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ME163 Mechanical Vibrations (Winter 2009)
Due Wednesday 2.4.09 7pm
(in the box or to a TA)
Worth 75 Extra Credit points
Answers emailed by 9pm, midterm the day after
1. Consider the system in Fgure below, with
m
eff
= 20 [kg], damping coe±cient
c
= 500 [Ns/m] and
spring sti²ness
k
= 2000 [N/m]. Suppose the system is forced by a sinusoidal force
f
(
t
) = 350cos
ωt
.
You may neglect gravity.
(a) Derive the equation of motion for the length of the spring (ie the distance between the mass
and the connection point for the spring/damper).
(b) Calculate the steady state response of the spring length at
ω
= 10 [rad/s] using phasors.
k
m
c
f
Let us call
F
d
the force of the damper, and
F
s
the force of the spring. ³urthermore, let us call
k
m
c
x_d
x_s
F_d
F_s
distances
x
d
and
x
s
as in Fgure, and let
x
s
0
be the unstretched length of the spring. Being damper
and spring in series, we have
F
d
=
−
F
s
. Then
m
(¨
x
d
+ ¨
x
s
) =
−
F
s
=
−
k
(
x
s
−
x
s
0
)
,
ME163 Mechanical Vibrations
1
HW4
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 Winter '08
 Mezic,I

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