Explanation Given information: The mass ( m A B ) of the slender rod AB is 3 kg. The mass ( m 0 ) of the disk is 5 kg. The spring constant ( k ) is 280 N/m . The length ( l ) of the rod is 300 mm. The radius ( r ) of the disk is 80 mm. Assuming the acceleration due to gravity ( g ) as 9.81 m/s 2 . Calculation: Show the position 1 and position 2 of the slender rod with disk as in the Figure 1. For position 1: Write the expression for mass moment of inertia of disk ( I d i s k ) : I d i s k = 1 2 m 0 r 2 Write the expression for mass moment of inertia of rod ( I A ) r o d : ( I A ) r o d = 1 3 m A B l 2 Express the term ( θ ˙ m ) : θ ˙ m = ω n θ m Write the expression for the kinetic energy ( T 1 ) : T 1 = 1 2 I d i s k θ ˙ m 2 + 1 2 ( I A ) r o d θ ˙ m 2 Substitute 1 2 m 0 r 2 for I d i s k and 1 3 m A B l 2 for ( I A ) r o d . T 1 = 1 2 ( 1 2 m 0 r 2 ) θ ˙ m 2 + 1 2 ( 1 3 m A B l 2 ) θ ˙ m 2 Substitute ω n θ m for θ ˙ m . T 1 = 1 2 ( 1 2 m 0 r 2 ) ω n 2 θ m 2 + 1 2 ( 1 3 m A B l 2 ) ω n 2 θ m 2 Write the expression for the potential energy ( V 1 ) : V 1 = 0 Position 2: Write the expression for the kinetic energy ( T 2 ) : T 1 = 0 Write the expression for the displacement ( y m ) for position 2: h = l 2 ( 1 − cos θ m ) ( ∵ ( 1 − cos θ m = 2 sin 2 θ m 2 ) ) = l 2 2 sin 2 θ m 2 For small oscillation 2 sin 2 θ m 2 ≈ θ m 2 2 . h = l 2 θ m 2 2 = l 4 θ m 2 Write the expression for the displacement ( h ) for position 2: x m = r θ m Write the expression for the potential energy ( V 2 ) : V 2 = 1 2 k ( x m ) 2 + m A B g h Substitute l 4 θ m 2 for h and r θ m for x m

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

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 19.3, Problem 19.80P

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The period of vibration (τn) of the system.

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Chapter 19.3, Problem 19.79P

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The period of vibration ( τ n ) of the system.

Solution Summary: The author shows the position 1 and position 2 of the slender rod with disk as in the Figure 1.

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The period of vibration (τn) of the system.

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