∆ ∆−∆+ 222

222

(4.24)

Let

−

∆−=

∆−

∆=

∆+

∆= k

tmTand

tc

tmR

tc

tmP 222

22

,2

=> ( )tFTxRxPx ttttt =++ ∆−∆+

Making ttPx ∆+ subject of formula, we have;

=> ( ) ttttt RxTxtFPx ∆−∆+ −−= (4.25)

This implies that to get ttx ∆+ , we need to have tx and ttx ∆− .

From the boundary condition, at 0=t , then 00 == tt xandx &

To have ttx ∆− , by the Taylor’s series expansion of degree two;

35

ttttt xtxtxx &&&2

2∆+∆−=∆−

At ,0=tx we have;

0

2

000 2xtxtxx t &&&

∆+∆−=∆− (4.26)

After getting 0xandx t∆− , then (4.25) becomes;

( ) ttt RxTxtFPx ∆−∆+ −−= 0

From equation (4.24), let us take integration constant;

c

tdat

ct

bt

a 12

,22,21,1 2

22 =∆

==∆

=∆

=∆

=

Then (4.26) becomes;

000 xdxtxx t &&& +∆−=∆−

Then (4.24) becomes;

( ) ( ) ( ) ( )tFxcmkxcbmaxbcma ttttt =−+−++ ∆−∆+ (4.27)

Also, from (4.27), we can now have three forms of matrix, namely;

(i) Mass matrix: This is a sparse matrix, that is, it is primarily populated with zero (Stoer and Bulirsch,

2002). This is; bcmaP +=

(ii) Stiffness matrix: This is a band matrix in which the non-zero elements are clustered near the diagonal.

This is; ( ) cmkkcmT −=−−=

(iii) Damping matrix: This is a symmetric matrix which is equal to its transpose that is jiij aa = . This is;

cbmaR +=

36

Let ( )tFPx tt =∆+

ð ( )tFPx tt1−

∆+ =

Therefore, the effective force vector is;

( ) ( ) tRxTxtFtF ∆−−−= 0

All the above expression can be summarise under the following algorithm;

A. Initial computation

1. Form stiffness [ ]K , mass [ ]M and damping [ ]C matrices.

2. Initialize [ ] [ ] [ ]000 , xandxx &&&

3. Select time step t∆ and calculate integration constants;

c

dact

bt

a 1,2,21,1

2 ==∆

=∆

=

4. Calculate [ ] [ ] [ ] [ ]000 xdxtxx t &&& +∆−=∆−

5. Form effective mass matrix [ ] [ ] [ ]cbmaP +=

B. For each time step;

1. Calculate effective force vector at time t;

[ ] [ ] [ ] [ ]ttt xRxTFF ∆−−−= 0

2. Solve the displacement at time tt ∆+

[ ] [ ][ ]ttt FPx 1−∆+ =

When solving most problems under structural dynamics, the following should be put in place, the

initial condition of the general equation of motion for dynamics system for the displacement and velocity at

0=t . After this, the next step of direct integration comes to place. In direct integration procedure, it

requires the value of the previous time tx , before getting ttx ∆+ and also to get tx , we must also have ttx ∆− .

37

4.3 NUMERICAL SOLUTION

Example 1: Find the displacement x by central difference method at time step 0.04 and;

=

−

−=

−

−=

=

tt

Fandkcm5.12sin2005.12sin500

51000120012003000

,123002800

2800700,

1000050

Solution

Let the displacement be

=

2

1

xx

x

From initial boundary condition, i.e. ,0=t then 0,0 == tt xx & and also 0=F since ( ) 005.12sin =

=

−

−+

−

−+

tt

xx

xx

xx

5.12sin2005.12sin500

51000120012003000

1230028002800100

1000050

2

1

2

1

2

1

&

&

&&

&&

=

−

−+

−

−+

00

00

51000120012003000

00

1230028002800100

1000050

2

1

xx&&

&&

ð

=

=00

02

1

txx&&

&&

From the question, 04.0=∆t

62504.011

22 ==∆

=t

a

( ) 5.1204.02

121

==∆

=t

b

125062522 =×=×= ac

0008.01250

1==d

For mass matrix P;

38

−

−=+=

−

−=

−

−=

=

=

216250350003500040000

15375035000350008750

1230028002800700

5.12

625000032150

1000050

625

bcmaP

bc

ma

For stiffness matrix T;

=

=

−

−=

125000062500

1000050

1250

51000120012003000

cm

k

−−−−

=−=740001200120059500

cmkT

For damping matrix R;

−

=

−

−−

=−=

91250350003500022500

15375035000350008750

625000032150

cbmaR

From the other initial condition at 04.0=∆t ,

( ) ( )

=

−−=

∆−∆−=

−

∆−

00

00

204.004.0

2

2

0004.0

0

2

00

xxx

xtxtxx t

&

&&&

39

To get ;1−P

−

−=

=−

216250350003500040000

1

P

PPadj

P

Determinant of P= ( ) ( )350003500021625040000 −×−−×=P

( ) ( )

7425000000

12250000008650000000

=

−=

To get Padj , we need the cofactors, 22211211 ,,, cccc

( ) ( )( ) ( )

( ) ( )( ) ( ) 40000400001

35000350001

400003500035000216250

35000350001

2162502162501

2222

1221

2112

1111

=−=

=−−=

=⇒

=−−=

=−=

+

+

+

+

c

c

Padj

c

c

6

66

661

1039.571.471.412.29

1039.51071.41071.41012.29

7425000000400003500035000216250

−

−−

−−−

×

=

××××

=

=P

To now get the displacement by 04.0=∆t ;

1. At 04.0=∆t

ttt RxTxFF ∆−−−= 0

40

[ ] [ ] 04.0004.004.0 −−−= RxTxFF

Where 04.0sec,/5.12 == tradω

( )( )

=

+

+

××

=

885.95713.239

00

00

04.05.12sin20004.05.12sin500

××

=

∴ −

885.95713.239

1039.571.471.412.29 6

04.02

1

xx

( ) ( )( ) ( )

3

6

10646.1432.7

10885.9539.5713.23971.4885.9571.4713.23912.29

−

−

×

=

×

×+××+×

=

2. At 08.0=∆t

[ ] [ ] 004.008.008.0 RxTxFF −−=

−

−×

−−−−

−

= −

00

91250350003500022500

10646.1432.7

740001200120059500

5.12sin2005.12sin500 3

tt

( )( )

( ) ( )( ) ( )

−×

×−+×−×−+×−

−

××

= −

00

10646.174000432.71200646.11200432.759500

08.05.12sin20008.05.12sin500 3

=

−

+

=

0164.2999142.864

00

7224.1301792.444

294.168735.420

41

( ) ( )( ) ( )

=

×

×+××+×

=

××

=

∴

−

−

00569.002680.0

100164.29939.59142.86471.40164.29971.49142.86412.29

0164.2999142.864

1039.571.471.412.29

6

6

08.02

1

xx

3. At 12.0=∆t

[ ] [ ] 04.008.012.012.0 RxTxFF −−=

310646.1432.7

91250350003500022500

00569.002680.0

740001200120059500

5.12sin2005,12sin500 −×

−

−

−−−−

−

=

tt

( )( )

( ) ( )( ) ( )

( ) ( )( ) ( )

310646.191250432.735000

646.135000432.722500

00569.0740000268.0120000569.0120002680.059500

12.05.12sin20012.05.12sin500

−×

×−+×

×+×

−

×−+×−×−+×−

−

××

=

−

=

+

=

9225.10983.224

719.6521755.2100

9225.10983.224

_22.453428.1601

4990.1997475.498

=

7965.5423455.1875

( ) ( )( ) ( )

=

×

×+××+×

=

××

=

∴

−

−

01176.005717.0

107965.54239.53455.187571.47965.54271.43455.187512.29

7965.5423455.1875

1039.571.471.412.29

6

6

12.02

1

xx

42

4. At 16.0=∆t

[ ] [ ]

( )( )

−

−

−−−−

−

××

=

−−=

00569.00268.0

91250350003500022500

01176.005717.0

740001200120059500

16.05.12sin20016.05.12sin500

08.012.016.016.0 RxTxFF

( ) ( )( ) ( )

( ) ( )( ) ( )

×−+×

×+×−

×−+×−×−+×−

−

=

00569.0912500268.03500000569.0350000268.022500

01176.07400005717.0120001176.0120005717.059500

8595.1816487.454

=

−

=

−

+

=

916.7015255.3055

7875.41815.802

7035.11206755.3857

7875.41815.802

844.9380262.3403

8595.1816487.454

( ) ( )( ) ( )

=

×

×+××+×

=

××

=

∴

−

−

01817.009213.0

10916.70139.55255.305571.4916.70171.464.305012.29

916.7015255.3055

1039.571.471.412.29

6

6

16.02

1

xx

5. At 2.0=∆t

[ ] [ ] 12.016.02.02.0 RxTxFF −−=

43

( )( )

−

−

−−−−

−

××

=01176.005717.0

91250350003500022500

01817.009213.0

740001200120059500

2.05.12sin2002.05.12sin500

−

+

=

85.927925.1697

136.1455539.5503

6944.1192361.299

−

=

85.927925.1697

8304.15747751.5802

=

9804.6468501.4104

××

=

∴ −

9804.6468501.4104

1039.571.471.421.29 6

2.02

1

xx

( ) ( )( ) ( )

6109804.64639.58501.410471.49804.64671.48501.410412.29 −×

×+××+×

=

=

02282.012258.0

6. At 24.0=∆t

[ ] [ ] 16.02.024.024.0 RxTxFF −−=

( )( )

−

−

−−−−

−

××

=01817.009213.0

91250350003500022500

02282.012258.0

740001200120059500

24.05.12sin20024.05.12sin500

( ) ( )( ) ( )

( ) ( )( ) ( )

×−+×

×+×−

×−+×−×−+×−

−

=

01817.09125009213.03500001817.03500009213.022500

02282.07400012258.0120002282.0120012258.059500

224.2856.70

−

+

=

5375.1566875.2708

776.1835894.7320

224.2856.70

44

−

=

5375.1566875.2708

1864454.7391

=

4625.297579.4682

××

=

∴ −

4625.297579.4682

1039.571.471.412.29 6

2

1

xx

( ) ( )( ) ( )

6

104625.29739.5579.468271.44625.29771.4579.468212.29 −

×

×+××+×

=

=

02366.013776.0

45

The table below gives the value of displacement and effective force for 25 time steps when 04.0=∆tfor central difference method and plotted below using Excel;

S/No Time step F1 F2 x1 x2

1 0.04 239.713 95.885 0.007432 0.001636

2 0.08 864.9142 299.0164 0.02680 0.00569

3 0.12 1875.3455 542.7965 0.05717 0.01176

4. 0.16 3055.5255 701.916 0.09213 0.01817

5. 0.20 4104.8501 646.9804 0.12258 0.02282

6. 0.24 4682.579 297.4625 0.13776 0.02366

7. 0.28 4492.9704 -361.9796 0.12953 0.01921

8. 0.32 3423.9858 -1237.0095 0.09388 0.00946

9. 0.36 1521.6719 -2163.4475 0.03426 -0.00449

10. 0.40 -889.7801 -2905.5079 -0.03960 -0.01985

11. 0.44 -3346.4902 -3266.3406 -0.11283 -0.03337

12. 0.48 -5307.3867 -3085.9716 -0.16909 -0.04163

13. 0.52 -6296.636 -2422.5145 -0.19477 -0.04271

14. 0.56 -6049.9987 -1143.1902 -0.18156 -0.03466

15. 0.60 -4498.247 324.5495 -0.12987 -0.01943

16. 0.64 -1957.7019 1797.0826 -0.04854 0.00047

17. 0.68 1113.8026 2955.6279 0.04635 0.02118

18. 0.72 4065.0002 3447.1512 0.13461 0.03773

19. 0.76 6232.8204 3233.8477 0.19673 0.04679

20. 0.80 7140.2979 2411.2443 0.21928 0.04663

21. 0.84 6599.1931 921.8543 0.19651 0.03605

22. 0.88 4669.7599 -716.2985 0.13261 0.01813

23. 0.92 1791.0999 -2262.6259 0.04150 -0.00376

24. 0.96 -1421.8235 -3322.7421 -0.05705 -0.02461

25. 1.00 -4259.3179 -3698.4643 -0.14145 -0.03990

46

Example 2: Find the displacement x by central difference method at time step 0.05 and;

=

−

−=

−

−=

=

tt

Fandkcm5.12sin2005.12sin500

51000120012003000

,123002800

2800700,

1000050

Solution

Let the displacement be

=

2

1

xx

x

From initial boundary condition, i.e. ,0=t then 0,0 == tt xx & and also 0=F since ( ) 005.12sin =

=

−

−+

−

−+

tt

xx

xx

xx

5.12sin2005.12sin500

51000120012003000

1230028002800100

1000050

2

1

2

1

2

1

&

&

&&

&&

=

−

−+

−

−+

00

00

51000120012003000

00

1230028002800100

1000050

2

1

xx&&

&&

ð

=

=00

02

1

txx&&

&&

From the question, 05.0=∆t

40005.011

22 ==∆

=t

a

( ) 1005.02

121

==∆

=t

b

80040022 =×=×= ac

00125.0800

1==d

For mass matrix P;

47

−

−=+=

−

−=

−

−=

=

=

163000280002800027000

12300028000280007000

1230028002800700

10

400000020000

1000050

400

bcmaP

bc

ma

For stiffness matrix T;

=

=

−

−=

800000040000

1000050

800

51000120012003000

cm

k

−−−−

=−=290001200120037000

cmkT

For damping matrix R;

−

=

−

−−

=−=

83000280002800013000

12300028000280007000

400000020000

cbmaR

From the other initial condition at 05.0=∆t ,

( ) ( )

=

−−=

∆−∆−=

−

∆−

00

00

205.005.0

2

2

0005.0

0

2

00

xxx

xtxtxx t

&

&&&

48

To get ;1−P

−

−=

=−

163000280002800027000

1

P

PPadj

P

Determinant of P= ( ) ( )280002800016300027000 −×−−×=P

( ) ( )

3617000000

7840000004401000000

=

−=

To get Padj , we need the cofactors, 22211211 ,,, cccc

( ) ( )( ) ( )

( ) ( )( ) ( ) 27000270001

28000280001

270002800028000163000

28000280001

1630001630001

2222

1221

2112

1111

=−=

=−−=

=⇒

=−−=

=−=

+

+

+

+

c

c

Padj

c

c

6

66

661

1046.774.774.706.45

1046.71074.71074.71006.45

3617000000270002800028000163000

−

−−

−−−

×

=

××××

=

=P

To now get the displacement by 05.0=∆t ;

1. At 05.0=∆t

ttt RxTxFF ∆−−−= 0

49

[ ] [ ] 05.0005.005.0 −−−= RxTxFF

Where 05.0sec,/5.12 == tradω

( )( )

=

+

+

××

=

0195.1175486.292

00

00

05.05.12sin20005.05.12sin500

××

=

∴ −

0195.1175486.292

1046.774.774.706.45 6

05.02

1

xx

( ) ( )( ) ( )

=

×

×+××+×

= −

00314.001409.0

100195.11746.75486.29274.70195.11774.75486.29206.45 6

2. At 1.0=∆t

[ ] [ ] 05.01.01.0 RxTxFF −−=

−

−

−−−−

−

=

00

83000280002800013000

00314.001409.0

290001200120037000

5.12sin2005.12sin500tt

( )( )

( ) ( )( ) ( )

−

×−+×−×−+×−

−

××

=00

00314.02900001409.0120000314.0120001409.037000

1.05.12sin2001.05.12sin500

=

−

+

=

7649.2975903.999

00

968.107098.525

7969.1894923.474

50

( ) ( )( ) ( )

=

×

×+××+×

=

××

=

∴

−

−

00996.004735.0

107649.29746.75903.99974.77649.29774.75903.99906.45

7649.2975903.999

1046.774.774.706.45

6.0

6

1.02

1

xx

3. At 15.0=∆t

[ ] [ ] 05.01.015.015.0 RxTxFF −−=

−

−

−−−−

−

=

00314.001409.0

83000280002800013000

00996.004735.0

290001200120037000

5.12sin2005.12sin500tt

( )( )

( ) ( )( ) ( )

( ) ( )( ) ( )

×−+×

×+×

−

×−+×−×−+×−

−

××

=

00314.08300001409.02800000314.02800001409.013000

00996.02900004735.0120000996.0120004735.037000

15.05.12sin20015.05.12sin500

−

=

+

=

9.13309.271

4772.5369449.2240

9.13309.271

_66.345902.1763

8172.1900429.477

=

5772.4028549.1969

( ) ( )( ) ( )

=

×

×+××+×

=

××

=

∴

−

−

01825.009188.0

107965.40246.78549.196974.77965.40274.78549.196906.45

5772.4028549.1969

1046.774.774.706.45

6

6

15.02

1

xx

51

The table below gives the value of displacement and effective force for 20 time steps when 05.0=∆t

for central difference method and plotted below using Excel;

S/No Time step F1 F2 x1 x2

1. 0.05 292.5486 117.0195 0.01409 0.00314

2. 0.10 999.5903 297.7649 0.04735 0.00996

3. 0.15 1969.8549 402.5772 0.09188 0.01825

4. 0.20 2826.2661 260.0804 0.12936 0.02382

5. 0.25 3117.7599 -208.5596 0.13887 0.02258

6. 0.30 2530.8653 -937.8683 0.10678 0.01259

7. 0.35 1056.6069 -1709.6984 0.03438 -0.00458

8. 0.40 -953.5581 -2228.2189 -0.06021 -0.02400

9. 0.45 -2243.7111 -2123.3685 -0.11754 -0.03321

10. 0.50 -2950.6916 -1416.8938 -0.14392 -0.03341

11. 0.55 -2628.298 -495.3304 -0.12226 -0.02404

12. 0.60 -1277.029 600.457 -0.05290 -0.00540

13. 0.65 780.4684 1400.5798 0.04601 0.01649

14. 0.70 2873.42 1691.3668 0.14257 0.03486

15. 0.75 4281.9507 1272.3655 0.20279 0.04263

16. 0.80 4452.8854 272.2338 0.20275 0.03650

17. 0.85 3249.5802 -1023.3334 0.13851 0.01752

18. 0.90 1004.24 -2102.7267 0.02898 -0.00791

19. 0.95 -1547.2188 -2746.2527 -0.09097 -0.03246

20. 1.00 -3593.2629 -2531.7384 -0.18151 -0.04672

52

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.2 0.4 0.6 0.8 1

Disp

lace

men

t

Time Steps

x1

x2

Figure 3: Displacement against time step of 0.05

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.2 0.4 0.6 0.8 1

Disp

lace

men

t

Time Step

x1

x2

Figure 2: Displacement against time step of 0.04

53

CHAPTER FIVE

5.1 CONCLUSION

From the above analysed problem and numerical examples, by using central difference method, it

shows that it is through the general equation of motion (equation 3.23) that displacement ( )ttx ∆+ can be

calculated as explained in the steps of central difference method, and it reveals that the sinusoidal vibration

of automobiles is a function of its inertial force, damping force, stiffness force and the external force which

combines to form a second order linear differential equation called the general equation of motion. And also

by using one of the methods to solve the problem, I conclude that the plotted graph (Figure 2 &3) are

conditionally stable because if exceeded, the displacement, velocity and acceleration grows without limit.

5.2 RECOMMENDATION

I recommend that;

1. A course that deals with the fundamental of dynamics should be introduced to the students to aid their

knowledge about dynamics.

2. A good and appropriate time step should be selected for meaningful and proper evaluation and

analysis of result based on direct integration method.

3. In a dynamics system, this method of step-by-step integration using central difference method should

be used to solve problems because it is advantageous and more accurate in predicting the response of

the dynamics system.

4. The damping is not high because if it is too high, the solution may not undergo core overflow.

54

5.3 REFERENCES

1. Balakumar Balachandran and Edward B. Magrab (2009). Vibrations, second edition.

2. Boyd D. Schimel, Jow-Lian Ding, Michael J. Anderson and Walter J. Grantham (1997),

Dynamic Systems Laboratory Manual, School of Mechanical and Materials Engineering,

Washington State University.

3. Sondipon Adhikar (2000). Damping Models for Structural Vibration. Trinity College, Cambridge.

September.

4. Indrajit Chowdhury & Shambhu P. Dasgupta (2009), Dynamics of Structure and Foundation – A

Unified Approach 1. Fundamentals, CRC Press/Balkema

5. Ankush Jalhotra, (2009), Study of vibration characteristics of different materials by sine sweep test.

M.Eng. Thesis, Mechanical Engineering department, Thapar University, Patiala, India.

6. Douglas Thorby, (2008), Structural Dynamics and Vibration in Practice, Butterworth-Heinemann

publications.