Assignment 2 (Free vibration and Forced vibration)
1. A boy riding a bicycle can be modelled as a spring-mass-damper system with an
equivalent weight, stiffness, and damping constant of 800 N, 50,000 N/m, and 1,000
N-s/m, respectively. The differential setting of the concrete blocks on the road caused
the level surface to decrease suddenly, as indicated in Fig. 2.109. If the speed of the
bicycle is 5m/s (18 km/hr), determine the displacement of the boy in the vertical
direction. Assume that the bicycle is free of vertical vibration before encountering the
step change in the vertical displacement.
2. Derive the equation of motion and find the steady-state response of the system shown
in the below Figure for rotational motion about the hinge O for the following data
k1=5000 N/m; k2=5000 N/m, a=0.25 m, b=0.5 m, l=1m, M=50 kg, m=10 kg, 𝐹0 =500
N, 𝜔 = 1000 𝑟𝑝𝑚
3. A video camera, of mass 2.0 kg, is mounted on the top of a bank building for
surveillance. The video camera is fixed at one end of a tubular aluminium rod whose
other end is fixed to the building as shown in Fig. 3.50. The wind-induced force acting
on the video camera, f(t), is found to be harmonic with f(t)=25 cos(75.398t) N.
Determine the cross-sectional dimensions of the aluminum tube if the maximum
amplitude of vibration of the video camera is to be limited to 0.005 m.
4. A single story building frame is subjected to a harmonic ground acceleration as shown
in Figure below. Find the steady state motion of the floor (mass m). Find the horizontal
displacement of the floor mass m of the building frame when the ground acceleration
is given by 𝑥̈𝑔 =100 sin(𝜔𝑡) mm/sec. Assume m=2000 kg, k=0.1 MN/m, 𝜔 =
25 𝑟𝑎𝑑/𝑠, and 𝑥𝑔 (𝑡 = 0) = 𝑥̇𝑔 (𝑡 = 0) = 𝑥(𝑡 = 0) = 𝑥̇ (𝑡 = 0) = 0
5. An automobile is modelled as a single-degree-of-freedom system vibrating in the
vertical direction. It is driven along a road whose elevation varies sinusoidally. The
distance from peak to trough is 0.2 m and the distance along the road between the
peaks is 35 m. If the natural frequency of the automobile is 2 Hz and the damping ratio
of the shock absorbers is 0.15, determine the amplitude of vibration of the automobile
at a speed of 60 km/hour. If the speed of the automobile is varied, find the most
unfavourable speed for the passengers.
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