This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Frequency Response We are interested in the solution(s) to the linear constantcoefficient differential equa tion x ( n ) + a n 1 x ( n 1) + + a 1 x + a x = b m u ( m ) + + b u (1) One approach is to use Laplace transforms and the resulting transfer function. Another approach is based on the classical solution to the ordinary differential equation itself. A particular solution x p ( t ) is any solution that satisfies the differential equation. We usually reserve the notation x p ( t ), however, for a solution that does not contain any part of the homogeneous, or complementary solution, x h ( t ), obtained by solving the equation with u 0. Note that if x p = sin t , then the lefthand side will be entirely in terms of sin t and cos t . Similarly, if x p is a polynomial or exponential in t , then the lefthand side will be a polynomial or exponential in t . So, if we have a certain form on the righthand side, we can try to get it by guessing a related form for x h . For example, suppose we have the thirdorder ODE x (3) + a 2 x + a 1 x + a x = K sin t (2) where the righthand side is the sinusoidal input used for frequency response analysis. We guess a particular solution of the form x p ( t ) = A sin t + B cos t (3) where A and B are undetermined coefficients, but are different from the undetermined coefficients that appear in the homogeneous solution. We differentiate the assumed so lution three times and substitute the results into the original differential equation. That is x p ( t ) = A sin t...
View
Full
Document
This note was uploaded on 01/01/2012 for the course AOE 5984 taught by Professor Devenport during the Fall '08 term at Virginia Tech.
 Fall '08
 DEVENPORT

Click to edit the document details