Category: Mathematics

The objective of this project is to develop a mathematical model for a vehicle, simulate the
response of the vehicle to the engine being shut off with MATLAB/Simulink, and design
appropriate stiffness values for the tire-and-wheel assembling.
Figure 1 shows the sketch of the side section of a vehicle. To simply the model, the following
assumptions are made:
(1) The entire mass of the system as concentrated at the center of gravity (c.g.).
(2) The input by the engine being shut off is modeled as an impulse moment applied to the
vehicle, which is 1500N*m;
(3) Only the motion of the vehicle in the x-y plane is considered. For the sake of
concentrating on the vibration characteristic of the vehicle, the rigid translation in the y
direction is ignored. So the motions of the vehicle in the x-y plane include the rotation in
the x-y plane (pitch) and up-and – down motion in the x direction (bounce).
(4) Each tire-and-wheel assembling is approximated as a simple spring-dashpot arrangement
as shown in Figure 1.
(5) All tire-and-wheel assembling in the vehicle are identical.
Figure 1 sketch of the side section of a vehicle
In the system, the total mass of the vehicle is 3500kg, the moment of inertia of the vehicle
around c.g. is J = mr2
(r2
= 0.64m2
), l1= 1.4m, and l2 = 0.9m, friction coefficient c for each tireand-wheel assembling is 2000N*s/m.
The steps in the design and simulation process should include (but would not be limited to):
(1) Draw the simplified physical model for the vehicle based on the above assumptions.
(2) Develop a mathematical model using the simplified physical model.
a. Draw the free-body diagram of the vehicle and write differential equations, inputoutput equations or state variable equations for the system.
b. The input is the impulse moment by the engine being shut off and the outputs are
the rotation and up-and-down motion in the x direction.
(3) Implement your mathematical model in a Simulink simulation and design appropriate
stiffness values for the tire-and-wheel assembling k with which the maximum amplitude
of the angular response is less than 0.08 radians and the maximum amplitude of the upand-down motion in the x direction is less than 0.02m.
a. Using the block diagram to simulate the response of the vehicle (both angular and
up-and- down motions) to the impulse moment by the engine being shut off.
b. Draw the block diagram using one of three different forms of equations
(differential equations, input-output equations, and state-variable equations).
c. Simulate the two responses (rotation and up-and-down motion) of the vehicle to
the impulse moment input with different values of k. Design appropriate k value
and display graphs of both the two responses in time domain.
(4) Write a final report describing your model and simulation. This is a formal report, which
should contain a clear and complete description of your physical system, the
mathematical model and the Simulink simulation.
The report must be typed and should contain, at a minimum, the following sections.
 Introduction. A description of the physical system. Use figures to assist your
description.
 Mathematical Model. A step-by-step description of the development of your
mathematical model.
 Simulink Simulation. A step-by-step description of the Simulink simulation and
design of the stiffness k. Discuss how the stiffness k affects the response of the system.
 Conclusions. Here you should describe the strengths and weaknesses of your
model and simulation. Describe any way in which the model could be improved.
This project is not like a homework problem, in that there is not one correct answer. You could
choose different block diagram models to do the simulation and design the stiffness value of k to
meet the requirement. You are expected to use your imagination in the design process and to try
to use all of the tools that you have learned. Also, you will be judged on quality of report as well
as quality of results.