Explanation Given information: The coefficient of damping is c . The amplitude of the motion is x m . The circular frequency of the harmonic force is ω f . Calculation: The expression for the steady state response ( x ) of the machine as follows: x = x m sin ( ω f t − ϕ ) Differentiate the above equation with respect to time ‘ t ’. d x d t = x m ω f cos ( ω f t − ϕ ) d x = x m ω f cos ( ω f t − ϕ ) d t Calculate the force transmitted ( F d ) to the machine for the dashpot using the relation: F d = c d x Substitute x m ω f cos ( ω f t − ϕ ) for dx . F d = c x m ω f cos ( ω f t − ϕ ) Calculate the work done in a complete cycle ( E ) using the relation: E = ∫ F d d x Here, F d is the force and dx is the distance. Substitute c x m ω f cos ( ω f t − ϕ ) for F d and x m ω f cos ( ω f t − ϕ ) d t for dx . E = ∫ c x m ω f cos ( ω f t − ϕ ) x m ω f cos ( ω f t − ϕ ) d t = ∫ c x m 2 ω f 2 cos 2 ( ω f t − ϕ ) d t Differentiate the above equation with respect to time ‘ t ’ in the limit from 0 to 2 π ω f . E = ∫ 0 2 π ω f c x m 2 ω f 2 cos 2 ( ω f t − ϕ ) d t [ cos 2 ( ω f t − ϕ ) = 1 − 2 cos ( ω f t − ϕ ) 2

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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 19.5, Problem 19.150P

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Show that the mechanical energy dissipated per cycle by the dashpot is E=πcxm2ωf.

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Chapter 19.5, Problem 19.149P

Chapter 19.5, Problem 19.151P

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Show that the mechanical energy dissipated per cycle by the dashpot is E = π c x m 2 ω f .

Solution Summary: The author explains that the mechanical energy dissipated per cycle by the dashpot is E=pi cxm2omega _f.

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Show that the mechanical energy dissipated per cycle by the dashpot is E=πcxm2ωf.

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