UConn ME 260W - Free Vibration with Viscous and Coulomb Damping (2 pages)

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Free Vibration with Viscous and Coulomb Damping



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Free Vibration with Viscous and Coulomb Damping

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Pages:
2
School:
University Of Connecticut
Course:
Me 260w - Measurement Techniques

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ME260W HW 3 Free Vibration with Viscous and Coulomb Damping Due time 2 00pm Mar 24 collect after Thursday lecture Background Free vibration of a single degree of freedom SDOF system with viscous damping can be described by the general equation of motion mx cv x kx 0 1 1 The natural frequency of the system n is given by k 1 2 m The critical damping cc damping ratio and the damped vibration frequency d are defined as n cc 2m n 2m k m cv cc 1 3 1 4 d 1 2 n 1 5 When 1 the response of the system shows oscillating behavior and the displacement amplitude decays exponentially i e the natural logarithm of the amplitude ratio for any two displacements separated in time by one or more periods of the damped vibration is a constant The logarithmic decrement is defined by A 2 ln k 1 n Td 1 6 1 2 Ak In above equation Ak is the amplitude at arbitrary time t and Ak 1 is the amplitude at 2 is the period of the damped vibration time t Td Td d Free vibration of an SDOF system with Coulomb friction damping can be described by the general equation of motion 1 7 mx cc sign x kx 0 It can be shown that the amplitude of vibration decrease linearly and the amount of decrement per oscillation is given by 4c Ak Ak 1 c 1 8 k Problem 1 Assume a compound pendulum oscillating with a small initial swing angle about 10 degrees The pendulum makes 1 complete oscillation per second and in 10 seconds its amplitude diminishes 50 Assume the damping is pure viscous in nature calculate the logarithmic decrement and determine the viscous damping constant cv and the damping ratio In what proportion would the period of vibration be decreased if the damping were removed Problem 2 Consider the same pendulum motion described in Problem 1 assume the damping is pure friction Coulomb in nature estimate the Coulomb damping coefficient cc Problem 3 A second order system is assumed to have pure viscous damping The amplitude of the response is observed to decrease from 1 to 0 138 after one complete cycle the ringing



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