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Unformatted text preview: Solutions: Homework Set 7 1. Problem 5.1. Draw FBDs of the two disks. Consider rotation, due to external moments M 1 and M 2 and torques in the two shafts. Equations of motion: I 1 1 + GJ 1 L 1 + GJ 2 L 2 1 GJ 2 L 2 2 = M 1 I 2 2 GJ 2 L 2 1 + GJ 2 L 2 2 = M 2 These can be placed in matrix form: I 1 I 2 1 2 + GJ 1 L 1 + GJ 2 L 2 GJ 2 L 2 GJ 2 L 2 GJ 2 L 2 1 2 = M 1 M 2 2. Problem 5.12. For free vibration, with I 1 = I 2 = I , GJ 1 = GJ 2 = GJ , and L 1 = L 2 = L , EOMs become I 1 1 1 2 + GJ L 2 1 1 1 1 2 = The eigenvalue problem becomes 2 1 1 1 1 2 = 1 2 where = 2 IL GJ . Characteristic equation: 2  1 1 1 = 2 3 + 1 = 0 Eigenvalues: 1 = 3 5 2 , 2 = 3 + 5 2 . Natural frequencies: 1 = 0 . 6180 r GJ IL , 2 = 1 . 6180 r GJ IL . Modal vectors satisfy 2 i 1 1 1 i u 1 i u 2 i = and are u 1 = u 11 1 1 . 6180 , u 2 = u 12 1 . 6180 . 1 3. Problem 5.18. For free vibration, let ( t ) = U ( t ) in EOMs, and multiply EOMs by U T to diagonalize both matrices: U T I 1 1 U + U T GJ L 2 1 1 1 U = Solutions of the resulting modal EOMs are i = A i cos i t + B i sin i t , with i s found for Problem 5.12. Evaluating at t = 0: 1 (0) 2 (0) = 1 . 5 = U (0) = U A 1 A 2 so A 1 A 2 = U 1 1 (0)...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Staff

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