Appendix

Underdamped oscillation: If \(\delta_{\tiny\mbox{d}}\) is real, Equations (\ref{eq:udsol}) and (\ref{eq:vdsol}) will consist of an exponential term with a negative decay constant which describes the damping, and a sinusoidal term describing the oscillation. This means that the system oscillates with its magnitude gradually approaching zero.
Critically damped oscillation: If \(\delta_{\tiny\mbox{d}}=0\) (or approaches zero), we can approximate the sine and cosine in Equation (\ref{eq:udsol}) by their Taylor expansions around zero
\begin{equation} \label{eq:cridamp1} \label{eq:cridamp1}u_{\tiny\mbox{d}}(t)=\exp(-\gamma_{\tiny\mbox{d}}t)(a_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}(\delta_{\tiny\mbox{d}}t-\frac{({\delta_{\tiny\mbox{d}}t})^{3}}{3!}+...)+b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}(1-\frac{({\delta_{\tiny\mbox{d}}t})^{2}}{2!}+...))+f_{{\tiny\mbox{u}}_{\tiny\mbox{d}}},\par \\ \end{equation}
and the same holds for \(v_{\tiny\mbox{d}}(t)\) of Equation (\ref{eq:vdsol}). For both the sine and the cosine expansion, all terms higher than the first are very small, and, thus, can be neglected. Furthermore, since \(a_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}\) can be abbreviated by \(a_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}=\frac{c_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}}{\delta_{\tiny\mbox{d}}}\) (see Equation (\ref{eq:uprop})), the general solution for critical damping is given by
\begin{equation} \label{eq:cridampsol} \label{eq:cridampsol}u_{\tiny\mbox{d}}(t)=\exp(-\gamma_{\tiny\mbox{d}}t)(c_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}t+b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}})+f_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}.\par \\ \end{equation}
In the critically damped case, the system returns to zero most rapidly without oscillations.
Overdamped oscillation: If \(\delta_{\tiny\mbox{d}}\) is imaginary, Equation (\ref{eq:udsol}) turns into
\begin{equation} u_{\tiny\mbox{d}}(t)=\exp(-\gamma_{\tiny\mbox{d}}t)(a_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}i\sinh(\delta_{\tiny\mbox{d}}t)+b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}\cosh(\delta_{\tiny\mbox{d}}t))+f_{{\tiny\mbox{u}}_{\tiny\mbox{d}}},\par \\ \end{equation}
and Equation (\ref{eq:vdsol}) accordingly. Substituting the hyperbolic sine and hyperbolic cosine with their Euler representations results in
\begin{equation} u_{\tiny\mbox{d}}(t)=\exp(-\gamma_{\tiny\mbox{d}}t)(a_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}i\frac{\exp(\delta_{\tiny\mbox{d}}t)-\exp(-\delta_{\tiny\mbox{d}}t)}{2}+b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}\frac{\exp(\delta_{\tiny\mbox{d}}t)+\exp(-\delta_{\tiny\mbox{d}}t)}{2})+f_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}.\par \\ \end{equation}
After rearranging the terms, we obtain
\begin{equation} u_{\tiny\mbox{d}}(t)=\exp(-(\gamma_{\tiny\mbox{d}}-\delta_{\tiny\mbox{d}})t)(\frac{b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}+ia_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}}{2})+\exp(-(\gamma_{\tiny\mbox{d}}+\delta_{\tiny\mbox{d}})t)(\frac{b_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}-ia_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}}{2})+f_{{\tiny\mbox{u}}_{\tiny\mbox{d}}}.\par \\ \end{equation}
The overdamped case consists of two exponential terms with two different decay constants, which means the oscillator after perturbation returns to the equilibrium state without oscillating. Depending on the relationship between \(\gamma_{\tiny\mbox{d}}\) and \(\delta_{\tiny\mbox{d}}\), the corresponding normal mode can be either overdamped unstable or overdamped stable (see Figure \ref{fig:chart}). Figure \ref{fig:unov} shows examples of an overdamped, an underdamped anda critically damped normal mode.