MATH319 Slides 115 Undamped harmonic oscillator 117 Resonance

116 Marginal stability

and the general solution of ${{d}\over{dt}}X=AX$ is

X=c_{1}\begin{bmatrix}\cos\nu t\\ -\nu\sin\nu t\end{bmatrix}+c_{2}\begin{bmatrix}\sin\nu t\\ \nu\cos\nu t\end{bmatrix},

for constants $c_{1},c_{2}$ . In particular, all these solutions are bounded, so we have marginal stability.

$\bullet$ For $U\neq 0$ and $\omega\neq\nu$ , the input has angular frequency different from the natural angular frequency, and the solution is the complementary function plus a particular integral

X=c_{1}\begin{bmatrix}\cos\nu t\\ -\nu\sin\nu t\end{bmatrix}+c_{2}\begin{bmatrix}\sin\nu t\\ \nu\cos\nu t\end{bmatrix}+{{U_{0}}\over{\nu^{2}-\omega^{2}}}\begin{bmatrix}% \cos\omega t\\ -\omega\sin\omega t\end{bmatrix};

here the complementary function oscillates at natural angular frequency $\nu$ ; whereas the particular integral oscillates at the input angular frequency $\omega$ . These solutions are all bounded. One can obtain these particular integrals by W3.2, or by guesswork.