The tensioner is the unknown

Every configuration on this site is the root of something. A four-bar’s is the root of its loop-closure equations; a spatial loop’s is the root of the logarithm of its closure transform; a gear train’s ratio is the null space of an integer matrix. The habit is deep enough that the interesting question about a new mechanism is usually which equation, and how many of them there are.

For a strand the answer is: one, whatever the mechanism looks like.

Why one #

A linkage closes when a vector comes back to where it started. Going round the loop link by link, the displacements must sum to zero, and that is two scalar equations in the plane and six in space. It is why a four-bar with one free joint angle is determined, why a five-bar needs two inputs, and why the solver behind every figure here is a Newton iteration on a residual with components.

A strand closes when a number comes back: its own length. The run over any number of bodies has one total, that total is fixed by the strand, and the equation is

$$\text{run}(\text{configuration}) = L_\text{strand}.$$

One equation. Not one per body, not one per span — one, for the whole mechanism, however many pulleys it goes round.

So a run with one free body is determined, and a run with two free bodies is not. A serpentine drive with two spring-loaded idlers has a one-parameter family of taut configurations — every one of them geometrically legitimate, with every wrap positive and every span a genuine tangent — and choosing between them requires knowing what each spring is pulling with. This field stops there, and it is worth being explicit that the stopping is a boundary rather than a difficulty: the geometry genuinely does not decide.

Solving it #

The run above has a crankshaft, a compressor, a pump and an alternator on fixed centres, and an idler carried on a 100 mm arm. Fitting a 1,020 mm belt gives an arm angle of 4.519762199 rad, with a residual of $1 \times 10^{-13}$ mm.

The method is bisection rather than Newton, and that is a decision rather than laziness. The length is a smooth function of the arm angle only while the idler is actually touching the strand. Swing the arm past the point where it stops touching and the length becomes constant — the belt simply runs past where the idler used to be — so the function has a flat shelf on either side of its useful interval, and a Newton step taken on a flat function goes to infinity. The bracket is where the answer is, and bracketing it is the honest statement of what a tensioner can do.

What a tensioner can take up #

The interval over which the idler is touching at all is 4.1224 to 4.7740 radians: 37.33 degrees of arm travel. Over that whole interval the run’s length goes from 1,013.373 mm to 1,030.991 mm.

Seventeen and a half millimetres. On a belt of just over a metre, a tensioner with a 100 mm arm swinging through thirty-seven degrees adjusts the required length by 1.7%.

That is the number a belt drive is designed around, and it is easy to get badly wrong by eye. The arm looks as though it sweeps a long way; the idler moves 65 mm along its arc; and the belt length changes by less than a third of that, because the idler is pushing into the run at a shallow angle and taking up strand from two spans at once.

The law that says how much #

The rate at which an idler takes up strand has a closed form, and it is the field’s neatest result.

The strand arrives at the idler along one direction and leaves along another, and those two differ by exactly the wrap angle. Move the idler’s centre by $d\mathbf{c}$ and the run’s length changes by $(\mathbf{u}_\text{in} - \mathbf{u}_\text{out})\cdot d\mathbf{c}$ — whose magnitude is

$$2\sin(\theta/2),$$

and whose direction bisects the two spans. Nothing else about the mechanism enters it: not the radii of the other pulleys, not the arm, not the number of bodies in the run.

Measured against the run’s own gradient across the contact range:

An idler that barely touches does nothing #

The design consequence is at the top of that table and it is stronger than “diminishing returns”.

At the edge of contact the idler’s wrap is a hundredth of a degree and its authority is 0.000245 mm of belt per millimetre of its own travel. It is not weak; it is doing nothing at all. Slide it a whole millimetre and the belt length changes by a quarter of a micron.

Authority vanishes exactly where contact does, and it vanishes linearly: $2\sin(\theta/2) \approx \theta$ for small wraps, so a tensioner with five degrees of wrap has one fifth the authority of one with twenty-five. The useful part of a tensioner’s travel is therefore strictly inside the part where it is touching, and the boundary between them is not marked by anything a drawing shows. An idler drawn just kissing the belt looks like a tensioner and is a decoration, for a reason with an exponent attached to it.

The other end of the table is worth noting too. At 78.964° of wrap the idler takes up 1.27 mm per millimetre, which is more than one: pressing an idler deep into a run takes up more strand than the idler moves, because it is bending two spans rather than displacing one.

The arm is a lever, and not a good one #

There are two rates here and they are easy to conflate. $2\sin(\theta/2)$ is the strand taken up per millimetre the idler’s centre moves, along the bisector of its two spans. What a designer sets is the arm angle, and the arm carries the idler along a circle rather than along the bisector.

Dividing one by the other gives the arm’s effective lever — the distance that, multiplied by the take-up rate, gives the take-up per radian:

The arm is 100 mm long and its effective lever is between 54 and 77 mm, because the direction it moves the idler is never the direction that takes up the most strand. That inefficiency is not a loss in any energetic sense — it is the cosine between the arm’s tangent and the spans’ bisector, and it changes as the arm swings because the bisector turns with the wrap.

It also runs the wrong way. The lever is largest where the authority is smallest, which flattens the length curve’s left-hand end less than the wrap alone would, and it is smallest where the authority is largest. Multiply the two and the take-up per radian still rises by a factor of fifty-seven across the travel.

Two idlers, and what the geometry declines to say #

The claim that a run with two free bodies is undetermined is worth measuring rather than asserting, because it is the field’s clearest boundary.

Put a second idler on the span between the crankshaft and the compressor, on its own arm, and sweep both arms over every position at which the run is taut. Of 1,681 sampled pairs, 861 give a legitimate run, and their lengths cover 1,013.43 to 1,041.61 mm — so a belt anywhere in that range fits, and the question is where the mechanism sits.

Fix the belt at 1,026.719 mm and the pairs that satisfy it are $(4.285, 0.540)$, $(4.300, 0.600)$, $(4.390, 0.840)$, $(4.420, 0.900)$, $(4.450, 0.960)$ — a curve through the two-dimensional space of arm angles rather than a point. Every position on it is geometrically perfect: every span a genuine tangent, every wrap positive where it should be, the turning number exactly one, the length exactly right.

The mechanism is somewhere on that curve and this field cannot say where. What decides it is the two springs, and a spring is a force.

That is not a gap to be apologised for. A single automatic tensioner is a mechanism whose position is decided by geometry, and a designer can compute it; two of them on one belt is a mechanism whose position is decided by a balance of forces, and no amount of drawing settles it. Knowing which of the two has been built is worth the arithmetic above.

Where the wrap it takes comes from #

An idler pressed against the outside of a run has a negative wrap, and the wraps of a closed run add to exactly one turn. So an idler taking 25.342° must be matched by 385.342° of positive wrap elsewhere — a tensioner creates wrap as well as taking up length.

Which pulleys receive it is a sharper question than the bookkeeping requires, and the answer is that the two pulleys either side of the idler take all of it. Across the whole contact range the crankshaft’s wrap goes from 109.089° to 120.399° and the alternator’s from 95.903° to 163.497°, while the compressor’s stays at 74.709° and the pump’s at 80.359° — unchanged to a thousandth of a degree at every position of the arm.

So there are two reasons to move a tensioner and they are different questions. Take-up is about the length equation and is answered by $2\sin(\theta/2)$; wrap is about the grip on a particular pulley and is answered by which span the idler sits on. A tensioner in the wrong span can be perfectly effective at the first and useless at the second.

Outside the interval, the geometry still answers #

The reason this mechanism needs a bracket rather than a solver is worth showing, because it is the failure this field’s routines exist to refuse.

Ask for an arm angle outside the contact interval and every step of the computation still succeeds. Tangent lines exist between the idler and its neighbours; the tangency points are real points on real circles; the wraps are legitimate arcs; the length is a number. The path that comes back wraps the idler by three hundred and forty-eight degrees instead of six, cuts through the alternator, and crosses itself once — and none of that is visible in a drawing of a curve that is already nearly closed.

Its turning number is two rather than one, and that is how the case was found.

Asked for a belt of 1,040 mm, the solver refuses with the interval in the message: the run spans 1,013.373 to 1,030.991 mm, and the strand is 1,040. There is no arm angle that takes up that belt, and the useful form of the answer is the range rather than a nearest-fit.

The comparison worth keeping #

A linkage and a strand run are both mechanisms whose configuration is the root of an equation, and almost everything else about them differs.

A linkage’s closure is a vector equation, so its solution set has a dimension that can be counted before anything is solved, and the count is the same everywhere. A strand’s closure is one scalar equation, so the number of free bodies it can determine is exactly one, however elaborate the mechanism gets.

A linkage’s solve either converges or refuses, and a configuration that cannot be assembled is one the solver cannot reach. A strand’s closure has no residual to report and no convergence to fail: the length is a formula, and a configuration the strand could not take is one that must be tested for afterwards, by asking whether the path is one a strand could lie on.

And a linkage’s singularities are where its Jacobian loses rank — the toggles and dead centres this site’s linkage field is built around. A strand run has an analogous place, and it is the edge of contact: the length function’s derivative goes to zero there, so the mechanism loses its ability to be adjusted while remaining perfectly assembled. It is the subject of an essay of its own.

What the fixed pulleys contribute #

One last piece of bookkeeping, because it explains why the take-up is so small.

Of the run’s 1,018.05 mm at mid-travel, 722.87 mm is straight spans and 295.18 mm lies on the five bodies. Swinging the arm changes both: the two spans either side of the idler lengthen as it presses in, and the wraps on its two neighbours grow while its own grows too. All of those are additions — every one of the five terms that changes gets larger as the idler swings in — which is why the length curve rises monotonically across the whole contact interval and why a tensioner can only ever ask for a longer belt than the bare four-pulley loop needs.

The bare loop over the four fixed pulleys — the hull of them — is 1,013.37 mm. That is the shortest belt this drive can use, and it is the length at which the idler is on the point of not touching. Every belt the mechanism can accept is longer than that, by up to 17.6 mm, and the whole design space of the tensioner lives in that band.

One equation, and what happens with two unknowns #

The count is stated above for one and two free bodies, and the general form is worth having because it turns an observation about a serpentine belt into the same accounting the rest of this site runs on.

A strand run has one closure equation, whatever its size. So a run with $n$ free bodies has $n$ unknowns and one equation, and its solution set has dimension $n - 1$. One free body: a point, determined. Two: a curve, which is the pair of arm angles measured above. Three: a surface. The geometry pins down exactly one degree of freedom’s worth of the arrangement, no matter how much arrangement there is.

That is the strand’s own mobility, and reading it that way puts this field beside the transmission field’s two-freedom gearsets rather than beside the linkages. A gearset with two freedoms and one input is not confused and not broken; it is a mechanism with a relation rather than a function, and it needs a second condition before it does anything definite. A run with two idlers is the same object: one relation, two unknowns, and a curve of configurations all of which are geometrically legitimate.

The parallel extends to what supplies the missing condition. A gearset gets a brake or a clutch — one more linear equation, chosen by hardware. A two-idler run gets a force balance: each arm carries a spring, each spring’s torque depends on the arm’s angle and on the belt tension, and the pair settles where both torques balance. That is one more equation and it is not geometric, which is precisely why it is outside this field.

So the boundary is not arbitrary and it is not a gap. This field can determine a run with one free body and no more, and it is exact about which runs those are: count the bodies whose position is not fixed, and if the count is one the geometry answers. That is a checkable precondition rather than a caveat, and a run that fails it should be reported as underdetermined rather than solved with an arbitrary choice made somewhere.

It also explains a piece of design practice that would otherwise look like conservatism. Real accessory drives have one tensioner, almost without exception, even on runs long enough that two would seem to help. A second one adds no geometric determinacy and makes the belt’s configuration depend on two spring rates and whatever friction is in two pivots — a mechanism whose resting position is decided by forces rather than by shape. One tensioner keeps the run in the region where the geometry decides, which is the region a designer can compute.

What is not modelled #

The belt has no stiffness, so it takes the tangent lines exactly and turns through its wrap angles as a corner would; a real belt has a bending radius and rides slightly proud of a small idler, which changes the effective radius and therefore the take-up. Nothing here knows what tension the belt is at, so nothing here says whether the tensioner is doing its job — the arm angle computed above is the one that makes the run’s length equal the belt’s unstretched length, and a real tensioner is set to a force rather than to a length. The spring on the arm is absent entirely, which is what makes the second idler undecidable.