29 - Response of 1st-order system to sinusoidal input x + x...

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Response of 1st-order system to sinusoidal input τx ˙ + x = f ( t ) x (0) = 0 initial condition f ( t ) = f 0 cos ω 0 t periodic forcing function ± ω 0 = angular frequency rad sec ± ν 0 = 2 ω π 0 = frequency [Hz] = sec 1 f 0 = amplitude Solution: x ( t ) = x h ( t ) + x p ( t ) = homogeneous + particular x h ( t ) = Ae t τ Conjecture: x p ( t ) = αf 0 cos ( ω 0 t + ψ ) Procedure: First calculate α , ψ , then A. τx ˙ p ( t ) + x p ( t ) = f ( t ) τω 0 αf 0 sin ( ω 0 t + ψ ) + αf 0 cos ( ω 0 t + ψ ) = f 0 cos ω 0 t trig substitution τω 0 αf 0 [sin ω 0 t cos ψ + cos ω 0 t sin ψ ] + αf 0 [cos ω 0 t cos ψ sin ω 0 t sin ψ ] = f 0 cos ω 0 t αf 0 ( ω 0 τ cos ψ +sin ψ ) sin ω 0 t + αf 0 ( ω 0 τ sin ψ +cos ψ ) cos ω 0 t = f 0 cos ω 0 t must be true for all t equate coefficients ² αf 0 ( ω 0 τ sin ψ + cos ψ ) = f 0 (1) ω 0 t cos ψ + sin ψ = 0 (2) From (2), tan ψ = ω 0 t f 0 1 1 From (1) tan ψ sin ψ + cos ψ = αf 0 cos ψ = α α = cos ψ = 1 = 1 1+tan
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29 - Response of 1st-order system to sinusoidal input x + x...

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