MATH319 Slides 108 Matrix form of the damped harmonic oscillator 110 Stability cases

109 Cases of the damped harmonic oscillator

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

\begin{pmatrix}{\hbox{solutions}}&\Delta<0&\Delta=0&\Delta>0\\ \beta<0&{\hbox{unbounded oscillations}}&{\hbox{exponential growth}}&{\hbox{exp% growth}}\\ \beta=0&{\hbox{periodic}}&{\hbox{constant}}&{\hbox{hyperbolic}}\\ \beta>0&{\hbox{decaying oscillations}}&{\hbox{critically damped}}&{\hbox{exp % decay}}\end{pmatrix}

The damped oscillator is exponentially stable if and only if $\beta>0$ and $\gamma>0$ . When $\beta=0$ and $\gamma>0$ , the oscillator is marginally stable. For $\beta<0$ , the oscillator is unstable.