MAE 30B – Dynamics and Vibrations
Spring 2020
Final Exam
Name: ____________________
Student ID: __________________
Directions: All work must be shown completely and neatly to receive credit for a problem.
[25 points] Problem 1
A pendulum consists of a 2.5 kg slender rod AB and a 5 kg square plate. The square plate has an edge
length of 0.5 m. The slender rod has a length of 2.5 m. A 15 g bullet of velocity 1000 m/s is fired at the
plate with an angle of φ, as shown in figure a. The bullet becomes embedded into the bottom edge of the
plate, on the centerline of the pendulum, as shown in figure b. The square plate is initially at rest.
Consider φ, in degrees, to be equal to the last two numerical digits of your student PID divided by 2
plus 5. For example, if your PID is A18752350, then φ=
50
2
+ 5 = 𝟑𝟎°.
Determine:
a) The angular velocity of the square plate just after the bullet becomes embedded into its edge.
b) The maximum angle θ that the pendulum will swing up when it stops. Find θ relative to a
vertical line going through the pendulum’s starting centerline (Refer to figure c).
[30 points] Problem 2
(Part 1 – 10 points)
The slender uniform rod of mass m is attached to two springs as shown in the figure. At the the vertical
position, the springs are at their natural length (unstretched length). The springs have constants of 𝑛 ∙ 𝑘
and 𝑘 as shown in the figure. Determine the natural frequency of the vibration in terms of the parameters
given (m, k, l, g).
Consider n to be equal to the last numerical digit of your student PID plus 1. For example, if your PID
is A18752350, then 𝒏 = 0 + 1 = 𝟏.
Problem 2 (Part 2 – 20 points)
The block-spring-damper system shown in the figure, has a mass of m = 9 kg, and a damping coefficient
of 𝒄 = 60 N ∙ s/m. The surface is smooth.
Consider spring stiffness k to be equal to the first three numerical digits of your student PID plus 200.
For example, if your PID is A18752350, then 𝒌 = 187 + 200 = 387 N/m.
x
F
m
k
c
The block is initially at rest when an initial velocity of 0.7 m/s is given to it. The system vibrates freely
after the initial velocity, with F = 0 (i.e. free vibration).
Determine
a) if the system overdamped or underdamped
b) the position of the block as a function of time for this free vibration (find the necessary constants)
The same mass-spring-damper system is now subjected to a harmonic force of 𝑭 = (𝟑𝟎 𝐬𝐢𝐧 𝟓. 𝟓𝒕)N. For
this part you may disregard the transient/free vibration response of the system. Determine:
c) the steady-state response of the mass (the position of the block as a function of time).
d) the maximum acceleration and maximum linear momentum of the block in this forced vibration
[25 points] Problem 3
The bent rod ABC shown in the figure has a uniform mass distribution of 1 kg/m and is initially kept at
rest by a cord attached to the wall.
Consider that the length of the rod AB, 𝒉𝑨𝑩 , is equal to the last digit of your student PID number
divided by 10 + 1. For example, if your PID is A14567851, then 𝒉𝑨𝑩 = 𝟏⁄𝟏𝟎 + 𝟏 = 𝟏. 𝟏 𝐦. The
length of the rod BC, 𝒉𝑩𝑪 , is equal to 1.5 m
At the instant that the cord is cut, determine:
a) The moment of inertia of the entire bent rod ABC with respect to point A (remember to use the
parallel axes theorem)
b) The angular acceleration of the bent rod ABC
c) The reaction forces at point A
(Hint for part c: find the location of the mass center for the bent rod ABC)
[20 points] Problem 4
Block 𝐷 of the linkage mechanism, as shown in the figure, is confined to move within the slot 𝐴𝐵. If the
bar 𝐶𝐷 is rotating at a constant rate of 𝜔𝐶𝐷 = 4 rad/s, determine the angular velocity and angular
acceleration of member 𝐴𝐵 at the instant shown (i.e. 𝐶𝐷 is horizontal).
[5 points] Bonus Problem
A cart with mass of m = 5 kg has a velocity 𝑣cart = 30 m/s just before it strikes the mass M, which is
attached to a spring of constant 𝑘 = 1500 N/m and a damper with coefficient 𝑐 = 200 N ∙ s/m. After the
collision, the two masses, m and M, couple with each other (become one mass).
𝑀 = 𝒏 ∙ 𝑚 where n is equal to the last numerical digit of your student PID plus 9. For example, if
your PID is A18752350, then 𝒏 = 0 + 9 = 𝟗. Therefor 𝑀 = 9 ∙ 5 = 45 kg
Determine the equation for the position of the coupled masses as a function of time after the coupling.
vcart
m
k
M
c