Explanation Given information: The length of the two vertical cable is l . The distance between the point AO is a 2 . The distance between the point OB is a 2 . Calculation: Show the displacement of the system as in the Figure (1). For position 1: Let θ m be the small angle between the bar AB with horizontal and α m small angle between the cable with vertical. Write the expression for the angle displacement: A A ' = B B ' a 2 θ m = l α m Rewrite the above expression for α m . α m = a 2 l θ m Express the vertical displacement ( y C ) : y C = l − l cos α m = l ( 1 − cos α m ) ( ∵ ( 1 − cos α m = 2 sin 2 α m 2 ) ) = l 2 sin 2 α m 2 For small oscillation 2 sin 2 α m 2 ≈ α m 2 2 . y C = l α m 2 2 Substitute a 2 l θ m for α m . y C = l ( a 2 l θ m ) 2 2 = a 2 8 l θ m 2 Write the expression for the kinetic energy ( T 1 ) : T 1 = 0 Write the expression for the potential energy ( V 1 ) : V 1 = m g y C Substitute a 2 8 l θ m 2 for y C . V 1 = m g ( a 2 8 l θ m 2 ) For position 2: Express the centroidal moment of inertia ( I ¯ ) of the system: I ¯ = 1 12 m a 2 Write the expression for the kinetic energy ( T 2 ) : T 2 = 1 2 I ¯ θ ˙ m 2 Substitute 1 12 m a 2 for I ¯ . T 2 = 1 2 ( 1 12 m a 2 ) θ ˙ m 2 Express the term ( θ ˙ m ) : θ ˙ m = ω n θ m Here, the natural circular frequency of oscillation is ω n

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ISBN: 9781259633133

Author: BEER

Publisher: MCG

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Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

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Chapter 19.3, Problem 19.95P

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The frequency (fn) of oscillation when the pipe is given a small rotation about the centroidal axis OO' and released.

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Chapter 19.3, Problem 19.94P

Chapter 19.3, Problem 19.96P

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VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

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 frequency ( f n ) of oscillation when the pipe is given a small rotation about the centroidal axis O O ' and released.

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The frequency (fn) of oscillation when the pipe is given a small rotation about the centroidal axis OO' and released.

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