Explanation Given information: The mass ( M ) of the cylinder is 6 kg. The mass ( m ) of the bar AB is 4 kg. The spring constant ( k ) is 3.5 kN/m . The coefficient of static friction ( μ s ) between the cylinder and the surface is 0.5. The acceleration due to gravity ( g ) is 9.81 m/s 2 . Calculation: Show the free body diagram of the system as in Figure (1). Write the below equation by adding the horizontal force in the bar AB : m x ¨ = 2 k x − P (1) Write the below equation by adding the horizontal force in the cylinder: M x ¨ = P − F (2) Write the force ( F ) equation: F r = 1 2 M r 2 θ ¨ Substitute x ¨ r for θ ¨ . F r = 1 2 M r 2 x ¨ r F = 1 2 M x ¨ Substitute 1 2 M x ¨ for F in equation (2). M x ¨ = P − 1 2 M x ¨ Rewrite the above equation for P . M x ¨ + 1 2 M x ¨ = P P = 3 2 M x ¨ Substitute 3 2 M x ¨ for P in equation (1). m x ¨ = 2 k x − 3 2 M x ¨ m x ¨ + 3 2 M x ¨ = 2 k x ( m + 3 2 M ) x ¨ = 2 k x (3) Compare the differential Equation (3) with the general differential equation of motion ( x ¨ + ω n 2 x = 0 ) and express the natural circular frequency of vibration ( ω n ) : ω n 2 = 2 k ( m + 3 2 M ) (4) Calculate the natural circular frequency ( ω n ) : Substitute 4 kg for m , 6kg for M , and 3.5 kN/m for k in equation (4). ω n 2 = 2 ( 3

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977251

Author: BEER

Publisher: MCG

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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 19.40P

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The maximum amplitude (xm) of the motion of point C which is compatible with the assumption of rolling.

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Chapter 19.2, Problem 19.39P

Chapter 19.2, Problem 19.41P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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The maximum amplitude ( x m ) of the motion of point C which is compatible with the assumption of rolling.

Solution Summary: The author shows the free body diagram of the system as in Figure (1).

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The maximum amplitude (xm) of the motion of point C which is compatible with the assumption of rolling.

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Chapter 19 Solutions