Explanation Write the expression to find the natural frequency of the system ( ω n ) . ω n = k m (I) Here, the spring constant is k and the mass is m . Write the expression to find the damping ratio ( ζ ) . ζ = c 2 m ω n (II) Here, the coefficient of damping is c . Write the expression to find the displacement for an overdamped system. x = A 1 e λ 1 t + A 2 e λ 2 t (III) Write the expression to find λ 1 . λ 1 = ω n ( − ζ + ζ 2 − 1 ) (IV) Write the expression to find λ 2 . λ 2 = ω n ( − ζ − ζ 2 − 1 ) (V) Conclusion: Substitute, 98 N / m for k and 2 kg for m in Equation (I). ω n = 98 N / m 2 kg ω n = 7 rad / s Substitute, 42 N ⋅ s / m for c , 7 rad / s for ω n and 2 kg for m in Equation (II). ζ = 42 N ⋅ s / m 2 ( 2 kg ) ( 7 rad / s ) ζ = 1.5 Substitute, 7 rad / s for ω n and 1.5 for ζ in Equation (IV). λ 1 = 7 ( − 1.5 + 1.5 2 − 1 ) λ 1 = − 2.674 Substitute, 7 rad / s for ω n and 1

8th Edition

ISBN: 9781118885840

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

Publisher: WILEY

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Chapter 8.2, Problem 42P

To determine

The displacement as a function of time.

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Chapter 8 Solutions

Engineering Mechanics: Dynamics

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