The transmission angle
Grashof’s condition says whether a linkage turns. It says nothing about whether it is any good, and the number that does is the transmission angle.
Call it μ. It is the angle at the joint where the coupler meets the output link — at B, between BA and BO₄. When μ is 90° the coupler pushes the rocker squarely and all of the force it delivers turns into torque about the output pivot. When μ approaches 0° or 180° the coupler is pushing nearly along the rocker rather than across it, almost all of the force goes into the output bearing, and almost none of it does anything.
Why it is a geometric quantity
The force the coupler carries acts along the coupler, because a two-force member pinned at both ends can carry load no other way. Resolve that force at B into a component perpendicular to the rocker, which produces torque, and a component along it, which produces nothing but a bearing reaction.
The perpendicular component is . That is the whole derivation, and it is why the useful fraction of the transmitted force is exactly the sine of the transmission angle — 100% at 90°, 71% at 45°, 17% at 10°.
Nothing about mass, speed, friction or material enters. This is a statement about directions, which is why it can be computed exactly from a solved configuration and why it is meaningful before anything about the machine’s duty has been decided.
The rule, and what kind of rule it is
The convention is to keep μ between 40° and 140°, which is the same as requiring .
That is a rule of thumb with a reason rather than a theorem. The reason is that the bearing reaction grows as the useful component shrinks, and a linkage with a poor transmission angle wastes force, wears its bearings, and — with friction, which this site does not model — can lock entirely rather than merely working badly.
Where the threshold sits is a judgement. Lightly loaded mechanisms run happily below 40°; heavily loaded ones are designed to stay near 90°. The number is a convention and this site says so wherever it is used, because every conclusion drawn from it inherits that status.
Where μ is worst
It is worth being precise about which configuration produces the minimum, because it is easily confused with a different one.
μ collapses when the coupler and the rocker approach collinearity. At that moment the coupler’s force has almost no component across the rocker.
That is not the same as a toggle, which is where the crank and the coupler become collinear. At a toggle the output momentarily stops and the mechanical advantage diverges — the mechanism is at its strongest, not its weakest.
Conflating them is common enough that it has its own essay. The short version: one is enormous force output, the other is force disappearing into bearings.
Computing it from the solve
Because every position on this site comes out of a solve, μ is a by-product: take the vectors BA and BO₄ from the solved snapshot, and take the angle between them.
The alternative is a closed form in the four lengths and the input angle, which exists and is standard. Using the solved positions instead has one practical advantage that matters more than elegance: it works unchanged for any mechanism the solver can position, including ones with no published formula, and it cannot drift out of step with the picture beside it. The angle in the caption is the angle in the drawing because they came from the same snapshot.
What a good linkage looks like
The extremes of μ occur when the crank is collinear with the frame — pointing directly towards or away from the far pivot — which is a useful thing to know because it means the worst transmission angle can be found by checking two positions rather than by sweeping.
Designers use that: given a required rocker swing, the four lengths are chosen to make the worse of those two extremes as large as possible. That is a small optimisation problem with a closed-form answer, and it is the reason real four-bars often have proportions that look arbitrary.
The cam equivalent
The same idea appears in cam design under a different name and with the complementary convention.
A cam’s pressure angle is measured between the follower’s direction of travel and the normal to the cam surface — so it is the complement of a transmission angle, and the rule is stated as a maximum rather than a minimum: keep it below about 30°.
The lever available in each case is different, and that difference is instructive. For a cam there is one dominant knob: make the cam bigger. For a linkage there is no single parameter; μ depends on all four lengths at once, which is why linkage design is a search and cam design is largely a calculation.
What it leaves out
μ is a statement about the direction of force transmission at an instant. It is silent about magnitude, about whether the mechanism will survive, and about friction.
That last omission is the significant one, because friction is what turns a poor transmission angle from wasteful into fatal. A joint with friction has a locking condition: below some transmission angle the mechanism will not move however hard it is pushed, and where that angle sits depends on the friction coefficient — a quantity this site does not have.
So the 40° rule is doing two jobs. It keeps the useful force fraction reasonable, which is the part computed here, and it keeps a margin against a locking condition that is not computed here at all. Anyone using the number for a real design needs the second part too.
Why 40° and not some other number
The usual rule is that the transmission angle should stay above about 40°, and it is worth being clear that this is not a theorem. Nothing fails at 39°. The number is a distillation of what the force components do as μ falls, combined with what real joints do to the ideal picture.
The component of the coupler force that drives the rocker goes as sin μ; the component that merely pushes on the bearings goes as cos μ. At μ = 90° the split is all drive and no waste. At 40° it is 0.64 to 0.77 — already more bearing load than useful force. At 20° it is 0.34 to 0.94, and at 10° it is 0.17 to 0.98.
Two things go wrong together as that ratio worsens. The bearing loads grow, which is a wear and sizing problem, and the mechanism’s sensitivity to clearance grows faster, which is an accuracy problem. A pin joint with a few hundredths of a millimetre of clearance produces an output error proportional to roughly 1/sin μ, so the same joint that is invisible at 80° is a visible slop at 20°.
Below about 30° a third effect appears: friction in the joints can exceed the driving component, and the mechanism sticks in a position it should pass through. That is the second sense of jamming, and where it starts depends on the coefficient of friction, which is why the boundary is a range rather than a value.
So 40° is a margin, chosen so that a linkage designed to it still works after the joints have worn, the loads have been underestimated and the lubrication has aged. Precision instruments use 60°. Hand-operated mechanisms with generous joints get away with 30°. And a toggle clamp deliberately runs to nearly 0°, because there the whole point is that the output cannot back-drive the input.
Where the extremes are, and why they are computed rather than looked up
For a crank-rocker four-bar the extremes of the transmission angle occur when the crank lies along the ground line — at 0° and 180° — and both can be written in closed form from the four link lengths by the cosine rule.
That closed form is genuinely useful and this site does not use it as the source of the plotted values. The numbers in the figures come from the solved configurations, the same ones that produce the positions and the velocities, and the closed form is used the other way round: as an independent route that must agree.
The reason is the habit the whole site runs on. A closed-form expression for μ is correct for the four-bar it was derived for and silently wrong for anything else — a linkage with an attached coupler point, a slider, a six-bar. Reading the angle off the solved geometry works for all of them, and checking it against the closed form where the closed form applies proves the general route is right.
The agreement is to within arithmetic noise, and the check earns its place because the two routes fail differently: the closed form fails if the algebra is wrong, and the geometric route fails if the joint indices are wrong. A single method that agrees with itself proves nothing at all, which is the point that the conjugate-action test had to learn twice.
What the angle looks like over a full turn
Plotted against crank angle, μ for a crank-rocker is a smooth closed curve with one maximum and one minimum per revolution, and its shape carries design information that a single worst-case number does not.
A linkage whose μ curve is a shallow ripple around 80° is a good linkage everywhere. One whose curve spends most of the turn near 90° and dips sharply to 35° is worse than its average suggests, because the dip is where the load usually is — the extremes of μ coincide with the crank along the ground line, which for many machines is exactly where the working stroke begins.
The design lever is the ground-link length. Lengthening it flattens the μ curve and narrows the rocker’s swing; shortening it widens the swing and deepens the dip. That trade — output range against force quality — is the central one in four-bar design, and there is no proportion that escapes it. It is why Grashof’s condition is a necessary condition and never a sufficient one: it says the crank turns fully, and says nothing about whether the turning is any use.
Why it is checked rather than assumed
The transmission angle is a derived quantity: it is read off configurations the solver produced, and it inherits whatever is true of them. That makes it a good place to state what “checked” means on this site, because μ is checkable in three independent ways at once.
Against the closed form. For a plain four-bar the cosine rule gives μ from the four link lengths and the crank angle, and the geometric reading off the solved joints must agree. It does, to arithmetic noise, across the sweep.
Against the velocity solution. Where μ is 90° the coupler force is entirely useful and the mechanical advantage is at a local extremum; where μ approaches 0° or 180° the mechanical advantage diverges. The μ curve and the mechanical-advantage curve are computed from different parts of the machinery — geometry and the velocity solve respectively — and their features must line up. They do, and the fact that the worst μ and the toggle turn out to be 222° apart is a result of that comparison rather than an assumption fed into it.
Against the residual. A μ computed from a configuration whose residual is not at noise level is a number about nothing. Every configuration behind these plots solves to 10⁻¹³ or better, which is why the tolerance is a check rather than a stopping rule.
The reason to bother is that μ is exactly the kind of quantity that gets drawn rather than measured. It is an angle in a diagram, it is easy to sketch, and a sketch of a linkage at a configuration it cannot reach will happily display a transmission angle that the mechanism never has. The angle is only as good as the configuration it is measured in, and the configurations here exist because the constraints were solved rather than drawn.
What a designer does with the number
The transmission angle is a diagnostic, and the useful question is what to change when the diagnosis is bad. There are only a few levers.
Lengthen the ground link. This is the strongest and the least obvious. A longer ground flattens the μ curve and raises its minimum, at the cost of a narrower rocker swing and a physically larger mechanism.
Shorten the crank. Same direction, same trade: less output travel for better force transmission throughout.
Move the working stroke. The extremes of μ occur where the crank lies along the ground line, and those positions are fixed by the geometry. If the load can be arranged to occur elsewhere in the revolution — by re-timing the machine rather than re-proportioning the linkage — the worst μ still exists and no longer matters.
Add a link. A six-bar has more freedom to satisfy both a motion requirement and a force one, which is most of why six-bars exist. It is the expensive answer.
Accept it. If the mechanism is lightly loaded and the joints are good, 25° is survivable, and precision is the thing that suffers first rather than function.
What a designer should not do is treat the 40° rule as satisfied by an average. μ is a function of configuration and its minimum is what the rule is about, which is the same error as quoting a linkage’s ratio as a number — a variation summarised into a constant, with the worst case lost in the summary.
The exception is the mechanism designed to reach μ = 0 on purpose, where the diagnosis is not bad news at all. A toggle is a low transmission angle put to work, and telling the two situations apart requires knowing which configuration the number came from — which is why every plot here is against crank angle rather than a single figure.