ME 163 Engineering Vibrations (W21)
February 10, 2021
1. The exam is open-book and open-notes.
2. You must complete the exam yourself. You are not allowed to search the answers online nor
consult a private tutor or anyone else in the class for help.
3. Please box important formulas and numerical results and ensure your solutions are legible.
4. The exam begins 2/10/2021 at 3:15 pm PT.
5. The exam ends 2/11/2021 at 3:15 pm PT.
6. The exam must be turned in via GauchoSpace before the end time.
Problem 1. Consider the spring-mass-damper system given in figure 1. Assume the mass is
m = 1 kg, the viscous damping coefficient c = 0.1Ns/m, the spring stifness is k = 1N/m. Note
that the forcing F (t) is NOT considered in parts (a)–(d).
(a) Write down the equations of motion of the system. (6 pts)
(b) What are the effective stiffness and damping of the system? (6 pts)
(c) Suppose we want to change the system so its damped natural frequency ωd is 1/2 of the
damped natural frequency obtained with the given stiffness and damping coefficients. Keeping
constant k, compute c needed to achieve this. (6 pts)
(d) With c = 0.1Ns/m and k = 1N/m, initial conditions x(0) = 1m, x˙(0) = 0, and F (t) = 0,
how many cycles will it take for the amplitude of oscillation to reach 0.01m? (6 pts)
(e) Assume a direct forcing F (t) = F0 sin(ωt), shown in figure 1, is going to be applied on the
system. Your engineer colleague tells you that they are worried about the response of the
system at high frequencies. You decide that to mitigate this, you will change the value of k
so that the peak response frequency, ωp (in rad/s), is 1/4 of the peak response frequency
obtained with the given values of c and k. Keeping constant c, what would k need to be to
achieve this? (6 pts)
Figure 1: Problem 1: Spring-mass-damper system.
Problem 2. Consider the cantilever with mass mb = 10 kg, effective stiffness kb = 10, 000 N/m
and an additional mass m = 20 kg added at length l/2, depicted in figure 2. Make the Rayleigh
approximation for the shape of the motion of the cantilever, the same as used in class. Note that
the vertical spring and damper in the figure are NOT considered in questions (a)-(d).
(a) Compute the kinetic energy of the system in terms of the displacement of the rightmost point
of the cantilever, ∆. (6 pts)
(b) What is the effective mass of the system? (6 pts)
(c) If we move the mass m to 2l/3, what is the change in the natural frequency of the beam, in
percentage? (6 pts)
(d) Assume the mass m is a point mass, i.e. you can place it anywhere on the beam from 0 to
l. What is the lowest natural frequency that this system can have if the cantilever mass mb
and its length l are kept constant, and what position of the mass m does it correspond to?
(e) Assume a vertical spring is added to the right end of the cantilever, and a damper at l/2. The
spring stiffness is k = 20, 000 N/m and the damping coefficient c = 34000 kg/s. What is
the damped angular frequency (i.e. the frequency of damped free oscillations) of the system
with the mass at l/2? (6 pts)
Figure 2: Cantilever with added mass. Left: Configuration for parts (a)-(d). Right: Configuration
for part (e).