Different Levels of Damping
Yield Different Solution Behavior
and Give Different Decay Rates


Fix the mass and the spring; i.e., let mass \(m\) and the spring constant \(k\) be fixed.

For different values of the damping coefficient \(\gamma\), the resulting vibrations will behave differently, in particular, will have different decay rates.

In the following, fix \(m=\frac{1}{2},k=8\), and consider different values of \(\gamma\).

The Diff Eq Eigenvalues
\(\displaystyle m\frac{d^2 y}{dt^2} +\gamma\frac{dy}{dt} +ky =0 \) \(\displaystyle \lambda=\frac{-\gamma\pm \sqrt{\gamma^2-4mk} }{2m} \)
\(\displaystyle \frac{1}{2}\frac{d^2 y}{dt^2} +\gamma\frac{dy}{dt} +8y =0 \) \(\lambda=-\gamma\pm \sqrt{\gamma^2-16}\)

An Animation of Vibrations for Different Damping Coefficients
The mass and the spring constant are fixed: \(m=\frac{1}{2},k=8\).
The values of the damping coefficients are: \(\gamma=0,1,4,7\).

Type of Damping Conditions Eigenvalues Solution Behavior Examples
No damping \(\gamma=0\) \(\begin{array}{l} \mbox{Purely imaginary:}\\[0.5ex] \lambda_{1,2}=\pm i \sqrt{\frac{k}{m}} \end{array}\) Periodic Oscillation
\(\begin{array}{l} m=\frac{1}{2},k=8,\gamma=0.\\[0.5ex] \frac{1}{2}\frac{d^2 y}{dt^2} +8y =0\\[0.5ex] y(t)=A\cos(4t)+B\sin(4t)\\[0.5ex] \mbox{No Decay.} \end{array}\)
Under-damping \(\begin{array}{l} \gamma>0,\\[0.5ex] \gamma^2-4mk<0 \end{array}\) \(\begin{array}{l} \mbox{Complex eigenvalues}\\[0.5ex] \mbox{with } \mbox{Re}\,\left(\lambda_{1,2}\right)<0 \end{array}\) Oscillatory decay
\(\begin{array}{l} m=\frac{1}{2},k=8,\gamma=1.\\[0.5ex] \frac{1}{2}\frac{d^2 y}{dt^2} +\frac{dy}{dt} +8y =0\\[0.5ex] y(t)=e^{-t}[A\cos(\sqrt{15}\ t)+B\sin(\sqrt{15}\ t)]\\[0.5ex] \mbox{Decay Rate: \(-1\)} \end{array}\)
Critical-damping \(\begin{array}{l} \gamma>0,\\[0.5ex] \gamma^2-4mk=0 \end{array}\) \(\begin{array}{l} \mbox{Repeated, real: }\\[0.5ex] \lambda_{1}=\lambda_{2}<0 \end{array}\) Monotonic Decay
\(\begin{array}{l} m=\frac{1}{2},k=8,\gamma=4.\\[0.5ex] \frac{1}{2}\frac{d^2 y}{dt^2} +4\frac{dy}{dt} +8y =0\\[0.5ex] y(t)=Ae^{-4t}+Bte^{-4t}\\[0.5ex] \mbox{Decay Rate: \(-4\)} \end{array}\)
Over-damping \(\begin{array}{l} \gamma>0,\\[0.5ex] \gamma^2-4mk>0 \end{array}\) \(\begin{array}{l} \mbox{Distinct, real:}\\[0.5ex] \lambda_{1}<0,\lambda_{2}<0 \end{array}\) Monotonic Decay
\(\begin{array}{l} m=\frac{1}{2},k=8,\gamma=7.\\[0.5ex] \frac{1}{2}\frac{d^2 y}{dt^2} +7\frac{dy}{dt} +8y =0\\[0.5ex] y(t)=Ae^{(-7+\sqrt{33})t}+Be^{(-7-\sqrt{33})t}\\[0.5ex] \mbox{Decay Rate: } \lambda_1=-7+\sqrt{33}\approx -1.26 \end{array}\)

Critical Damping (red), Under-Damping (green), and Over-Damping (blue).
QUESTION: Compare the three graphs. Which one decays fastest?