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bodeplots

# bodeplots - Frequency Response We are interested in the...

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Frequency Response We are interested in the solution(s) to the linear constant-coefficient differential equa- tion x ( n ) + a n - 1 x ( n - 1) + · · · + a 1 ˙ x + a 0 x = b m u ( m ) + · · · + b 0 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 ωt , then the left-hand side will be entirely in terms of sin ωt and cos ωt . Similarly, if x p is a polynomial or exponential in t , then the left-hand side will be a polynomial or exponential in t . So, if we have a certain form on the right-hand side, we can try to get it by guessing a related form for x h . For example, suppose we have the third-order ODE x (3) + a 2 ¨ x + a 1 ˙ x + a 0 x = K sin ωt (2) where the right-hand side is the sinusoidal input used for frequency response analysis. We guess a particular solution of the form x p ( t ) = A sin ωt + B cos ωt (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

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