Animated Equations

Mass on a spring

The motion of a mass on a spring is governed by the differential equation

m x´´ - b x´ - k x = 0
where m is the mass at the end of the spring, x is the displacement of the spring from its rest position, b is a damping factor, and k is the spring constant. This type of system is described in Section 3.4 of Edwards & Penney (1996).

This second-order system can be rewritten as two first-order equations

x´ = v
v´ = -[b v + k x] / m
where v is the velocity of the mass.

The animation at the top of the page shows the behaviour of the system with m = 1, b = 0, k = 5 and initial conditions x(0) = -1 and v(0) = 0. The motion of the mass was calculated using a Runge-Kutta method.

Damping

If b is non-zero then the system is said to be damped. If b2 < 4 k m then the system is underdamped. The next animation shows what happens when the damping factor is changed to b = 0.5.

Overdamping

If the damping factor b > 4 k m then the system does not oscillate, and the system is said to be overdamped. The final animation shows the behaviour of the system when the damping factor is increased to b = 10.