5.2 - 64 5.2 Response to Time-Varying Functions Response of...

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Unformatted text preview: 64 5.2 Response to Time-Varying Functions Response of 1st Order Systems to F(t) Governing equation ( ) dx x F t dt + = General solution { } ( ). t t x e C e F t d t = + Particular Solutions 1. F(t) = K t x K C e = + t x K 65 General case: step function When t = , 1 1 1 0.368 0.632 x e = = = Rule o' thumb: Following a stepwise change in forcing, F ( t ), response is 63% complete when t = Corollary: can be measured from step response t >= 0, F(t) = 1 t >=0, x(t) = 1 - exp(-t/ ) 0 1 x(t) t 66 2. F(t) = A i sin t sin( ) x(t) and F(t) are both simple harmonics is constant but (some attenuation or gain) and 0 (some phase shift) t o o i x A t Ce A A = + x t Solution of pure sine-wave forcing using complex arithmetic Let, ( ) i t F t Ae = , then, ( ) t i t x t Ce Be = + . When ( ) i t F t A e = , then ( ) i t x t Be = ....
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This note was uploaded on 10/12/2009 for the course AME 341AL at USC.

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5.2 - 64 5.2 Response to Time-Varying Functions Response of...

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