Explanation Draw the free body diagram. Write the relation between the displacement of point A ( x A ) and the displacement of mass m 1 ( x 1 ) . x A = x 1 (I) Write the relation between the displacement of point B ( x B ) and the displacement of mass m 2 ( x 2 ) . x B = x 2 (II) Write the relation between the displacement of point A ( x A ) and the displacement of point B ( x B ) . x A a = x B b x 1 a = x 2 b x 2 = b a x 1 (III) Write the force balance in x -direction for mass m 1 . ∑ F x 1 = m 1 x ¨ 1 − F A − k 1 x 1 = m 1 x ¨ 1 F A = − ( m 1 x ¨ 1 + k 1 x 1 ) (IV) Here, the force at point A is F A , the spring constant of spring 1 is k 1 . Write the force balance in x -direction for mass m 2 . ∑ F x 2 = m 2 x ¨ 2 F B − k 2 x 2 = m 2 x ¨ 2 F A = m 2 x ¨ 2 + k 2 x 2 (V) Here, the force at point B is F B , the spring constant of spring 2 is k 2 . Write the expression obtained by taking moment about point O. ∑ M O = I O θ ¨ ( F A − c 1 x ˙ 1 ) a − ( F B + c 2 x ˙ 2 ) b = m 3 k O 2 ( x ¨ 1 a ) (VI) Here, the viscous damping coefficient of damper 1 is c 1 and the viscous damping coefficient of damper 2 is c 2 . Conclusion: Differentiate Equation (III). x ˙ 2 = b a x ˙ 1 (VII) Differentiate Equation (VII). x ¨ 2 = b a x ¨ 1 (VIII) Substitute, Equation (III), Equation (IV), Equation (VII) and Equation (VIII) in Equation (VI)

8th Edition

ISBN: 9781119047315

Author: Bolton

Publisher: VST

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Chapter 8.4, Problem 87P

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