A mass-spring-damper system is second order. It has the differential equation:

where y(t) is the displacement, x(t) is the applied force, m is the mass, c is the damping coefficient, and k is the spring coefficient. The Laplace Transform between the force and the displacement is:

The qualitative behavior of the system depends on the damping ratio. To find the damping ratio, consider a general system of the form:

The damping ratio, zeta, is found for the mass-spring-damper system by comparing denominators.

If 0<zeta<1, the system is underdamped and will respond to a step input with damped oscillations. If zeta = 0, the system is marginally stable and will respond with undamped oscillations. If zeta > 1, the system is overdamped and will respond to a step input with no oscillations. If zeta = 1, the system is marginally damped and will respond to a step input without oscillations.