ME163 Mechanical Vibrations (Winter 2009)
HW6 Due 2.19.09 in class
25% if late within 24 hours
50% if later than that
1.
Multiple Frequency Excitation.
Consider the a second order model of the transmission of sound
through a interior wall in your home:
m
e
¨
x
+
c
e
˙
x
+
k
e
x
=
f
(
t
)
where
x
(
t
) is the sound level in your room and
y
(
t
) is the sound level in your roommates room
(
m
e
= 10 kg,
k
e
= 100 N/m,
c
e
= 16 Ns/m). Your roommate is playing very loud music, so loud that
it is getting to the maximum limit that her speakers can handle. Because of this there is harmonic
distortion in the sound
f
(
t
) that is being transmitted through your wall. Assume that the distorted
sound is such that it looks like
f
(
t
) =
A
1
sin
ωt
+
A
2
sin 3
ωt
.
(a) Using superposition, Fnd the steadystate solution for the system. (Hint: Write the assumed
solution in the form
x
(
t
) =
X
1
f
1
(
ω, t, φ
1
) +
X
2
f
2
(3
ω, t, φ
2
), where
X
1
,
X
2
,
f
1
(
·
) and
f
2
(
·
) have
to be determined, there will be amplitudes and phases!)
(b) Total harmonic distortion is calculated by:
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 Winter '08
 Mezic,I
 Fourier Series, Dirac delta function, harmonic distortion

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