Explanation Given information: The frequency of oscillation in rpm when no oil is present ( N ) is 200 rpm. The frequency of oscillation in rpm when oil is present ( N o i l ) is 180 rpm. The mass moment of inertia ( I ¯ ) is 0.3 kg ⋅ m 2 . Calculation: The expression for the moment balance equation of differential equation of motion for the torsional pendulum as follows: K θ + C θ ˙ = − I ¯ θ ¨ I ¯ θ ¨ + K θ + C θ ˙ = 0 (1) Here, θ is the initial twist given to the shaft, K is the torsional stiffness of the pendulum, K θ is the torsional force acting on the shaft, C θ ˙ is the couple due to damping of oil, C is the damping constant of the oil, θ ˙ is the derivative of angular displacement, I ¯ θ ¨ is the restoring couple that act on the pendulum, I ¯ is the moment of inertia of the pendulum and θ ¨ is the second derivative of angular displacement. The expression for light damping condition for the solution of the differential equation of motion as follows: θ = e − λ t ( C 1 sin ω d t + C 2 cos ω d t ) Here, λ is the parameter equal to the ratio of damping constant and twice the moment of inertia, ω d is the frequency of the damped vibration and t is the time. The expression for the parameter ( λ ) as follows: λ = C 2 I ¯ (2) The expression for the frequency of damped vibration ( ω d ) as follows: ω d = ω n 2 − λ 2 (3) Here, ω n is the natural circular frequency of vibration. Calculate the natural circular frequency of vibration ( ω n ) using the formula: ω n = 2 π N 60 Substitute 200 rpm for N . ω n = 2 π ( 200 rpm ) 60 = 20.944 rad / s Calculate the frequency of damped vibration ( ω d ) using the formula: ω d = 2 π N o i l 60 Substitute 180 rpm for N o i l

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977251

Author: BEER

Publisher: MCG

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Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

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Chapter 19.5, Problem 19.133P

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Find the damping constant for the oil (C).

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Chapter 19.5, Problem 19.132P

Chapter 19.5, Problem 19.134P

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Find the damping constant for the oil ( C ).

Solution Summary: The author calculates the moment balance equation of differential equations of motion for the torsional pendulum.

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Find the damping constant for the oil (C).

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Chapter 19 Solutions