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

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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

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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 19.5, Problem 19.140P

Chapter 19.5, Problem 19.142P

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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