18.03 Lecture Notes - Lecture 12: Vibration, Damping Ratio, Dashpot
Document Summary
Name* m,b,k relation char. roots exp. sol"s basic real solns. Overdamped b^2/4m > k two diff. real e^{r1 t}, e^{r2 t} same roots, r1, r2. Critically b^2/4m = k repeated root e^{rt} e^{rt}, te^{rt} damped r = - b/2m. Underdamped b^2/4m < k -b/2m +- iw e^{r1 t}, e^{r2 t} e^{-bt/2m}cos(w t) e^{-bt/2m}sin(w t) ( w = omega_d = sqrt{ (k/m) - (b/2m)^2 } ) * the names here are 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. We are studying equations of the form mx" + bx" + kx = 0 (*) which model a mass, dashpot, spring system without external forcing term. We found that (*) has an exponential solution e^{rt} exactly when r is a root of the "characteristic polynomial" p(s) = ms^2 + bs + k .
