hwk06_soln

hwk06_soln - Homework 6.1 For this problem: 1. Derive the...

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Unformatted text preview: Homework 6.1 For this problem: 1. Derive the mass and stiffness matrices for the system in terms of the absolute generalized coordinates x 1 and x 2 . Assume the disk to be homogeneous. 2. Determine the natural frequencies for the system. Leave your natural frequencies in terms of p k/m and . 3. Determine the beat period for the free response of the system corresponding to << 1. T = 1 2 (3 m ) x 2 1 + 1 2 I C 2 = 1 2 (3 m ) x 2 1 + 1 2 3 2 mR 2 x 2 R 2 = 1 2 (3 m ) x 2 1 + 1 2 3 2 m x 2 2 Therefore, [ M ] = 3 m 2 2 0 0 1 U = 1 2 (2 k ) x 2 1 + 1 2 ( k )( x 2- x 1 ) 2 + 1 2 ( k ) x 2 2 = 1 2 (2 + ) kx 2 1- kx 1 x 2 + 1 2 (1 + ) kx 2 2 Therefore, [ K ] = 2 U x i x j = k 2 + - - 1 + Eigenvalue problem:- 2 + 2 + - - - + 1 + X 1 X 2 = where = (3 m/ 2 k ) 2 . Natural frequencies: 0 =- 2 + 2 + - - - + 1 + = (- 2 + 2 + )(- + 1 + )- 2 = 2 2- (4 + 3 ) + 2 + 3 Therefore, 1 , 2...
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hwk06_soln - Homework 6.1 For this problem: 1. Derive the...

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