Simulation of a Spring Mass Damper System Using Matlab

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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 1

viii

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

1.5

Objectives

1.6

Justification

1.7

Structure and Layout of Report

6 6 6 6 2

LITERATURE REVIEW

4

2.1

Modeling of physical systems

8 2.1.1

2.2

Modeling a spring mass damper system 2.1.1.1

Single-degree-of-freedom system

2.1.1.2

Multi degree of freedom system

Common practical examples of mass spring damper systems 2.2.1

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Automobile suspension - Passive suspension - Semi-active suspension - Active suspension

2.3

Quarter car model

2.4

Tuned mass damper

17

18

3

METHODOLOGY 3.1

Modeling of a One Degree of Freedom Spring

21 Mass Damper system 3.2

Modeling of a Three Degree of Freedom Spring Mass Damper System

24

5

3.3 4

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RESULTS AND DISCUSSION 4.1

5

Simulation

Results and discussion 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

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CONCLUSION AND RECOMMENDATION FOR FUTURE WORK 5.1

Conclusion

44

5.2

Recommendations

44

REFERENCES

45

6

LIST OF FIGURES FIG. NO.

TITLE

PAGE

2.1

Typical One-degree-of freedom system

2.2

Two-degree-of-freedom system

10

2.3

Three-degree-of-freedom system

11

2.4

Passive suspension system

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2.5

Semi-active suspension system

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2.6

A low bandwidth or soft active suspension system

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2.7

A high bandwidth or stiff active suspension system

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2.8

A Quarter car model

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2.9

Quarter car suspension

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2.10

A cantilever beam with a tuned mass damper at the tip

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9

7

2.11

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

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the building (bottom) 3.1

Damped spring mass

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3.2

3-degree-of-freedom system

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3.3

Forces acting on m1

3.4

Forces acting on m2

3.5

Forces acting on m3

4.1

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

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4.2


32

4.3


33

4.4


34

4.5


34

4.6


35

4.7


36

4.8


36

4.9


37

4.10


38

4.11


38

4.12


39

25 26 26

8

4.13


40

4.14


40

4.15


41

4.16


42

4.17


42

4.18


43

LIST OF TABLES TABLE NO.

TITLE

2.1

Significance of m, c, and k in Different Systems

PAGE 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

Simulation Tool: MATLAB®

1.3

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 i.

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

ii.

help one learn about and use all of its features. The MATLAB® online help provides task-oriented and reference information about MATLAB® features. MATLAB® is easily available. Downloadable demo versions can be obtained from

iii.

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

iv.

properties of systems and also easily provides analytical representation of such systems. MATLAB® is compatible with most operating systems and is based on open

v.

standards, i.e. it can be used in conjunction with other programs such as Java, C, Microsoft Excel, etc. MATLAB® is built with the ability to manipulate direct computer memory thereby

vi.

allowing it to run faster than most other renowned programs like Java, C, FORTRAN, etc which have an indirect link to computer memory. MATLAB® has a feature, SIMULINK, which is visual and allows one to bypass

vii.

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 5

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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 secondorder 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

2.1.1.1

Three-degree-of-freedom system

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)

where the dots above x denote first and second derivatives respectively

(2.1)

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Table 2.1

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

Vibrating System 1. Translatory

m Mass (kg)

spring-mass-damper

c

k

Variable x

Viscous damping

Spring stiffness

Linear

constant (N.s/m)

(N/m)

displacement (m)

Torsional damping

Torsional spring

Angular

constant (m.N.s/rad)

stiffness (m.N/rad) displacement

system, Fig. 2.1a 2. Rotational spring-

Mass

mass-damper system, moment of Fig. 2.1c

inertia

(rad)

(kg.m2) 3. Swinging

Moment of

Damping constant of

Angular stiffness

Angular

pendulum, Fig. 2.1b

inertia of bob

surrounding medium

constant due to

displacement

(kg.m2)

(m.N.s/rad)

gravity (N.m/rad)

(rad)

4. Transversely

Mass at end

Damping constant

Flexural stiffness

Transverse

vibrating cantilever

of beam (kg)

due to surrounding

of beam (N/m)

displacement of

beam, Fig 2.1d

medium (N.s/m)

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 ndegree-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-degreeof-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)

32 mx=inertia force cx'=damping force kx=spring force

(7a)

x+cmx+kmx=0

Or

(7b)

[D2+cmD+km]x=0

Where D=d/dt and D2=d2/dt Equation (7b) is an ordinary differential equation of the second order. Its characteristic equation is (8)

D2+cmD+km=0

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 (12)

c2m=ccc×cc2m=ℶωn cm=2ℶωn

(13)

Thus D1,2=-ωn±ℶωn2-ωn2 = -ℶωn±ωnℶ2-1

(14)

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Hence, equation (7a) can be rewritten as (15)

x+2ℶωnx+ωn2x=0

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.

34 m F3213214 K C X

Figure 3.2

3-degree-of-freedom system

We isolate the individual masses: k2x1-x2 m1x1 k1x1 c1x1 c2x1-x2 x1 m, 1x1, x1

Figure 3.3

Forces acting on m1

35

m m2x2 k2x1-x2 c2x3-x1 c3x2-x3 k3x2-x3 x22, x2, x2

Figure 3.4

Forces acting on m2

Figure 3.5

Forces acting on m3

m3x3 m k3x3-x2 c3x3-x2 c4x3 k4x3 x3, x3, x3 3

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

obtain the following E.O.M.

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-inputsingle-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 whole system.

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

41

6 5.5

Displacement, y1(t)(m)

5 4.5 4 3.5 3 2.5 2 1.5

0

20

40

Figure 4.1

60

80

100 Time, t(sec)

120

140

160

180

200


7.5


7 6.5 6 5.5 5 4.5 4 3.5

0

20

40

Figure 4.2

60

80

100 Time, t(sec)

120

140

160

180

200


42

8.5 8


7.5 7 6.5 6 5.5 5 4.5 4

0

20

40

Figure 4.3

4.1.2

Scenario 2-

60

80

100 120 Time, t(sec)

140

160

180

200


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 approaches time 40 seconds the overall damping effect of c4 on

only reason why the displacement of the masses subsides when it (more clearly observed in the case of mass 3) is because of the whole system.

43

6 5.5


5 4.5 4 3.5 3 2.5 2 1.5 1 0

20

40

Figure 4.4

60

80

100 Time, t(sec)

120

140

160

180

200


7.5 7


6.5 6 5.5 5 4.5 4 3.5 3 0

20

40

Figure 4.5

60

80

100 Time, t(sec)

120

140

160

180

200


44

8.5 8


7.5 7 6.5 6 5.5 5 4.5 4 0

20

40

Figure 4.6

4.1.3

Scenario 3-

60

80

100 Time, t(sec)

120

140

160

180

200


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 measuring instruments, racing cars etc.)

require early damping of the vibration (e.g.

45

6


5.5

5

4.5

4

3.5

3

2.5

0

20

40

Figure 4.7

60

80


140

160

180

200



7

6.5

6

5.5

5

4.5

0

20

40

6 0

80

100 T im e,t(se c)

120

140

160

180

200

46

Figure 4.8


8


7.5

7

6.5

6

5.5

0

20

40

Figure 4.9

4.1.4

Scenario 4-

60

80


140

160

180

200


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 begins

200 seconds. However, the third mass becomes steady not long after the process since k3=0 and k4=1.

47

6


5 .5 5 4 .5 4 3 .5 3 2 .5 2 0

2 0

4 0

Figure 4.10

6 0

8 0

1 0 0 T im e ,t(se c)

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0


9.5


9

8.5

8

7.5

7 0

20

40

Figure 4.11

60

80

100 Time, t(sec)

120

140

160

180

200


48

13 12.5


12 11.5 11 10.5 10 9.5 9 8.5 8

0

20

40

Figure 4.12

4.1.5

Scenario 5-

60

80


140

160

180

200


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 the vibrations.

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

49

6 5.5


5 4.5 4 3.5 3 2.5 2 0

20

40

Figure 4.13

60

80

100 Time, t(sec)

120

140

160

180

200


7


6.5

6

5.5

5

4.5

4

3.5

0

20

40

60

80

100 T im e, t(sec)

120

140

160

180

200

50

Figure 4.14


8


7.5

7

6.5

6

5.5

0

20

40

Figure 4.15

4.1.6

Scenario 6-

60

80


140

160

180

200


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

6


5 4 3 2 1 0 -1 -2

0

20

40

60

80

100 Time, t(sec)

120

140

160

180

200

Figure 4.16 Displacement vs. Time (for Mass 1, scenario 6)

7 6


5 4 3 2 1 0 -1 -2 -3

0

20

40

60

80

100 Time, t(sec)

120

140

160

180

200

52

Figure 4.17


8 7 6


5 4 3 2 1 0 -1 -2

0

20

40

Figure 4.18

60

80

100 120 Tim e , t(s e c )

140

160

180

200


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)

Simulation of a Spring Mass Damper System Using Matlab

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