The mass that makes another stand still

A machine with a rotating part shakes at the frequency it turns at. If that frequency is near one of the machine’s own natural frequencies, the shaking is amplified — by a factor of ten, or fifty, or whatever the damping allows — and something eventually breaks. The obvious repairs are to change the machine’s natural frequency, to change the running speed, or to add damping. There is a fourth repair which is stranger than any of them: attach a second, much smaller oscillator, tuned to the offending frequency, and the machine stops moving altogether.

Exactly zero is the claim, and it is worth being careful about what kind of claim it is. It is not a statement that the response is small, or that it is zero to some order in a small parameter. In the undamped calculation the amplitude of the protected mass at the tuned frequency is algebraically zero, and it is zero for any absorber mass whatever — a gram or a tonne, tuned to the same frequency, gives the same perfect protection.

What is being cancelled #

The bare machine’s response is the familiar one.

Write the steady state as a pair of simultaneous equations. With the machine’s mass $m_1$ and stiffness $k_1$, the absorber’s $m_2$ and $k_2$, and a force $F_0\cos\omega t$ applied to the machine:

$$(k_1 + k_2 - \omega^2 m_1)X_1 - k_2 X_2 = F_0,$$

$$-k_2 X_1 + (k_2 - \omega^2 m_2)X_2 = 0.$$

Solving by Cramer’s rule puts the second equation’s diagonal term in the numerator of $X_1$:

$$X_1 = \frac{(k_2 - \omega^2 m_2)F_0}{(k_1 + k_2 - \omega^2 m_1)(k_2 - \omega^2 m_2) - k_2^2}.$$

The numerator vanishes when $\omega^2 = k_2/m_2$ — the absorber’s own natural frequency, and nothing else. That is the whole derivation, and the algebra has a physical reading that is much more useful than the formula.

So the mechanism is a force cancellation, not a dissipation. The absorber is a device for generating a sinusoidal force of a chosen amplitude and phase, and it generates one that is minus the drive. The machine then has no net force on it, and a body with no net force on it in a steady sinusoidal state does not move.

The absorber’s own amplitude follows from that. It has to supply a force $F_0$, and it supplies it through a spring of stiffness $k_2$, so its amplitude is $F_0/k_2$. A light absorber has a soft spring and must therefore move a long way — with a mass ratio of a tenth and equal tuning, it moves ten times the static deflection of the machine. Halving the absorber’s mass doubles its stroke. That, and not any loss of protection, is what limits how small an absorber can be made: eventually the thing has nowhere to go.

One consequence is worth stating plainly because it sounds wrong. The protection does not improve as the absorber is made heavier. A one-gram absorber correctly tuned holds a one-tonne machine exactly as still as a hundred-kilogram absorber correctly tuned — the zero is a zero either way. Everything the extra mass buys is bought elsewhere: a shorter stroke, a wider useful band, and more tolerance of mistuning. Anyone who expects “more mass, more protection” is thinking of an absorber as a sponge, and it is not one.

The one peak that became two #

Nothing is free. The denominator of $X_1$ is a quadratic in $\omega^2$ with two roots, and those roots are the natural frequencies of the two-mass system.

For equal tuning the two roots are

$$\left(\frac{\omega_\pm}{\omega_1}\right)^2 = 1 + \frac{\mu}{2} \pm \sqrt{\mu + \frac{\mu^2}{4}}, \qquad \mu = \frac{m_2}{m_1},$$

which for a ten per cent absorber gives 0.854 and 1.171. Both are resonances, and in the undamped calculation both are infinite. The machine has gone from having one frequency it must avoid to having two, with a safe window between them.

That square root is the design constraint, and the shape of it — two levels pushed apart by a coupling, never allowed to meet — is the crossing that never happens in a system with no quantum mechanics in it. A one per cent absorber protects perfectly at exactly one frequency and has a taller-than-original peak five per cent away on either side; a machine whose speed wanders by more than that is worse off than before it was fitted. The safe band is roughly $\sqrt\mu$ wide in fractional frequency, so a band of ten per cent needs a mass ratio of about one per cent and a band of thirty per cent needs about ten.

The same zero in a circuit #

The mechanism has an electrical twin that is worth meeting, because it makes the strangeness go away.

Put an inductor and a capacitor in series and hang the pair across a signal path. At the frequency where their reactances are equal and opposite, the series pair has zero impedance — it is a short circuit — and the voltage at that point of the path is driven to zero however hard the source pushes. That is a notch filter, it removes one frequency completely, and nobody finds it mysterious. Nothing is absorbing the signal; a branch has simply been provided that takes all of the current at that one frequency, and a path with something across it that will take any current at zero voltage cannot have a voltage.

The mechanical absorber is that circuit, term for term. The absorber’s mass is the inductor, its spring is the capacitor, and the series pair hangs across the machine. At the tuned frequency the branch presents zero impedance to the applied force, takes all of it, and leaves nothing to move the machine. The reason its own amplitude is large is the reason the current in a series resonant branch is large, and the reason damping fills in the zero is that a resistor in the branch stops the impedance from reaching zero.

The general name for the phenomenon is an antiresonance, and it is worth having, because it names what is otherwise easy to describe as a resonance behaving oddly. A resonance is a pole of the response; an antiresonance is a zero of it. Adding a degree of freedom to a system adds one of each, which is exactly what the figures show: one new peak and one new hole, arriving together.

Where the absorber has to be bolted #

The claim that any mass works, however small, needs one qualification that decides every real installation: the mass ratio in the formulas is not a ratio to the machine’s mass but to the modal mass — how much of the structure actually participates in the mode being suppressed, weighted by how much it moves.

That has two consequences. An absorber must be mounted where the mode has amplitude; put it at a node and it is a lump of metal doing nothing, because the mode’s motion cannot reach it through the spring. And the mass ratio it achieves is far better than a ratio to the whole structure suggests, because a structure swaying in its first mode has most of its material barely moving.

The tall-building case is the one everybody has seen. A pendulum of a few hundred tonnes hung near the top of a skyscraper weighing hundreds of thousands of tonnes looks like a mass ratio of a thousandth, which by the square-root rule would give a useless band. Against the modal mass of the first sway mode it is a per cent or two, and the band it protects is a few per cent wide — which is enough, because a building’s sway frequency is a fixed property of the structure rather than something that wanders like a machine’s running speed.

The bridge case is the one that made the technique famous, and it is the same arithmetic with a different drive. A footbridge that swayed laterally under the crowd that was walking on it had absorbers tuned to the sway frequency added afterwards, along with dissipative dampers, and the combination is what the figures above describe: the tuned masses provide the cancellation and the dampers keep the two new modes from being resonances of their own.

The history is a ship #

The device was patented in 1909 by Frahm, for rolling ships, and the first version had no springs in it at all: two tanks of water connected by a duct, arranged so that the water sloshed from side to side at the ship’s own roll period and pushed back at the right phase. The tuning was set by the geometry of the duct, and the “mass” was a few per cent of the ship’s displacement.

Den Hartog worked out the damped optimum in the following decades, and it is his fixed-point argument the figures above measure. The reason the analysis took twenty years after the device is instructive: the undamped version is a two-line calculation and gives an infinite answer at two frequencies, which is obviously wrong and obviously not fatal, and the question of what the best compromise is has no answer until somebody notices that all the curves pass through two points.

What damping buys, and what it costs #

The two infinite peaks are an artefact of assuming no losses, and a real system has some. But damping placed in the absorber does something more interesting than rounding off the peaks: it destroys the zero.

The reason the zero goes is direct: a damped absorber’s spring-and-dashpot force is no longer exactly opposite the drive, because a dashpot’s force is a quarter cycle out of step with a spring’s. The cancellation becomes partial, and how partial depends on how much damping there is.

There is a way to see why damping must destroy the zero without any algebra. The zero exists because two forces cancel exactly, and exact cancellation of two sinusoids requires them to be equal in size and opposite in phase. A spring’s force is in phase with a displacement and a dashpot’s is a quarter cycle away from it, so a spring-and-dashpot combination produces a force at some intermediate phase — never exactly opposite the drive, whatever its size. The cancellation is then partial by an amount set by the phase error, which is set by the damping ratio, and the depth of the hole is the price of the phase error.

The fixed points are the useful structure. Every curve in that figure crosses at the same two frequencies whatever the damping, including the undamped one and the infinitely damped one, so no amount of damping can bring the response below their height anywhere near them. The best possible absorber therefore brings both peaks down to exactly that height, and the damping that achieves it is the classical optimum — around 0.16 of critical for a ten per cent absorber, which is what the drawn curves measure. Everything above that height is avoidable and everything below is not.

Damping applied to the machine itself changes how a disturbance decays and lowers the resonant peak, and it can never produce a zero of the response. That is the distinction the whole essay turns on: a damper takes energy out and an absorber cancels a motion, and only the second can bring a particular frequency’s response to nothing. Adding damping to a structure makes it less bad everywhere; adding an absorber makes it perfect at one frequency and worse at two others.

There is a second lesson buried in the existence of the fixed points, and it is a general one about design. A quantity that cannot be moved by the parameter being tuned is a bound, and finding it is usually more valuable than optimising. The two fixed points here are visible on the figure before any optimisation is attempted, they set the best achievable performance immediately, and they tell a designer whether the whole approach is worth pursuing before any effort is spent choosing a damping ratio. If the fixed points are already too high, no absorber of that mass ratio will do, and the answer is more mass rather than better tuning.

The quarter cycle that makes it work #

The phase is where the mechanism is easiest to see and hardest to guess.

The phase is the part that makes it work, and it is a quarter cycle. Below its natural frequency an oscillator moves in step with the drive; above it, in opposition; and exactly at it, a quarter cycle behind. Tune the absorber so that the machine’s driving frequency is the absorber’s natural one, and the absorber’s motion is a quarter cycle behind the drive — which puts the force it returns exactly opposite to the drive, and the two cancel.

At the tuned frequency the absorber is being driven exactly at its own resonance, so it responds a quarter cycle behind the motion of its base — and its base is not moving, which is the paradox that makes people distrust the calculation. The resolution is that the absorber is not being driven by the base’s motion. It is being driven by the base’s force, transmitted through the spring, and a spring transmits force through zero displacement of one end perfectly well as long as the other end is moving.

In the transient, energy passes back and forth between the two masses: the drive puts it into the machine, the machine hands it to the absorber, and the absorber hands it back. At steady state the handing-over has settled into a pattern where the machine keeps none of it — which is the honest description of what an absorber does, and it explains why the absorber’s own amplitude is large. It is doing all the moving.

The transient behaviour also explains a practical detail that is otherwise puzzling. Absorbers are routinely fitted with a small amount of damping even where the drive frequency is known precisely and the tuning is stable, and the reason is not the steady state at all: an undamped two-mass system rings for ever at both of its new natural frequencies once anything excites them, and everything excites them — a switch-on, a load change, a gust. The damping is there to make the transients go away, and it costs a little of the steady-state protection to do it.

Where the model stops #

Two masses is one degree of freedom each, and a real machine has many. An absorber tuned to one mode of a structure is invisible to the others only if it happens to sit at a node of them, which is rarely arranged. In practice the absorber couples to every mode whose shape is non-zero at the mounting point, and the design problem is a matrix one.

Everything here is linear. A real spring is not, especially at the amplitude an absorber runs at — ten static deflections is a large stroke — and a nonlinear absorber’s resonance leans, so the frequency it protects depends on how hard it is driven. Absorbers built for large amplitudes are deliberately made nonlinear for exactly this reason, which trades a sharp zero at one frequency for a shallow one over a range.

The tuning is assumed to hold. The whole device rests on $k_2/m_2$ matching the drive, and both drift: springs age and soften, temperature changes stiffness, and the machine’s own frequency moves as it wears. A one per cent absorber whose tuning has drifted by two per cent is not a slightly worse absorber, it is a resonance.

And the analysis is a steady state. Everything computed here is the response after transients have died away, which for a lightly damped two-mass system can take a great many cycles. Switch a machine on and the absorber does nothing useful for a while — the response builds up through beats between the two new modes, and the peak transient amplitude of the machine during that build-up can exceed anything in the steady-state curves.

What the pictures cannot show #

The figures draw amplitude against frequency, which is a summary of infinitely many steady states and not a picture of anything that happens. Nothing in them shows time, and the two most practically important facts about an absorber — how long it takes to become effective, and what happens while a machine runs up through the lower resonance on its way to operating speed — are both facts about time.

Nor is the force the absorber applies to the machine visible in any curve but the motion one. It is the whole mechanism, and on the frequency-response figures it appears only as the absence of a response.

The smallest one in service #

The largest absorbers are the ones photographed — a several-hundred-tonne sphere hung inside a tower — and the most numerous are the size of a fist.

Look at a high-voltage transmission line where it leaves a pylon and there is usually a small dumbbell clamped a short distance along the conductor: two weights on a short length of flexible cable. It is this page’s device, built for the vibration a steady crosswind excites when vortices shed from the conductor at a frequency near one of the span’s own modes. Left alone, that vibration is small in amplitude and never stops, and the conductor eventually fails from fatigue where it is clamped.

The dumbbell is tuned so that its own resonance sits in the band the wind excites, and it moves instead of the line. Nothing about the physics differs from the tower’s; only the mass, which is a couple of kilograms rather than a couple of hundred tonnes, and the failure being prevented, which is a crack rather than a complaint.

Where the ladder goes next #

This ladder began with the frequency that gets an answer and continued with the swing that is pumped rather than pushed. This rung is the third way to change a resonance: not by driving it differently, but by adding a degree of freedom whose response cancels the drive. The rungs after it are the optimally tuned damped absorber worked out properly, with the fixed points located and the tuning ratio that puts them at equal heights; multi-mode absorbers; and the continuous limit, in which many absorbers at many frequencies become a band gap — the same zero, repeated, turning into a range of frequencies a structure will not transmit at all.

The habit worth carrying away is the distinction between opposing and cancelling. A damper opposes a motion and can only ever make it smaller; a second oscillator cancels a force and can make the motion zero. The two produce curves of the same general shape, and only one of them can produce a hole.