Simulation of a Spring Mass Damper System Using Matlab, Nov. 2009

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SIMULATION OF A SPRING MASS DAMPER SYSTEM

USING MATLAB

A Project work in partial fulfillment of the requirements for award of B.Sc Engineering

Department of Mechanical Engineering

Faculty of Engineering

University of Lagos, Akoka

Yaba, Lagos

Nigeria

November 2009

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ABSTRACT

The spring mass damper can be built or represented on the computer instead of going to the

workshop to fabricate such system and its performance under various conditions can also be

observed without having to subject the real system to these conditions hence, you save materials

and money, since the system can be used countless times. Energy is also saved because such

system is more easily built on a computer than physically. Moreover, it may be very difficult to

measure some outputs of some systems such as displacement but such values can be measured

with ease through simulation.

With this project, we aim to investigate the performance of a spring mass damper system, under

various conditions, through modeling, without having to subject the real system to these

conditions. The results are obtained in visual forms so that they can be readily interpreted and

discussed.

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TABLE OF CONTENTS

CHAPTER TITLE PAGE

ACKNOWLEDGEMENT iii

ABSTRACT iv

TABLE OF CONTENTS v

LIST OF FIGURES viii

1 INTRODUCTION

1

1.1 Background

1.2 Mechanical Vibration

2

1.3 Simulation Tool – MATLAB®

3

1.3.1 Why? MATLAB®

1.3.2 The MATLAB® system

1.4 Problem Statement

6

1.5 Objectives

6

1.6 Justification

6

1.7 Structure and Layout of Report

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2 LITERATURE REVIEW

4

2.1 Modeling of physical systems

8

2.1.1 Modeling a spring mass damper system

2.1.1.1 Single-degree-of-freedom system

2.1.1.2 Multi degree of freedom system

2.2 Common practical examples of mass spring damper systems 13

2.2.1 Automobile suspension

- Passive suspension

- Semi-active suspension

- Active suspension

2.3 Quarter car model

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2.4 Tuned mass damper

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3 METHODOLOGY

3.1 Modeling of a One Degree of Freedom Spring

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Mass Damper system

3.2 Modeling of a Three Degree of Freedom Spring Mass 24

Damper System

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3.3 Simulation 27

4 RESULTS AND DISCUSSION

4.1 Results and discussion 31

4.1.1 SCENARIO 1

4.1.2 SCENARIO 2

4.1.3 SCENARIO 3

4.1.4 SCENARIO 4

4.1.5 SCENARIO 5

4.1.6 SCENARIO 6

5 CONCLUSION AND RECOMMENDATION FOR FUTURE WORK

5.1 Conclusion 44

5.2 Recommendations 44

REFERENCES 45

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LIST OF FIGURES

FIG. NO. TITLE PAGE

2.1 Typical One-degree-of freedom system 9

2.2 Two-degree-of-freedom system 10

2.3 Three-degree-of-freedom system 11

2.4 Passive suspension system 14

2.5 Semi-active suspension system 15

2.6 A low bandwidth or soft active suspension system 16

2.7 A high bandwidth or stiff active suspension system 16

2.8 A Quarter car model 17

2.9 Quarter car suspension 18

2.10 A cantilever beam with a tuned mass damper at the tip 19

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2.11 Taipei-101’s tuned mass damper (top) and its placement in 20

the building (bottom)

3.1 Damped spring mass 22

3.2 3-degree-of-freedom system 25

3.3 Forces acting on m1 25



4.1 Displacement vs. Time (for Mass 1, scenario 1) 32












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LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Significance of m, c, and k in Different Systems 12

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CHAPTER 1

10 INTRODUCTION

1.1 Background

Springs usually occur physically as a coil of metal, and their idealizations have pretty simple

behavior: compressing the spring will result in the spring pushing back, and stretching the spring

will have it trying to pull back towards the start position, so any displacement along the axis of

the spring will be countered by an opposite force that will tend to move the spring back to it's

original position (Beer and Johnston, 2002). The fundamental spring equation is given as:

F = -kx

Where k is the spring constant (how loose or springy the spring is), x is the difference between

the springs current length and its rest length, and F is the force on both endpoints of the spring.

Usually one endpoint is fixed, the other is the one that bounces around- which is usually what

happens: an initial impulse displaces the spring, the unfixed end of the spring acquires some

velocity moving back, but it passes through the zero-displacement point, is pulled back in the

other direction, and may bounce perpetually in the absence of any dampening forces. Physical

springs have more complex behavior(like the transverse vibration and accompanying sound

when they're bent away from their axis) and could be described by more complex models but

we'll start from the simplest model.

Dampers

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Ideally, one could assume that all vibrating systems are free of damping. However, in actuality,

all vibrations are damped to some degree by friction forces. These forces can be caused by dry

friction, or Coulomb friction, between rigid bodies, by fluid friction when a rigid body moves in

a fluid, or by internal friction between the molecules of a seemingly elastic body. These all fall

under the category of free, damped vibrations. Hence, we have dampers of the viscous type,

Coulomb type or hysteresis type. The equation of motion (E.O.M) for viscously damped free

vibration is given by:

mx + cx + kx = 0

The equation of motion (E.O.M) for Coulomb damped free vibration is given by:

mx+kx+F=0

The area of concentration is on the area of dampers (forced damped vibration). If the system is

considered to be subjected to a periodic force P of magnitude P =Pm sinwft, the E.O.M becomes:

mx + cx + kx = Pm sinwft

A damper is kind of the opposite of a spring, except it operates on relative velocity rather than

displacement (Appleyard, M. and Wellstead, 1995). Spring endpoints moving away from each

other will have forces imparted from the damper that will act against that motion (only on the

spring axis, however), as well as endpoint moving towards each other. This will tend to return

the spring to a static position. Also endpoints moving in unison will not be affected (the damper

won't act as drag), and one endpoint unmoving and the other moving will average out to both

moving slower than the one endpoint.

1.2 Mechanical Vibrations

Mechanical systems may undergo free vibrations or they may be subjected to forced vibrations.

The vibrations are damped when friction forces are present and un-damped otherwise. The

suspension system of an automobile, for example, consists essentially of a spring and a shock

absorber (damper), which will cause the body of the car to undergo damped forced vibrations

when the car is driven over an uneven road.

Most vibrations in machines and structures are undesirable because of the increased stresses and

energy losses which accompany them. They should therefore be eliminated or reduced as much

as possible by appropriate design.

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The analysis of vibrations has become increasingly important in recent years owing to the current

trend toward higher-speed machines and lighter structures.

The analysis of vibration is a very extensive subject. In this project we will briefly look at a

simple case of vibration –the spring mass damper system, a one degree freedom system of

bodies. After a brief overlook of the simple system, we will take a complex case study – A 3

degree of freedom sysytem

1.3 Simulation Tool: MATLAB®

We need to see the performance of the system under various conditions without actually having

to subject the real system to these conditions, hence we simulate.

The simulation tool that is made use of is the MATLAB®.

The name MATLAB® stands for matrix laboratory (The MathWorks Inc, 2007). MATLAB®

was originally written to provide easy access to matrix software developed by the LINPACK and

EISPACK projects. Today, MATLAB® engines incorporate the LAPACK and BLAS libraries,

embedding the state of the art in software for matrix computation. It integrates computation,

visualization, and programming in an easy-to-use environment where problems and solutions are

expressed in familiar mathematical notation. Typical uses include:

• Math and computation

• Algorithm development

• Data acquisition

• Modelling, simulation, and prototyping

• Data analysis, exploration, and visualization

• Scientific and engineering graphics

• Application development, including graphical user interface building.

1.3.1 Why MATLAB®?

i. MATLAB® is an interactive system whose basic data element is an array that does

not require dimensioning. This allows you to solve many technical computing

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problems, especially those with matrix and vector formulations, in a fraction of the

time it would take to write a program in a scalar non-interactive language such as C

or FORTRAN.

ii. MATLAB® provides extensive documentation, in both printed and online format, to

help one learn about and use all of its features. The MATLAB® online help provides

task-oriented and reference information about MATLAB® features.

iii. MATLAB® is easily available. Downloadable demo versions can be obtained from

their website or one can buy the full version with license key also through their online

website. This is not the same with MATHEMATICA® which is very similar to

MATLAB®.

iv. MATLAB® possesses a rich library of functions and data structure that mimic the

properties of systems and also easily provides analytical representation of such

systems.

v. MATLAB® is compatible with most operating systems and is based on open

standards, i.e. it can be used in conjunction with other programs such as Java, C,

Microsoft Excel, etc.

vi. MATLAB® is built with the ability to manipulate direct computer memory thereby

allowing it to run faster than most other renowned programs like Java, C, FORTRAN,

etc which have an indirect link to computer memory.

vii. MATLAB® has a feature, SIMULINK, which is visual and allows one to bypass

complex mathematical calculations by using its block symbols to represent such

calculations hence saving time. With SIMULINK, a system can be constructed and

tested easily by varying parameters with the output available graphically and

pictorially.

1.3.2 The MATLAB® System

The MATLAB® system consists of five main parts:

• Development Environment. This is the set of tools and facilities that facilitate

MATLAB® functions and files. Many of these tools are graphical user interfaces. It

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includes the MATLAB® desktop and Command Window, a command history, an

editor and debugger, and browsers for viewing help, the workspace, files, and the

search path.

• The MATLAB ® Mathematical Function Library. This is a vast collection of

computational algorithms ranging from elementary functions, like sum, sine, cosine,

and complex arithmetic, to more sophisticated functions like matrix inverse, matrix

Eigen values, Bessel functions, and fast Fourier transforms.

• The MATLAB ® Language. This is a high-level matrix/array language with control

flow statements, functions, data structures, input/output, and object-oriented

programming features. It allows both "programming in the small" to rapidly create

quick and dirty throw-away programs, and "programming in the large" to create large

and complex application programs.

• Graphics. MATLAB® has extensive facilities for displaying vectors and matrices as

graphs, as well as annotating and printing these graphs. It includes high-level

functions for two-dimensional and three-dimensional data visualization, image

processing, animation, and presentation graphics. It also includes low-level functions

that allow full customization of the appearance of graphics as well as to build

complete graphical user interfaces on MATLAB applications.

• The MATLAB ® Application Program Interface (API). This is a library that allows

writing C and FORTRAN programs that interact with MATLAB. It includes facilities

for calling routines from MATLAB (dynamic linking), calling MATLAB as a

computational engine, and for reading and writing MAT-files.

1.4 Problem statement

A physical system is to be replaced by a mathematical model in order to predict its vibration

behavior. The accuracy of the predicted behavior depends on the level of difficulty associated

with the mathematical model.

The model must account for the four basic phenomena associated with the physical system,

namely, the elasticity, inertia, excitation or input energy, and damping or dissipation of energy.

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The mathematical model should not be too complex and overly sophisticated to include more

details of the system than are necessary.

1.5 Objective

To investigate the performance of a spring mass damper system, under various conditions,

through modeling, without having to subject the real system to these conditions.

1.6 Justification

The spring mass damper can be built or represented on the computer instead of going to the

workshop to fabricate such system and its performance under various conditions can also be

observed without having to subject the real system to these conditions hence, you save materials

and money, since the system can be used countless times. Energy is also saved because such

system is more easily built on a computer than physically. Moreover, it may be very difficult to

measure some outputs of some systems such as displacement but such values can be measured

with ease through simulation.

1.7 Structure and Layout of Report

This report is organized into five chapters.

Chapter 1 gives the background of the spring mass damper system and the objectives of the

project.

Chapter 2 discusses the literature review of the spring mass damper system.

In Chapter 3, the methodology of the simulation is presented.

Chapter 4 discusses the performance evaluation of the results by means of computer simulation

in MATLAB.

The summary of the results and future research based on this study will be presented in Chapter

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CHAPTER 2

2.0 LITERATURE VIEW

2.1 Modeling of physical (dynamic) systems

A mathematical model of a dynamic system is defined as a set of equations that represents the

dynamics of the system accurately or, at least, fairly well (Ogata, 2002). A mathematical model

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is not unique to a given system. A system may be represented in many different ways and,

therefore, may have many mathematical models, depending on one’s perspective.

The dynamics of many systems, whether they are mechanical, electrical, thermal, economic,

biological, and so on, may be described in terms of differential equations. Such differential

equations may be obtained by using physical laws governing a particular system, for example,

Newton’s laws for mechanical systems and Kirchhoff’s laws for electrical systems. It must be

kept in mind that deriving reasonable mathematical models is the most important part of the

entire analysis of control systems.

Mathematical models may assume many different forms. Depending on the particular system and

the particular circumstances, one mathematical model may be better suited than other models

(Ogata, 2002). For example, in optimal control problems, it is advantageous to use state-space

representations. On the other hand, for the transient-response or frequency-response analysis of

single-input-single-output, linear, time-invariant systems, the transfer function representation

may be more convenient than any other. Once a mathematical model of as system is obtained,

various analytical and computer tools (e.g. MATLAB) can be used for analysis and synthesis

purposes.

2.1.1 Modeling a spring mass damper system

Based on the nature of the mathematical model used, the system may be called a discrete (or

lumped) system or a continuous (or distributed) system (John Wiley & Sons, Inc, 2006). In the

discrete model, the physical system is assumed to consist of several rigid bodies (usually

considered as point masses) connected by springs and dampers. The springs denote restoring

forces that tend to return the masses to their respective undisturbed (or equilibrium) states. The

dampers provide resistance to velocity and dissipate the energy of the system. In the continuous

model, the mass, elasticity, and damping are assumed to be distributed throughout the system.

The equations of motion of a discrete system are in the form of a system of n coupled second-

order ordinary differential equations, where n denotes the number of masses (discrete masses or

rigid bodies). The number of independent coordinates needed to describe the configuration of a

system at any time during vibration defines the degrees of freedom of the system. For example,

Figs. 2.1, 2.2 and 2.3 denote typical one-, two-, and three-degree-of-freedom systems,

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respectively. A point mass can have three translational degrees of freedom while a rigid body can

have three translational and three rotational degrees of freedom. Many mechanical and structural

components and systems such as bars, beams, plates, and shells have distributed mass, elasticity,

and damping. The equation of motion of a continuous system is in the form of a partial

differential equation. A continuous system can be modeled either as a discrete- or

lumpedparameter system with varying number of degrees of freedom or as a continuous system

with infinite number of degrees of freedom, as illustrated for a cantilever beam in Fig. 2.4.

Figure 2.1 Typical One-degree-of freedom systems

The oscillatory motion of a body may be harmonic, periodic, or nonperiodic in nature.

If the time variation of the displacement of the mass is sinusoidal, the motion will be harmonic.

The number of cycles of motion per unit time defines the frequency, and the maximum

magnitude of motion is called the amplitude of vibration. If the periodic variation of motion is

not harmonic, the motion will be periodic. In this case, the periodic motion can be expressed as a

sum of harmonic motions of different frequencies. If the time variation of the displacement of the

mass is arbitrary (nonperiodic), the motion is said to be nonperiodic.

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If the nonperiodic motion can be described either by an equation or by a set of tabulated values,

the motion is considered to be deterministic. On the other hand, if the motion cannot be

described by any equation or tabulated values, it is said to be random or probabilistic.

When an external force or excitation is applied to a mechanical or structural system, the

amplitude of the resulting vibration can become very large when a frequency component of the

applied force or excitation approaches one of the natural frequencies of the system, particularly

the fundamental one. Such a condition, known as resonance, and the attendant stresses and

strains might cause a failure of the system. Because of this, designers should have a means of

determining the natural frequencies of mechanical and structural systems using analytical or

experimental approaches.

Figure 2.2 Two-degree-of-freedom system

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Figure 2.3 Three-degree-of-freedom system

2.1.1.1 Single-degree-of-freedom system

A study of the vibration characteristics of a single-degree-of-freedom-system is extremely

important in the study of vibration and shock because the approximate or qualitative response of

most systems can be determined by using a single-degree-of-freedom model for the system

(Appleyard M. and Wellstead, 1995).

A general single-degree-of-freedom system consists of a mass m, a spring of stiffness k, and a

viscous damper with a damping constant c, as shown in Fig. 2.1a, the significance of the

quantities m, c and k for different types of systems is given in Table 2.1

The equation of motion is given by:

mx +cx+kx=F(t) (2.1)

where the dots above x denote first and second derivatives respectively

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Table 2.1 Significance of m, c, and k in Different Systems (John Wiley & Sons, 2006)

Vibrating System m c k Variable x

1. Translatory

spring-mass-damper

system, Fig. 2.1a

Mass (kg) Viscous damping

constant (N.s/m)

Spring stiffness

(N/m)

Linear

displacement (m)

2. Rotational spring-

mass-damper system,

Fig. 2.1c

Mass

moment of

inertia

(kg.m2)

Torsional damping

constant (m.N.s/rad)

Torsional spring

stiffness (m.N/rad)

Angular

displacement

(rad)

3. Swinging

pendulum, Fig. 2.1b

Moment of

inertia of bob

(kg.m2)

Damping constant of

surrounding medium

(m.N.s/rad)

Angular stiffness

constant due to

gravity (N.m/rad)

Angular

displacement

(rad)

4. Transversely

vibrating cantilever

beam, Fig 2.1d

Mass at end

of beam (kg)

Damping constant

due to surrounding

medium (N.s/m)

Flexural stiffness

of beam (N/m)

Transverse

displacement of

mass at end of

cantilever (m)

2.1.1.2 Multi-degree-of-freedom system

Most mechanical and structural systems have distributed mass, elasticity, and damping (John

Wiley & Sons, 2006). These systems are modeled as multi- (n-) degree-of-freedom systems to

facilitate analysis of their vibration behavior. Several methods are available to construct an n-

degree-of-freedom model from a continuous system. These include the physical lumping or

modeling method, finite element method, finite difference method, modal analysis method,

Rayleigh–Ritz method, Galerkin method, and many others (Karnopp, 1994). In most cases, the

number of degrees of freedom (n) to be used in the model depends on the frequency range. If the

system is expected to undergo significant deformations at higher frequencies, the model should

include enough number of degrees of freedom to cover all the important frequencies. Most

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vibration characteristics of a n-degree-of-freedom system are similar to those of a single-degree-

of-freedom system. An n-degree-of-freedom system will have n natural frequencies, its free

vibrations denote exponentially decaying motions, its forced vibrations exhibit resonance

behavior, etc. However, there are some vibration characteristics that are unique to an n-degree-of

-freedom system which are absent in single-degree-of-freedom systems. For example, the

existence of normal modes, orthogonality of normal modes, and decomposition of the response

of the system (free or forced) in terms of normal modes are unique to multi-degree-of-freedom

systems.

2.2 Common practical examples of mass spring damper system

These include:

• Automobile suspension system

• Quarter car model

• Tuned mass damper

• Muscles and tendons in the human body

2.2.1 Automobile suspension system

The suspension system can be categorized into passive, semi-active and active suspension

system according to external power input to the system and/or a control bandwidth (Appleyard

and Wellstead, 1995). A passive suspension system is a conventional suspension system consists

of a non-controlled spring and shock-absorbing damper as shown in figure 2.1. The semi-active

suspension as shown in figure 2.2 has the same elements but the damper has two or more

selectable damping rate. An active suspension is one in which the passive components are

augmented by actuators that supply additional force. Besides these three types of suspension

systems, a skyhook type damper has been considered in the early design of the active suspension

system. In the skyhook damper suspension system, an imaginary damper is placed between the

sprung mass and the sky. The imaginary damper provides a force on the vehicle body

proportional to the sprung mass absolute velocity

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Passive Suspension System

The commercial vehicles today use passive suspension system to control the dynamics of a

vehicle’s vertical motion as well as pitch and roll. Passive indicates that the suspension elements

cannot supply energy to the suspension system. The passive suspension system controls the

motion of the body and wheel by limiting their relative velocities to a rate that gives the desired

ride characteristics. This is achieved by using some type of damping element placed between the

body and the wheels of the vehicle, such as hydraulic shock absorber.

Figure 2.4 Passive suspension system

Semi-Active Suspension System

In early semi-active suspension system, the regulating of the damping force can be achieved by

utilizing the controlled dampers under closed loop control, and such is only capable of

dissipating energy (Williams, 1994). Two types of dampers are used in the semi- active

suspension namely the two state dampers and the continuous variable dampers. The two state

dampers switched rapidly between states under closed-loop control. The disadvantage of this

system is that while it controls the body frequencies effectively, the rapid switching, particularly

when there are high velocities across the dampers, generates high-frequency harmonics which

makes the suspension feel harsh, and leads to the generation of unacceptable noise.

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The continuous variable dampers have a characteristic that can be rapidly varied over a wide

range. When the body velocity and damper velocity are in the same direction, the damper force is

controlled to emulate the skyhook damper. When they are in the opposite directions, the damper

is switched to its lower rate, this being the closest it can get to the ideal skyhook force. The

disadvantage of the continuous variable damper is that it is difficult to find devices that are

capable in generating a high force at low velocities and a low force at high velocities, and be able

to move rapidly between the two.

Figure 2.5 Semi-active suspension system

Active Suspension System

Active suspensions differ from the conventional passive suspensions in their

ability to inject energy into the system, as well as store and dissipate it.

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Figure 2.6 A low bandwidth or soft active suspension system

Figure 2.7 A high bandwidth or stiff active suspension system

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2.3 The Quarter Car Model

A quarter car model is a well-known model for simulating one-dimensional vehicle suspension

performance. In its simplified form, the suspension consists of a spring of stiffness K and a

damper with damping coefficient C. The spring performs the role of supporting the static weight

of the vehicle while the damper helps in dissipating the vibrational energy and limiting the input

from the road that is transmitted to the vehicle(Ahmet Naci Mete, Sandip D Kulkarni, Michael

Gerbracht, Noah Fehrenbacher).

The values for the stiffness and damping coefficient have to be chosen to optimize vehicle

performance under a certain range of vehicle load and road conditions. For a passive system with

a highly uneven input, there is an inherent conflict between system stability and passenger

comfort. For an extremely stiff suspension, the system will be highly stable, but acceleration of

the sprung mass will be high, and the passenger comfort will be low. For a non-stiff suspension,

passenger comfort will increase, but the vehicle becomes unstable.

From past research, active damper systems have proved to be very effective in improving the

comfort and handling. However, when the vehicle is moving over a rough terrain the active

systems do not have the reliability of a passive damper system. A failed active system can

become dangerous if not coupled with a passive system. Hence, semi active dampers are used for

off-road vehicle suspensions. A semiactive system gives fail-safe damping control, better

performance than passive systems and requires lesser power than active systems.

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Figure 2.8 A Quarter car model

The dynamic behaviour of the quarter car is given by the equation:

where m1=(m1+∆m1) represents the real mass of the quarter car, composed by a nominal

parameter m1 and an uncertain one ∆m1.

Figure 2.9 Quarter car suspension

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2.4 The tuned mass damper

The use of tuned mass dampers (TMD) is another widely used passive vibration damping

treatment. These devices are viscously damped 2nd order systems appended to a vibrating

structure. Proper selection of the parameters of these appendages, tunes the TMD to one of the

natural frequencies of the underdamped flexible structure, resulting in the addition of damping to

that resonance (R. Kashani, Ph.D. 2007).

Unlike dashpot which is most effective in adding damping to the first mode, TMD can target any

mode, including the first, and add considerable amount of damping to it. Another distinction

between TMD and dashpot is that TMD is a single point device and can simply be attached to a

structure at one end with its other end being free.

TMD consists of mass, which moves relatively to the structure and is attached to it by a spring

and a viscous damper in parallel as shown in figure 2.10. The structural vibration generates the

excitation of the TMD. As a result, the kinetic energy is transferred from the structure to the

TMD and is absorbed by the damping component of the device. The MD usually experience

large displacements.

TMD incorporated into a structure where the first mode of the structural response dominates, it is

expected to be very effective. The optimum tuning and damping ratios that result in the

maximum absorbed energy have been studied by several investigators. TMDs have been found

effective in reducing the response of structures to winds and harmonic loads and have been

installed in a number of buildings.

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Figure 2.10 A cantilever beam with a tuned mass damper at the tip

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Figure 2.11 Taipei-101’s tuned mass damper (top) and its placement in the building

(bottom)

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CHAPTER 3

3.0 METHODOLOGY

3.1 one degree of freedom spring mass damper system

If we take an ordinary spring that resists compression as well as extension and suspend it

vertically from a fixed support and at the end of the lower spring, we attach a body of mass m

(assume m to be so large that we may disregard the mass of the spring), when we pull the body

down a certain distance and then release it, it undergoes motion. We assume that the body moves

strictly vertically. The motion of this mechanical system is to be determined. This motion is

governed by Newton’s second law

Mass x Acceleration = mx = Force (1)

Where “Force” is the resultant of all the forces acting on the body. Here, x = d2x/dt2, where x(t)

is the displacement of the body and t is time.

We choose the downward direction as positive thus regarding downward forces positive and

upward forces negative.

The spring is first un-stretched. When we attach the body, the latter stretches the spring by an

amount s0. This causes an upward force F0 in the spring given as

F0=-ks0 (2) (Hooke’s law)

This force balances the weight of the body, i.e. W + F0 = mg -ks0= 0. This is called static

equilibrium.

If the body is pulled downward, it further stretches the spring by some amount x > 0 (the

distance we pull it down). By Hooke’s law, this causes an (additional) upward force F1 in the

spring such that

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F1= -kx

Figure 3.1 Damped spring mass

F1 is a restoring force. It has the tendency to restore the system, that is, to pull the body back to x

= 0.

If we connect the mass to a dashpot, we have to take the corresponding viscous damping into

account. The corresponding damping force has the direction opposite to the instantaneous

motion. We assume that it is proportional to the velocity x'= dx/dt of the body. This is generally

a good approximation, at least for small velocities. Thus, the damping force is of the form

F2= -cx' (3)

c is called the damping constant. The resultant forces acting on the body now is

F1+ F2=-kx -cx' (4)

Hence, by Newton’s second law,

mx= -kx -cx' (5)

This shows that the motion of the damped mechanical system is governed by the linear

differential equation with constant coefficients

mx+kx+cx'=0 (6)

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mx=inertia force

cx'=damping force

kx=spring force

x+cmx+kmx=0 (7a)

Or [D2+cmD+km]x=0 (7b)

Where D=d/dt and D2=d2/dt

Equation (7b) is an ordinary differential equation of the second order. Its characteristic equation

is

D2+cmD+km=0 (8)

Its roots are

D1,2=-c2m±c2m2-km (9)

For critical damping, the term under the square root sign is equal to zero, and the damping

coefficient is called the critical damping coefficient (cc). Thus,

Cc2m2-km=0

Cc2m=km=ωn=natural circular frequency inrads (10)

ωn2=km

Cc=2mωn=2km (11)

Damping ratio, ℶ=c/cc

c2m=ccc×cc2m=ℶωn (12)

cm=2ℶωn (13)

Thus D1,2=-ωn±ℶωn2-ωn2

= -ℶωn±ωnℶ2-1 (14)

33

Hence, equation (7a) can be rewritten as

x+2ℶωnx+ωn2x=0 (15)

From Laplace transforms, we get

s2Xs-sx0--x'X(0-)+2ℶωnsXs-x0-+ωn2Xs=0 (16)

Mass position = x0- = x0 (m)

Mass velocity = x'= 0 (m/s)

s2Xs-sx0+2ℶωnsXs-x0+ωn2Xs=0 (17)

When we solve by using Laplace, we obtain

xt=x0e-ℶωnt1-ℶ2sin(ωn1-ℶ2t+θ) (18)

This is the system’s response, i.e. displacement at any point in time, t. The system is

underdamped when ℶ<1, overdamped when ℶ>1 and critically damped when ℶ=1.

3.2 Three degree of freedom spring mass damper system

We now consider the three-degree-of-freedom system consisting of three masses m1, m2, and

m3(kg); three forces F1, F2 and F3(N) acting on the masses; four springs with stiffnesses k1, k2,

k3 and k4(N/m); and four viscous dampers with damping constants c1, c2, c3 and c4(Ns/m) as

shown in Fig. 3.2. The mass mi subjected to the force Fi(t) undergoes a displacement xi (t), i = 1,

2, 3.

Assumptions made

We are assuming that there is negligible friction between the surfaces of the masses and the

surface of the ground. Therefore, there will be no considerations for friction in our mathematical

modeling and simulation.

F1

F2

F3

K1

C1

C2

C3

K2

K3

C4

X1

X2

X3

m1

m2

m3

k2x1-x2c2x1-x2c1x1k1x1m1x1x1, x1, x1

34

Figure 3.2 3-degree-of-freedom system

We isolate the individual masses:

Figure 3.3 Forces acting on m1

m1

35



mx +cx+kx=F is the general equation governing the system. When we isolate each mass, we

obtain the following E.O.M.

k3x2-x3c3x2-x3c2x3-x1k2x1-x2m2x2x2, x2, x2 m2

k4x3c4x3c3x3-x2k3x3-x2m3

m3x3x3, x3, x3

36

m1x1+k1x1+k2x1-x2+c1x1+c2x1-x2=F1

m2x2+k2x2-x1+k3x2-x1+c2x2-x1+c3x2-x3=F2

m3x3+k3x3-x2+k4x3+c3x3-x2+c4x3=F3

3.3 Simulation

We must remember that computer language is garbage in, garbage out (GIGO), hence what we

input into the program needs to be readable and intrepreted in the right manner by the program.

This was a big challenge in solving this problem.

After vigorous efforts, search and study through MATLAB’s various commands, we obtained a

solution by programming using the equivalent state space model of the system.

The state space modeling is a modern control theory. The modern trend in engineering systems is

toward greater complexity, due mainly to the requirements of complex tasks and good accuracy.

Complex systems may have multiple inputs and multiple outputs and may be time varying.

Because of the necessity of meeting increasingly stringent requirements on the performance of

control systems, the increase in system complexity, and easy access to large scale computers,

modern control theory, which is a new approach to the analysis and design of complex control

systems, has been developed since around 1960 (Ogata, 2002). This new approach is based on

the concept of state. The concept of state by itself is not new since it has been in existence for a

long time in the field of classical dynamics and other fields.

Modern Contol Theory Versus Conventional Control Theory

Modern control theory is contrasted with conventional control theory in that the former is

applicable to multiple-input-multiple-output systems, which may be linear or nonlinear, time

invariant or time varying, while the latter is applicable only to linear time-invariant single-input-

single-output systems. Also, modern control theory is essentially a time-domain appoach, while

conventional control theory is a complex frequency-domain approach.

37

So, we obtained equations for y1, y2, y3, y4, y5, y6in our three degree of freedom system

using state space model theory where y1=x1, y2= x1, y3=x2, y4=x2, y5=x3, y6=x3

as follows:

y1= y2

y2= -k1+k2m1y1-c1+c2m1y2+(k2/m1)y3+(c2/m1)y4+F1/m1

y3= y4

y4= (k2/m2)y1+c2/m2y2-k2+k3m2y3-c2+c3m2y4+(k3/m2)y5+ (c3/m2)y6+F2/m2

y5= y6

y6= (k3/m3)y3+(c3/m3)y4-k3+k4m3y5-c3+c4m3y6+F3/m3

Then, using values of masses 1, 2 , 3 as 6, 9, 5kg (respectively), spring stiffness’s 1, 2, 3, 4 as 6,

7, 4, 1N/m (respectively), dampers 1, 2, 3, 4 as 1, 0.2, 0.1, 2Ns/m (respectively), forces 1,

2, 3 as 3, 9, 12N (respectively), we inputted these into the program and coded as follows

in the MATLAB Editor:

function dydt = massspring(t,y)

m1 = 6;

38

m2 = 9;

m3 = 5;

k1 = 6;

k2 = 7;

k3 = 4;

k4 = 1;

c1 = 1;

c2 = 0.2;

c3 = 0.1;

c4 = 2;

F1 = 3;

F2 = 9;

F3 = 12;

dydt = [ y(2)

-(((k1+k2)/m1)*y(1))- (((c1+c2)/m1)*y(2))+(((k2)/m1)*y(3))+(((c2)/m1)*y(4))+ (F1/m1)

y(4)

(((k2)/m2)*y(1))+ (((c2)/m2)*y(2))- (((k2+k3)/m2)*y(3))- (((c2+c3)/m2)*y(4))+

(((k3)/m2)*y(5))+ (((c3)/m2)*y(6))+ (F2/m2)

y(6)

(((k3)/m3)*y(3))+ (((c3)/m3)*y(4))- (((k3+k4)/m3)*y(5))- (((c3+c4)/m3)*y(6))+

(F3/m3)];

Then, we wrote another program on a new page, invoking the first program in this new one.

% TO SOLVE THE SYSTEM OF NON-LINEAR ODE's FOR THE SPRING MASS DAMPER

clc;

[t,y] = ode45(@massspring,[0:1: 200],[6;0;7;0;8;0]);

figure(1)

plot (t,y(:,1))

39

figure(2)

plot (t,y(:,3))

figure(3)

plot (t,y(:,5))

We then varied some of the inputs while keeping the others constant and generated different

displacement-time graphs in order to observe the system’s performance.

40

CHAPTER 4

4.1 RESULTS AND DISCUSSION

After the mathematical model had been inputted into and solved by MATLAB, we went put our

simulation to use by testing various conditions of the system. As was incorporated into our

programming commands, MATLAB provided us with visual representations (plotted graphs) of

these various conditions of the system which we went on to interpret.

Below are the results we obtained and our discussions.

4.1.1 Scenario 1- c1=1, c2=0.2, c3=0.1, c4=2

Here inputted values for c1, c2, c3 and c4 (dampers) and MATLAB produced the

graph shown below. It is observed that the body (mass 1) is displaced to and fro

its original position for the first 40 – 50 seconds before the damping starts to take full

effect, and it comes to rest (stabilizes) at 80 seconds. This could be described as a

‘damped’ vibration.

For this scenario, both masses 2 and 3 have similar displacement-time graphs as mass 1.

All the masses are both affected by their own individual damping and that of the

whole system.

41

0 20 40 60 80 100 120 140 160 180 2001.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time, t(sec)

Dis

pla

cem

ent, y

1(t)(m

)

Figure 4.1 Displacement vs. Time (for Mass 1, scenario 1)

0 20 40 60 80 100 120 140 160 180 2003.5

4

4.5

5

5.5

6

6.5

7

7.5

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)


42

0 20 40 60 80 100 120 140 160 180 2004

4.5

5

5.5

6

6.5

7

7.5

8

8.5

Time, t(sec)

Dis

plac

emen

t, y

3(t)

(m)


4.1.2 Scenario 2- c1=c2=c3=0, c4=2

In this scenario, we set c1, c2 and c3=0 (no damping or negligible), while leaving c4 as equal to

2NS/m. As can be observed from the graphs for masses 1, 2 and 3 below, because there is

little or no damping, the masses seem to never come to rest even at a time of 200

seconds. In fact, the only reason why the displacement of the masses subsides when it

approaches time 40 seconds (more clearly observed in the case of mass 3) is because of

the overall damping effect of c4 on the whole system.

43

0 20 40 60 80 100 120 140 160 180 2001

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time, t(sec)

Dis

pla

cem

ent, y

1(t)(m

)


0 20 40 60 80 100 120 140 160 180 2003

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)


44

0 20 40 60 80 100 120 140 160 180 2004

4.5

5

5.5

6

6.5

7

7.5

8

8.5

Time, t(sec)

Dis

pla

cem

ent, y

3(t)(m

)


4.1.3 Scenario 3- c1=10, c2=9, c3=15, c4=2

In this third case, we tried to see the effect of over-damping by raising the values of c1, c2, and

c3 to very high values. As can be observed from the graphs below, the masses achieve

high displacement, and then a state of rest almost immediately after, reflecting how

heavily damped the system is. This is clearly a state of stiff spring coefficient,

usually the case in devices that require early damping of the vibration (e.g.

measuring instruments, racing cars etc.)

45

0 20 40 60 80 100 120 140 160 180 2002.5

3

3.5

4

4.5

5

5.5

6

Time, t(sec)

Dis

plac

em

ent

, y1

(t)(

m)


0 20 40 60 80 100 120 140 160 180 2004.5

5

5.5

6

6.5

7

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)

46


0 20 40 60 80 100 120 140 160 180 2005.5

6

6.5

7

7.5

8

Time, t(sec)

Dis

plac

emen

t, y

3(t)

(m)


4.1.4 Scenario 4- K1=3, K2=2, K3=0, K4=1

In this case, we tried to see the effect of reducing the spring stiffness’s. As can be observed from

the graphs below, the masses 1 and 2 move to and fro and do not still come to a steady

state after 200 seconds. However, the third mass becomes steady not long after the process

begins since k3=0 and k4=1.

47

0 20 40 60 80 100 120 140 160 180 2002

2.5

3

3.5

4

4.5

5

5.5

6

Time, t(sec)

Dis

pla

cem

ent, y

1(t)(m

)


0 20 40 60 80 100 120 140 160 180 2007

7.5

8

8.5

9

9.5

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)


48

0 20 40 60 80 100 120 140 160 180 2008

8.5

9

9.5

10

10.5

11

11.5

12

12.5

13

Time, t(sec)

Dis

plac

emen

t, y

3(t)

(m)


4.1.5 Scenario 5- m1=1, m2=3, m3=0.5

In this case, we tried to see the effect of reducing the masses. As can be observed from the graphs

below, the system comes to rest faster than that of scenario 1 where the values of the

masses are higher, so obviously, the less heavy the masses, the easier it is to control

the vibrations.

49

0 20 40 60 80 100 120 140 160 180 2002

2.5

3

3.5

4

4.5

5

5.5

6

Time, t(sec)

Dis

pla

cem

ent, y

1(t)(m

)


0 20 40 60 80 100 120 140 160 180 2003.5

4

4.5

5

5.5

6

6.5

7

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)

50


0 20 40 60 80 100 120 140 160 180 2005.5

6

6.5

7

7.5

8

Time, t(sec)

Dis

plac

emen

t, y

3(t)

(m)


4.1.6 Scenario 6- F1=1, F2=1, F3=1

The effect of reducing the forces acting on the masses is observed in this sixth and final case. The

system here also stabilizes faster than that of scenario 1 which implies that the lesser the

force on a system, the faster it stabilizes, i.e. lesser vibration on the system.

51

0 20 40 60 80 100 120 140 160 180 200-2

-1

0

1

2

3

4

5

6

Time, t(sec)

Dis

pla

cem

ent, y

1(t)(m

)


0 20 40 60 80 100 120 140 160 180 200-3

-2

-1

0

1

2

3

4

5

6

7

Time, t(sec)

Dis

pla

cem

ent, y

2(t)(m

)

52


0 20 40 60 8 0 10 0 120 140 16 0 1 80 200-2

-1

0

1

2

3

4

5

6

7

8

Tim e , t (s ec )

Dis

plac

emen

t, y3

(t)(m

)


So, in like manner as above, we can change the values of our input parameters and see the effect

on the system.

53

CHAPTER 5

5.0 CONCLUSION AND RECOMMENDATION

5.1 Conclusion

From the results achieved above in chapter 4, we conclude that a spring mass damper system,

which is widely used in mechanical applications, can be well represented and simulated on a

computer to reproduce real-life situations and accurately predict different conditions and outputs

desired.

Thus it can be used to design systems which have not been manufactured for testing.

5.2 Recommendation

We recommend the following for future work:

I. A mathematical model of the system, considering the friction forces (i.e. a more complex

system).

II. The use of SIMULINK which is a circuit-like representation of systems and VIRTUAL

REALITY (both incorporated into MATLAB) for more visual representation of the

system, so that even a layman (as in the case of VIRTUAL REALITY) can easily

interpret.

54

REFERENCES

Ferdinand P. Beer & E. Russell Johnston (1997). Vector Mechanics for Engineers, Sixth

Edition.

Pgs. 1172 – 1174.

Katsuhiko Ogata (2002). Modern Control Engineering, Fourth Edition. Pgs. 53-54, 70-90.

Allen S. Hall, Alfred R. Holowenko, Herman G. Laughlin (2002). Schaum’s Outlines Machine

Design. Pgs. 89-92

John Wiley & Sons, Inc. Edited by Myer Kutz (2006). Mechanical Engineers’ Handbook:

Materials and Mechanical Design, Volume 1, Third Edition. Pgs. 1204-1209.

www.matlabcentral.com – The official MATLAB® website.

The MathWorks Incorporated (2007) – MATLAB® product help.

Ahmet Naci Mete, Sandip D Kulkarni, Michael Gerbracht, Noah Fehrenbacher (2005).

“Quarter car model using a semi-active MRF damper”.

Yahaya Md. Sam PhD. (2006). “Robust Control of Active Suspension System for a quarter car

model”Project for Department of Control and Instrumentation Engineering, Universiti

Teknologi, Malaysia, 81310 UTM Skudai. Pgs. 6-21

Appleyard M. and Wellstead P.E. (1995). Active Suspension: some background. IEEE Proc.

Control Theory Application. 142(2): 123-128.

Karnopp, D. (1990). Design Principles for Vibration Control Systems using Semi-Active

Dampers. ASME Journal of Dynamic Systems, Measurement and Control. 112:448-455.

R. Kashani, Ph.D. (www.deicon.com)

http://www.matlabcentral.com/

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Simulation of a Spring Mass Damper System Using Matlab, Nov. 2009

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