ASE 365 - HW 7 Solutions

ASE 365 - HW 7 Solutions - Solutions: Homework Set 7 1....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

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.

Page1 / 5

ASE 365 - HW 7 Solutions - Solutions: Homework Set 7 1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online