The drawer that jams when pulled by one handle

There is a particular kind of furniture failure that everyone has met. A long, shallow drawer in a desk or a kitchen unit, with a handle at each end, or a single knob placed off to one side, is pulled by that one handle. It comes out a few centimetres, turns slightly, and stops. Pulling harder only makes it stick more firmly. Pushing it back in a little and pulling from the middle of its front brings it out at once with hardly any effort. Nothing broke and nothing is in the way, and the drawer was free a moment ago.

The explanation is a free-body diagram with two contact points in it, and it says something stronger than that off-centre pulling is inefficient. Below a certain ratio of lengths, off-centre pulling cannot work at all, however much force is applied. The ratio involves the friction coefficient and two distances, and it is the same one that governs machine slides, sash windows, the tailstock of a lathe and a robot pushing a peg into a hole.

Two corners take the load #

Look down on the drawer from above. It sits between two runners with a little clearance at each side. Pull on a handle a distance $e$ to one side of its centreline and the pull does two things: it tries to move the drawer out, and it tries to turn it. The drawer turns until a front corner touches one runner and the diagonally opposite back corner touches the other. From then on, those two contacts carry everything.

Each contact pushes on the drawer with a normal force $N$, sideways, and the two are equal and opposite, because nothing else is pushing sideways. They are separated along the drawer’s length by the engaged length $L$, the length over which the drawer’s sides overlap the runners, so together they form a couple of size $NL$. That couple is the only thing available to stop the drawer turning, and the pull’s turning effect is $Fe$. So

$$N = \frac{Fe}{L}.$$

Each contact also has friction along the runner, of at most $\mu N$, opposing the motion, the ceiling that the force that takes what it needs described. The two friction forces act at opposite sides of the centreline, one at each runner, so they turn the drawer in opposite directions by equal amounts and their moments cancel exactly. That is the one subtle step, and it is what makes the answer so clean. Friction then enters only through the force balance along the drawer:

$$F = 2\mu N + \mu_0 mg,$$

where the last term is the ordinary drag of the drawer’s weight on its runners, which a centred pull also has to overcome.

The pull that cannot win #

Substitute the normal force into the force balance:

$$F = \frac{2\mu e}{L}F + \mu_0 mg, \qquad F = \frac{\mu_0 mg}{1 - 2\mu e/L}.$$

The pull needed is the centred pull divided by $1 - 2\mu e/L$. When $L$ is long compared with $2\mu e$ the penalty is small. As $L$ shrinks towards $2\mu e$ the denominator goes to zero and the required force goes to infinity. Below that, the formula gives a negative force, which means that no positive pull satisfies the balance: the friction that the pull generates is always larger than the pull.

This is self-locking, and it is different in kind from a drawer that is merely stiff. A stiff drawer needs a larger force. A locked drawer needs a different force — in a different place. Every newton added at the handle adds $e/L$ newtons to each corner and $2\mu e/L$ newtons of friction, more than one newton when $L < 2\mu e$. The harder it is pulled, the harder the corners bite. The cure is to push it back slightly, which unloads the corners, and pull nearer the middle.

The slope and the forces on it met the simplest self-locking device, a wedge or a screw whose angle is below the friction angle: such a screw will hold a load without any brake, because the force trying to turn it back produces more friction than turning force. The drawer is the same idea with geometry in place of an angle. The ratio $L/e$ plays the part of the wedge’s slope, and the friction coefficient sets the threshold in both.

Why it stops partway out #

A drawer’s engaged length is not fixed. Closed, its sides overlap the runners along most of its depth. As it comes out, the overlap shrinks, and $L$ falls steadily towards the length of the stops. So a drawer pulled at one end will slide when closed and lock at a definite point on the way out.

That is the whole of the familiar behaviour. The first 15 cm of travel feel almost normal, with the pull gradually stiffening. Then over the last few centimetres before the lock, the force needed climbs steeply, and at 24.5 cm it is infinite. A person pulling feels the drawer go from free to stuck in a short distance and concludes, reasonably but wrongly, that something has caught. Pull from the centre and $e$ drops to a few centimetres, $2\mu e$ to a few centimetres too, and the lock moves beyond the end of the travel.

The numbers also say which drawers have the problem. A deep, narrow drawer has long runners and little room for an offset, so $L/e$ is large and it never locks. A wide, shallow drawer — a cutlery tray, a map chest, a keyboard shelf — has short runners and a lot of width, and a handle near one end gives it an offset of a third of a metre. Cabinet makers learned the remedies long before anyone wrote the inequality down: two handles, or one in the middle; wax on the runners, which lowers $\mu$; a central guide under the drawer, which shortens the effective width; or metal slides with ball bearings, which take $\mu$ to almost zero so that no offset can lock them.

A rule written for machines #

The condition can be put as a single boundary between sliding and locking.

Machine designers who use plain sliding bearings — bronze bushes on shafts, a slide in a guide, a quill in a drill press — work to a rule of thumb sometimes called the binding ratio, or the two-to-one rule: the distance from the guide to the line of the driving force should not exceed twice the length of the bearing. It is usually given without derivation, as experience. It is the boundary in the picture at a friction coefficient of a quarter, a typical value for a lubricated metal pair. A designer who keeps to two-to-one keeps a margin against the friction rising as the lubricant ages; one who uses a dry plastic bushing with $\mu$ of 0.4 needs the ratio nearer 1.2. The rule is not arbitrary. It is the drawer’s inequality with a typical coefficient filled in.

Robotics met the same boundary from another direction. When a robot pushes a peg into a closely fitting hole, the peg can tilt and touch the hole at two points, one near the mouth and one deeper in, and whether it then slides in or sticks depends on the line of the pushing force relative to those contacts. Daniel Whitney’s analysis of peg-in-hole assembly in 1982 drew the region of forces that produce motion and the region that produce jamming, and its boundary is this same balance of a couple against friction. Compliant wrists, which let the peg turn so that the pushing force passes nearer the hole’s axis, were designed to keep $e$ small for exactly the reason a drawer should be pulled from the middle.

Precision is not the cure #

The obvious remedy for a jamming drawer is a better fit, on the reasoning that a drawer which cannot turn cannot jam. The calculation says otherwise.

With clearance, the drawer turns through a small angle before its corners bite, and that angle brings the two contacts slightly closer together along the length, shortening the couple’s arm from $L$ to $L\cos\theta - W\sin\theta$. That is the only effect, and it is small: the angle is the clearance divided by the diagonal, a degree or so at most. A perfect, zero-clearance fit still has two contact points on opposite sides that can press as hard as the pull demands, and so it still locks at $L = 2\mu e$. A tighter fit makes a drawer feel more solid. It does not stop it jamming, and by raising the chance of rubbing along the whole side, it can raise the friction and make things worse.

That is a typical lesson of a free-body diagram. The intuition that slop causes jamming is about the motion: a sloppy drawer visibly twists, so twisting looks like the cause. The diagram is about forces, and the forces are set by where the pull is and how long the guide is. Slide or topple found the same separation for a block pushed on a floor, where whether it slides or tips depends on the height of the push against the width of the base and on the friction coefficient, and not on how hard the push is.

Friction that feeds on its own load #

Self-locking is one member of a family of mechanisms in which friction raises the very force that produces it. The family is worth laying side by side, because the feedback takes a different form in each and the form decides what a person can do about it.

In the drawer the feedback is linear. The pull produces normal forces in proportion to itself, so the friction is a fixed multiple of the pull, $2\mu e/L$. Either that multiple is less than one and the drawer always moves, or it is more and the drawer never does. There is no force at which behaviour changes; only the geometry changes it. In a rope wrapped round a post, which the turn that multiplies what a hand can hold followed, the feedback is continuous along the rope, each element’s tension pressing the next against the post, and the result is exponential: the holding force grows as $e^{\mu\theta}$ with the angle of wrap. There the geometry is chosen precisely so that friction runs away, and a sailor makes three turns to hold a load a hand could never hold.

A brake shoe that is dragged into its drum by the drum’s own rotation, a leading shoe, is a third member. Its friction pulls it harder against the drum and raises the normal force, so it brakes more strongly than a trailing shoe for the same push — and if its pivot is placed badly, it grabs and locks the wheel, which is the drawer’s lock arriving in a moving machine. Designers of drum brakes keep the shoe’s geometry inside the region where the self-energising effect helps without running away, by an inequality of the same shape as the drawer’s.

The useful lesson across all three is that a mechanism held by friction should be checked for the sign of its feedback before its size. A stiff mechanism with negative feedback, where the friction falls as the force rises, only needs more effort. One with positive feedback above one, the drawer below its critical length, needs a change of geometry, and no amount of effort replaces it.

When more than two points touch #

The analysis used two contacts, and two contacts are exactly enough to hold a drawer against turning and sliding sideways, so the forces could be found from the equilibrium equations alone. A drawer with no clearance at all, rubbing along both sides at once, has more contacts than it needs, and then statics cannot say how the load is shared among them. The table statics cannot settle met the same indeterminacy for a four-legged table, where the share each leg takes depends on the floor’s unevenness and the legs’ stiffness rather than on balance.

For the drawer this sets a limit on the clearance result. With a little clearance, the corners decide and the answer is clean. With an interference fit, the sides are pressed along their length by the wood’s own stiffness, every contact carries friction whether or not the pull is off-centre, and the drawer is simply stiff all the way out. That stiffness is not the lock described here, and adding precision past the point of zero clearance converts one problem into the other.

Fingers, drawers and a cone #

The drawer and the fingers that always meet in the middle are close relatives. Both are a rigid body held at two contacts, with friction at each and normal forces set by geometry. Under the rule, the geometry made the moving contact heavier as it moved, so the motion stopped itself and found a balance. In the drawer, the geometry makes the friction proportional to the pull, so the motion either always works or never does, with nothing in between except a stiffening approach to the boundary.

A neat way to see both is the friction cone. At each contact, the total force the surface can exert — normal plus friction — must lie inside a cone around the normal, of half-angle $\arctan\mu$. A body is held when some combination of forces inside the two cones can balance whatever is applied. For the drawer, the pull is held exactly when its line of action passes through the region where the two cones overlap, and that overlap reaches out from the runners by a distance set by the cone angle and the engaged length. A pull whose line passes through it is resisted by the contacts without limit. A pull that misses it — one nearer the centreline — can never be fully balanced, so the drawer slides. The inequality $L < 2\mu e$ is that geometric statement turned into arithmetic. It is also the picture behind the rod that friction cannot decide about, where the cone at a single sliding contact, made too wide by too much friction, left rigid-body mechanics with no consistent answer at all.

What the picture cannot show #

The model is a rigid drawer between rigid runners, with two point contacts and one coefficient of friction. Real drawers are made of wood that bends, and a long, thin drawer front bows when pulled at one end, which moves the line of the pull. Runners are not continuous rails; many drawers ride on a few blocks or rollers, and the contacts sit wherever those are rather than at the corners. Wooden runners swell in damp weather, which raises the friction coefficient — the grip that is not a coefficient explained why that number moves with moisture, load and time in contact — and a coefficient of 0.35 for waxed wood can become 0.6 for swollen bare wood, nearly doubling the locking length. That is why a drawer can be fine in winter and stick in summer.

The analysis is also quasi-static. A jerk on the handle can sometimes free a locked drawer, because the impulse arrives before the corners have been loaded, and the dynamic situation differs from the static one. A drawer pulled briskly from one end may come out further than the static figure says before it locks. The weight of the contents is taken to be centred; a drawer loaded on one side has its own turning moment, which adds to or subtracts from the pull’s.

And the vertical direction has been left out. A drawer can also tilt downward as it comes out and press its back edge up against the carcase above, with its own couple and its own friction. That is the same inequality in the vertical plane, with the drawer’s height in place of its width, and in a drawer pulled out most of the way it is often the one that decides. The figures here show only the horizontal case, where the offset is the one a person chooses.

Still open: how a gripper should feel a jam coming #

The self-locking condition is exact for rigid bodies. For a robot inserting a part, what matters is detecting that the forces are heading for the locked region before they arrive, and backing off — which is what a person does with a drawer without thinking. Force sensors in a robot’s wrist measure the forces and torques at the hand, and in principle the jamming diagram tells the controller whether the current force is on the safe or the locked side. In practice the contacts are not points, the friction is not constant, and the measurements are noisy, and fast, reliable insertion of tight parts by robots without precise fixtures is still an active problem in assembly automation. A child pulling a sticky drawer from the middle has already solved the version of it that matters at home.

Pulled a distance $e$ off its centreline, a drawer bears on two diagonal corners with normal forces $Fe/L$, and since the two frictions cancel in moment, the pull needed is the centred pull over $1 - 2\mu e/L$ — infinite at an engaged length of $2\mu e$, 17.5 cm for a 60 cm drawer pulled at one end with μ = 0.35, which locks after 24.5 cm of travel at any force; clearance barely moves the lock, and at μ = 0.25 the boundary is the machine designers’ two-to-one rule. The habit worth carrying is to ask, before pulling harder, whether the force being added is also adding to what resists it — because when it is, strength is the wrong tool.