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 α . 3. Determine the beat period for the free response of the system corresponding to α << 1. T = 1 2 (3 m ) ˙ x 2 1 + 1 2 I C ˙ θ 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 ( αk )( x 2- x 1 ) 2 + 1 2 ( k ) x 2 2 = 1 2 (2 + α ) kx 2 1- αkx 1 x 2 + 1 2 (1 + α ) kx 2 2 Therefore, [ K ] = ∂ 2 U ∂x i ∂x j = k 2 + α- α- α 1 + α Eigenvalue problem:- 2 μ + 2 + α- α- α- μ + 1 + α X 1 X 2 = where μ = (3 m/ 2 k ) ω 2 . Natural frequencies: 0 =- 2 μ + 2 + α- α- α- μ + 1 + α = (- 2 μ + 2 + α )(- μ + 1 + α )- α 2 = 2 μ 2- (4 + 3 α ) μ + 2 + 3 α Therefore, μ 1 , 2...
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hwk06_soln - Homework 6.1 For this problem 1 Derive the...

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