Two springs stand in for a parallelogram

The lamp that stays where it is put balanced a two-arm desk lamp exactly. The trick was the spring of no free length. Its energy is half its stiffness times the square of the distance between its ends, and when one end is a distance $a$ above a pivot and the other a distance $b$ along an arm, that square contains $-2ab\cos\theta$, with θ the arm’s angle from the vertical. The arm’s weight stores $mgl\cos\theta$. Choose $kab = mgl$ and the two cosines cancel at every angle, so the arm stays wherever it is left, held by nothing.

For the upper arm that needed a parallelogram. A spring attached on the lower arm would see the angle between the two arms, not the upper arm’s own angle from the vertical, and the two balances would interfere. A parallelogram carried the upper arm’s angle down to a crank at the base, and a spring attached there saw it alone.

That essay ended on the part of a real lamp it left out. The head tilts on the end of the upper arm, and its weight hangs off that wrist. The wrist’s torque is a cosine of the head’s angle from the vertical, and it asks for the same kind of spring. The question was whether the only way to give a spring that angle is a second parallelogram in series with the first, which is what some drafting-machine arms carry. It is not. What a parallelogram supplies is the vertical, and the vertical can be brought to the head by springs as well as by links, provided every term a spring brings that is not wanted is taken away by another.

The head asks for its angle from the vertical #

The lamp here is the earlier essay’s, with its head separated out. Its lower arm is 0.4 m long, carrying 0.4 kg at its middle. Its upper arm is 0.35 m, carrying 0.3 kg at its middle. At the upper arm’s end, on a pin of its own, is a head of 0.5 kg whose weight acts 0.08 m out from that wrist. Its working range is wide. The lower arm swings from 60° one side of vertical to 60° the other, the upper arm from 10° past vertical to 150°, and the head from 60° to 200° — from pointing up and out to hanging past straight down.

The head’s weight has a moment of $0.5 \times 9.81 \times 0.08 = 0.392$ N·m about the wrist, and the wrist’s torque from it is that moment times the sine of the head’s angle from the vertical. With no spring, the wrist’s own friction has to hold all of that at the worst angle. A stiff pin buys grip with release measured what a pivot’s friction costs when a mechanism must also move freely. A head held by its wrist’s friction is stiff to tilt by exactly as much as it is heavy to hold.

To balance the head with a spring of no free length, the spring’s energy must contain a term in the cosine of the head’s angle from the vertical, $\theta_3$, with a coefficient of −0.392 N·m. And it must contain nothing else that changes.

The bookkeeping every spring obeys #

Every point of the lamp is a sum of short vectors: the lower arm’s direction $u_1$ times some length, the upper arm’s $u_2$, the head’s $u_3$. For anything that keeps the base’s orientation — the base itself, or a bracket a parallelogram holds upright — there is the vertical, $e_y$. A spring between two points stores half its stiffness times the square of the difference, and the square of a sum of such vectors is simple. Each pairing of $e_y$ with $u_m$ gives a term in $\cos\theta_m$, the angle from the vertical. Each pairing of $u_m$ with $u_n$ gives a term in $\cos(\theta_m - \theta_n)$, the angle between the two bodies. Nothing else appears.

Gravity has only the first kind: the height of every mass is a sum of cosines of its bodies’ angles from the vertical. So the lamp is balanced exactly when the springs’ $\cos\theta_m$ terms cancel gravity’s, one for each body, and the springs’ cross terms add up to nought, one condition for each pair of bodies. There are three of the first kind and three of the second. What each design below does is satisfy the six, and the question is only what it spends to do so.

A spring across the wrist, from a point on the upper arm to a point on the head, pairs $u_2$ with $u_3$ and nothing else. It has one term, in $\cos(\theta_3 - \theta_2)$. It never mentions $\theta_3$ by itself, and so it cannot balance the head.

A spring across the wrist balances at one arm angle #

That is worth seeing quantitatively, because a spring across the wrist is what the simplest lamp would try first. With the upper arm at one particular angle, the angle between arm and head is the head’s angle from the vertical plus a constant. Aim the spring for that arm angle and it balances the head there exactly. Move the arm and the balance slides.

For a range of upper-arm angles Δ, a search over the spring’s stiffness and aim finds the least worst torque, and it is $\sin(\Delta/2)$ of the head’s moment. It sits on that curve at every range from nought to 160°. The reason is two vectors: the head’s torque is the difference between the weight’s moment, fixed in the vertical, and the spring’s, which turns with the arm, and the largest difference between a vector and one turned by up to $\Delta/2$ either way is least when the spring is a factor $\cos(\Delta/2)$ weaker than a full balance. Over 40° of arm travel the spring leaves 0.34 of the head’s moment. Over this lamp’s 160° it leaves 0.98, and the wrist’s friction has to do almost everything it would have done with no spring at all.

Two parallelograms, one spring #

The straightforward exact design carries the vertical out to the wrist. A parallelogram in the lower arm holds a bracket upright at the elbow. A second parallelogram in the upper arm, from that bracket, holds another bracket upright at the wrist. A spring from a point above the wrist on the wrist bracket to a point on the head then pairs $e_y$ with $u_3$ and nothing else, like the base spring of a single arm, and balances the head exactly.

Over the whole three-joint range the lamp’s energy is then flat to $1.6 \times 10^{-13}$ of the head’s moment. Its three springs stay what the separate balances make them. The lower arm’s spring, carrying its arm and everything beyond the elbow, reaches 50 N at its longest, and the head’s spring reaches only 14 N. The price is four links in parallelograms rather than two, and every pin in them is another place for play. A parallelogram a micron wrong measured what a real parallelogram does when it is not quite one.

One parallelogram, two head springs #

Now take the second parallelogram away. The only upright body left out along the arm is the elbow’s bracket. A spring from the top of that bracket to a point on the head spans the whole upper arm, and its vector pairs $e_y$ with $u_3$, which is the term the head wants. But it also pairs $e_y$ with $u_2$, since the upper arm lies between its ends, and $u_2$ with $u_3$. The first of those extra terms is a cosine of the upper arm’s angle from the vertical, which the upper arm’s own spring was already providing; that spring simply takes on less. The second is a cross term in the angle between the upper arm and the head, and nothing else in the lamp has one.

A spring across the wrist has exactly one term, in that angle. Sized to the cross term and aimed straight along the upper arm, it cancels it, and the two head springs together see the head’s angle from the vertical and nothing else.

The figure below is that lamp. The elbow bracket is carried by the parallelogram in the lower arm. The long spring from the bracket’s top reaches to a point 0.15 m out on the head, and a short one crosses the wrist. Over the lamp’s whole range the energy is flat to $2 \times 10^{-12}$ of the head’s moment, at every pose the figure can be dragged through.

The stiffnesses are all positive, and that is the one real constraint. The long spring’s term in the upper arm’s angle must not be larger than the upper arm’s own need, or the upper arm’s spring would have to push rather than pull. It is not, because the head spring is hung 0.15 m out on the head and so needs only a modest stiffness; hung at 0.06 m the upper arm’s spring would have come out negative. The upper arm’s spring drops from 413 N/m to 244 N/m, since the long spring carries some of that arm’s weight. The wrist spring reaches 62 N, the largest force in the lamp: it is short and must be stiff to cancel a term that the long spring built from the whole length of the arm.

Six springs and no parallelogram at all #

The same bookkeeping says a lamp needs no parallelogram at all, only springs enough to cancel what the base’s springs bring.

The base is the only upright body left, so every spring that is to see an angle from the vertical must start there. A spring from a post on the base to the head pairs $e_y$ with all three arms and brings a cross term with each pair. Two more base springs, to the lower arm and to the upper arm, make up what each arm’s weight needs. Then the cross terms. A spring from a point on the lower arm beyond the elbow to the head supplies a negative term in the angle between the lower arm and the head, cancelling the base spring’s, and adds to the other two. A spring across the elbow and one across the wrist then cancel what is left in their own angles. Every column of the table sums to nought.

Each spring in that table is there because of a term another spring brought. The base-to-head spring brings +1.046 N·m in the angle between the lower arm and the head. The lower-arm-to-head spring takes away exactly 1.046, adds 3.662 in the upper arm–head angle, and the wrist spring’s −4.578 cancels that and the base spring’s own 0.916. The pattern is general. Any body that must see the vertical can be reached by a spring from the base, and every joint the spring crosses on the way costs one spring across that joint.

The cost is in the forces. The two cancelling springs across the elbow and the wrist are the stiffest, 412 and 763 N/m, and reach 82 N and 122 N, twice the largest force in the two-parallelogram lamp. They are cancelling terms that come from the lengths of whole arms with springs attached a few centimetres from a pin. Longer attachment points would lower them, and a real lamp would have to find the room.

Along one motion #

The six designs can be compared along one motion that swings all three joints through the range. The lower arm goes from one side to the other, the upper arm rises to 150° and comes back, and the head sweeps from 200° to 60°.

With no head spring the energy climbs by 1.44 head moments as the head comes up from hanging to pointing out, which is the head’s weight lifted through its 0.08 m arm. With a spring across the wrist it climbs further, to 1.6. The spring was aimed for the middle of the arm’s range, and at the ends of this motion the arm is far from that, so the spring pulls the head the wrong way. The other four lie on the axis. They differ in what they spend, and not in what they achieve.

What each exact design pays #

There is a clean exchange along the table. Two parallelograms cost four links and use three springs, none working hard. One parallelogram costs two links and uses four springs, one of them stiff. No parallelogram costs no links and uses six springs, two of them working twice as hard as anything in the first design. The balance is exact in all three, and what moves along the row is where the lamp’s complexity sits: in pins that can have play, or in springs that must be made to a stiffness and attached to a position.

The fourth exact design spends mass. A counterweight behind the wrist, of 0.8 kg at 0.05 m, puts the head’s centre of mass on the wrist’s axis, so the head’s weight has no moment about its pin and needs no spring. The arms then carry that mass as well as the head’s, and their springs must be stiffer to match.

The counterweight’s mass is the head’s times 0.08 m over d, so it falls as d grows. Every kilogram of it rides on both arms. At 0.05 m the lamp carries 0.8 kg more, the lower arm’s spring must be 1.8 times as stiff and the upper arm’s 2.23 times, and the upper arm’s spring reaches 134 N. At 0.12 m it is 0.33 kg and 1.33 and 1.51 times. A long tail behind the head is the price of a light counterweight, which is why lamps that use one usually hang it on a stalk. The balance is exact at every d; the counterweight changes what the springs must do, never whether they can.

What the bookkeeping settles #

It settles which designs can be exact, and it does so before a single spring is sized. A spring balances a body only if one of its ends is on something that keeps the base’s orientation; every such spring that reaches across joints brings a cross term for each pair of bodies it spans, and each cross term needs a spring across a joint to cancel it. That is the whole content of the choice between parallelograms and springs. A parallelogram brings the vertical closer, so that a spring from it crosses fewer joints; a spring across a joint pays for a joint crossed.

It also settles why the earlier lamp needed its parallelogram in the form it did. A parallelogram carries an angle from one end of a mechanism to the other. What a balanced lamp needs carried is the vertical, and a parallelogram — the four-bar with its opposite sides equal — whose one side is the base is a way of carrying the vertical. The mechanism Grübler says cannot move is the same shape used the other way: its parallel bar adds no freedom and a count says it removes one. Here it adds no freedom and delivers an angle, which springs can deliver instead at a cost in force.

What it does not settle #

Springs of no free length. Every spring here stores exactly half its stiffness times its length squared. The lamp that stays where it is put measured what a real spring’s free length costs a single arm, a share $L_0/s$ of its weight’s torque. With six springs, each with its own free length and some of them short, those errors add in ways that are not measured here.

Where the springs go. The attachment points were chosen so that every stiffness comes out positive, and the forces depend on them. No search for the attachment positions that make the largest force least was made, and the no-parallelogram design in particular might do much better than 122 N with longer levers.

Parallelograms as perfect. Every parallelogram here is exact. A parallelogram with play at its pins passes the vertical on with an error, and a spring from its bracket sees that error as a tilt of the head. A length error is undone by its own size measured how a parallelogram built a thousandth wrong behaves, and none of that is carried into the balance here.

One lamp. The masses, lengths and ranges are one lamp’s. The rules — the $\sin(\Delta/2)$ of a wrist spring, the one spring per joint crossed — do not depend on them, but the forces do.

Still open: a spring that must push #

The bookkeeping required every stiffness to be positive, and in two designs the attachment points had to be moved to make it so. With the elbow-bracket spring hung close to the wrist, the upper arm’s spring came out negative. The design would then need a spring that pushes, or a spring attached on the other side of the pin, which reverses its term.

The distinct argument there would be the set of attachment positions that give every spring a positive stiffness, drawn as a region, and whether that region is empty for some lamps. A lamp with a light upper arm and a heavy head is the likely case: its upper arm’s own need is small, and the spring that brings the head’s vertical might always supply more than the arm can absorb. The measurement would map the region for each design over the ratio of head mass to arm mass, find where it closes, and ask whether the no-parallelogram design, with more springs and so more freedom, stays open where the one-parallelogram design does not.