hw_6 - ME163 Mechanical Vibrations(Winter 2009 HW-6 Due...

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ME163 Mechanical Vibrations (Winter 2009) HW-6 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, find the steady-state 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: THD = P 2 + P 3 +
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