final_soln_cover

final_soln_cover - ME 563 – Fall 2011 Final Exam Grade...

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Unformatted text preview: ME 563 – Fall 2011 Final Exam Grade distribution (in %) 97.5 97.5 97.5 97.5 96.3 95.0 93.8 92.5 92.5 90.0 90.0 88.8 88.8 88.8 87.5 85.0 85.0 85.0 85.0 83.8 80.0 78.8 78.8 77.5 76.3 75.0 73.8 72.5 70.0 68.8 68.8 68.8 58.8 58.8 45.0 43.8 42.5 35.0 y t ( ) = y sin ! t x R k no slip c m O Problem No. 1 – vibration isolation Consider the base-excited, single-DOF system shown below. Assume the disk to be homogeneous with its center of mass at O and with its centroidal mass moment of inertia being I O = 1 2 mR 2 . The spring is unstretched when x = y = . Determine the range of values for the stiffness k for which the amplitude of steady- state response, x P t ( ) , is less than y . Express your answer in terms of, at most, the parameters m, c, k and . SOLUTION T = 1 2 I C ! ! 2 = 1 2 3 mR 2 2 " # $ % & ' ! x R " # $ % & ' 2 = 3 4 m ! x 2 U = 1 2 k x ! y ( ) 2 R = 1 2 c ! x ! ! y ( ) 2 EOM: 3 2 m !! x + c ! x + kx = c ! y + ky = c ! y cos ! t + k y sin ! t or, !! x + 2 !" n ! x + " n 2 x = 2 !" n y cos # t + " n 2 y sin # t where ! n ¡ = ¡ k 3 m . From lecture: x p t ( ) = 1 + 2 ! " / # n ( ) 2 1 $ " / # n ( ) 2 % & ' ( ) * 2 + 2 ! " / # n % & ( ) 2 y sin " t $ + ( ) = T " ( ) y sin " t $ + ( ) From lecture we know that T ! ( ) < 1 for ! > 2 " n , or: 2 ¡ n 2 = 4 k 3 m < ¢ 2 £ k < 3 4 m ¢ 2 ! Problem No. 2 – vibration absorption An undamped three-DOF system has the following EOMs in terms of an unknown stiffness variable K: M ! " # $ ! "" x + K ! " # $ ! x = ! f sin % t where M ! " # $ = 20 10 30 ! " % % % # $ & & & kg K ! " # $ = 300 % 200 % 200 200 + K % K % K 100 + K !...
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final_soln_cover - ME 563 – Fall 2011 Final Exam Grade...

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