Explanation Given information: The weight of the machine ( W ) is 500 lb. The spring constant ( k ) is 250 lb/inch. The coefficient of damping ( c ) is 10 lb ⋅ s / in . . The periodic force is applied with a frequency ( f f ) is 2 cycles. The magnitude of the periodic force ( P m ) is 20 lb. The amplitude of the steady state vibration ( x m ) is 0.05 inch. The acceleration due to gravity ( g ) is 32 .2ft/s 2 . Calculation: The periodic force is applied with a frequency of 2.0 cycles per second or 2.0 Hz. Calculate the circular frequency of the periodic force ( ω f ) using the formula: ω f = f f 2 π Here, f f is the periodic force in Hz. Substitute 2.0 Hz for f f . ω f = 2 π ( 2.0 Hz ) = 12.5664 rad / s The system consists of two springs each with a spring constant of 250 lb / in . Thus, the spring constant of the system is given by two times the spring constant of 250 lb / in . . Calculate the spring constant of the system ( k ) for two springs using the relation: k = 2 k s p r i n g Here, k is the spring constant of the system and k s p r i n g is the spring constant of each spring. Substitute 250 lb / in . for k s p r i n g . k = 2 ( 250 lb / in . ) = 500 lb / in . = 500 lb 1 in . × 1 ft 12 in . = 6000 lb / ft Calculate the natural circular frequency ( ω n ) using the formula: ω n = k m Substitute 6000 lb / ft for k and 15.5280 lb ⋅ s 2 / ft for m . ω n = 6000 lb / ft 15.5280 lb ⋅ s 2 / ft = 19.657 rad / s Calculate the critical damping coefficient ( c c ) using the formula: c c = 2 m ω n Substitute 15.5280 lb ⋅ s 2 / ft for m , and 19.657 rad / s for ω n . c c = 2 ( 15.5280 lb ⋅ s 2 / ft ) ( 19.657 rad / s ) = 610.47 lb ⋅ s / ft The periodic force is applied with a frequency of 2

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 19.5, Problem 19.140P

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The coefficient of damping (c).

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Chapter 19.5, Problem 19.139P

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Vector Mechanics for Engineers: Statics and Dynamics

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The coefficient of damping ( c ).

Solution Summary: The author explains the coefficient of damping, which is underset_21.4, the spring constant, and the circular frequency of the periodic force.

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