Description
Question 1 (3 marks)
Consider a mass m attached by a massless spring to a piston, as illustrated in Fig. 1. The attachment point
of the spring to the piston can be described by the co-ordinate X while the mass is described by co-ordinate
x.
(a) If the motion of the mass is subject to a damping force Fd = −mνx˙, show that it is described by the
second-order differential equation
x¨ + νx˙ + ω
2
0x = F0(t), (1)
where ω0 =
p
k/m and F0(t) = ω
2
0X(t).
(b) Assume the piston is initialized at X(0) = 0 and is driven according to,
X(t) = X0e
αtcos(ωt). (2)
Determine the particular solution of Eq. (1) in this case.
(c) What is the resonance frequency ωR? Does its existence depend on the sign of α?
Question 2 (3 marks)
When a car drives along a bumpy road, periodic ripples in the road surface can force the wheels to oscillate
on the suspension (e.g., springs).
(a) If the spacing between ripples on the road is about 2 m, at what speed will the cars suspension be
driven into resonance?
(b) Estimate a realistic damping constant provided by the shock absorbers so that the car’s suspension
does not catastrophically fall apart.
Note: This question has no unique quantitative solution. You will be marked primarily on your approach
and the application of physical principles. However, your answer should provide reasonable estimates for
relevant parameters to enable you to arrive at a concrete solution. Things to ponder include: How high are
the ripples in the road? How far does a car drop in height when 4 adults hop inside?
Question 3 (4 marks)
Consider the nonlinear damped-driven system
x¨ + (x
2 − x˙
2 − 1) ˙x + x = 0, (3)
that describes an harmonic oscillator with m = ω = 1.
(a) Find an expression for the change in energy E˙
.
(b) For the polar co-ordinates (r, θ) given by the transformation x = r cos(θ) and ˙x = r sin(θ), derive the
equations of motion:
r˙ = r(1 − r
2
) sin2
(θ) (4)
˙θ = (1 − r
2
) sin(θ) cos(θ) − 1. (5)
(c) Construct a rough phase portrait of the system in terms of the co-ordinates (x, x˙). Your diagram
should include at least three trajectories that characterize motion in the system, one of which is an
attractor. Hint: Look at your solutions in (a) and (b) for insight.