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hw2_spring_09_soln

# hw2_spring_09_soln - EML 6267 Assignment#2 Spring 2009...

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EML 6267 Assignment #2 Spring 2009 Please remember to put your name on your assignment. Due: 1/28/09 1. (20 pts.) Consider a 2DOF lumped parameter, chain-type model undergoing forced vibration. k 1 = 2x10 5 N/m k 2 = 5.5x10 4 c 1 = 60 N-s/m c 2 = 16.5 m 1 = 2.5 kg m 2 = 1.2 a. (5 pts.) Show that proportional damping exists. = 2 . 1 0 0 5 . 2 m = 5 . 16 5 . 16 5 . 16 5 . 76 c × × × × = 4 4 4 5 10 5 . 5 10 5 . 5 10 5 . 5 10 55 . 2 k For proportional damping, [ ] [ ] [ ] k m c β α + = . True if β = 3x10 -4 , α = 0. b. (5 pts.) Determine the natural frequencies and eigenvectors if a force F 1 = F 0 e i ω t is acting on coordinate x 1 of the system. 57 . 177 1 = n ω rad/s 04 . 341 2 = n ω rad/s = 2032 . 3 1 1 ψ = 65045 . 0 1 2 ψ c. (10 pts.) Determine the FRFs Q 1 /R 1 , Q 2 /R 2 , X 2 /F 1 and X 1 /F 1 . Express them mathematically and plot the Real and Imaginary parts of these FRFs (please include your computer code, use a frequency range of 0 to 100 Hz, and use units of m/N for your vertical axes). Use the standard form for Q 1 /R 1 , Q 2 /R 2 , where ζ q1 = 0.0266, ζ q2 = 0.0512, k q11 = 4.6697x10 5 N/m and k q22 = 3.4982x10 5 . X 2 /F 1 = p 1 * Q 1 /R 1 + p 2 * Q 2 /R 2 X 1 /F 1 = Q 1 /R 1 + Q 2 /R 2

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0 10 20 30 40 50 60 70 80 90 100 -2 -1 0 1 2 x 10 -5 Real (m/N) 0 10 20 30 40 50 60 70 80 90 100 -4 -3 -2 -1 0 x 10 -5 Frequency (Hz) Imag (m/N) Q 1 /R 1 Q 2 /R 2 0 10 20 30 40 50 60 70 80 90 100 -5 0 5 x 10 -5 Real (m/N) 0 10 20 30 40 50 60 70 80 90 100 -10 -5 0 x 10 -5 Frequency (Hz) Imag (m/N) X 1 /F 1 X 2 /F 1
2. (20 pts.) Determine the [m], [c], and [k] matrices in local coordinates for the model shown below. Hint: your matrices should be symmetric. The equations of motion from the free body diagrams are: ( ) 0 3 3 2 2 1 3 2 1 1 1 1 1 = + + + + x k x k x k k k x c x m & & & ( ) 0 3 4 1 2 3 2 2 4 2 2 2 2 2 = + + + x k x k x c x k k x c x m & & & & ( ) ( ) 0 2 4 1 3 2 2 3 5 4 3 3 3 2 3 3 = + + + + + x k x k x c x k k k x c c x m & & & & The corresponding matrices are: = 3 2 1 0 0 0 0 0 0 m m m m + = 3 2 2 2 2 1 0 0 0 0 c c c c c c c m 2 k 1 x 1 x 2 c 1 m 1 c 2 m 3 x 3 k 2 k 3 k 4 k 5 c 3 x 1 m 1 k 1 x 1 1 1 x c & 1 1 x m & & k 1 ( x 1 - x

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