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hw1_spring_09_soln - EML 6267 Assignment#1 Spring 2009...

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EML 6267 Assignment #1 Spring 2009 Please remember to put your name on your assignment. Due: 1/21/09 1. (25 pts.) For a given single degree-of-freedom (SDOF) lumped parameter system (undamped), the free vibration, x ( t ), is shown below. a) (20 pts.) If the motion was initiated with a velocity of 2 π mm/s and zero initial displacement, express the motion in the forms show below (use units of mm): i) (5 pts.) x ( t ) = A cos( ω n t ) + B sin( ω n t ) Ans. ( ) ( ) ( ) ( ) ( ) ( ) B x A x t B t A x t B t A x n n n n n n n ω π ω ω ω ω ω ω = = = = + = + = 2 0 0 0 cos sin sin cos & & From the plot, the period of the signal is 0.1 s, so the frequency in Hz is 1/0.1 = 10 Hz. Therefore, ω n = 2 π *10 rad/s. You can now solve for B = 2 π / ω n = 0.1. ( ) t x 10 2 sin 1 . 0 = π mm 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Time (s) x (mm) HW 1, problem 1
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This matches the plot (zero phase shift sine wave with amplitude of 0.1 mm). ii) (5 pts.) x ( t ) = C cos( ω n t + Φ c ) Ans. ( ) ( ) ( ) ( ) ( ) ( ) c n c c n n c n C x C x t C x t C x φ ω π φ φ ω ω φ ω sin 2 0 cos 0 0 sin cos = = = = + = + = & & Initial conditions satisfied for C = B and φ c = 270 deg = -90 deg. = + = 2 10 2 cos 1 . 0 2 3 10 2 cos 1 . 0 π π π π t t x mm Note the 90 deg phase shift between sin and cos. iii) (5 pts.) x ( t ) = C sin( ω n t + Φ s ) Ans. Same as part i). ( ) t x 10 2 sin 1 . 0 = π mm iv) (5 pts.) x ( t ) = De Ee i t i t n n ( ) ( ) ω ω + . Ans. ( ) ( ) ( ) ( ) ( ) ( ) ( ) E D i x E D x Ee i De i x Ee De x n t i n t i n t i t i n n n n = = + = = = + = ω π ω ω ω ω ω ω 2 0 0 0 & & Multiply x (0) expression by i ω n and sum with initial velocity condition. Solve for D = π /i ω n , then E = -D . Substitute ω n = 2 π *10 to obtain: ( ) ( ) t i t i e i e i x 10 2 10 2 20 20 + = π π mm b) (5 pts.) Given your expression for x ( t ) in part a-i), write an expression for the velocity, dx ( t )/ dt , (mm/s). Plot your result over the same time interval (0 to 0.5 s) shown above. Verify
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