A spring that must push is hung in the wrong place

The lamp that stays where it is put balanced a two-arm desk lamp exactly, at every pose, with springs of no free length. A spring of no free length stores energy in proportion to the square of its length, and the square of a distance between two points on a linkage is a constant plus cosines of the links’ angles. The weight’s energy is also a sum of cosines, so balance is a matter of matching coefficients. Two springs stand in for a parallelogram added a head that tilts on the upper arm’s end and balanced it too, with one parallelogram and two head springs, or with none and six.

That essay recorded a nuisance. The balance equations are linear in the stiffnesses and always have a solution, but the solution does not care about signs. With the head spring hung close to the wrist, the upper arm’s spring came out negative, a spring that would have to push. Springs of no free length are made by pre-stretching a coil and hiding its free length behind a pulley or a lever. They pull, and a pushing one is a different and worse mechanism. The attachment points were moved until every stiffness was positive, and the essay ended by asking whether that is always possible. A lamp with a light upper arm and a heavy head looked like the case where it might not be.

It is always possible, and the reason is short enough to state as a rule for where to hang a spring.

The stiffnesses in closed form #

Each spring’s energy, written in the links’ angles, contributes to a few of six coefficients. They are the cosines of the lower arm’s, the upper arm’s and the head’s angles from the vertical, and the cosines of their three differences. Gravity contributes to the first three only. Balance is the six columns summing to nought.

For the one-parallelogram design the columns fall in a staircase. The lower arm’s spring, from a post on the base to the arm, contributes only to the lower arm’s cosine. The upper arm’s spring, from the parallelogram’s upright bracket at the elbow, contributes only to the upper arm’s. The head spring, from the top of that bracket to the head, contributes to the upper arm’s cosine, the head’s cosine and the cosine of their difference. The wrist spring, across the wrist from the upper arm to the head, contributes only to that difference. So the equations can be solved one at a time, from the head down, and each stiffness comes out in closed form:

$$k_{\text{head}} = \frac{G_3}{a_3 b_3},\qquad k_{\text{wrist}} = \frac{k_{\text{head}} L_2 b_3}{(a_4 - L_2)\, b_4},\qquad k_{\text{upper}} = \frac{G_2 - G_3 L_2/b_3}{a_2 b_2},\qquad k_{\text{lower}} = \frac{G_1}{a_1 b_1}.$$

Here $G_1$, $G_2$ and $G_3$ are gravity’s coefficients for the three angles, $a$ and $b$ are each spring’s attachment distances along the bodies it joins, and $L_2$ is the upper arm’s length. The formulas were checked against the balance found by solving the whole term table numerically, which knows nothing of the staircase; they agree to $10^{-9}$ at every attachment point swept.

Three signs are free, one is not #

Three of the four stiffnesses are positive whenever the springs are where the design puts them. The lower arm’s and the head’s have positive numerators and positive attachment distances. The wrist spring’s is positive when its upper-arm attachment point sits beyond the wrist, $a_4 > L_2$, on a short extension of the arm, which is how the design was built: the spring has to cancel a cross term the head spring brings, and only an attachment point past the joint gives it the right sign.

The fourth is the upper arm’s, and its numerator is a difference. $G_2$ is what the upper arm’s balance needs: the upper arm’s own weight and the head’s, carried at the end of the arm. $G_3 L_2/b_3$ is what the head spring already supplies to the upper arm. The head spring runs from the elbow bracket to the head, so it pulls on the upper arm as well as the head, and the closer it meets the head to the wrist, the larger its stiffness has to be to hold the head and the more it pulls on the arm. If the head spring meets the head too close to the wrist, it oversupplies the upper arm, and the upper arm’s own spring has to take the excess back. That is a push.

The figure shows it on the lamp as built. The lower arm’s and the wrist’s stiffnesses do not depend on the head attachment at all. The head spring’s falls as one over the attachment point’s distance. The upper arm’s climbs from deeply negative to positive and crosses nought at one place, 0.0615 m from the wrist. At 0.03 m it is −434 N/m, a spring that must push. At 0.08 m, the head’s centre of mass, it is +95 N/m.

Why the head spring pulls on the arm #

The oversupply has a mechanical reading that does not need the term table. The head spring runs from the top of the parallelogram’s upright bracket at the elbow to a point on the head. When the upper arm swings, the wrist moves, the head spring’s head end moves with it, and the spring’s length changes even if the head’s own angle does not. So the head spring does work on the upper arm’s swing. In the energy that is its term in the upper arm’s cosine, $k_{\text{head}} a_3 L_2$, and with $k_{\text{head}} = G_3/(a_3 b_3)$ it is $G_3 L_2/b_3$. That term pulls the upper arm up in the same sense as the arm’s own spring, so the two springs share the arm’s balance.

The share the head spring takes grows as its head attachment moves towards the wrist, because a spring attached close to the wrist must be stiff to hold the head’s moment, and a stiff spring between the elbow and the wrist end of the arm works hard on the arm. When that share exceeds the arm’s whole need, the arm’s own spring has to give back the difference. Moving the head attachment outwards softens the head spring, shrinks its share, and hands the balance back to the arm’s spring. The threshold is the attachment point at which the head spring’s share is exactly the arm’s need, and since the arm’s need includes the arm’s own weight, which the head spring never sees, the head spring can never take all of it once it is fixed at the head’s centre of mass.

How large the margin is #

On the lamp as built the threshold is 0.0615 m from the wrist and the head’s centre of mass is at 0.08 m, a margin of 18.5 mm. Near the threshold the upper arm’s spring is weak and changes quickly with the attachment point: its stiffness changes by 6.7 N/m for every millimetre the attachment point moves at the threshold, from nought. At the head’s centre of mass it is 95 N/m and changes by 4.0 N/m per millimetre, about 4% of itself.

That is a second reason to hang the head spring at the centre of mass rather than just beyond the threshold. A spring hung just past the threshold has a small, positive stiffness that a few millimetres of manufacturing error can turn negative, and a negative stiffness is a lamp that falls. A parallelogram a micron wrong found how unforgiving a linkage can be of small errors in its lengths. A spring balance near a sign threshold is the same kind of design, and the centre of mass is the place where it is forgiving.

Where the threshold lies #

Setting the upper arm’s numerator to nought gives the threshold in terms of the lamp’s masses and lengths:

$$b_3^* = \frac{G_3 L_2}{G_2} = l_3 \cdot \frac{m_3 L_2}{m_2 l_{2g} + m_3 L_2},$$

with $m_2$ the upper arm’s mass at $l_{2g}$ from the elbow and $m_3$ the head’s at $l_3$ from the wrist. The fraction multiplying $l_3$ has the head’s term in its numerator and the head’s term plus something positive in its denominator. It is less than one for every lamp. So the threshold is always short of the head’s centre of mass.

That is the whole answer to the question the earlier essay asked. Hang the head spring’s head end at the head’s centre of mass, or beyond it, and the upper arm’s spring pulls, whatever the lamp weighs.

The threshold was found a second way that shares nothing with the term table. For each lamp, the stiffnesses that make the lamp’s total energy flattest over its whole range of poses were found by least squares on a grid, and the head attachment was bisected on the sign of the upper arm’s stiffness. Across ten lamps, from a head a twentieth of the upper arm’s mass to fifty times it, the bisection agrees with the formula to $10^{-4}$ m.

The figure answers the specific worry, a light upper arm under a heavy head. As the head grows the threshold moves out towards the head’s centre of mass, 91% of the way for a head five times the arm’s mass and 99% for fifty, and it never arrives. The worry was right about the direction and wrong about the conclusion. A heavy head makes the margin small; it never makes it negative.

Without a parallelogram #

The no-parallelogram design is harder to read, because its six springs fill all six columns and the staircase becomes a lattice. Three springs run from the base, to the lower arm, the upper arm and the head. A fourth runs from an extension of the lower arm to the head. The last two cross the elbow and the wrist. Solved one column at a time from the head down, the signs again come down to attachment points. The base-to-head spring is positive from its own column. The base-to-upper-arm spring is positive under the same threshold as before, the head attachment beyond $b_3^*$. The lower-arm-to-head spring and the two joint springs are positive when their attachment points sit beyond the joints they cross. And the lower arm’s spring is positive when the base-to-upper-arm spring meets the upper arm far enough out, at the wrist or beyond.

The term table shows why the lattice still has the staircase’s logic. The base-to-head spring is the only one that sees the head’s own angle, so its stiffness is fixed by gravity’s head column alone, as before. Every other spring is fixed by a column in which the springs already solved have left a remainder, and the remainder’s sign is decided by where those springs are attached. Reading the table from the head column outward is the same sequence of signs the closed forms gave for the simpler design.

With those attachment points the design balances every lamp tried with every spring pulling. For a head a twentieth of the arm’s mass the stiffnesses run from 1 to 142 N/m, and for a head fifty times it from 92 to 22,890 N/m. The energy is flat to $10^{-11}$ of the head’s moment or better in every case. A heavy head needs stiff springs, and the wrist spring of the heaviest lamp is stiff enough to be a practical problem. It does not need a pushing one.

The rule already in the simplest lamp #

The rule was present in the simplest balanced lamp, in a form too simple to notice. The lamp that stays where it is put balanced a single arm with a spring of no free length from a post above the arm’s pivot. The arm’s weight puts a positive coefficient in front of the cosine of the arm’s angle, and the spring’s coefficient is the product of its two attachment distances with a sign set by which side of the pivot the post is on. Above the pivot, the spring’s coefficient is negative and can cancel the weight’s with a positive stiffness. Below it, the same cancellation needs a negative one. So even the one-arm lamp has a spring that must push, if its post is on the wrong side.

Every sign condition in this essay is that observation repeated with more bodies. A spring’s coefficient in a column is a product of attachment distances, its sign is the side of a joint or a threshold each attachment point is on, and the stiffness that cancels the column’s remainder is positive exactly when that sign opposes the remainder’s. The parallelogram, a four-bar whose coupler stays parallel to its frame, matters only because it gives some springs an upright end to pull from, which removes columns from the problem. What remains is always a question of where each attachment point sits, never of how heavy the lamp is.

A rule for where to hang a spring #

The pattern across both designs is general enough to state as a rule. A negative stiffness in an exact spring balance is not a demand for a compression spring. It means one spring is oversupplying a body that another spring also acts on, and the cure is to move the oversupplying spring’s attachment point until its share of the work on that body falls below what the body needs.

For a lamp with a tilting head the rule has a concrete form. Every spring that crosses a joint is attached beyond it, and every spring that reaches the head meets it at or beyond its centre of mass. Under that rule every stiffness in both designs is positive for every combination of masses, and the only cost of an extreme lamp is the size of its springs. A parallelogram carries an angle is what lets the one-parallelogram design keep its head spring’s other end upright. The rule says where on the head the spring should end.

What signs do not settle #

How large the springs are. A spring’s stiffness is only half its size. The other half is the force it carries at its longest, and a spring of no free length stretched across a long distance carries a large force. The rule keeps every stiffness positive and says nothing about whether the springs fit in the lamp. On the heaviest lamp they would not.

Springs with free length. Real springs have free lengths, and a spring of no free length is an idealisation reached with pulleys. A free length adds a term that is not a cosine and spoils the exact balance. A length error is undone by its own size measured how much for a single arm. Signs aside, every design here is exact only for ideal springs.

Friction in the joints. A real lamp’s joints have friction, and a lamp balanced to within its joints’ friction stays put even when its springs are slightly off. That makes the small, sensitive stiffness near the threshold less dangerous than the arithmetic suggests, and it makes a lamp balanced at the centre of mass better than it needs to be. The joints are treated here as ideal pins; why a hinge works treats them as they are.

The head’s range. The head’s centre of mass is taken as fixed on the head. A lamp whose shade can be rotated or whose bulb can be changed has a centre of mass that moves, and an attachment point at it moves too.

Still open: the smallest springs that balance the head #

The rule makes every spring pull, and among the designs that obey it there is still a choice of attachment points. Each choice gives a set of stiffnesses and spring forces, and a builder wants the smallest springs that fit in the lamp.

The distinct argument would be that optimisation, with the signs as constraints. It would minimise the largest spring force over the lamp’s whole range of poses, over the attachment points of the one-parallelogram design, subject to every stiffness positive and every attachment point within the lamp’s own outline. The measurement would say how much smaller the best design’s springs are than the ones here, whether the best design sits on the threshold or well beyond it, and whether a design with no parallelogram can ever have smaller springs than one with a parallelogram, given that it has six of them.