1
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 Engineering University of Lagos, Akoka Yaba, Lagos Nigeria
November 2009
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ABSTRACT
The spring mass damper can be built bu ilt 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 ssystem ystem 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 v ery 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
2
viii
1
INTRODUCTION
1.1
Background
1.2
Mechanical Vibration
2
1.3
Simulation Tool – Tool – MATLAB 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
6
LITERATURE REVIEW
2.1
Modeling of physical systems
2.1.1
8
Modeling a spring mass damper system
2.1.1.1
Single-degree-of-freedom system
2.1.1.2
Multi degree of freedom system
4
2.2
Common practical examples of mass spring damper systems
2.2.1
13
Automobile suspension - Passive suspension - Semi-active suspension - Active suspension
3
2.3
Quarter car model
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2.4
Tuned mass damper
18
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
24
Damper System
3.3
4
Simulation
27
RESULTS AND DISCUSSION
4.1
Results and discussion
4.1.1
SCENARIO 1
4.1.2
SCENARIO 2
31
5
5
4.1.3
SCENARIO 3
4.1.4
SCENARIO 4
4.1.5
SCENARIO 5
4.1.6
SCENARIO 6
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
15
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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2.11
Taipei-101’s Taipei-101’s tuned mass damper (top) and its placement in
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9
the building (bottom)
3.1
Damped spring mass
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3.2
3-degree-of-freedom system
25
3.3
Forces acting on m1
25
7
3.4
Forces acting on m2
3.5
Forces acting on m3
26
4.1
Displacement vs. Time (for Mass 1, scenario 1)
32
4.2
Displacement vs. Time (for Mass 2, scenario 1)
32
4.3
Displacement vs. Time (for Mass 3, scenario 1)
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4.4
Displacement vs. Time (for Mass 1, scenario 2)
34
4.5
Displacement vs. Time (for Mass 2, scenario 2)
34
4.6
Displacement vs. Time (for Mass 3, scenario 2)
35
4.7
Displacement vs. Time (for Mass 1, scenario 3)
36
4.8
Displacement vs. Time (for Mass 2, scenario 3)
36
4.9
Displacement vs. Time (for Mass 3, scenario 3)
37
4.10
Displacement vs. Time (for Mass 1, scenario 4)
38
4.11
Displacement vs. Time (for Mass 2, scenario 4)
38
4.12
Displacement vs. Time (for Mass 3, scenario 4)
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4.13
Displacement vs. Time (for Mass 1, scenario 5)
40
4.14
Displacement vs. Time (for Mass 2, scenario 5)
40
4.15
Displacement vs. Time (for Mass 3, scenario 5)
41
4.16
Displacement vs. Time (for Mass 1, scenario 6)
42
4.17
Displacement vs. Time (for Mass 2, scenario 6)
26
42
8
4.18
Displacement vs. Time (for Mass 3, scenario 6)
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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
1.0 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 spring pushing back, and stretching the spring will have it trying to pull back towards the start position, so any displacement along the ax is 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). 2002). The fundamental spring equation equ ation 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 end points 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 u nfixed 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
Ideally, one could assume that all vibrating s ystems 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
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a fluid, or by internal friction between the molecules of a seemingly elastic body. These Th ese 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:
̈ ̇
m + c + k = 0 The equation of motion (E.O.M) for Coulomb damped free vibration is given by:
̈ = 0
m
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 magnitude P = = P m sinwf t, t, the E.O.M becomes:
̈ ̇
m + c + k = P = P m sinwf t 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 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 undergo free vibrations or vibrations or they may be subjected to forced to forced vibrations. vibrations. The vibrations are damped when when friction forces are present and un-damped otherwise. otherwise. The suspension system of an automobile, for example, con sists essentially of a spring and a shock absorber (damper), which will cause the body bod y of the car to undergo damped damp ed forced vibrations when the car is driven over an uneven road.
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Most vibrations in machines and structures are undesirable because o f the increased stresses and energy losses which accompany them. They should therefore be eliminated or reduced as much as possible by appropriate design. 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 ex tensive subject. In this project we will briefly look at a simple case of vibration – vibration – the the spring mass damper system, a one degree de gree freedom system of bodies. After a brief overlook of the simple system, we will take a complex case study – study – A 3 degree of f r eedom eedom sysytem sysytem
Simulation Tool: MATLAB ®
1.3
We need to see the performance p erformance of the system under various conditions without actuall y having to subject the real system to these conditions, cond itions, 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). 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 an d 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 wh ere 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
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1.3.1
i.
Application development, including graphical user interface building.
Why MATLAB®? MATLAB® is an interactive system whose basic data element is an array that does
not require dimensioning. This allows you to solve man y technical computing problems, especially those with matrix and vector formulations, in a fraction of the 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 ve rsions can be obtained from
their website or one can buy bu y the full version with license key also through their on line website. This is not the same with MATHEMATICA ® which is very similar to MATLAB®.
iv.
structure that mimic the MATLAB® possesses a rich library of functions and data structure properties of systems and also easily provides analytical 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 Jav a, 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 s ymbols to represent such calculations hence saving time. With SIMULINK, S IMULINK, a system can be constructed and tested easily by varying parameters with the output a vailable graphically and pictorially.
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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 a re graphical user interfaces. It 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" 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 m atrices 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). (API). This is a library library that allows writing C and FORTRAN programs that interact with MAT LAB. It includes facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and an d writing MAT-files.
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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 d issipation of energy. 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 bu ilt 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 ssystem ystem 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 v ery 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.
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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 ba sed 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 s ystem is defined as a set of equations that represents the dynamics of the system accurately or, at least, fairly well (Ogata, 2002). 2002). A mathematical model is not unique to a given system. A system may be represented in ma ny different ways and, therefore, may have many mathematical models, depending on one’s perspective.
The dynamics of many systems, whether they the y are mechanical, electrical, thermal, economic, biological, and so on, may be described in terms of differential equations. Such differential equations may be obtained by b y 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 b e better suited than other models (Ogata, 2002). 2002). For example, in optimal control co ntrol 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 s ynthesis purposes.
2.1.1
Modeling a spring mass damper system
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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). 2006). In the discrete model, the physical system is assumed to con sist 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. s ystem. 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 de grees 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, respectively. A point mass can have three th ree translational degrees of freedom while a rigid bod y 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 mode led either as a discrete- or lumpedparameter system with varying number of degrees o f freedom or as a continuous system s ystem with infinite number of degrees of freedom, as illustrated for a cantilever beam in Fig. 2.4.
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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 o f vibration. If the periodic variation of motion is not harmonic, the motion will be periodic. p eriodic. In this case, the periodic motion can be expressed as a sum of harmonic motions of different frequencies. If the time variation of t he displacement of the mass is arbitrary (nonperiodic), the motion is said to be n onperiodic. 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 c annot 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 me ans of determining the natural frequencies of mechanical and structural systems using analytical or experimental approaches.
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2.1.1.1
Figure 2.2
Two-degree-of-freedom system
Figure 2.3
Three-degree-of-freedom system
Single-degree-of-freedom Single-degree-of-f reedom 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). 1995).
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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 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:
̈ ̇ =
(2.1)
where the dots above x above x denote first and second derivatives respectively
Table 2.1
Significance of m, c, and k in Different Systems ( John 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)
system, Fig. 2.1a 2. Rotational spring-
Mass
Torsional damping
Torsional spring
Angular
mass-damper system,
moment of
constant (m.N.s/rad)
stiffness (m.N/rad)
displacement
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)
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2.1.1.2
Multi-degree-of-freedom Multi-degree-of-f reedom system
Most mechanical and structural systems have distributed mass, elasticity, and damping (John Wiley & Sons, 2006). 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 Ritz method, Galerkin method, and many others (Karnopp, 1994). 1994). In most cases, the number of degrees of freedom (n (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 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 v ibrations 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
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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 o ne in which the passive components c omponents are augmented by actuators that supply additional force. Besides these three types of suspension systems, a skyhook type damper has been b een considered in the early design of o f 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
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. P assive indicates that the suspension 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 t ype of damping element placed between the body and the wheels of the vehicle, such as hydraulic shock absorber.
Figure 2.4
Passive suspension system
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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.
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.
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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.
Figure 2.6
A low bandwidth or soft active suspension system
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Figure 2.7
2.3
A high bandwidth or stiff active suspension system
The Quarter Car Model
A quarter car model is a well-known model for simulating one-dimensional vehicle suspension performance. In its simplified form, form, the suspension consists of a spring of stiffness K and a damper with damping coefficient C. The spring sp ring performs the role of supporting the static weight of the vehicle while the damper d amper helps in dissipating the vibrational energy a nd 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 s ystem stability and passenger comfort. For an extremely stiff suspension, the system will be h ighly 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 p roved 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 p assive 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:
̅ = ∆ represents the real mass of the quarter car, composed by a nominal parameter and an uncertain one ∆ . where
Figure 2.9
Quarter car suspension
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2.4
The tuned mass damper
The use of tuned mass dampers da mpers (TMD) is another widely used passive vibration damping treatment. These devices are viscously damped 2n d order systems appended to a vibrating structure. Proper selection of the parameters of these appenda ges, tunes the TMD to one of o f 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 an y mode, including the first, and add considerable amount of damping to it. Another An other 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 k inetic 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 ver y 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 Taipei- 101’s tuned mass damper (top) and its placement in the building (bottom)
CHAPTER 3 3.0
METHODOLOGY
3.1
one degree of freedom spring mass damper system
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If we take an ordinary spring that resists compression as well as ex tension 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 dete rmined. This motion is governed by Newton’s second law Mass x Acceleration =
̈ = Force
(1)
̈ x/d, where x(t)
Where “Force” is the resultant of all the forces the forces acting on the body. Here, = 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
. This causes an upward force in the spring given as =
amount
(2) (Hooke’s law)
= mg = 0. This is called static
This force balances the weight of the body, i.e. W + 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 Hooke ’s law, this causes an (additional) upward force spring such that
=
in the
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Figure 3.1
Damped spring mass
is a restoring force. It has the tendency tendenc y to restore the system, that is, to pull the bod y back to x = 0. If we connect the mass to a dashpot, we have to ta ke 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
= / of the body. This is ′
generally a good approximation, at least for small velocities. Thus, the damping force is of the form
= ′
(3)
c is called the damping constant. The resultant forces acting on the body now is
= ′
(4)
Hence, by Newton’s second law,
̈ = ′
(5)
This shows that the motion of the damped mechanical system is governed by the linear differential equation with constant coefficients
= 0 ̈ ′
̈ =
(6)
32
= = ̈ ̇ = 0 Or [ ] = 0 Where D=d/dt and = / ′
(7a) (7b)
Equation (7b) is an ordinary differential equation of the second order. Its characteristic equation is
= 0
(8)
Its roots are
, = ±
(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 ( ). Thus,
= 0 = = = =
= 2 = 2√ Damping ratio, ℶ = / = × = ℶ = 2ℶ
(10)
(11)
(12) (13)
33
Thus
, = ± ℶ = ℶ ± √ℶ 1
(14)
Hence, equation (7a) can be rewritten as
̈ 2ℶ 2ℶ̇ = 0
(15)
From Laplace transforms, we get
0− 0−2ℶ( 0−) = 0 (16) Mass position = 0− = (m) Mass velocity = = 0 (m/s) 2ℶ = 0 ′
′
(17)
When we solve by using Laplace, we obtain
ℶ = √ −ℶ sin√1ℶ −ℶ sin
(18)
This is the system’s response, i.e. displacement at any point in time, t. The system is
ℶ
ℶ
underdamped when <1, overdamped when >1 and critically damped when
3.2
ℶ =1.
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 k 1, 1, k 2, 2, k 3 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 2, 3.
subjected to the force F (t ) undergoes a displacement (t ),), i = 1, i
34
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. F2
F
F3
K2
K1
K3
m1
m2
m3
C2
C1
C3
X1
X2
Figure 3.2
3-degree-of-freedom system
We isolate the individual masses:
̇ ̇
m1
̇
̈
̇ , ̈ ,
Figure 3.3
m2
̇ 3 ̇
Forces acting on m 1
3 3 3̇ ̇ 3
̈
C4
X3
35
Figure 3.4
Forces acting on m 2
m3
43 4̇ 3
33 3̇ 3 ̇
3̈ 3 3, ̇ 3, ̈ 3 Figure 3.5
Forces acting on m 3
g overning the system. When we isolate each mass, ̈ ̇ = is the general equation governing we obtain the following E.O.M.
̈ ̇ ̇ ̇ = ̈ 3 ̇ ̇ 3̇ ̇ 3 = 3̈ 3 33 43 3̇ 3 ̇ 4̇ 3 = 3 3.3
Simulation
36
We must remember that computer language is garbage ga rbage 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 mod el 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 requ irements of complex tasks and good accuracy. ac curacy. 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 ex istence 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 con trol 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, Also, modern control theory is essentially a time-domain appoach, while conventional control theory is a complex frequency-domain approach.
̇ , ̇ , 3̇ , 4̇ , 5̇ , 6̇ in our three degree of freedom system using state space model theory where = , = ̇ , 3 = , 4 = ̇ , 5 =3 , 6 = 3̇ as follows:
So, we obtained equations for
37
̇ = ̇ = + + /3 /4 / 3̇ = 4 4̇ = / / + 3 + 4 3/5 3/6 / 5̇ = 6 6̇ = 3/33 3/34 + 5 + 6 3/3 Then, using values of masses 1, 2 , 3 as 6, 9, 5k g (respectively), (respectively), spring stiffness’s 1, 2, 3, 4 as 6, 7, 4, 1N/m (respectively), dampers 1, 2, 3, 4 as 1, 0.2, 0.2 , 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; m2 = 9; m3 = 5; k1 = 6; k2 = 7; k3 = 4; k4 = 1;
38
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) (((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 NO N-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))
figure(2) plot (t,y(:,3))
39
figure(3) plot (t,y(:,5))
We then varied some of the inputs inpu ts while keeping the others constant and generated different displacement-time graphs in order to observe the the system’s performance. system’s performance.
CHAPTER 4
40
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 inco rporated 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 a nd fro its original position for the first 40 – 40 – 50 50 seconds before the damping d amping 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 individu al damping and that of the whole system.
41
6 5.5
)
5 4.5 )t
m( y
1
(
4 ,t n e
m
e
3.5 c la si
p
3 D
2.5 2 1.5
0
20
40
Figure 4.1
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 1, scenario 1)
7.5 7 6.5 t(
(m)
)
6 2 n
,t
y
5.5 m
e e la
c
5 is
p D
4.5 4 3.5
0
20
40
Figure 4.2
60
80 100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 1)
42
8.5 8 7.5 )
7 m( )t ( 3 y
6.5 ,t n e m e
6 c al p si D
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
Displacement vs. Time (for Mass 3, scenario 1)
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 o f mass 3) is because of the overall damping effect of c4 on the whole system.
43
6 5.5 5 )
4.5 t)(
4 n
,t
y
1
m(
3.5 m
e al
c
e
3 p D
is
2.5 2 1.5 1
0
20
40
Figure 4.4
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 1, scenario 2)
7.5 7
)
6.5 6 )t
m( y
2
(
5.5 ,t n e
m
e
5 c al D
si
p
4.5 4 3.5 3
0
20
40
Figure 4.5
60
80 100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 2)
44
8.5 8 7.5 )
7 m( )t ( 3 y
6.5 ,t n e m e
6 c al p is D
5.5 5 4.5 4
0
20
40
Figure 4.6
4.1.3
Scenario 3-
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 3, scenario 2)
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 c ase in devices that require early damping of the vibration (e.g. me asuring instruments, racing cars etc.)
45
6
5.5
5 ) m( )t y
1
(
4.5 ,t n e
m
e
4 al
c p D
si
3.5
3
2.5 0
20
40
Figure 4.7
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 1, scenario 3)
7
6.5 ) (
)t
m(
6 y
2 t, n e m la
c
e
5.5 p is D
5
4.5
0
20
40
Figure 4.8
60
80
100 120 Time, Time, t(sec )
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 3)
46
8
7.5 ) m( )t (
7 3 y ,t n e m e c la
6.5 p si D
6
5.5
0
20
40
Figure 4.9
4.1.4
Scenario 4-
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 3, scenario 3)
K1=3, K2=2, K3=0, K4=1
In this case, we tried to see the effect e ffect of reducing the spring stiffness’s. 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 p rocess begins since k3=0 and k4=1.
47
6 5.5 5 t(
(m)
)
4.5 1 n
,t
y
4 m
e e al
c
3.5 si
p D
3 2.5 2
0
20
40
Figure 4.10
60
80 100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 1, scenario 4)
9.5
9
t(
(m)
)
8.5 y
2 ,t n e m al
c
e
8 p si D
7.5
7
0
20
40
Figure 4.11
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 4)
48
13 12.5 12 11.5 ) m( )t (
11 3 y ,t n
10.5 e m e c al
10 p is D
9.5 9 8.5 8
0
20
40
Figure 4.12
4.1.5
Scenario 5-
60
80
100 120 Time, t (sec)
140
160
180
200
Displacement vs. Time (for Mass 3, scenario 4)
m1=1, m2=3, m3=0.5
In this case, we tried to see the effect e ffect of reducing the masses. As can be b e 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 heav y the masses, the easier it is to control the vibrations.
49
6 5.5 5 t(
(m)
)
4.5 1 n
,t
y
4 m
e e la
c
3.5 si
p D
3 2.5 2
0
20
40
Figure 4.13
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 1, scenario 5)
7
6.5
6 ) (m) 2
t(
5.5 t,
y n e
m
e
5 la
c p D
is
4.5
4
3.5
0
20
40
Figure 4.14
60
80
100 120 Time, Time, t(sec )
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 5)
50
8
7.5 ) m( )t (
7 3 y t, n e m e c al
6.5 p is D
6
5.5
0
20
40
Figure 4.15
4.1.6
Scenario 6-
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 3, scenario 5)
F1=1, F2=1, F3=1
The effect of reducing the forces acting on the masses is observed in this sixth and an d 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 s ystem.
51
6 5 4 )t
m(
)
3 1
( n
,t
y
2 m
e e la
c
1 si
p D
0 -1 -2
0
20
40
60
80
100 120 Time, t(sec)
140
160
180
200
Figure 4.16 Displacement vs. Time (for Mass 1, scenario 6)
7 6 5 4 ) (
)t
m(
3 n
t,
y
2
2 m
e al
c
e
1 p D
si
0 -1 -2 -3
0
20
40
Figure 4.17
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 2, scenario 6)
52
8 7 6 5 ) m( )t (
4 3 y ,t n
3 e m e c la
2 p si D
1 0 -1 -2 0
20
40
Figure 4.18
60
80
100 120 Time, t(sec)
140
160
180
200
Displacement vs. Time (for Mass 3, scenario 6)
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 con ditions 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 – 1172 – 1174. 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 (2006). Mechanical Engineers’ Handbook: Materials and Mechanical Design, Volume 1, Third Edition. Pgs. 1204-1209. www.matlabcentral.com – www.matlabcentral.com – The The official MATLAB® website. The MathWorks Incorporated (2007) – (2007) – MATLAB MATLAB® product help. Ahmet Naci Mete, Sandip D Kulkarni, Michael Gerbracht, Noah Fehrenbacher (2005). “Quarter car model using a semi-active semi-active MRF damper”. Yahaya Md. Sam PhD. (2006). “ Robust Control of Active Suspension System for a quarter car model ”Project ”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)