Explanation Given information: The radius of the disk is r . The mass of the disk is m . The length of the rod AB is L 2 . The length of the rod BC is L 2 . Calculation: Write the expression for mass moment of inertia ( I ¯ ) : I ¯ = 1 2 m r 2 The angle α is equal to angular acceleration L 2 r θ ¨ . Write the expression for tangential acceleration ( a t ) : a t = r α Substitute L 2 r θ ¨ for α . a t = r × L 2 r θ ¨ = L 2 θ ¨ Show the free body diagram equation as in Figure (1). Take moment about D in the system for external forces. ( ∑ M D ) e x t e r n a l = − m g r θ − k L θ r 2 Take moment about D in the system for effective forces. ( ∑ M D ) e f f e c t i v e = I ¯ α + m r a t Substitute 1 2 m r 2 for I ¯ , L 2 r θ ¨ for α , and L 2 θ ¨ for a t . ( ∑ M D ) e f f e c t i v e = ( 1 2 m r 2 × L 2 r θ ¨ ) + m r ( L 2 θ ¨ ) = m r L 4 θ ¨ + m r L 2 θ ¨ Equate the moment about C in the system for external and effective forces. ( ∑ M O ) e x t e r n a l = ( ∑ M O ) e f f e c t i v e − m g r θ − k L θ r 2 = m r L 4 θ ¨ + m L 2 r θ ¨ − m r L 4 θ ¨ − m L 2 r θ ¨ − m g r θ − k L r θ 2 = 0 ( m r L 4 + m r L 2 ) θ ¨ + ( m g r + k L r 2<

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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 19.57P

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The frequency (fn) of the resulting vibration in term of m, L, k, and g.

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Chapter 19.2, Problem 19.56P

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The frequency ( f n ) of the resulting vibration in term of m , L , k , and g .

Solution Summary: The author calculates the frequency (f_n) of the resulting vibration in terms of m, L, k, and g.

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The frequency (fn) of the resulting vibration in term of m, L, k, and g.

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