midterm_1_solu

midterm_1_solu - Name ME 163 Mechanical Vibrations First...

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Name: ME 163 - Mechanical Vibrations First Midterm 50 points total, One 8.5 x 11” sheet of notes! Show all your work for full credit By signing your name you agree to the UCSB Honor Code Question 1 (17pts): Consider the differential equation of motion from a typical m,c,k vibratory system x + 3 . 2 ˙ x + 16 x = 20 cos(1 . 2 t ) Part A (2 pts): Calculate the natural frequency ω n . The natural frequency is radicalbig 16 / 4 = 2 . 0[ rad s ] Part B (2 pts): Calculate damping factor ζ . The damping factor is ζ = c 2 m = 3 . 2 / (2 * 4 * 2) = 0 . 2 Part C (2 pts): Calculate damped natural frequency ω d . The damped natural frequency is ω d = ω n radicalbig 1 - ζ 2 = 2 . 0 1 - 0 . 2 2 = 1 . 9596[ rad s ] Part D (4 pts): Figure 1 has a normalized transfer function plot of k | g ( ω ) | = k | X F 0 | on the left hand side and phase on the right hand side. Using this plot, determine the magnitude of the transfer function g ( ω ) at the forcing frequency 1 . 2 [ r s ]. Reading from the plot we get 1 . 463 and k from above is 16 so g (0 . 6) = 1 . 463 / 16 = 0 . 0914.
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