Explanation Given information: The amplitude of the collar ( x C ) is δ m sin ω f t . Assume the static deflection δ m is small compared with the length l of the pendulum. Calculation: Show the system at before and after moving the collar horizontally as in Figure (1). Refer Figure 1, θ is the angular displacement of the pendulum, m is the mass of the pendulum bob, and mg is the gravitational force due to the mass of the bob. Before giving horizontal movement, the force mg is equal to the tension in the pendulum. The expression for the force balance equation for initial condition as follows: T = m g Calculate the value of sin θ by referring the Figure (1). sin θ = x − x c l The expression for the force balance equation in x -direction in the displaced condition as follows: ∑ F x = m a x Here, ∑ F x is the sum of forces in x-direction and a x is the acceleration in the x -direction. The only force in the x -direction is the tension component T sin θ and it is opposite in direction to that of the acceleration. x ¨ is the acceleration in the x -direction due to the linear displacement. Substitute − T sin θ for ∑ F x and x ¨ for a x . − T sin θ = m x ¨ Substitute mg for T and x − x c l for sin θ . − m g ( x − x c l ) = m x ¨ − g ( x − x c l ) = x ¨ − g x l + g x c l = x ¨ x ¨ + g l x = g l x c Substitute δ m sin ω f t for x c . x ¨ + g l x = g l δ m sin ω f t (1) Here, t is the time. Compare the differential equation (1) with the general differential equation of motion for forced vibration ( x ¨ + ω n 2 x = ω n 2 δ m sin ω f t ) . The expression for the natural circular frequency of forced vibration ( ω n ) as follows: ω n 2 = g l ω n = g l The expression for the amplitude of the forced vibration ( x m ) as follows: x m = δ m 1 − ( ω f ω n ) 2

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ISBN: 9781260804638

Author: BEER

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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 19.4, Problem 19.109P

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Find the range of values of (ωf) for which the amplitude of the motion of the bob is less than δm.

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Chapter 19.4, Problem 19.108P

Chapter 19.4, Problem 19.110P

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Find the range of values of ( ω f ) for which the amplitude of the motion of the bob is less than δ m .

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Find the range of values of (ωf) for which the amplitude of the motion of the bob is less than δm.

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