ME163 Mechanical Vibrations (Winter 2009)
HW8 Due Wednesday 4.04.09 7pm
(in the box or to a TA)
Answers on web by 9pm, midterm the day after
1.
2DOF Modeling.
Consider the system shown in Figure 1 which is a crude model of a coal cart
with a scoop attached to its side. The cart has a mass
M
and is propelled along a horizontal track
by a force
f
. The scoop hangs o± the side and is actuated by a torque T (the scoop has a total mass
m
and moment of inertia
I
about the center of mass
C
). Obtain the equations of motion using
x
1
and
θ
as coordinates and
f
and
T
as the inputs. Consider gravity.
T
M
f
g
m
C
ϑ
a
y
x
1
x
2
Figure 1: Coal cart with attached scoop.
2.
2DOF Oscillator.
Consider the system in Figure 2. Suppose
k
1
= 10
4
N/m,
k
2
=
k
3
= 2
k
1
, and
that
m
1
=
m
2
= 20 kg (there is no gravity).
(a) Derive equations of motion. Write down the mass and sti±ness matrix.
(b) Consider
f
(
t
) = 0. Calculate the frequencies
ω
1
,
ω
2
without using a computer. Also, calculate
mode shapes and draw them.
(c) Now, consider
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 Winter '08
 Mezic,I
 Mass, Frequency, Mode shape

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