# 8.2.3 The friction damper

Consider a mass-spring oscillator, whose motion is resisted by ideal Coulomb friction. If the displacement of the mass is $x(t)$, there are three different equations that might govern the next stage of motion, depending on the value of the velocity $\dot{x}$. If $\dot{x}>0$, the mass is sliding forwards so it is resisted by a friction force. But for Coulomb friction that force is simply a constant, independent of the motion of the mass, so the governing equation can be written in the form

$$\ddot{x}+p^2 x =-F\mathrm{~~~~~~~}(\dot{x}>0) \tag{1}$$

where $p$ is the resonance frequency of the oscillator in the absence of the friction force, and $F$ describes the constant friction force. The solution of this equation must take the form

$$x=A \cos p(t-t_0) – F/p^2\mathrm{~~~~~~~}(\dot{x}>0) \tag{2}$$

for some values of the constants $A$ and $t_0$. If $\dot{x}<0$, the mass is sliding the other way. The equation is the same, except that the friction force is reversed:

$$\ddot{x}+p^2 x =F\mathrm{~~~~~~~}(\dot{x}<0). \tag{3}$$

This time the solution must take the form

$$x=B \cos p(t-t_0) + F/p^2\mathrm{~~~~~~~}(\dot{x}<0) \tag{4}$$

for some values of the constants $B$ and $t_0$. The third possibility is that $\dot{x}=0$ so the mass is stationary. This is possible if the spring force lies within the limits of sticking friction: $p^2 x$ must lie between $\pm F$, so $x$ must lie between $\pm F/p^2$.

Suppose we start the motion with the mass at rest at $x=0$, and we apply a positive initial velocity $V$ so that sliding commences with $\dot{x}>0$. The solution follows eq. (2), with suitable values of $A$ and $t_0$ to satisfy the initial conditions. The resulting sine wave is shown in the dashed line in Fig. 1. Notice the asymmetric placement of the sine wave: it is centred on $-F/p^2$, shown as the lower of the two dash-dot lines in the plot. But the actual motion will only follow this while $\dot{x}>0$, so in fact the only part that is relevant is the portion shown in red.

Now $\dot{x}<0$, so the motion of the mass now follows eq. (4) with suitable values of the constants $B$ and $t_0$ so that it joins on to the red curve. This time the sine wave, shown in Fig. 2, is centred on $F/p^2$, shown as the upper of the two dash-dot lines. Again, this solution is only relevant while the velocity remains negative, so the part we want from this sine wave is the portion plotted in blue.
After that, the pattern repeats. Each alternate half-cycle of sine wave is centred around the upper or lower dash-dot line. Sooner or later, the point of zero velocity at the end of one of these half-cycles of motion will lie between those two lines, and at that point the motion stops. The mass simply sits still thereafter, held in place by the friction force. The case plotted in Figs. 1 and 2 is quite extreme, to make the pattern clear: the motion will stop soon after the portion plotted in red and blue. The case shown in Fig. 11 of section 8.2 had a lower friction force $F$, so that the motion continued for more cycles and the linear envelope of decay was clearly visible.