This document analyzes the suspension system of an automobile modeled as a two-degree-of-freedom spring-mass-damper system. Equations of motion were derived using Lagrange's equations and modeled in SIMULINK. Natural frequencies were found to be 5.1 rad/s and 6.5 rad/s. Increasing damping reduced bounce by 3x10-3 m to 1x10-3 m and pitch by 5x10-4 m to less than 1x10-4 m. MATLAB modal analysis verified results and natural frequencies were compared.
The report was done to develop a MATLAB model of a suspension system of a 4 wheel vehicle going over an uneven road. A simulink model was developed to simulate the various forces acting on the suspension of the vehicle.
This is Part 2 of a 10 Part Series in Automotive Dynamics and Design, with an emphasis on Mass Properties. This series was intended to constitute the basis of a semester long course on the subject.
The report was done to develop a MATLAB model of a suspension system of a 4 wheel vehicle going over an uneven road. A simulink model was developed to simulate the various forces acting on the suspension of the vehicle.
This is Part 2 of a 10 Part Series in Automotive Dynamics and Design, with an emphasis on Mass Properties. This series was intended to constitute the basis of a semester long course on the subject.
Simulation methods to assess the brake noise behavior of passenger vehicle br...Altair
An important aspect of vehicle comfort is the silence of the brake system. Especially low velocity braking maneuvers are sometimes accompanied by heavy noise occurrences. Although these noise occurrences do not affect the brake system negatively with respect to braking performance they mostly result in customer insecurity. Therefore, most OEMs invest much time and money on developing silent brake systems. The presentation will illustrate the physical reasons behind such brake noise events. Practical methods for vibration damping will be discussed as well. Different analytical approaches for early estimation of noise potentials of brake systems based on the Finite-Element-Method as well as Multi-Body-Dynamics will be presented. Finally, future steps for the investigation of brake noise occurrences by analytical methods will be offered.
Speakers
Daniel Scharding, Adam Opel AG
MODELLING SIMULATION AND CONTROL OF AN ACTIVE SUSPENSION SYSTEM IAEME Publication
Conventional passive suspension systems lag in providing the optimum level of performance. Passive suspensions are a trade-off between the conflicting demands of comfort and control. An active suspension system provides both comfort and control along with active roll and pitch control during cornering and braking. Thus it gives a ride that is level and bump free over an incredibly rough terrain. This paper is a review the active suspension system and the modelling, simulation and control of an active suspension system in MATLAB/Simulink. The performance of the system is
determined by computer simulation in MATLAB/Simulink. The performance of the system can be controlled and improved by proper tuning a proportional-integral-derivative (PID) controller.
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
Solution manual for the finite element method in engineering, fifth edition ...physicsbook
https://unihelp.xyz/solution-manual-finite-element-method-in-engineering-rao/
Solution Manual for The Finite Element Method in Engineering – Fifth Edition
This manual cover the chapters 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 of the text.
Design, Development & Analysis of Loopwheel TechnologyABHISHEKPUND
In today’s world, Bicycles are the most favorite choice when it comes to causes like health, pollution, and the environment. Researches have been done in order to make the ride comfortable. This undertaking report introduces the Loop wheel. The purpose of our project was to reduce shocks on uneven roads, improve shock Absorption & take a smooth ride. Loop Wheel is a suspension system, Built to Experience a smooth ride on uneven roads by reducing shocks! So we replaced Spokes by 3 carbon springs. If we are riding on uneven roads, the spring can move in between Hub and Rim. As it's gone past a bump or bad road then the spring which is been touched to the surface will get compressed and others get to expand! So the whole impact power gets distribute in the wheel and the rider will feel nothing about that impact.
Longitudinal Vehicle Dynamics
-Maximum tractive effort of two-axle and track-semitrailer vehicles.
-The braking force of a two-axle vehicle.
-Acceleration time and distance.
-Relationship between engine torque and thrust force.
-Relationship between engine speed and vehicle speed
Electromagnetic Braking System using Ultrasonic SensorAI Publications
The concept of our paper is based on the automatic braking system. As we know that most time the accident takes place due to loss of control, drunk & drive and much more breaking reason.In our system there is a sensor that detects the speed of movement of the vehicle and an ultrasonic sensor, that sense the distance of the object in front of the vehicle. The sensor system will sense the speed of the vehicle or object in the front while the microcontroller calculates the distance required to stop the moving vehicle. The whole system is automatic and the braking application will take place without any manual input therefore it will decrease the rate of error hence the rate of road accidents.
A short presentation on Aerodynamic optimization techniques in design of Formula 1 cars.
The major performance gain in a Formula 1 car is its Aerodynamic performance, as
engine and mechanical tweaks to the car only provide marginal gains. Thus, it has become
the key to success in this sport, resulting in teams spending millions of dollars on research
and development in this field each year. Even though the aerodynamics in formula 1 is at an
advanced stage, there is potential for further development. With the under-body
aerodynamics banned by the FIA, the only significant changes that can be made to improve
the aerodynamic performance of the car are by modifying the front and rear wings crosssections,
i.e. airfoils, or by developing new diffuser to modify the air flow underneath the car.
Design of airfoil is one of the important factors to consider while designing the car. Design of
the optimum airfoils is track-dependent since each track has different aerodynamic
requirements. The development of the F1 car is regulated by the rules sanctioned by the FIA.
In recent years, the FIA has reduced the allowable operational hours for development at the
wind-tunnel by a F1 team. This study, focuses on the fundamentals of aerodynamics in a F1
car and the various techniques that were and are used by in the design of an F1 car by
different teams on the chassis, the drag reduction techniques on the front and rear wings etc.
for achieving the best lap times possible around a particular track. This will also effectively
focus the area of development in aerodynamics for the car and testing methods both software
and real time for evaluating the design tweaks.
Simulation methods to assess the brake noise behavior of passenger vehicle br...Altair
An important aspect of vehicle comfort is the silence of the brake system. Especially low velocity braking maneuvers are sometimes accompanied by heavy noise occurrences. Although these noise occurrences do not affect the brake system negatively with respect to braking performance they mostly result in customer insecurity. Therefore, most OEMs invest much time and money on developing silent brake systems. The presentation will illustrate the physical reasons behind such brake noise events. Practical methods for vibration damping will be discussed as well. Different analytical approaches for early estimation of noise potentials of brake systems based on the Finite-Element-Method as well as Multi-Body-Dynamics will be presented. Finally, future steps for the investigation of brake noise occurrences by analytical methods will be offered.
Speakers
Daniel Scharding, Adam Opel AG
MODELLING SIMULATION AND CONTROL OF AN ACTIVE SUSPENSION SYSTEM IAEME Publication
Conventional passive suspension systems lag in providing the optimum level of performance. Passive suspensions are a trade-off between the conflicting demands of comfort and control. An active suspension system provides both comfort and control along with active roll and pitch control during cornering and braking. Thus it gives a ride that is level and bump free over an incredibly rough terrain. This paper is a review the active suspension system and the modelling, simulation and control of an active suspension system in MATLAB/Simulink. The performance of the system is
determined by computer simulation in MATLAB/Simulink. The performance of the system can be controlled and improved by proper tuning a proportional-integral-derivative (PID) controller.
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
Solution manual for the finite element method in engineering, fifth edition ...physicsbook
https://unihelp.xyz/solution-manual-finite-element-method-in-engineering-rao/
Solution Manual for The Finite Element Method in Engineering – Fifth Edition
This manual cover the chapters 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 of the text.
Design, Development & Analysis of Loopwheel TechnologyABHISHEKPUND
In today’s world, Bicycles are the most favorite choice when it comes to causes like health, pollution, and the environment. Researches have been done in order to make the ride comfortable. This undertaking report introduces the Loop wheel. The purpose of our project was to reduce shocks on uneven roads, improve shock Absorption & take a smooth ride. Loop Wheel is a suspension system, Built to Experience a smooth ride on uneven roads by reducing shocks! So we replaced Spokes by 3 carbon springs. If we are riding on uneven roads, the spring can move in between Hub and Rim. As it's gone past a bump or bad road then the spring which is been touched to the surface will get compressed and others get to expand! So the whole impact power gets distribute in the wheel and the rider will feel nothing about that impact.
Longitudinal Vehicle Dynamics
-Maximum tractive effort of two-axle and track-semitrailer vehicles.
-The braking force of a two-axle vehicle.
-Acceleration time and distance.
-Relationship between engine torque and thrust force.
-Relationship between engine speed and vehicle speed
Electromagnetic Braking System using Ultrasonic SensorAI Publications
The concept of our paper is based on the automatic braking system. As we know that most time the accident takes place due to loss of control, drunk & drive and much more breaking reason.In our system there is a sensor that detects the speed of movement of the vehicle and an ultrasonic sensor, that sense the distance of the object in front of the vehicle. The sensor system will sense the speed of the vehicle or object in the front while the microcontroller calculates the distance required to stop the moving vehicle. The whole system is automatic and the braking application will take place without any manual input therefore it will decrease the rate of error hence the rate of road accidents.
A short presentation on Aerodynamic optimization techniques in design of Formula 1 cars.
The major performance gain in a Formula 1 car is its Aerodynamic performance, as
engine and mechanical tweaks to the car only provide marginal gains. Thus, it has become
the key to success in this sport, resulting in teams spending millions of dollars on research
and development in this field each year. Even though the aerodynamics in formula 1 is at an
advanced stage, there is potential for further development. With the under-body
aerodynamics banned by the FIA, the only significant changes that can be made to improve
the aerodynamic performance of the car are by modifying the front and rear wings crosssections,
i.e. airfoils, or by developing new diffuser to modify the air flow underneath the car.
Design of airfoil is one of the important factors to consider while designing the car. Design of
the optimum airfoils is track-dependent since each track has different aerodynamic
requirements. The development of the F1 car is regulated by the rules sanctioned by the FIA.
In recent years, the FIA has reduced the allowable operational hours for development at the
wind-tunnel by a F1 team. This study, focuses on the fundamentals of aerodynamics in a F1
car and the various techniques that were and are used by in the design of an F1 car by
different teams on the chassis, the drag reduction techniques on the front and rear wings etc.
for achieving the best lap times possible around a particular track. This will also effectively
focus the area of development in aerodynamics for the car and testing methods both software
and real time for evaluating the design tweaks.
Using real interpolation method for adaptive identification of nonlinear inve...IJECEIAES
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ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.
A fuzzy logic controllerfora two link functional manipulatorIJCNCJournal
This paper presents a new approach for designing a Fuzzy Logic Controller "FLC"for a dynamically multivariable nonlinear coupling system. The conventional controller with constant gains for different operating points may not be sufficient to guarantee satisfactory performance for Robot manipulator. The Fuzzy Logic Controller utilizes the error and the change of error as fuzzy linguistic inputs to regulate the system performance. The proposed controller have been developed to simulate the dynamic behavior of A
Two-Link Functional Manipulator. The new controller uses only the available information of the input-output for controlling the position and velocity of the robot axes of the motion of the end effectors
—This paper presents a new image based visual servoing (IBVS) control scheme for omnidirectional wheeled mobile robots with four swedish wheels. The contribution is the proposal of a scheme that consider the overall dynamic of the system; this means, we put together mechanical and electrical dynamics. The actuators are direct current (DC) motors, which imply that the system input signals are armature voltage applied to DC motors. In our control scheme the PD control law and eye-to-hand camera configuration are used to compute the armature voltages and to measure system states, respectively. Stability proof is performed via Lypunov direct method and LaSalle's invariance principle. Simulation and experimental results were performed in order to validate the theoretical proposal and to show the good performance of the posture errors. Keywords—IBVS, posture control, omnidirectional wheeled mobile robot, dynamic actuator, Lyapunov direct method.
Optimal FOPI-FOPD controller design for rotary inverted pendulum system using...TELKOMNIKA JOURNAL
The rotary inverted pendulum (RIP) has been used in various control application areas. This system can be represented as two degree of freedom (2-DOF), consisting of a rotating arm and rotating pendulum rod. RIP is an excellent example of designing a single-input multi-output (SIMO) system. Due to unstable RIP system dynamics, and its nonlinear model, multiple control techniques have been used to control this system. This paper uses integer and fractional order proportional integral-proportional derivative (PI-PD) controllers to stabilize the pendulum in the vertical direction. Constrained optimization approaches, such as the grey wolf optimization (GWO) methodology, are utilized to estimate the parametric values of the controllers. The simulation results showed that the fractional order PI-PD controller outperforms the integer order PI-PD controller with and without disturbance signal existence. A multiple results comparison has illustrated the superiority of fractional order controller over a previous work.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
Kinematics Analysis of Parallel Mechanism Based on Force Feedback DeviceIJRES Journal
Kinematic analysis of mechanism is the fundamental work of force feedback device research.The
composition of Delta mechanism based on Omega.7 force feedback device was illustrated in this paper.The
kinematic loop equations of Delta mechanism was established according to its geometric relationship,also the
inverse kinematics solution of Delta mechanism were obtained. And the numerical forward kinematics were
calculated by Newton iteration algorithm.Finally,The analysis of velocity and acceleration was carried out
through matrix operations.Kinematic analysis of Delta mechanism provides a theoretical basis for following
study.
Dynamics and control of a robotic arm having four linksAmin A. Mohammed
Abstract The manipulator control is an important problem
in robotics. To work out this problem, a correct dynamic
model for the robot manipulator must be in hand. Hence, this
work first presents the dynamic model of an existing 4-DOF
robot manipulator based on the Euler–Lagrange principle,
utilizing the body Jacobian of each link and the generalized
inertia matrix. Furthermore, essential properties of the
dynamic model are analyzed for the purpose of control. Then,
a PID controller is designed to control the position of the
robot by decoupling the dynamic model. To achieve a good
performance, the differential evolution algorithm is used for
the selection of parameters of the PID controller. Feedback
linearization scheme is also utilized for the position and trajectory
tracking control of the manipulator. The obtained
results reveal that the PID control coupled with the differential
evolution algorithm and the feedback linearization
control enhance the performance of the robotic manipulator.
It is also found out that increasing masses of manipulator
links do not affect the performance of the PID position control,
but higher control torques are required in these cases.
Keywords Robot control · PID · Differential evolution ·
Feedback linearization
Body travel performance improvement of space vehicle electromagnetic suspensi...Mustefa Jibril
Electromagnetic suspension system (EMS) is mostly used in the field of high-speed vehicle. In this paper, a space exploring vehicle quarter electromagnetic suspension system is modelled, designed and simulated using linear quadratic optimal control problem. Linear quadratic Gaussian and linear quadratic integral controllers are designed to improve the body travel of the vehicle using bump road profile. Comparison between the proposed controllers is done and a promising simulation result have been analyzed.
Comparative Analysis of Multiple Controllers for Semi-Active Suspension SystemPrashantkumar R
International Conference on Emerging Research in Computing, Information, Communication and Applications, ERCICA 2014, held on 01-02 Aug 2014 in Nitte Meenakhsi Institute of Technology, Bangalore, India
Paper ID: 600
1. Analysis of an Automobile Suspension
by
Derek Maxim
Hieu Nguyen
Ryan Parent
Eric Twiest
School of Engineering
Grand Valley State University
EGR 350 – Vibrations
Section A
Instructor: Dr. Ali Mohammazadeh
August 4, 2006
2. Introduction
Modeling the suspension of an automobile is of interest for many automotive and
vibrations engineers. Of importance for these engineers are the ride quality of the vehicle
traversing over broken roads and control of body motion. When traveling over rough terrain, the
vehicle exhibits bounce (up and down), pitch (rotation about the center of gravity along the
vehicle's length) and roll (rotation about the center of gravity along the vehicle's width) motions.
For this project, the bounce and pitch motion of the car over rough roads are of interest and will
be analyzed in this report.
Assumptions
For the analysis, it will be assumed that the vehicle is a rigid body with a suspension that
will be modeled as a two-degree-of-freedom (DOF) system. The setup of the suspension will
consist of equivalent springs in which the stiffness of the tire and the spring are combined, and
equivalent dampers that account for the shock absorber and the damping of the tire.
Theory
Figure 1 shows the two DOF system schematic that was used to determine the equations
of motion of the vehicle.
Figure 1: Spring-mass-damper model of the vehicle
3. To determine the equations of motion, Lagrange's equations, also known as the energy
method, were utilized. Equation (1) shows the general form of Lagrange's equations
(1)
where L is the sum of the kinetic and potential energies, or
(2)
where T is the kinetic energy, U is the potential energy of the system. The terms qi and Qi from
Eq. (1) represents a degree of freedom and the non-conservative work for each DOF (subscript i
denoting the first and second degrees of freedom); represents the derivative of qi.
To derive the equations of motion using Lagrange, the degrees of freedom i needs to be
defined. This is shown in Eqns. (3) and (4).
(3)
(4)
Next, the kinetic energy of the system is shown in Eq. (5),
(5)
where M is the mass of the body, is the bounce velocity of the body about its center of
gravity, J is the polar moment of inertia, and is the angular acceleration of the body.
The potential energy of the system is shown in Eq. (6)
(6)
where k1 and k2 are the equivalent spring rates of the front and rear suspension, xCG is the
displacement of the body's center of gravity, l1 and l2 are the distances from the center of gravity
4. to the front suspension and rear suspensions, and y1 and y2 are the input functions of the road for
the front and rear of the system.
Combining Eqs. (5) & (6) produces the energy equation, Eq. (7)
(7)
The equations for non-conservative work for both degrees of freedom are shown in Eqs.
(8) & ( 9)
(8)
(9)
where Q1 and Q2 are non-conservative work for q1 and q2, c1 and c2 are the damping coefficients
of the system and and are the time derivatives of the road input function.
Finally, taking the derivatives of the q terms and combining all of the equations into the
form of Eq. (1), the equations of motion for the system are
(10)
(11)
The parameters of the system are as follows: k1 = k2 = 30000 N/m, c1 = c2 = 3000 N*s/m,
M = 2000 kg, J = 2500 kg*m2, l1 = 1 m, and l2 = 1.5 m. Substituting these values and expanding
Eqs. (10) & (11) yields Eqs. (12) & (13)
(12)
(13)
The car is traveling at 13.88 m/s over road that is assumed to be sinusoidal in cross-
5. section with an amplitude of 10 millimeters (0.01 meters) and having a wavelength of 5 meters.
With this information, the input functions y1 and y2 are defined in Eqs. (14) & (15)
(14)
(15)
Where, t is the time traveled and π is the time shift that accounts for the time that it takes for the
rear suspension to negotiate the "bump" that the front suspension had negotiated.
Results
SIMULINK
The system was simulated using MATLAB's SIMULINK program. Figure 2 shows the
schematic that was used for analysis.
Figure 2: SIMULINK model of the two-degree-of-freedom system
The schematic shown in Figure 2 was used to determine the natural frequencies ω1 and ω2 of the
system. Using MATLAB, the modes of vibration, which are due to the system possessing two
different natural frequencies, were calculated to determine ω1 and ω2 in SIMULINK. From
MATLAB, the first and second modes of vibration were 0.477 and -0.596 (see MATLAB
6. results). Figures 3 and 4 show the plots produced by SIMULINK, which contains the natural
frequencies, and verify the MATLAB results. From Figures 3 and 4, the natural frequencies
were determined from the "Power Spectral Density" plots (middle graphs) and were ω1 = 5.1 rad/
s and ω2 = 6.5 rad/s.
Figure 3: SIMULINK plot results for the first mode of vibration showing the bounce (left graph)
and pitch response (right graph); Power Spectral Density graph used to determine natural
frequency ω1
Figure 4: SIMULINK plot results for the second mode of vibration showing bounce (left plot)
and pitch (right plot) response where natural frequency ω2 can be determined from the Power
Spectral Density graph
In addition, SIMULINK was used to model the response of the system to the road
conditions. Once road conditions were modeled, the SIMULINK model was modified using a
7. slider gain to reduce the pitch motion of the vehicle. Figures 5 and 6 show the response of the
system under the given car parameters and Figures 7 and 8 show the response when the gains on
the dampers in the system were modified to achieve the most desirable results. Comparing
Figures 5 and 6 to Figures 7 and 8 the figures, it was easy to see that by increasing damping gain
by a factor of 10, pitching motion decreases from 5x10-4 meters to less than 1x10-4 meter.
Bounce motion also decreases from 3x10-3 meters to 1x10-3 meters.
Figure 5: Bounce (left) and pitch motion (right) plot results for the unadjusted modeling of the
system under original conditions
Figure 6: SIMULINK model used to determine the response of the system
8. Figure 7: Bounce (left) and pitch motion (right) response plot results for the system with
higher viscosity (increased gain) dampers
Figure 8: SIMULINK model with slider gain block included to reduce the pitching motion of the
system
MATLAB
MATLAB, a mathematical processing software, was used to compare and verify the
model analyzed in SIMULINK. The program was also used to compare the responses of the
system using a function known as "lsim" and modal analysis. Attached at the end of this report
are the codes used to run lsim and the modal analysis.
Before the analysis of the system was performed using the lsim function, the modes and
9. natural frequencies of the system were determined. Figure 9 shows the plot of the modes
produced in MATLAB. From Figure 9, mode 1 is seen to have an oscillation of lower amplitude
than mode 2, which has an oscillation of higher amplitude. Using modal analysis, the
displacement degrees of freedom due to natural frequencies ω1 and ω2 were u1 = [-0.0197,
0.0094] meters and u2 = [0.0105, 0.0176] meters. From these results, it can be concluded that
mode 2 has a greater effect on the system than mode 1.
Figure 9: Plot of the modes of the system; mode 1 is shown to have an oscillation with a smaller
frequency than mode 2
To use the lsim function in MATLAB. To convert the equations into transfer functions,
the equations themselves must undergo a Laplace transformation. The generic equation for the
transfer function is shown in Eq. (16), whereas the specific transfer functions of the system, after
undergoing the Laplace transformation, are shown in Eqs. (17)-(20) (see Appendix A for
derivation of these equations).
(16)
(17)
(18)
(19)
10. (20)
With these transfer functions entered into MATLAB, the frequency response plots of the bounce
and pitching motion were created and are shown in Figures 10 and 11.
Figure 10: Bounce motion plot resulting from the analysis of the system using the lsim function
Figure 11: Pitching motion plot of the system resulting for the use of the lsim function
Comparing Figures 10 and 11 to Figure 5 (SIMULINK plot of the system), it can be seen that
both models show similar behavior to the road input, with small differences in amplitude. The
response of the front and rear suspensions to the road using lsim analysis are shown in Figures
12-13 and Figures 14-15. Figure 12 shows the front suspension response to bounce, Figure 13
11. shows the pitching response of the same suspension, Figure 14 shows the rear suspension
response to bounce, and Figure 15 shows the pitching motion response.
Figure 12: Front suspension response to bounce using the lsim function
Figure 13: Pitching response of the front suspension to the road input
Figure 14: Bounce response of the rear suspension to road input
12. Figure 15: Pitching response of the rear suspension to road input
Modal analysis was performed using MATLAB to compare the response of the system to
the lsim analysis and the matrices needed to perform the analysis can be seen in Appendix A.
However, it was not completed at the time of writing, so it cannot be proved in this report that
the response from the use of the lsim function is similar to the response resulting from modal
analysis. It is expected that the results would be similar, assuming that the matrices included in
this report from modal analysis were correct and the parameters and input functions were
transformed correctly.
Conclusions
Using MATLAB to model the suspension system (albeit simplified two-degree-of
freedom compared to a system that can be modeled with as much as ten degrees of freedom), it
was found that the suspension with front and rear spring rates of 30,000 Newton per meter, front
and rear dampers of a rate of 3,000 Newton-second per meter for a 2,000-kg vehicle quells the
excitation produced by the road in approximately 1.5 seconds. The second mode of vibration
was found to contribute the bounce and pitch motion of the vehicle more than the first mode of
vibration. The response of the system using modal analysis was also performed to verify the
response of the system. Though the eigenvalues and eigenvectors were determined using
MATLAB, unfortunately, the response of the system from the analysis was incomplete at the
time of writing.
SIMULINK was also used to model the suspension system and it was found to be within
agreement with the MATLAB model. Using the slider gain to increase or decrease the damping
rate on the SIMULINK model, it was found that by increasing the damping gain (and therefore
13. damping rate), the bounce and pitch motions of the vehicle decreased by a factor of
approximately 5 and 3, respectively.