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Unformatted text preview: ME163 Mechanical Vibrations (Winter 2009) HW-6 Due 2.20.09 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 fre- quencies through a interior wall in your home: m e ¨ x + c e ˙ x + k e x = f ( t ) (where 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 speaker is such that the forcing sound 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 ) + X 2 f 2 (3 ω,t ), where X 1 , X 2 , f 1 ( · ) and f 2 ( · ) have to be determined, there will be two phases!) The steady-state solution is given by: x ( t ) = X 1 sin ( ωt + φ 1 ) + X 2 sin (3 ωt + φ 2 ) , where X 1 = A 1 k radicalbig (1- Ω 2 ) 2 + (2 ζ Ω) 2 , X 2 = A 2 k radicalbig (1- 9Ω 2 ) 2 + (6 ζ Ω) 2 , φ 1 =- tan- 1 parenleftBig 2 ζ Ω 1- Ω 2 parenrightBig , φ 2 =- tan- 1 parenleftBig 6 ζ Ω 1- 9Ω 2 parenrightBig , with Ω = ω ω n . (b) The way harmonic distortion is measured is typically: THD = P 2 + P 3 + P 4 + ... + P n P 1 = A RMS , 2 + A RMS , 3 + A RMS , 4 + ... + A RMS ,n A RMS , 1 where P 1 is the signal power for the first harmonic, and P 2 ... P n are the powers in the higher order harmonics, and A RMS ,i are RMS signals strengths ( A RMS = 1 √ 2 A Peak ). In our case we)....
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- Winter '08
- Fourier Series, Sin, Dirac delta function, Ωn