(a) Explanation Given information: The mass of the slender rod AB ( m A B ) is 3 kg. The mass of the uniform disk ( m d i s k ) is 5 kg. The damping coefficient ( c ) is 9N ⋅ s/m . The radius of the disk ( r ) is 100 mm. The length of the slender rod ( l ) is 400 mm. The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Calculate the weight of the slender rod ( W A B ) using the formula: W A B = m A B g Substitute 3 kg for m A B and 9.81 m / s 2 for g . W A B = 3 ( 9.81 ) = 29.43 kg ⋅ m/s 2 Calculate the moment of inertia of the disk ( I ¯ d i s k ) using the formula: I ¯ d i s k = 1 2 m d i s k r 2 Substitute 5 kg for m d i s k and 100 mm for r . I ¯ d i s k = 1 2 × 5 × ( 100 mm × 1 1000 m mm ) 2 = 0.025 kg ⋅ m 2 Calculate the moment of inertia of the slender rod ( I ¯ A B ) using the formula: I ¯ A B = 1 12 m A B l 2 Substitute 3 kg for m A B and 400 mm for l . I ¯ A B = 1 12 × 3 × ( 400 mm × 1 1000 m mm ) 2 = 0.040 kg ⋅ m 2 Assume the disk and rod assemble be rotated through a small angle of ( θ ) in counterclockwise direction. Show the free body diagram of the disk and slender rod as in Figure (1). Refer Figure (1), Take the moment about A . ∑ M A = ∑ ( M A ) e f f Here, M A is the moment at A and ( M A ) e f f is the effective moment about A . − W A B x − F d r = I ¯ d i s k θ ¨ + I ¯ A B θ ¨ + m A B l 2 θ ¨ × ( l 2 ) Substitute l 2 sin θ for x and c r θ ˙ for F d (b) To determine Find the damping factor ( c c c ) .

12th Edition

ISBN: 9781260804638

Author: BEER

Publisher: MCG

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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, Problem 19.164RP

(a)

To determine

The differential equation of motion for small oscillations.

(b)

To determine

Find the damping factor (ccc).

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Chapter 19, Problem 19.163RP

Chapter 19, Problem 19.165RP

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The differential equation of motion for small oscillations.

Solution Summary: The author explains the differential equation of motion for small oscillations.

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The differential equation of motion for small oscillations.

Find the damping factor (ccc).

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Chapter 19 Solutions