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14:650:443:B1 Vibration & Control
Homework 2
(Due Thursday, 06/11/2015 in class)
1. A 6-kg block is suspended from a spring with stiffness 300 N/m. The block is acted
upon by a vertical force 10sin(8 ) NF t= , where t is in seconds, g = 10 m/s2 (don't
forget gravity). At the beginning, the block is pulled down 0.3 m from the original
length of the spring and released from rest at t = 0. Determine the position of block
x(t). Suppose when the spring has its original length, the position of the block is x = 0
and positive direction is downward.
2. A uniform rod has a mass of m and length L. It is pinned at point O and connected
to two springs with stiffness k at the middle point. If it is acted upon by a harmonic
force at the end with 0 sinF F t= , determine the amplitude of the steady-state
vibration.
3. A mass-damper-spring system is acted upon by a constant force F = 20 N. Suppose
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when the spring has its original length, the position of mass is x = 0 and positive
direction is right. At t = 0, the mass is released from rest at x = 0.25 m. And its x(t)
plot is shown below. The unit for x is m, and for t is s.
Based on the information in the plot, determine the value for mass m, damping
coefficient c and spring stiffness k. The coordinate of the first peak (after t = 0) is
(0.893, 0.1564) and the coordinate of the fourth peak is (4.465, 0.0585). You may
need to find additional information from the plot.
4. A mass-damper-spring system has m = 10 kg, c = 80 Ns/m, k = 60 N/m. Suppose
when the spring has its original length, the position of mass is x = 0 and positive
direction is right. The block is displaced to x = 50 mm and released from rest,
determine the time required for it to return to the position x = 2 mm.
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5. An automobile's suspension system is critical to its response to input from the
uneven road surface. Figure 3.45 shows a drawing of a suspension system that
includes a coil spring, a shock absorber, and connecting links to the car frame and the
wheel. Next to this artist's rendering is a simple model that can be used to estimate the
vertical oscillations of the system. From a static state, the system is given an initial
velocity of v(0) = 15 m/s, while x(0) = 0. Calculate the response x(t) of the simplified
system for the following parameter values: m = 40 kg, k = 2.5 kN/m, and c = 2.4
kNs/m. There are no external forces acting on the wheel.
6. A mass-spring system rests on a rough surface with coefficient of kinetic friction
. Suppose when the spring has its original length, the position of mass is x = 0 and
positive direction is right. At t = 0, the mass is given an initial velocity of v(0) = 0.7
m/s, while x(0) = 0.
Now 0.02, 0.04, 0.08 = respectively. For each value of , use computer program
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to plot the x(t) curve from t = 0 until the time when the mass fully stops. (Print all the
3 graphs and your source codes)