Machine Kinematics
solved, then drawn
Concept

Mechanical advantage — where it appears

The ratio of a mechanism's output effort to its input effort, which for a linkage is the reciprocal of its velocity ratio. It becomes unbounded at a toggle, which is what a clamp is built to exploit.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

Two things that are not the same configuration. The mechanical advantage of a four-bar, from the velocity solution: for a lossless mechanism it is the reciprocal of the output-to-input speed ratio, so wherever the output momentarily stops the force ratio diverges. It reaches 1673 at 222° — a toggle, where crank and coupler line up, and the reason a knee-joint clamp holds with almost no effort. The transmission angle is at its worst somewhere else entirely: 46.5° at 0°, 222° away, where coupler and rocker line up instead. Both get called "the mechanism jamming" and they are opposite situations — one is enormous force output, the other is force going into the bearings.

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

wrong · Misconception The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

The transmission angle

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

linkages · Fourbar One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

A drum is a size, a wrap is a shape

A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.

strands · Strand A two-arm lamp balanced at every pose. A two-arm desk lamp: a lower arm 0.4 m long pivoted on its base, an upper arm carrying a 0.8 kg head 0.35 m out. A parallelogram — the lower arm, a short crank at the base, a link parallel to the lower arm, and the upper arm's first few centimetres — keeps the base crank parallel to the upper arm, so the upper arm's spring can run from a post above the base pivot to that crank and see the upper arm's angle alone. The lower arm's spring runs from the same post to the lower arm. Both springs have no free length and stiffnesses matched to their arms' weights. At every pose the lamp's total potential energy is the same: here it differs from the pose with both arms upright by 3e-16 of the head's moment, and the joint torques are 0 and 0 N·m — rounding. Dragging swings the lamp.

The lamp that stays where it is put

A desk lamp that stays wherever it is left is balanced, not held: its springs store exactly the energy its weight gives up, at every pose. For one arm that takes a spring with no free length, fixed above the pivot, whose energy has a cos θ term that cancels the weight's to 10⁻¹⁵. A spring with a free length L₀ leaves an unbalance of exactly L₀/s of the weight's torque, and a stiffness a fraction ε off leaves 2ε of its moment. For two arms it takes a parallelogram: without one, the upper spring sees the angle between the arms and the lamp's energy varies by twice the head's moment; with one, both springs sit at the base and the lamp is flat everywhere.

applied · Fourbar A tilting head balanced with one parallelogram and two springs. A desk lamp with a lower arm 0.4 m long, an upper arm of 0.35 m and a 0.5 kg head that tilts on the upper arm's end, its weight 0.08 m from that wrist. A parallelogram in the lower arm keeps a bracket at the elbow upright. Four springs of no free length: the lower arm's from a post on the base, the upper arm's from the elbow's bracket, and two for the head — one from the top of the elbow's bracket to a point on the head, which sees the head's angle from the vertical, and one across the wrist from the upper arm, which cancels the term the first brings in the angle between upper arm and head. At this pose, lower arm 0°, upper arm 80°, head 130° from the upward vertical, the lamp's energy differs from the first pose's by 5e-15 of the head's moment, and the torque at the wrist is 9e-10 N·m. Dragging moves all three joints.

Two springs stand in for a parallelogram

A lamp's head tilts on the end of its arm, and its weight's moment is a cosine of the head's angle from the vertical, so a spring can balance it only if the spring sees that angle. A second parallelogram carries the vertical out to the wrist and one spring does it. So do two springs with one parallelogram, or six with none: each spring that reaches across a joint brings a term in the joint's angle, and another spring across that joint takes it away. A spring across the wrist alone balances the head at one arm angle and leaves sin(Δ/2) of its moment over a range Δ.

applied · Fourbar Where the head spring hangs decides whether the arm's spring must push. The tilting-head desk lamp with one parallelogram: lower arm 0.4 m, upper arm 0.35 m, a 0.5 kg head with its weight 0.08 m beyond the wrist, and four springs of no free length. The spring from the top of the elbow's bracket to the head is hung 0.03 m along the head (the slider). Solved for an exact balance, its stiffness is 131 N/m, the wrist spring's 687, the lower arm's 561, and the upper arm's −434, negative: a spring that would have to push (drawn red). The upper arm's spring pulls exactly when the head attachment point is beyond 0.0615 m, 76.9% of the way to the head's centre of mass.

A spring that must push is hung in the wrong place

Balancing a lamp with a tilting head exactly, with springs of no free length, sometimes asks one spring for a negative stiffness, a spring that would have to push. It looked like a property of the lamp: a light upper arm under a heavy head might admit no design in which every spring pulls. It is a property of one attachment point. The upper arm's spring pulls exactly when the head spring meets the head beyond a point that is always short of the head's centre of mass, 77% of the way on this lamp, and at most 99% even with a head fifty times the arm's mass. Hang the head spring at its centre of mass and every spring pulls on every lamp.

applied · Fourbar

Named alongside it

The objects these essays reach for when they reach for this one.

Design ruleApproximationFour-barParallelogramParallelogram linkageToggleTransmission angleRatioConstraintConstraint jacobianContact normalDead centre

All concepts