ch03_9 - Ch 3.9 Forced Vibrations We continue the...

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Ch 3.9: Forced Vibrations We continue the discussion of the last section, and now consider the presence of a periodic external force: t F t u k t u t u m ϖ γ cos ) ( ) ( ) ( 0 = + +
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Forced Vibrations with Damping Consider the equation below for damped motion and external forcing funcion F 0 cos ϖ t . The general solution of this equation has the form where the general solution of the homogeneous equation is and the particular solution of the nonhomogeneous equation is t F t ku t u t u m ϖ γ cos ) ( ) ( ) ( 0 = + + ( 29 ( 29 ) ( ) ( sin cos ) ( ) ( ) ( 2 2 1 1 t U t u t B t A t u c t u c t u C + = + + + = ϖ ϖ ) ( ) ( ) ( 2 2 1 1 t u c t u c t u C + = ( 29 ( 29 t B t A t U ϖ ϖ sin cos ) ( + =
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Homogeneous Solution The homogeneous solutions u 1 and u 2 depend on the roots r 1 and r 2 of the characteristic equation: Since m , γ , and k are are all positive constants, it follows that r 1 and r 2 are either real and negative, or complex conjugates with negative real part. In the first case, while in the second case Thus in either case, m mk r kr r mr 2 4 0 2 2 - ± - = = + + γ γ γ 0 ) ( lim = t u C t ( 29 , 0 lim ) ( lim 2 1 2 1 = + = t r t r t C t e c e c t u ( 29 . 0 sin cos lim ) ( lim 2 1 = + = t e c t e c t u t t t C t μ μ λ λ
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Transient and Steady-State Solutions Thus for the following equation and its general solution, we have Thus u C ( t ) is called the transient solution . Note however that is a steady oscillation with same frequency as forcing function. For this reason, U ( t ) is called the steady-state solution , or forced response . ( 29 ( 29 t B t A t U ϖ ϖ sin cos ) ( + = ( 29 0 ) ( ) ( lim ) ( lim 2 2 1 1 = + = t u c t u c t u t C t ( 29 ( 29 , sin cos ) ( ) ( ) ( cos ) ( ) ( ) ( ) ( ) ( 2 2 1 1 0 e e e e e e e e e e e e e e e e t U t u t B t A t u c t u c t u t F t ku t u t u m C ϖ ϖ ϖ γ + + + = = + +
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Transient Solution and Initial Conditions For the following equation and its general solution, the transient solution u C ( t ) enables us to satisfy whatever initial conditions might be imposed. With increasing time, the energy put into system by initial displacement and velocity is dissipated through damping force. The motion then becomes the response U ( t ) of the system to the external force F 0 cos ϖ t . Without damping, the effect of the initial conditions would persist for all time. ( 29 ( 29 ) ( ) ( 2 2 1 1 0 sin cos ) ( ) ( ) ( cos ) ( ) ( ) ( t U t u t B t A t u c t u c t u t F t ku t u t u m C ϖ ϖ ϖ γ + + + = = + +
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Rewriting Forced Response Using trigonometric identities, it can be shown that can be rewritten as It can also be shown that where ( 29 δ ϖ - = t R t U cos ) ( ( 29 ( 29 t B t A t U ϖ ϖ sin cos ) ( + = 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 0 2 2 2 2 2 0 2 0 ) ( sin , ) ( ) ( cos , ) ( ϖ γ ϖ ϖ ϖ γ δ ϖ γ ϖ ϖ ϖ ϖ δ ϖ γ ϖ ϖ + - = + - - = + - = m m m m F R m k / 2 0 = ϖ
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Amplitude Analysis of Forced Response
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