Explanation Given information: The mass ( m ) of the sport car is 1,300 kg. The period ( τ n 1 ) of oscillation is 4.04 sec when the L is 4 m. The period ( τ n 2 ) of oscillation is 3.54 sec when the L is 3 m. Calculation: Show the free-body-diagram equation as in Figure (1). The external forces in the system are force due mass of the car. The effective force in the system is m a ¯ c and m a ¯ t and the effective restoring couple is I ¯ θ ¨ . Take moment about O in the system for external forces. ( ∑ M O ) e x t e r n a l = ( m g × ( L − h ) sin θ ) = m ( L − h ) g sin θ As the angular displacement is very small, take the value of sin θ as θ . Thus, the moment of external forces is m ( L − h ) g θ . Take moment about O in the system for effective forces. ( ∑ M O ) e f f e c t i v e = ( m a ¯ t × ( L − h ) ) + ( I ¯ θ ¨ ) = − m ( L − h ) a ¯ t − I ¯ θ ¨ The tangential component of acceleration a ¯ t is equal to the product of distance ( L - h ) and angular acceleration ( ( L − h ) θ ¨ ) . Thus express the moment about O due to the effective forces: ( ∑ M O ) e f f e c t i v e = − m ( L − h ) ( ( L − h ) θ ¨ ) − I ¯ θ ¨ = − ( m ( L − h ) 2 + I ¯ ) θ ¨ Equate the moment about O 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 ( L − h ) g θ = − ( m ( L − h ) 2 + I ¯ ) θ ¨ θ ¨ = − m ( L − h ) g m ( L − h ) 2 + I ¯ θ θ ¨ + m ( L − h ) g m ( L − h ) 2 + 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 oscillation ( ω n ) : ω n 2 = m ( L − h ) g m ( L − h ) 2 + I ¯ (2) Write the expression moment of inertia ( I ¯ ) of the car: I ¯ = m k ¯ 2 Substitute m k ¯ 2 for I ¯ in Equation (2). ω n 2 = m ( L − h ) g m ( L − h ) 2 + I ¯ ω n 2 = m ( L − h ) g m ( L − h ) 2 + ( m k ¯ 2 ) ω n 2 = m ( L − h ) g m ( ( L − h ) 2 + k ¯ 2 ) ω n 2 = ( L − h ) g ( L − h ) 2 + k ¯ 2 ( ( L − h ) 2 + k ¯ 2 ) ω n 2 = ( L − h ) g k ¯ 2 + ( L − h ) 2 = g ω n 2 ( L − h ) (3) Express the time period ( τ ) of oscillation: τ = 2 π ω n (4) Calculate the natural circular frequency ( ω n ) corresponding to L of 4 m

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ISBN: 9781265566296

Author: BEER

Publisher: MCG

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Free undamped vibrations

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Chapter 19.2, Problem 19.58P

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The value of h and the centroidal radius of gyration (k¯).

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The value of h and the centroidal radius of gyration ( k ¯ ) .

Solution Summary: The author shows the free-body-diagram equation as in Figure (1). The external forces in the system are force due mass of the car.

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The value of h and the centroidal radius of gyration (k¯).

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