A damping coefficient can be added to the matrix system of a simple harmonic oscillator. This causes the phase plane to spiral into the origin. The matrix form of this system would be:
$$\begin{bmatrix}
x’ \\ v’
\end{bmatrix}=
\begin{bmatrix}
0 & 1 \\ -2 & -0.2
\end{bmatrix}
\begin{bmatrix}
x \\ v
\end{bmatrix}$$
The eigenvalues and eigenvectors are:
$$\lambda_{1,2} = -0.1 \pm 1.41i$$
The negative real components represent a decaying oscillation.
[[Mechanical Time Constant]] – equation includes damping constant
- zachstarApplicationsEigenvectorsEigenvalues2019
Backlinks
[[Phase Plane]]
[[Simple Harmonic Oscillator]]
[[Uses of Eigenvalues and Eigenvectors]]