Candidates should answer BOTH compulsory questions.
Time Allowed: 1 hour, plus 15 minutes reading time during which notes may be made (on the
question paper only) but NO ANSWER MAY BE BEGUN.
Only calculators that conform to the list of models approved by the School of Engineering may
be used in this examination. The Engineering Data Book and standard graph paper will be
provided.
Read carefully the instructions on the answer book and make sure that the particulars required are
entered on each answer book.
1. Figure 1 shows a mechanical system with a mass M that is connected to a fixed surface via a
damper with co-efficient B. The input to the system is the force f(t) and the output of the system
is the velocity v(t) of the mass.
v(t)
B
f(t)
M
Figure 1
a. Draw a free-body diagram of the mass, showing the forces acting on the mass. Label the
diagram with the equations for each element. (4 marks)
b. Show that the output of the system (the velocity of the mass) is related to the input force by
the following differential equation:
𝑀 d𝑣 1
+ 𝑣 = 𝐵 𝑓(𝑡).
𝐵 d𝑡
(3 marks)
c. Derive a transfer function for the system in the Laplace domain with F(s) as the input and
V(s) as the output. Assume zero initial conditions. (4 marks)
d. With a step input to the system of magnitude h (at time t = 0) show that the system output is
as shown below (hint – you may wish to use partial fractions):
𝑡𝐵
ℎ
𝑣(𝑡) = 𝐵 [1 − 𝑒 − 𝑀 ].
(7 marks)
e. With particular values for the parameters of M = 1kg and B = 2 N s m-1 sketch the response
of the system to a step input of f(t) = 10N at time t = 0, assuming zero initial conditions.
Indicate the final value of the response, and also show the time at which the output response
has reached 63% of its final value. (7 marks)
Total 25 Marks
_______________________________________________________________________________
Continued
2
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