Explanation Given information: The period ( τ 1 ) of oscillation of the platform when the platform is empty is 2.2 sec. The period ( τ 2 ) of oscillation of the platform with the object A is 3.8 sec. The wire has a torsional constant ( K ) is 27 N ⋅ m/rad . Calculation: Show the free body equation diagram as in Figure (1). The external force in the system is tension in the wire and the effective force in the system is the couple I ¯ θ ¨ . Take moment about G in the system for external forces. ( ∑ M G ) e x t e r n a l = − K θ Take moment about G in the system for effective forces. ( ∑ M G ) e f f e c t i v e = I ¯ θ ¨ Equate the moment about G in the system for external and effective forces. ( ∑ M G ) e x t e r n a l = ( ∑ M G ) e f f e c t i v e − K θ = I ¯ θ ¨ − I ¯ θ ¨ − K θ = 0 I ¯ ( θ ¨ + K I ¯ θ ) = 0 θ ¨ + K I ¯ θ = 0 (1) Compare the differential Equation (1) with the general differential equation of motion ( x ¨ + ω n 2 x = 0 ) and express the natural circular frequency of vibration: ω n 2 = K I ¯ (2) Calculate the natural circular frequency ( ω n 1 ) when the plate is empty using the relation: ω n 1 = 2 π τ 1 Substitute 2.2 sec for τ 1 . ω n 1 = 2 π 2.2 = 2.856 rad/s Calculate the centroidal moment of inertia ( I ¯ 1 ) when the platform is empty: Substitute 27 N ⋅ m/rad for K and 2.856 rad/s for ω n 1 in Equation (2)

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

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Chapter 19.2, Problem 19.63P

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The centroidal (I¯A) moment of inertia of object A.

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The centroidal ( I ¯ A ) moment of inertia of object A .

Solution Summary: The author shows the free body equation diagram as in Figure (1). The external force is tension in the wire and the effective forces are the couple.

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