Explanation Given information: Mass per unit length m = 0.4 Κ g / m Displacement θ = 2 ° Time t = 5 sec Radius of the rod r = 0.2 m Concept used: Let, the total mass of the ring is m 1 = 0.4 ( 2 π r ) m 1 = 0.4 ( 2 × 3.14 × 0.2 ) m 1 = 0.5024 Κ g And, mass of the pivoted rod is m 2 = 0.4 ( r ) m 2 = 0.4 ( 0.2 ) m 2 = 0.08 Κ g Now, compute the distance h between point A and G, h = m 2 ( r 2 ) m 1 + m 2 h = 0.08 ( 0.2 2 ) 0.5024 + 0.08 h = 0.01373 m Calculation: Taking moment of inertia about point A, I A = m 1 r 2 + 1 3 m 2 r 2 I A = 0.5024 ( 0.2 ) 2 + 1 3 0.08 ( 0.2 ) 2 I A = 0.021163 Κ g . m 2 Now written the equation in the form of equation of un-damped vibration: I A θ ¨ + M g h θ = 0 θ ¨ + ( m 1 + m 2 ) g h I A θ = 0 θ ¨ + ( 0.5024 + 0.08 ) 9.81 × 0.01373 0.021163 θ = 0 θ ¨ + ( 3.70666792 ) θ = 0 Compare the above equation with un-damped equation of vibration; M θ ¨ + ω n 2 θ = 0 Natural frequency: ω n 2 = 3

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Free Damped Vibrations

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 19.2, Problem 19.62P

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The angular velocity and acceleration at a given period of time.

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The angular velocity and acceleration at a given period of time.

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The angular velocity and acceleration at a given period of time.

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