The third of itself a spring carries

Every minimum is a parabola found that any system near a stable equilibrium oscillates harmonically, at a frequency set by the curvature of its potential and the mass that moves. The two pendulums that swap coupled two such oscillators, why heating a perfect spring changes nothing found what the parabola leaves out at large amplitude, and the spectrum that weighs a bond used the frequency of a molecule’s vibration to measure the stiffness of a chemical bond, with the reduced mass of the two atoms standing in for “the mass that moves”.

That last phrase hides a question in the most ordinary oscillator of all. A mass hangs on a spring, and the spring has mass too. When the mass bounces, the spring’s coils move as well — the bottom coils with the mass, the top coils hardly at all — and their motion carries kinetic energy that the formula $\omega = \sqrt{k/M}$ leaves out. The standard repair is to add a third of the spring’s mass to the load. This essay asks where the third comes from, how good it is, and what happens to a spring heavy enough that no fixed fraction describes it: it stops being a single oscillator and becomes a column along which waves run, with notes of its own.

Where the third comes from #

The argument is Lord Rayleigh’s, from his Theory of Sound of 1877. Suppose the spring stretches uniformly as the mass moves, so that a coil a fraction $y$ of the way down the spring moves at $y$ times the mass’s speed $v$. A short length $dy$ of spring has mass $m_s\,dy$, and its kinetic energy is $\tfrac{1}{2}m_s\,dy\,(yv)^2$. Adding up the whole spring,

$$T_{\text{spring}} = \tfrac{1}{2}\,m_s v^2 \int_0^1 y^2\,dy = \tfrac{1}{2}\,\frac{m_s}{3}\,v^2.$$

The spring’s kinetic energy is exactly that of a mass $m_s/3$ moving with the load. Its potential energy is unchanged — a uniformly stretched spring stores $\tfrac{1}{2}kx^2$ however heavy it is — so the system is an oscillator of stiffness $k$ and mass $M + m_s/3$. The third is the average of $y^2$ along the spring: most of the coils move less than the load, the top ones barely at all, and they count in proportion to the square of their share of its motion.

That is an assumption about how the spring moves, and it needs checking. A spring is a continuous elastic column, and when it vibrates it is carrying compression waves up and down its length; the uniform stretch is only the shape it takes when those waves are slow compared with the time a wave takes to run along it.

The shape the spring actually takes #

The exact problem is a column of mass $m_s$ and stiffness $k$, fixed at the top and loaded at the bottom by $M$. Its slowest mode has the coils moving as a sine of their position, $\sin(\beta y)$, where $\beta$ is fixed by the load through

$$\beta \tan \beta = \frac{m_s}{M},$$

and the frequency is $\beta\sqrt{k/m_s}$.

The figure draws the shape for four springs. A spring a tenth of the load’s mass stretches almost exactly uniformly: its curve lies on the straight line. A spring as heavy as the load bows outward slightly, and one ten times heavier bows further, until a spring with nothing on it moves as a quarter of a sine wave — a node at the fixed top, an antinode at the free bottom, like the air in a pipe closed at one end. As the shape bows outward every coil moves further, relative to the load, than the straight line allows, so more of the spring is moving and the true effective share rises above a third.

The two descriptions meet in the limit of a light spring. When $m_s/M$ is small, $\beta$ is small, $\sin\beta y \approx \beta y$ is a straight line, and expanding $\beta\tan\beta$ in powers of $\beta$ gives exactly $\omega^2 = k/(M + m_s/3)$, with corrections of order $(m_s/M)^2$. The third is the first term of an exact answer.

How good the third is #

The figure compares the three answers across four decades of spring mass. The massless-spring formula is a horizontal line and drifts away from the truth as soon as the spring weighs anything: a spring a tenth of the mass already lowers the frequency by 1.7 per cent. The third tracks the exact curve almost perfectly until the spring becomes as heavy as the load, and even then — the dashed and solid curves separating only slightly — it is 0.66 per cent high.

On a logarithmic scale the difference in quality is stark. Ignoring the spring’s mass makes an error proportional to the spring’s mass; counting a third of it makes an error proportional to its square, so every halving of the spring’s mass quarters the error. For the springs of a school laboratory — twenty grams carrying a hundred — the textbook formula is three per cent fast and the corrected one is right to a few parts in ten thousand. Only when the spring is several times heavier than the load does the third fail badly, and a spring on its own, where there is no load at all, gives $\sqrt{3}$ in units where the truth is $\pi/2$: ten per cent high.

Why the estimate is always high #

The third is always an overestimate of the frequency, never an underestimate, and that is not a coincidence of this problem. It is a theorem, and it is the reason the estimate is so good.

Rayleigh’s principle says that if a vibrating system is assumed to move in some shape, and its frequency is computed by equating the maximum kinetic energy of that shape to its maximum potential energy, the result can never be below the true lowest frequency, and equals it only if the shape is exactly right. The true shape is the one that makes the frequency stationary — a variational principle of the same family as least action — and near a stationary point a quantity changes only in second order. An assumed shape that is wrong by a small amount therefore gives a frequency wrong by the square of that amount. The straight-line shape differs from the true sine by an amount proportional to $m_s/M$, and the frequency it predicts is wrong by an amount proportional to $(m_s/M)^2$, which is what the error figure shows.

That is a general lesson about approximations of this kind. A rough guess at a shape gives a good estimate of an energy or a frequency, because the energy is insensitive to small changes of shape at its extremum; the same principle lets a trial wavefunction give a good estimate of an atom’s ground-state energy, and it is why engineers compute natural frequencies of bridges and aircraft wings from guessed deflection shapes and get them right to a per cent. The price is that the estimate says nothing reliable about the shape itself, which can be wrong by far more than the frequency.

When a spring is a spring, and when it is a wave #

There is a single number that says whether the lumped picture — a mass and a massless spring, with a third of the spring’s mass added — can be trusted, and it has nothing specifically to do with springs. A disturbance at one end of the spring takes a time $\sqrt{m_s/k}$ to travel to the other, because the column’s wave speed is its length times $\sqrt{k/m_s}$. The bounce takes $2\pi\sqrt{M/k}$ or so. Their ratio is $\sqrt{m_s/M}/2\pi$. For a spring a tenth of the load’s mass a wave crosses the spring in five per cent of a bounce, twenty times each period, and the spring has time to share out any stretch evenly before the load has moved appreciably: it stretches uniformly, which is Rayleigh’s assumption. For a spring as heavy as the load a wave takes a sixth of a bounce to cross, and the stretch cannot keep up with the load; the shape bows, and the third starts to fail.

The same criterion separates lumped from distributed descriptions everywhere. An electrical circuit can be drawn with separate inductors and capacitors as long as the signals’ wavelengths are much longer than the circuit, so that the current is the same all along each wire; at frequencies where a wavelength fits inside the circuit the wires become transmission lines, carrying waves with reflections at every junction. A building sways as a single mass on a spring in a gentle wind and as a column of travelling waves in an earthquake. In each case the “mass” and the “spring” are idealisations of one continuous body, valid when that body’s own waves are fast compared with the motion being described, and the correction for their being slow begins with a fraction of the body’s mass, like the third.

A measurement that finds the third #

The correction is visible in the most elementary laboratory measurement. Hang a series of masses on a spring, time the oscillations, and plot the square of the period against the load. The massless formula predicts a straight line through the origin, $T^2 = 4\pi^2 M/k$. The measured points lie on a straight line with the right slope, but it does not pass through the origin: it crosses the axis at a negative load, $M = -m_s/3$. Extrapolated, the experiment says that a mass of minus a third of the spring’s own would make the period vanish, which is the spring’s effective mass read off as an intercept.

With a spring of 30 grams the intercept is at minus 10 grams, easily measured with kitchen scales and a stopwatch. A student who plots period against mass and forces the line through zero gets a stiffness a few per cent off and a residual pattern that curves; one who lets the intercept float measures the third directly, and can check it by weighing the spring. It is one of the few experiments in which an approximation appears as a number on an axis.

The notes above the bounce #

A spring heavy enough that the third fails is telling something more interesting than a correction: it is not one oscillator but many.

The equation $\beta\tan\beta = m_s/M$ has a root in every interval of length $\pi$, and each is a mode: a standing compression wave with one, two, three more nodes along the coils. With a heavy load the second and third are far above the bounce — ten and twenty times its frequency for a spring a tenth of the load’s mass — because in them the load hardly moves and the waves run between the fixed top and an almost-fixed bottom, like a string clamped at both ends. As the spring gets heavier they come down, and for a spring alone they settle at three and five times the fundamental, the odd harmonics of a column closed at one end, the same series how many ways there are to vibrate counted for a pipe. In between they are not whole multiples of anything, which is why a heavy spring struck sounds like a clank rather than a note.

These modes are called surge, and they matter wherever a spring is driven fast. The valves of a car engine are closed by coil springs and opened by a camshaft, and the cam’s motion, repeated once every two turns of the crankshaft, contains many harmonics. When one of them reaches the valve spring’s first surge frequency, waves run up and down the coils, the force the spring exerts on the valve fluctuates, and at high engine speed the valve can bounce off its seat or fail to follow the cam at all — valve float. Engine designers fight surge with springs whose coils are wound closer together at one end, so that the closely wound coils touch as the spring compresses and the spring’s stiffness and surge frequency change during the stroke, and with nested pairs of springs with different surge frequencies. The same modes carry road noise through a car’s suspension springs. A spring engineer thinks of a coil spring as a waveguide, which is what the overtone figure says it is.

The same share, in other shapes #

The idea of an effective mass reaches well beyond coil springs.

Any flexible part carrying a load at one point adds some fraction of its own mass to the load, and the fraction is set by the shape it moves in. For a cantilever — a diving board, a tuning-fork tine, the tiny silicon beam of an atomic-force microscope — Rayleigh’s method with the beam’s static bending shape gives $33/140$ of the beam’s mass, a little under a quarter, because most of a cantilever’s mass is near its clamped root and barely moves. The exact first mode of a cantilever with nothing on its tip gives 0.2427. That number is used every day to calibrate atomic-force microscopes: the stiffness of a cantilever is found by measuring its thermal vibration or its resonant frequency, and converting frequency to stiffness needs its effective mass. The dent that raises the note used the same logic in reverse: a mass added to a vibrating system changes its frequency in proportion to how much that point moves, which is exactly the weighting the effective mass expresses.

The effective mass is also different from the reduced mass of the spectrum that weighs a bond, and the difference is instructive. A reduced mass accounts for two bodies both moving, about their common centre of mass, each fully; an effective mass accounts for one distributed body moving by different amounts at different places. Both answer the same question — what single mass, placed at the point whose motion is being followed, would store the same kinetic energy — and both are exact only for the motion they assume.

Where the model stops #

The calculation treats the spring as a uniform elastic column that moves only along its length. A real helical spring is a coiled wire, and a wave travelling along it twists and bends the wire as well as compressing it; for most springs the compression wave dominates, but for springs with few coils or large coil angles the helical geometry couples compression to rotation of the spring’s end and shifts the surge frequencies. At large amplitudes the coils can touch — coil clash — which makes the spring stiffer suddenly and the motion non-sinusoidal. Real springs dissipate energy internally and at their mountings, which broadens the surge resonances without moving them much.

Gravity, perhaps surprisingly, does not affect any of the results for an ideal spring. A heavy spring hanging vertically is stretched more at the top than at the bottom, because each coil supports all the coils beneath it, but for a spring that obeys Hooke’s law gravity only moves the equilibrium position of each coil; the oscillations about that equilibrium are the same as they would be on a horizontal frictionless table.

What the pictures cannot show #

The figures show amplitudes and frequencies, the time-independent parts of the motion. A real spring started by pulling the load down and letting go is not in a single mode: it starts in a mixture, mostly the bounce with a little of each surge mode, and the higher modes appear as a fast shimmer on the coils that dies away while the bounce continues. The figures also cannot show what happens in a spring that is released rather than oscillating — a stretched spring let go at the top, whose bottom does not move at all until the wave of release, running down the coils, arrives at it. That is the same wave the surge modes are made of, seen as a single front rather than as a standing pattern.

Still open: weighing things by their effect on a resonator #

The effective mass has become a measuring instrument in its own right. Nanomechanical resonators — beams and membranes a few micrometres long, vibrating at millions of cycles a second — change their frequency measurably when a single molecule or virus lands on them, and the shift is the particle’s mass divided by twice the resonator’s effective mass, weighted by how much the landing point moves. Resonators of this kind have weighed individual proteins and in ideal conditions resolved a mass about that of a single proton. What limits them is partly noise and partly the effective mass itself: a particle’s contribution depends on where it lands, the landing point is not known, and extracting both mass and position requires reading several modes at once and inverting their shifts. How far this can be pushed — whether molecular masses can be read to the precision of conventional mass spectrometry without ionising the molecule — is being worked on in several laboratories.

The habit worth carrying away is to ask what shape an estimate assumes, and whether the answer is at a stationary point. A spring’s own mass adds a third of itself to the load because its coils move in proportion to their distance from the fixed end, and the estimate is good to the square of the spring-to-load ratio because frequency is stationary with respect to shape — where the third finally fails, the spring has become a column of waves with notes of its own. The same reasoning weighs cantilevers in microscopes and molecules on resonators, and warns an engine designer when a spring is about to sing.