Final Examination ISE 2142-212 Dynamics and Vibrations 2nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT
Name... ID...CR No……….
1
Problem 1 (10 pts, 30 min)
Calculate the response of a damped system to the periodic excitation f(t) depicted in the figure. The system damping ratio ζ is 0.1, the driving frequency is 1/4 of the system natural frequency, and the spring stiffness is k. Express f(t) as sum of harmonics by Fourier expansion first.
( ) ( )
( ) ( )
1
0
0
Fourier series
cos sin , 2
2
Fourier coefficients
2 cos , 0,1, 2,
2 sin , 1, 2, 3,
o
n o n o o
n
T
n o
T
n o
f t a a n t b n t
T
a f t n t dt n
T
b f t n t dt n
T
ω ω ω π
ω ω
∞
=
= + + =
= =
= =
∑
∫
∫
K K
Final Examination ISE 2142-212 Dynamics and Vibrations 2nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT
Name... ID...CR No……….
3
Problem 2 (10 pts, 30 min)
A vehicle traveling at a constant speed v in the horizontal direction encounters a road bump as shown.
Determine the response of the vehicle in the vertical direction.
Hint: This is a base excitation where the road bump y(t) can be written as a step function where 0
) (t =
y m for t <0 s and y(t)=0.05 m for t≥0 s,. Initial conditions are x0 = x&0 =0 so no need to find xh. Use convolution integral x t =
∫
t F τ h t−τ dτ t ≥0
0 when )
( ) ( )
( .
x m
k
Final Examination ISE 2142-212 Dynamics and Vibrations 2nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT
Name... ID...CR No……….
5
Problem 3 (20 pts, 60 min)
A system can be modeled as two blocks connecting with the spring. Each block has the mass m and the spring constant is k. Friction between the block and the ground can be neglected.
(a) Let x1 and x2 is the displacement of the blocks from the rest position. Suppose there is no external force applied to the system. Draw the free body diagram and derive the equations of motion of this system using Newton’s law. Arrange the equations into the standard matrix form;
0 Kx x
M&&+ = .
(b) Determine the natural frequencies and the mode shape of this system.
(c) If both blocks were initially at rest but the second one was displaced by x2o, determine the response of the system.
Hints: No need to normalize mode shapes.
Final Examination ISE 2142-212 Dynamics and Vibrations 2nd semester, 2007 International School of Engineering Chulalongkorn University NAV and PPT
Name... ID...CR No……….
7
Problem 4 (20 pts, 60 min)
In the figure shown below a vehicle is modeled as a two-DOF system with bounce and pitch motion.
Determine 1. EOM
2. natural frequencies and mode shapes
3. response by modal analysis after the engine has been shut off.
Hint: model it as an impulse moment applied to θ(t) and assume zero initial conditions. Reasonable parameters for the trucks are m=4000 kg, c1 =c2 =2000 Ns/m, k1 =k2 =20000 N/m, l1 =0.9 m,
4 .
2 =1
l m, and the radius of gyration r is 0.8 m. Icg =mr2
