The swing that dies in a straight line

The three ways of coming to rest found that an oscillator with a resisting force proportional to its speed can swing and fade, return without overshooting, or crawl back slowly, and that one number — the damping ratio — decides which. In every case the approach to rest is exponential: the swings shrink by the same factor every cycle, so they never quite reach zero, and the quality factor, the number of cycles it takes the energy to fall by a factor of $e^{2\pi}$, describes the whole decay.

That is the right description of a pendulum in air, a tuning fork, a quartz crystal and every oscillator whose losses come from something that pushes back in proportion to how fast it is moving. It is the wrong description of a mass sliding on a dry surface. The force that takes what it needs found that dry friction between two surfaces is not proportional to speed at all: once sliding, the friction is very nearly a constant force, set by the load and the surfaces and independent of how fast they slide, and before sliding it is whatever is needed, up to a limit, to stop them sliding. An oscillator damped by such a force has a different decay, a different way of stopping, and a different relation between where it starts and where it ends. Charles-Augustin de Coulomb, who measured the constancy of sliding friction in 1781, has given his name to it: Coulomb damping.

A fixed force, not a fixed fraction #

Take a mass on a spring of stiffness $k$, resting on a surface with sliding friction $F$. Pull it out and let go. The same constancy is what lets a rope wrapped round a post hold a ship: the turn that multiplies what a hand can hold found the friction on each element of the rope proportional to the force pressing it on the post and independent of how fast it slides. While it moves, the friction is a constant force opposed to the motion — so, during each half-swing, the mass feels the spring plus a constant push. A constant push on a spring does not change how it oscillates; it only moves the centre of the oscillation, by $F/k$. Each half-swing is therefore an exact half of a cosine, at exactly the undamped frequency, about a centre shifted by $F/k$ towards the side the mass came from.

Call $d = F/k$ the dead-band width. A mass released from rest at a distance $A_0$ swings about a centre at $d$ on its own side, so it reaches the far side at a distance $A_0 - 2d$. On the way back the friction reverses, the centre moves to $-d$, and the next turning point is at $A_0 - 4d$. The amplitude falls by exactly $2d$ every half-swing:

$$A_n = A_0 - 2nd.$$

The figure sets the two kinds of damping side by side, chosen to lose about the same amount on the first swing. The viscous swing shrinks along an exponential and is still oscillating long after the dry one has stopped. The dry swing shrinks along a straight line — the dashed line through its turning points — and stops. It does so without slowing its rhythm: every swing of the dry-friction mass takes exactly the undamped half-period, $\pi\sqrt{m/k}$, right up to the last.

Where the energy goes #

The straight line can be read off the energy as well. The spring stores $\tfrac{1}{2}kA^2$ at a turning point. On the way to the next turning point the mass slides a distance of about $2A$, and the friction takes $F$ times that distance, the work the floor that does no work contrasted with the forces that only redirect motion: friction against sliding is the one force in the problem that always takes. Setting the energy lost equal to the work done gives a loss in amplitude of $2F/k$ per half-swing, the same as the shifted-centre argument, with no reference to cosines at all.

The comparison with viscous damping is then a comparison of two powers. A force proportional to speed takes energy at a rate proportional to the square of the speed, so its loss per cycle grows as the square of the amplitude — the same fraction of the stored energy every cycle. A constant force takes energy in proportion to the distance slid, so its loss per cycle grows only as the amplitude itself — a fraction of the stored energy that grows as the swing shrinks. Engineers who want a quality factor for a frictional system anyway use an equivalent viscous damping, the viscous damping that would remove the same energy per cycle at the current amplitude; for Coulomb friction it is $2d/\pi A$, a damping ratio that rises without limit as the amplitude falls, which is the curve in the logarithmic plot below written as a number.

Why it stops, and where #

The mass stops at the first turning point that lands inside the band $|x| \le d$. At a turning point it is momentarily at rest, and from rest it can start moving again only if the spring’s pull, $k|x|$, exceeds the friction that has to be overcome. Inside the band the spring pulls with less than $F$, static friction supplies whatever is needed to hold it, and it stays.

So the motion ends after a definite number of half-swings, roughly $A_0/2d$, at a definite time. The bounces that add up to a stop found a bouncing ball also stopping in finite time, because each bounce lost a fixed fraction of the speed while the time between bounces shrank with it, so the infinitely many bounces added to a finite total. The dry-friction oscillator stops in finite time for a simpler reason: it has only finitely many swings to make, since each removes a fixed amount from a finite store.

The same motion drawn in phase space, as velocity against displacement, makes the geometry exact. An undamped oscillator traces a circle about the origin for ever. A viscously damped one traces a smooth spiral that winds inwards for ever. The dry-friction oscillator traces half a circle about the point $(+d, 0)$, then half a circle about $(-d, 0)$, then about $(+d, 0)$ again, each half-circle two widths smaller than the last — a spiral made of semicircles, with corners where the friction reverses — and stops dead when it reaches the band.

A friction that keeps time #

One property of the dry-friction oscillator is easy to miss in the figures. Every half-swing takes exactly the undamped half-period, from the first to the last, because during each half-swing the friction is a constant force and a constant force moves the centre of a spring’s oscillation without touching its frequency. A viscously damped oscillator runs slightly slow, by a fraction that grows with the square of its damping ratio. The dry-friction oscillator keeps perfect time while it dies.

That was no comfort to clockmakers, who wanted the opposite: an amplitude that stayed constant, not a frequency, since a pendulum’s period changes slightly with its amplitude, as the curve that does not ask where it started found Huygens trying to cure with cycloidal cheeks. A clock replaces the energy lost each swing through its escapement, and a pivot with sliding friction makes the loss depend on the pivot’s condition, its oil and its wear. The best pendulum clocks of the nineteenth and twentieth centuries hung their pendulums from thin spring strips instead of pivots, so that the only losses left were the air’s viscous drag and the spring’s internal friction — losses that scale smoothly with the swing and that an escapement can replace evenly.

A decay that curves #

The difference shows most plainly when the size of each swing is plotted on a logarithmic scale. A fixed fraction lost per swing is a straight line on such a plot; that is what viscous damping does, and its slope is the damping ratio. A fixed amount lost per swing is a curve that bends down: the fraction lost grows as the swings shrink, from a fifth of the first swing to nearly half of the fourth to all of the last. Plotted on ordinary axes the dry decay is the straight line and the viscous one the curve. Neither plot is more correct; each makes one law look simple and the other look complicated, and the choice of axes is the choice of which question the record is being asked.

That is the practical test. An engineer who records a structure’s free vibration after a knock, and finds that its logarithm falls in a straight line, has a damping that behaves like viscosity, and can quote a quality factor. One who finds the logarithm curving down towards a sudden stop has friction in the joints, and no quality factor will describe it: the structure is damped heavily at small amplitude and lightly at large, which matters when deciding whether a small disturbance will die out. Bolted and riveted structures, whose joints slip slightly over their contact surfaces, behave in between, with the micro-slip that the contact that slips before it slides found at the edges of every clamped contact giving a loss that grows faster than linearly with amplitude.

The stopping place remembers the start #

The mass stops at $A_0 - 2nd$ for the first $n$ that brings it within the band, on whichever side it happens to be. Released a little further out, it stops a little further from the centre, until one more half-swing becomes possible and it stops on the other side. The rest position is a zigzag across the band, determined by the release point modulo two band-widths, and it is at the spring’s natural position only by coincidence.

That is the reason an aneroid barometer is tapped before it is read, and why a spring balance with a sticky pivot gives a reading that depends on whether the load was added slowly from below or let down from above. Each is an oscillator with friction in its mechanism, and each comes to rest somewhere in its dead band. A tap gives the needle enough energy to leave the band and settle again — not necessarily at the true value, but somewhere new in the band, and a few taps show how wide the band is. The same zigzag explains a familiar annoyance with mechanical bathroom scales and analogue meters: step on and off twice and the reading differs, by up to the width of the band, because the needle approached from a different overshoot each time. The error is not random noise to be averaged away by repetition; it is a definite function of how the needle was brought to rest, and repeated readings taken the same way agree with each other while disagreeing with the truth. Precision instruments are designed to make the band small compared with the reading’s resolution: jewelled bearings, flexure pivots with no sliding at all, or a suspension by a taut fibre, which replaces friction with a slight elastic restoring torque.

When both act #

Real oscillators usually have both. A door closer, a car’s suspension, a seismometer’s pendulum and a violin string all have some loss proportional to speed and some constant friction. At large amplitude the viscous loss, growing with the speed, is the larger: the decay starts as a nearly straight line on a logarithmic plot. At small amplitude the constant friction is the larger, and the decay curves down to a stop. A single decay record, read from the large swings to the small, separates them: the early slope gives the viscous damping, the late curvature the dry friction.

The same separation is used to measure friction itself. A torsion pendulum whose bob rests on the surface being tested, set swinging and allowed to decay, gives the sliding friction from the straight-line part of its decay and the static friction from where it stops, without any force gauge at all.

Why the quietest instruments avoid it #

The kind of damping matters for more than how a swing dies. An oscillator in thermal equilibrium jiggles with a thermal energy of $kT$, and the way that jiggle is spread over frequencies is fixed by the way the oscillator loses energy: a mechanism that takes energy can also give it back, randomly, and the loss and the noise are two faces of one coupling to the surroundings. The noise a high Q moves out of the way found that a resonator with very low loss concentrates its thermal motion into a narrow band at its resonance and leaves the rest of the spectrum quiet. Friction in a sliding joint does the opposite. It is a loss that switches on and off with stick and slip, it scatters energy into a broad band of frequencies, and it produces sudden jumps rather than a smooth jiggle.

That is why the most sensitive mechanical instruments built — the suspended mirrors of gravitational-wave detectors, the torsion balances that test the equivalence principle — contain no sliding contact anywhere in the parts that matter. Their test masses hang from fibres of fused silica welded in place, whose losses are tiny, internal and smooth, and the designers’ first rule is that nothing in the suspension may rub. A single sliding contact, with its dead band and its sudden slips, would add more noise than every other source together.

What the clean picture leaves out #

Static and sliding friction differ. The force needed to start sliding is usually somewhat larger than the force that resists steady sliding. That widens the band where the mass sticks at a turning point to $F_s/k$, slightly larger than the $d$ governing the decay, and it is the difference that makes stick-slip oscillation possible when the surface is driven rather than the mass released; the chatter a stiffer holder removes found that kind of self-excited vibration in a cutting tool.

Sliding friction depends a little on speed. The force usually falls slightly as the speed rises from zero and then rises again, as the grip that is not a coefficient found for rubber. A falling friction at low speed feeds energy into the oscillation rather than removing it and can destabilise it; the constant-force model is the middle of a more complicated curve.

The normal force must be steady. The friction is the normal force times a coefficient. If the oscillation itself changes the normal force — a pendulum whose pivot load varies with the swing — the loss per half-swing varies too, and the straight line bends.

Still open: what friction does at the smallest swings #

The constant-force law is an excellent description of sliding between ordinary surfaces, and it fails, in a particular way, at very small amplitudes. Before two surfaces slide, their contact deforms elastically: the asperities touching each other bend, and a small displacement is resisted by a spring-like force rather than by friction at all. That pre-sliding regime, a few micrometres or less for most engineering contacts, means that the mass in the dead band is not stuck rigidly but held by a stiff, slightly lossy spring, and vibrations smaller than the pre-sliding displacement pass through the contact without ever making it slide.

How the transition from elastic pre-sliding to full sliding happens — over what distance, with what loss, and how it depends on the surfaces’ roughness and the time they have been in contact — matters for the positioning of precision machines, for the behaviour of faults in the Earth’s crust between earthquakes, and for the damping of bolted structures, and it is described by a family of empirical friction models rather than by any single law. The picture drawn here is the limit of large swings, where those details are invisible. In that limit it is exact: a constant force removes a constant amount, so a swing damped by dry friction dies in a straight line, at a countable time, at a place that remembers where it began.