10.6: The Mass-Spring-Damper System

Figure 1. Mass, spring, damper system

If one provides an initial displacement, , and velocity, , to the mass depicted in Figure then one finds that its displacement, at time satisfies

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 write Equation as

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 i.e., it follows that

If is real, i.e., if then both and are negative real numbers and decays to 0 without oscillation. If, on the contrary, is imaginary, i.e., , then

and so decays to 0 in an oscillatory fashion. When Equation holds the system is said to be overdamped while when Equation governs then we speak of the system as underdamped. It remains to discuss the case of critical damping.

• . 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 , we deduce from Equation that