MATH319 Slides 136 Partial fractions 138 Stability for systems and transfer functions

137 Poles of the transfer function of the damped harmonic oscillator

Consider $1/(s^{2}+\beta s+\gamma)$ with $\gamma>0$ and $\beta$ real with poles at $\lambda_{\pm}=(1/2)(-\beta\pm\sqrt{\Delta})$ where $\Delta=\beta^{2}-4\gamma.$ Then

\begin{pmatrix}{\hbox{poles}}&\beta&\Delta<0&\Delta=0&\Delta>0\\ &&\lambda_{+}=\bar{\lambda}_{-}&\lambda_{+}=\lambda_{-}&{\hbox{distinct real % roots}}\\ {\hbox{unstable}}&\beta<0&\Re\lambda_{\pm}>0&\lambda_{\pm}>0&0<\lambda_{-}<% \lambda_{+}\\ {\hbox{marginal}}&\beta=0&\Re\lambda_{+}=0&\lambda_{\pm}=0&\lambda_{-}<0<% \lambda_{+}\\ {\hbox{stable}}&\beta>0&\Re\lambda_{\pm}<0&\lambda_{\pm}<0&\lambda_{-}<\lambda% _{+}<0\end{pmatrix}

For a damped harmonic oscillator, we have $\beta,\gamma>0$ , so only the last row matters. The last row gives the stable cases.