Lect06b-SecondOrderSystems

Lect06b-SecondOrderSystems - M.D. Bryant ME 340 notes...

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M.D. Bryant ME 340 notes 9/21/10 Second Order Systems 2 energy storage elements => Resonance: inductance & capacitance trade energy Example : RLC circuit + - V R C L + - + - + - i V R V V C L V + iR + q C + L di dt = 0 since i = dq / dt L d 2 q dt 2 + R dq dt + q C = V ( t )
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M.D. Bryant ME 340 notes 9/21/10 Standard form, 2 nd order system: ˙ ˙ x + 2 ζω n ˙ x + ω n 2 x = f ( t ) Rearrange, make coefficient of 2 nd order term unity: Compare equations f ( t ) = V ( t ) L , 2 n = R L , n 2 = 1 LC n = 1 LC , ζ = R 2 L n = R 2 L / C System parameters natural frequency n damping ratio d 2 q dt 2 + R L dq dt + 1 LC q = V ( t ) L
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M.D. Bryant ME 340 notes 9/21/10 Example: Oscillator without damping no damping (no power loss) Ã R = 0 Solution by superposition x = x complementary + x particular complementary differential equation ( set forcing term f ( t )=0 ) ˙ ˙ x + ω n 2 x = 0 Homogeneous solution: x c Guess : x c = A cos n t + B sin n t Test (substitute) : 0 = ˙ ˙ x c + n 2 x c = d 2 ( A cos n t + B sin n t ) dt 2 + n 2 ( A cos n t + B sin n t ) = 0 ˙ ˙ x + n 2 x = f ( t )
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M.D. Bryant ME 340 notes 9/21/10 Particular Solution DC voltage f o switched in at t = 0 f(t) =f o Differential equation ( t > 0 ) : ˙ ˙ x + ω n 2 x = f o Particular solution (guess)
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Lect06b-SecondOrderSystems - M.D. Bryant ME 340 notes...

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