Lect06b-SecondOrderSystems

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

This preview shows pages 1–6. Sign up to view the full content.

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 )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 02/14/2012.

### Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online