To determine The equation of motion, the natural frequency and the damping ratio of the system. Explanation Draw the free body diagram. Write the relation between the displacement of mass m 1 ( x ) and the displacement of mass m 2 ( x 2 ) . 3 x = 2 x 2 x 2 = 3 2 x (I) Write the force balance in x -direction for mass m 1 . ∑ F x = m 1 x ¨ 3 T − k 1 x − c 1 x ˙ = m 1 x ¨ 3 T = m 1 x ¨ + c 1 x ˙ + k 1 x 6 T = 2 m 1 x ¨ + 2 c 1 x ˙ + 2 k 1 x (II) Here, the tension in the cable is T , the spring constant of spring 1 is k 1 and the viscous damping coefficient of damper 1 is c 1 . Write the force balance in x -direction for mass m 2 . ∑ F x 2 = m 2 x ¨ 2 − 2 T − k 2 x 2 − c 2 x ˙ 2 = m 2 x ¨ 2 − 2 T = m 2 x ¨ 2 + c 2 x ˙ 2 + k 2 x 2 − 6 T = 3 m 2 x ¨ 2 + 3 c 2 x ˙ 2 + 3 k 2 x 2 (III) Here, the spring constant of spring 2 is k 2 and the viscous damping coefficient of damper 2 is c 2 . Conclusion: Differentiate Equation (I). x ˙ 2 = 3 2 x ˙ (IV) Differentiate Equation (IV). x ¨ 2 = 3 2 x ¨ (V) Adding Equation (II) and Equation (III). 6 T − 6 T = 2 m 1 x ¨ + 2 c 1 x ˙ + 2 k 1 x + 3 m 2 x ¨ 2 + 3 c 2 x ˙ 2 + 3 k 2 x 2 ( 2 m 1 x ¨ + 3 m 2 x ¨ 2 ) + ( 2 c 1 x ˙ + 3 c 2 x ˙ 2 ) + ( 2 k 1 x + 3 k 2 x 2 ) = 0 (VI) Substitute, Equation (I), Equation (IV) and Equation (V) in Equation (VI)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Forced undamped vibrations

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Chapter 8.2, Problem 37P

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Chapter 8.2, Problem 36P

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Engineering Mechanics: Dynamics

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