The lamp that stays where it is put

A balanced-arm desk lamp can be pushed to any position within its reach and let go, and it stays there. It is not held there by stiff joints — its joints turn freely — and it is not locked. It is balanced: at every pose the springs store exactly the energy the weight of the arms and head has given up, so the lamp’s total potential energy is the same everywhere and no pose is downhill from any other. A machine that stays where it is put is a machine whose potential energy is a constant function of its configuration.

Like the horsehead on a pumping unit, where a shape was chosen so that a force’s arm came out right at every angle rather than at one, it is a design for a whole range rather than a point. And it is a kinematic statement about an energy, and the whole design follows from two geometric facts: what makes one arm’s spring energy exactly the reverse of its weight’s, and what lets two arms be balanced without interfering. The first is a spring’s length squared. The second is a parallelogram, the linkage that a parallelogram carries an angle found transmits an orientation from one place to another unchanged — the same property that keeps a delta robot’s platform flat, and that the site’s first essay on the four-bar meets as the special case whose opposite links are equal.

One arm, one spring, one cosine #

Take one arm pivoted at its base, at an angle $\theta$ from the upward vertical, carrying a weight $mg$ at distance $l$. The weight’s potential energy is $mgl\cos\theta$: highest with the arm upright, lowest hanging down, varying as a cosine.

Now fix a spring between a point a distance $a$ above the pivot and a point a distance $b$ along the arm. The spring’s length $s$ follows from the triangle they make with the pivot, $s^2 = a^2 + b^2 - 2ab\cos\theta$. If the spring has no free length — if its tension is proportional to its whole length rather than to its stretch beyond some natural length — its energy is $\tfrac12 k s^2$, and that is

$$\tfrac12 k\,(a^2 + b^2) - k\,a\,b\cos\theta.$$

A constant, and a cosine with the opposite sign to the weight’s. Choose $k\,a\,b = mgl$ and the two cosines cancel: the total energy is a constant, at every angle, with no approximation anywhere.

For an arm carrying 1.2 kg at 0.45 m, with the spring fixed 0.08 m above the pivot and attached 0.12 m along the arm, the balancing stiffness is 551.8 N/m. Measured over half a turn, the total energy varies by $10^{-15}$ of the weight’s moment $mgl$ — nothing. A spring five per cent stiffer leaves a spread of 0.10 of it.

The fixing point has to be directly above the pivot, and the same arithmetic says why. Fix the spring’s end on a line at an angle $\varphi$ from the vertical instead and its length squared becomes $a^2 + b^2 - 2ab\cos(\theta - \varphi)$, which expands into a $\cos\theta$ term and a $\sin\theta$ term. The weight has no $\sin\theta$ term to cancel it, so a tilted post leaves an unbalance proportional to $\sin\varphi$ at every angle. The lamp’s post is vertical for the same reason its springs have no free length.

A zero-length spring sounds like an idealisation, and it is what a lamp’s springs are. A coil spring wound with an initial tension — the coils pressed together so that it takes a definite force before it starts to open — behaves, once open, as if its natural length were shorter than its closed length. Wind in an initial tension of $k$ times the closed length and the spring’s force is $k$ times its whole length over its entire working range. The other common construction runs a cord from the spring’s end over a small pulley, so that the geometric point the energy depends on is where the cord leaves the pulley, and the spring itself can have any length.

What a free length costs #

A real spring with a free length $L_0$ stores $\tfrac12 k(s - L_0)^2$, which is the zero-length energy less $kL_0\,s$ plus a constant. The extra term is not a cosine. Differentiated, it gives a leftover torque $kL_0\,ds/d\theta = kL_0\,ab\sin\theta/s$, and since the weight’s torque is $mgl\sin\theta = kab\sin\theta$,

$$\frac{\text{unbalanced torque}}{\text{weight's torque}} = \frac{L_0}{s}.$$

The unbalance is the spring’s free length over its current length, at every angle, and it acts in the weight’s direction: a spring with a free length under-balances, and the arm droops.

The law is checked by differencing the total energy numerically at every ten degrees, for free lengths of 5, 10, 20 and 40 mm, and it agrees with $L_0/s$ to $2 \times 10^{-9}$. What the figure shows is where it bites. The spring is shortest when the arm points up towards its fixing point, and there a 20 mm free length on a spring only 44 mm long leaves nearly half the weight’s torque unbalanced; with the arm horizontal the same spring leaves 14%, and hanging down 10%. A lamp balanced with an ordinary spring would stay put near the horizontal, if its joints had a little friction, and sag when raised — which is exactly how a badly made one behaves.

Where to hang the spring #

The balance fixes only the product $ab$ — how far above the pivot the spring is fixed, times how far along the arm it is attached. How that product is split is free, and the split decides everything else about the spring.

The spring’s length runs from $|a - b|$, with the arm pointing straight up at its fixing point, to $a + b$, with it hanging straight down. So the split sets the spring’s working range, and with it the range of force it has to deliver, $k$ times each end. For the arm here, with $ab = 0.0096\ \text{m}^2$ and $k = 551.8$ N/m: split as 0.08 and 0.12 m, the spring works between 40 and 200 mm and pulls between 22 and 110 N; split as 0.06 and 0.16, between 100 and 220 mm and 55 to 121 N; split as 0.04 and 0.24, between 200 and 280 mm and 110 to 155 N. Each balances the arm exactly. The more unequal the split, the less the spring’s length and force vary — and the higher its lowest force, which the zero-length construction has to supply from initial tension.

The split also decides how much a free length costs, because $L_0/s$ is worst where the spring is shortest, at $s = |a - b|$. With a 10 mm free length the worst unbalance is a quarter of the weight’s torque on the first split, a tenth on the second and a twentieth on the third. And a split with $a$ and $b$ nearly equal — 0.096 and 0.1 m, the same product — lets the spring shrink to 4 mm with the arm upright, where no real spring can go and a 10 mm free length leaves it slack altogether. That is the design rule a lamp maker works to without stating it: hang the spring so that its shortest working length is large against whatever free length it cannot be rid of.

What a stiffness error costs #

The other departure a real lamp has is a spring of the wrong rate. With the stiffness off by a fraction $\varepsilon$, the spring’s cosine term is $(1+\varepsilon)$ times the weight’s, and the one left over is $\varepsilon\,mgl\cos\theta$, which swings from $+\varepsilon\,mgl$ to $-\varepsilon\,mgl$ over half a turn.

So the spread is exactly $2|\varepsilon|$ of the weight’s moment, to $3 \times 10^{-16}$ across errors from −10% to +10%. A five per cent spring leaves a tenth of the moment unbalanced, and the joint’s friction has to hold it. That makes the design rule concrete in a way “a well-balanced lamp” is not: a lamp meant to stay put with a joint friction of, say, 5% of the head’s moment needs its springs matched to $\pm 2.5\%$ and their free lengths under a twentieth of their shortest working length.

Two arms, and why the second spring needs the parallelogram #

A lamp with one arm reaches along a circle. A useful lamp has two arms — a lower one pivoted on the base and an upper one pivoted at the elbow carrying the head — and two angles, $\theta_1$ and $\theta_2$, each measured from the vertical.

The weight is easy to account for. The head’s height is $l_1\cos\theta_1 + l_2\cos\theta_2$, so its energy is a cosine in each angle separately, with the head’s weight counted at the elbow for the lower arm’s term. Two springs, each balancing one cosine, would do — if each spring’s energy depended on one angle only.

That is where the obvious construction fails. The natural place to fix the upper arm’s spring is on the lower arm, a little above the elbow, and run it to a point on the upper arm. But a spring fixed on the lower arm sees the angle between the two arms, $\theta_2 - \theta_1$, and its energy is a cosine of the difference. No spring anywhere can cancel $\cos(\theta_2 - \theta_1)$ against $\cos\theta_2$ at every pair of angles.

Measured over the working range — lower arm from −60° to 60°, upper arm from −10° to 150° — the lamp with its upper spring fixed on the lower arm has a total energy that varies by 2.00 times the head’s moment.

The parallelogram removes the difference. Add a short crank at the base, pivoted on the same axis as the lower arm, and a link from its end parallel to the lower arm, to a point on the upper arm the same distance from the elbow. The four points form a parallelogram, so the crank at the base stays parallel to the upper arm at every pose — a parallelogram carries an angle — and the upper arm’s spring can run from a post above the base pivot to that crank. It then sees $\theta_2$ alone, its energy is $-k_2 a_2 b_2\cos\theta_2$ plus a constant, and it cancels the head’s $\cos\theta_2$ term exactly as the single arm’s did. The lower arm’s spring, from the same post to the lower arm, cancels the $\cos\theta_1$ term, which carries the lower arm’s own weight and the head’s weight at the elbow.

With the parallelogram, the energy spread over the same working range is $10^{-15}$ of the head’s moment.

The parallelogram brings one restriction of its own, and it is a kinematic one. A parallelogram is the four-bar that sits exactly on the walls between the eight kinds of four-bar, and it has change points: positions where its four links lie in a line and it can pass from the parallelogram into the crossed, anti-parallelogram assembly. Here that happens when the upper arm lines up with the lower one, so that the base crank lies along the lower arm. A lamp that let its elbow straighten fully would reach that point, and past it the base crank would no longer follow the upper arm. So the lamp’s working range keeps the elbow bent — the swings drawn here keep the upper arm at least twenty degrees from the line of the lower — and the balance holds everywhere inside that range. The energy map above includes elbow angles outside it, as a calculation; a real lamp cannot get there. The two balances no longer interfere, because the parallelogram has moved the second spring to where it can only see the second angle.

The torques a bad lamp would feel #

The energy map is the complete answer, but the joint torques are what a user feels when letting go.

Along a swing from folded to reaching, lower arm from −30° to 55° and upper from 150° to 80°, the design without the parallelogram pushes each joint with a torque of up to the whole of the head’s moment — 2.75 N·m for a 0.8 kg head 0.35 m out — which is the lamp springing up or sagging wherever it is released. The design with the parallelogram leaves both joints at nought along the whole swing, to $10^{-9}$ N·m, which is the error of differencing the energy numerically.

A balance, not a lock #

It is worth separating this from the site’s other way of making a mechanism stay put. Locked on purpose parked a toggle a few degrees past its dead centre, where the linkage cannot be back-driven because its output is second order in its input; the mechanism is held in one position and resists being moved from it. A balanced lamp does the opposite. It resists nothing — the smallest push moves it — and it has no preferred position at all, because every position is equally preferred.

The two are different answers to different requirements. A toggle holds a load against disturbance, by driving the transmission angle to nought so that the load cannot turn the input; a balance lets a light hand move a heavy thing and then leaves it alone. What they share is that each is a fact about a function of the configuration — a transmission ratio going to nought for the toggle, a potential energy going flat for the balance — and each is exact in the idealisation and has a measurable departure in practice.

That departure, for the lamp, is the design’s two tolerances. The free length and the stiffness each leave a small cosine or a small $L_0/s$ torque behind, and joint friction covers them. And a parallelogram a micron wrong is a reminder that the parallelogram itself has tolerances too: a crank not quite the length of the upper arm’s matching segment carries the angle with a small periodic error, and the upper spring then sees a slightly wrong angle.

What the balance leaves out #

Only the potential energy. The lamp is balanced statically. Swung quickly, its inertia carries it on; the balance says only that it will come to rest wherever friction stops it.

Other arms that carry things. The same two ideas — a spring whose energy is a cosine, and a parallelogram that carries an angle to where the spring is — are what hold up camera-support arms and the stands of surgical microscopes, where the load is heavier and the tolerance on the balance tighter. Nothing here depends on the load being a lamp.

Masses as points. The arms’ own weight is lumped at a point on each arm, and the springs are massless. A distributed arm mass adds to the same cosine terms and is balanced the same way; a spring’s own mass is not.

The balance’s range. The zero-length law holds while the spring is in tension. A spring wound with initial tension opens only once the load exceeds that tension, and a lamp folded so far that a spring would have to push rather than pull is outside the balance.

Still open: balancing a head that tilts #

A real lamp’s head also tilts on the end of the upper arm, and its weight’s position then depends on three angles. A third spring could balance the tilt if its energy depended on the head’s angle alone — which, by the argument here, means carrying that angle to the base through a second parallelogram in series with the first, as some drafting-machine arms do.

The distinct argument would be whether that chain of parallelograms is the only way. A head’s weight moment about the tilt joint is a cosine of the head’s absolute angle, and there are three ways to give a spring that angle alone: a second parallelogram to the base, a spring mounted on the upper arm acting through a mechanism that subtracts the arm’s angle, or a spring on the head itself working against a point that moves with the upper arm’s parallelogram. The measurement would take each and ask the same question as here — how flat the total energy is over a three-angle working range — and whether any of them balances exactly without the parts count of a full second parallelogram.