10.6: The Mass-Spring-Damper System
( \newcommand{\kernel}{\mathrm{null}\,}\)
Figure 1. Mass, spring, damper system
If one provides an initial displacement,
where prime denotes differentiation with respect to time. It is customary to write this single second order equation as a pair of first order equations. More precisely, we set
and note that Equation becomes
Denoting
We recall from The Matrix Exponential module that
We shall proceed to compute the matrix exponential along the lines of The matrix Exponential via Eigenvalues and Eigenvectors module. To begin we record the resolvent
The eigenvalues are the roots of
We naturally consider two cases, the first being
. In this case the partial fraction expansion of yields
and so
If
and so
. In this case, and so we need only compute and . As there is but one and the are known to sum to the identity it follows that . Similarly, this equation dictates that
On substitution of this into this equation we find
Under the assumption, as above, that