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

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.

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

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)