MTH320
Mathematical Methods II
Tutor-Marked Assignment
July 2021 Presentation
MTH320
Tutor-Marked Assignment
TUTOR-MARKED ASSIGNMENT (TMA)
This assignment is worth 20% of the final mark for MTH320 Mathematical Methods II.
The cut-off date for this assignment is 28 October 2021 (Thursday), 23 55 hours.
Note to Students:
You are to include the following particulars in your submission: Course Code, Title of the
TMA, SUSS PI No., Your Name, and Submission Date.
For example, ABC123_TMA01_Sally001_TanMeiMeiSally (omit D/O, S/O). Use underscore
and not space.
Question 1
Figure Q1 below shows a (quarter) car chassis of mass m supported by a suspension system
where the spring has a stiffness of k N/m and the damper has a damping constant of c Ns/m.
The purpose of the suspension system is to isolate (to some extent) the road disturbances
imposed on the wheel/tire during the motion of the car. An example of such a mathematical
model of a (quarter) vehicular suspension has been demonstrated in Chapter 3 of SU3.
As a design engineer, you shall investigate the displacement response of the (quarter) chassis
to other types of road disturbances in this assignment. Assume that the wheel remains in contact
with the ground at all times.
Figure Q1 - A simplified model of a car suspension system (a quarter)
Source: Chegg.com
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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MTH320
(a)
Tutor-Marked Assignment
Suppose that the car is initially travelling at a constant velocity of v m/s with no
vertical movement of the car chassis. Then, at t = 0 (x = 0), the wheel encounters a rise
of the road surface described below (wheel not drawn to scale):
Figure Q1(a)
The road disturbance w(x) has the function:
.
By applying Newton’s 2nd Law to the mass m taking reference from a static
equilibrium position, show that the vertical displacement y(t) is governed by the IVP:
(10 marks)
(b)
Consider the suspension system to be critically damped with mass m = 400 kg and
spring stiffness k = 40,000 N/m. Hence, show that the IVP from Question 1(a) can be
specified as:
(5 marks)
(c)
Solve the IVP in Question 1 (b) using the method of undetermined coefficients.
Assume r = av ≠ 10.
(12 marks)
(d)
Let height h = 0.1 m, a = 1 and v = 5 m/s (18 km/h).
In one graph, plot the displacement functions for w(t) and y(t) solved in Question 1(c)
and explain your observations. What happens when a and/or v is increased? (High
marks are awarded for practical reasonings that connect well with the mathematical
findings.)
(10 marks)
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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MTH320
Tutor-Marked Assignment
(e)
Decompose the equation of motion in Question 1(b) into a system of ODEs and solve
for the displacement response y(t) again, this time using Duhamel’s principle. You need
not evaluate y’(t).
(20 marks)
(f)
In another scenario, suppose that the car is initially travelling at a constant velocity of
v m/s with no vertical movement of the car chassis. Then, at t = 0 (x = 0), the wheel
encounters a ramp of height h described below (wheel not drawn to scale):
Figure Q1(f)
The ramp w(x) has the function:
If the values of m, c and k are those defined in Question 1(b), show that the chassis
displacement y(t) is governed by the equation of motion:
where τ = b/v.
(8 marks)
(g)
Using Laplace transform, solve the equation of motion in Question 1(f) for y(t).
(15 marks)
(h)
Let the ramp height h = 0.1 m and length b = 1 m. Plot the displacement functions for
w(t) and y(t) solved in Question 1(g), for three values of the car velocity v given below.
Explain and discuss the effect of v on the displacement response y(t) when the car
crosses the ramp. (High marks are awarded for practical reasonings that connect well
with the mathematical findings.)
v1 = 1 m/s,
v2 = 2 m/s, v3 = 4 m/s
(10 marks)
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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MTH320
Tutor-Marked Assignment
The remaining 10 marks will be awarded based on the quality of report writing and
presentation, based on criteria such as typed or handwritten report, clarity and logic of
explanations, good use of charts/figures and spelling/grammatical errors etc.
Note that you will have to keep the size of your document to 4 MB or less upon submission.
---- END OF ASSIGNMENT ----
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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