# c13 - 18.03 Class 13, March 8, 2006 Summary of solutions to...

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18.03 Class 13, March 8, 2006 Summary of solutions to homogeneous second order LTI equations; Introduction to inhomogneneous equations. [1] We saw on Monday how to solve x" + bx' + kx = 0. Here is a summary table of unforced system responses. One of three things must happen to solutions of x" + bx' + kx = 0 . Name* b,k relation Char. roots Basic real solutions Overdamped b^2/4 > k Two diff. real r1, r2 e^{r1 t}, e^{r2 t} Critically damped b^2/4 = k Repeated root r = -b/2 e^{rt}, te^{rt} Underdamped b^2/4 < k Non-real roots a +- ci e^{at} cos(ct), e^{at} sin(ct) * The name here is appropriate under the assumption that b and k are both non-negative. The rest of the table makes sense in general, but it doesn't have a good interpretation in terms of a mechanical system. If b > 0 and k > 0 , then all solutions die off. The are "transients." In the underdamped case, the roots are -b/2 +- i sqrt(k - (b/2)^2). The imaginary part of the roots is +- omega_d where omega_d = sqrt(k - (b/2)^2) is the "damped circular frequency," and the real part of the roots is the "growth rate" -b/2 : -b/2 +- i omega_d The basic solutions are e^{-bt/2} cos(omega_d t) , e^{-bt/2} sin(omega_d t) , and in "polar form" the general solution is x = A e^{-bt/2) cos(omega_d t - phi) (*) Some people prefer to call omega_d the "pseudofrequency" of (*) , since unless b = 0 this is not a periodic function and so properly speaking doesn't have a frequency. Notice that as you increase damping, the pseudofrequency decreases, slowly at first, but faster as the damping approaches critical damping. At that instant, the pseudoperiod becomes infinite and you don't get

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## This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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c13 - 18.03 Class 13, March 8, 2006 Summary of solutions to...

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