Explanation Calculation: The expression for magnification factor for the damped forced vibration ( x m P m k ) as follows: x m P m k = 1 [ 1 − ( ω f ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω f ω n ) ] 2 Take square on both sides to eliminate the square root. ( x m P m k ) 2 = ( 1 [ 1 − ( ω f ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω f ω n ) ] 2 ) 2 = 1 [ 1 − ( ω f ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω f ω n ) ] 2 Differentiate the above equation with respect to ( ω f ω n ) 2 . d ( x m P m k ) 2 d ( ω f ω n ) 2 = − 1 ( [ 1 − ( ω f ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω f ω n ) ] 2 ) 2 × [ 2 ( 1 − ( ω f ω n ) 2 ) × − 1 ] + [ 2 ( c c c ) ] 2 ( ∵ 1 x = − 1 x 2 ) = − [ − 2 ( 1 − ( ω f ω n ) 2 ) ] + 4 c 2 c c 2 ( [ 1 − ( ω f ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω f ω n ) ] 2 ) 2 = 2 ( 1 − ( ω f ω n ) 2 ) − 4 c 2 c c 2 ( [ 1 − ( ω f ω n ) 2 ] 2

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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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…

Question

Chapter 19.5, Problem 19.141P

To determine

The range of values of damping factor (ccc) for which the magnification factor will always decrease as the frequency ratio (ωfωn) increases.

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Chapter 19.5, Problem 19.140P

Chapter 19.5, Problem 19.142P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The range of values of damping factor ( c c c ) for which the magnification factor will always decrease as the frequency ratio ( ω f ω n ) increases.

Solution Summary: The author explains the range of values of damping factor for the damped forced vibration, which will always decrease as the frequency ratio increases.

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The range of values of damping factor (ccc) for which the magnification factor will always decrease as the frequency ratio (ωfωn) increases.

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Chapter 19 Solutions