Four-bar_linkage - Wikiparency

# Four-bar linkage A **four-bar linkage**, or four-bar, is the simplest movable closed-chain [[Linkage_(mechanical)|linkage]]: four rigid bodies, called links or bars, joined in a loop by four one-degree-of-freedom [[Kinematic_pair|joints]]. When the joints are arranged so that the links move in parallel planes the assembly is a planar four-bar linkage, the form treated in most of this article; spherical and spatial four-bars exist and are used in practice. One link is held fixed and called the ground or frame. A grounded link that can turn through a complete revolution is a crank, one that cannot is a rocker, and the link that joins the two without touching ground is the coupler or floating link. The whole chain has one [[Degrees_of_freedom_(mechanics)|degree of freedom]] — by the planar mobility count `M = 3(L − 1) − 2J₁ − J₂`, four links and four pin joints give `M = 3(4 − 1) − 2 × 4 = 1` — so one input angle determines every other position, which is what makes the four-bar the workhorse of mechanism design. A single motor at the crank produces the whole motion, and a point carried on the coupler traces a curve far richer than a circle: approximate straight lines, figure eights, and the D-shaped strokes of a [[Pumpjack|pumpjack]] or a windscreen wiper. Two questions decide whether a proposed four-bar is any good. Whether the crank can turn all the way round is settled by the Grashof condition on the four link lengths. Whether it can transmit useful force is settled by the transmission angle at the driven pin, which falls toward zero at a toggle position and takes the mechanism's [[Mechanical_advantage|mechanical advantage]] to infinity — and its output force to uselessness — at the same instant. The framework microsim *The four-bar linkage: one crank, one loop, two ways to close it* turns the crank of a four-bar built from the sub-manual's shared geometry: the reader changes any of the four lengths and watches the vector loop close on the elbow-up branch or on the ghosted crossed branch, the coupler point draw its curve, the transmission angle swing at the rocker pin, and the Grashof class change under the linkage as a length crosses the condition. ## Planar four-bar linkage A planar four-bar is built from four links connected in a loop by four joints of one degree of freedom each. Only two such joints exist in the plane: the [[Revolute_joint|revolute joint]], a pin or hinge, denoted R, and the [[Prismatic_joint|prismatic joint]], a slider, denoted P. The naming follows from which link is held: the fixed one is the ground link, a grounded link that rotates fully is a [[Crank_(mechanism)|crank]], one that oscillates through a limited range is a rocker, and a link joined to ground by a prismatic joint is a slider — usefully thought of as a crank whose pivot has been moved infinitely far away, perpendicular to the direction of travel. The link that connects two other links without reaching ground is the coupler; in a slider-crank it is normally called the [[Connecting_rod|connecting rod]]. Three combinations of the two joint types exhaust the planar cases. Four revolute joints (RRRR) give the planar quadrilateral linkage, which includes the double-crank, the crank-rocker used in pumpjacks, the double-rocker used in [[Ackermann_steering_geometry|Ackermann steering]], the parallelogram and antiparallelogram, and the deltoid and trapezium forms. Three revolutes and one prismatic (RRRP and its inversions) give the [[Slider-crank_linkage|slider-crank]] of every [[Internal_combustion_engine|internal combustion engine]], and, by fixing a different link, the Whitworth and slotted-lever quick-return mechanisms of early [[Shaper|shaping machines]]. Two revolutes and two prismatics (PRRP) give the double slider, whose members are the [[Scotch_yoke|Scotch yoke]], the Oldham coupling and the [[Ellipsograph|elliptical trammel]]. Because the mobility is one in every case, these linkages can be driven by a single actuator and analysed as a one-parameter family of positions. That, with the fact that four links and four pins are cheap, stiff and easy to lubricate, is why planar four-bars are the base mechanism in so many machines, and why their [[Kinematics|kinematics]] and [[Dynamics_(mechanics)|dynamics]] are standard material in [[Mechanical_engineering|mechanical engineering]]. ## Planar quadrilateral linkage The planar quadrilateral linkage, the 4R or RRRR case, has four turning joints. One link is fixed as the frame; the two links attached to it are the grounded links, which serve as input and output; and the fourth is the coupler or floating link. Given the frame, each grounded link is one of four kinds according to the range of angle it can sweep: a [[Crank_(mechanism)|crank]], which turns a full 360 degrees; a rocker, whose range includes neither 0° nor 180°; a 0-rocker, whose range includes 0° but not 180°; and a π-rocker, whose range includes 180° but not 0°.[^mccarthy2010] Many authors distinguish only cranks from rockers, which is enough for most design work and gives the familiar names crank-rocker, double-crank and double-rocker. The configuration of the quadrilateral itself can be convex, concave or crossing. In the convex and concave arrangements no two links overlap, and the difference is whether one internal angle exceeds 180 degrees; in the crossing arrangement two links cross each other, the assembly of the antiparallelogram. For given link lengths and a given crank angle, the loop closes in two ways — the two intersections of a circle about the crank pin with a circle about the rocker pivot — and these are the two assembly branches, or circuits, of the mechanism. A physical [[Mechanism_(engineering)|mechanism]] is built on one branch and stays there: it cannot pass to the other without being taken apart, a fact worth stating because a [[Numerical_analysis|numerical]] solution will happily jump between them if nothing holds it. ### Grashof condition The Grashof condition answers whether any link can turn completely. Writing S for the shortest link, L for the longest, and P and Q for the other two, the condition is `S + L ≤ P + Q`, and when it holds the shortest link can rotate fully with respect to its neighbour.[^jensen201] Which link is shortest then decides the type: with the shortest link grounded the result is a double-crank, with the shortest adjacent to ground it is a crank-rocker, with the shortest as the coupler it is a double-rocker. When the inequality fails no link makes a complete turn and every grounded link is a rocker; when the two sides are equal the linkage can fold into a straight line — a [[Dead_centre_(engineering)|change point]], where the branches meet and the [[Mechanism_(engineering)|mechanism]] can pass from one to the other. The sim's default geometry is the shared illustrative set used throughout the design sub-manual — crank `a = 15`, coupler `b = 35`, rocker `c = 30`, ground `d = 40` mm — chosen to sit comfortably inside the Grashof window rather than transcribed from any particular machine.[^linkspec] Its shortest and longest links sum to `15 + 40 = 55` mm against `35 + 30 = 65` mm for the other two, so the condition holds with margin, and because the shortest link is the grounded input, the mechanism is a crank-rocker: the [[Crank_(mechanism)|crank]] turns continuously while the rocker sweeps back and forth. *Try: lengthen the coupler b from 35 to 60 mm and watch `S + L = 75` cross `P + Q = 70` — the class readout flips to non-Grashof, the crank stalls part-way round and becomes a rocker, and the rocker's travel closes down onto its limits; then step back to 35 mm and switch branches to see the same four lengths close as the crossed assembly, with the coupler curve replaced by its mirror.* ### Classification A finer classification sorts every quadrilateral linkage into eight cases from the signs of three sums. Let a, b, g and h be the lengths of the input crank, the output crank, the ground link and the floating link — the lettering of the classical table, which is not the sim's a, b, c, d for crank, coupler, rocker and ground. Then `T₁ = g + h − a − b`, `T₂ = b + g − a − h`, `T₃ = b + h − a − g`, and the signs of the three terms determine both whether the linkage is Grashof and what each grounded link can do.[^mccarthy2010] Four sign patterns are Grashof — `(−, −, +)` crank-crank, `(+, +, +)` crank-rocker, `(+, −, −)` rocker-crank and `(−, +, −)` rocker-rocker — and four are not, giving the pairs of 0-rockers and π-rockers. The cases where one term is exactly zero are the folding [[Linkage_(mechanical)|linkages]], in which the four links can lie along a line; counting these separately multiplies the classes considerably. The scheme is easy to apply. For the sim's geometry the input [[Crank_(mechanism)|crank]] is `a = 15`, the output crank (rocker) `b = 30`, the ground `g = 40` and the floating coupler `h = 35`, so `T₁ = 40 + 35 − 15 − 30 = +30`, `T₂ = 30 + 40 − 15 − 35 = +20` and `T₃ = 30 + 35 − 15 − 40 = +10`. The pattern `(+, +, +)` is the Grashof crank-rocker row of the table, which agrees with what the `S + L ≤ P + Q` test gave above and with the motion the sim draws. The value of the three-term form is that it names the input and the output separately: the shortest-link rule says that some link turns fully, while the sign pattern says which one, without a drawing. ## Design of four-bar mechanisms Designing a four-bar — synthesis, as opposed to the analysis above — means choosing link lengths that produce a wanted output motion from a given input motion, and doing it with the simplest mechanism that will serve. The classical division is into three problems: function generation, in which the output angle must follow a prescribed function of the input angle; path generation, in which a coupler point must pass through prescribed points; and motion generation or rigid-body guidance, in which the coupler must reach prescribed positions and orientations. Choosing the link lengths is dimensional synthesis, and in practice it is often an iterate-and-analyse loop, because exact procedures exist only for special cases and for small numbers of prescribed positions.[^mccarthy2010] One number governs the result more than any other. The transmission angle, the angle at the pin between the coupler and the driven link, measures how much of the coupler's force pushes the rocker along rather than into its bearing; design practice keeps it above roughly 40 degrees through the whole cycle, and a mechanism that violates this is stiff, noisy and heavy on its pins even when its geometry is otherwise correct.[^jensen209] Computed by the law of cosines from the diagonal that closes the loop, the sim's default geometry gives a minimum transmission angle of 44.4° when the crank points toward the rocker pivot and a maximum of 64.6° half a turn later — inside the usable window at every crank angle, which is what makes it a serviceable teaching geometry rather than merely a legal one.[^linkspec] The limit of the argument is the toggle or [[Dead_centre_(engineering)|dead-centre]] position, where the coupler and the driven link fall into line. There the mechanism's mechanical advantage — which can be read off the ratio of the distances from the two pivots to the [[Instant_centre_of_rotation|instant centre]] — becomes infinite, and the output force it can exert against a load goes with it, while the output velocity falls to zero.[^jensen215] Toggles are exploited deliberately in clamps, crushers and locking knee joints, and avoided everywhere else; at a toggle the mechanism is also momentarily indifferent about which branch to take next, which is why toggling linkages need a spring, a flywheel or an offset to break the ambiguity. ### Time ratio A four-bar driven at constant crank speed does not generally take the same time for the forward and return strokes of its output, and that asymmetry is measured by the time ratio: the time for the slower stroke divided by the time for the faster one. A mechanism with a time ratio of 1 is balanced; anything larger is a quick-return mechanism, valuable wherever the working stroke should be slow and powerful and the idle stroke fast, as in the [[Shaper|shaper]] and the slotted-lever drive. The geometry behind the number is the imbalance angle α, the angle between the two crank positions that put the output at its extremes. The crank sweeps `180° + α` during one stroke and `180° − α` during the other at the same speed, so the time ratio is `Q = (180 + α)/(180 − α)`; an imbalance angle of 30° gives `210/150 = 1.4`, a working stroke 40 percent longer than the return. In an in-line [[Slider-crank_linkage|slider-crank]] α is zero and the strokes are equal; offsetting the slider from the crank axis makes α non-zero, which is the cheapest way to buy a quick return. ### Timing charts A timing chart plots what each mechanism of a machine is doing against a common cycle angle, marking where each is dwelling, advancing or returning. Its purpose is [[Synchronization|synchronisation]]: on a machine with several linkages driven from one shaft — a press with a feed, a clamp and a ram — the chart is where interference is found before metal is cut, and where the phase of each cam or crank on the line shaft is fixed. The chart is also a first estimate of how hard the motion is on the machine. Treating a stroke of length ΔR completed in time Δt as a symmetric accelerate-and-decelerate motion gives `ΔR = ½ v_peak Δt` for the peak velocity and `ΔR = ¼ a (Δt)²` for the constant acceleration that produces it — elementary kinematics, and consistent with each other since `v_peak = a Δt/2`. A 50 mm stroke in 0.2 s therefore implies a peak velocity of 500 mm/s and an acceleration of about 5 m/s². Multiplied by the moving mass, that acceleration is the inertia force the pins and the frame must carry, and it is the number that usually decides the top speed of the machine rather than any strength of the links themselves. ## Slider-crank linkage The slider-crank is the four-bar with three revolute joints and one prismatic joint: a crank, a connecting rod and a slider constrained to a straight path. It converts rotation to reciprocation and back, which makes it the mechanism of the [[Internal_combustion_engine|internal combustion engine]], the reciprocating pump and the power press. Driven at the crank it produces a stroke of `2r` for crank radius r; driven at the [[Piston|piston]], as by gas pressure, it turns the crank. Its geometry is one equation. With crank radius r, connecting rod length l and the crank angle θ measured from the position of greatest extension, the in-line slider's distance from the crank centre is `x = r cos θ + √(l² − r² sin²θ)`. For a crank of 40 mm and a rod of 140 mm the stroke is 80 mm, and at θ = 90° the slider stands at `√(140² − 40²) = 134.2` mm rather than at the 140 mm a very long rod would give — the difference that makes a piston's motion non-sinusoidal and puts a second-order component into its acceleration, which engine balancing must then address. There are two kinds. An in-line slider-crank has the slider's line of travel passing through the crank's pivot, and its two strokes are symmetric. An offset slider-crank does not, so the slider moves faster one way than the other: a [[Quick_return_mechanism|quick-return mechanism]] whose time ratio follows from the imbalance angle as above. Fixing a link other than the frame gives the inversions — among them the Whitworth quick return and the oscillating-cylinder engine — which are the same chain viewed from a different link. ## Spherical and spatial four-bar linkages If the four joint axes are angled so that they all intersect at a single point, each link moves on the surface of a sphere centred there, and the assembly is a spherical four-bar linkage. Link lengths become arc angles, and the analysis is the plane one rewritten in [[Spherical_trigonometry|spherical trigonometry]]. The [[Universal_joint|universal joint]] is the familiar member of the family, and spherical four-bars appear wherever a rotation must be transmitted between shafts that meet at an angle, or where an output must be steered over a sphere rather than in a plane. A spatial four-bar has joint axes that neither lie in parallel planes nor meet at a point. Most such assemblies are rigid, since the mobility count in space, `M = 6(L − 1) − 5J₁` for four links and four one-freedom joints, gives `18 − 20 = −2`. The exceptions are the overconstrained mechanisms, whose special proportions allow motion the general count forbids: Bennett's linkage, described in 1903, is a spatial four-bar with revolute joints whose axes are angled and whose link lengths obey a particular relation, and it moves.[^bennett1903] The input-output equations of the spherical four-bar carry over to spatial linkages when the angle variables are replaced by [[Dual_number|dual numbers]], which is the standard algebraic route into spatial [[Kinematic_synthesis|kinematic synthesis]].[^mccarthy2010] ## Examples The four-bar is everywhere once its outline is recognised, and the same four links do quite different work depending on which is grounded and where the coupler point sits. Two groups are worth separating. The first is the named linkages, classical four-bars proportioned by a particular person for a particular coupler curve, which are studied for the curve itself and reused wherever that curve is wanted. The second is the applications, ordinary [[Machine|machines]] that turn out to be four-bars once the pin joints are found — a suspension arm, a wiper, a door closer — where the designer's interest is in a stroke, a clearance or a force rather than in the curve's mathematics. A great many mechanisms belong to both groups at once, since a machine that needs an approximate straight line usually gets it from one of the named linkages. What follows is a sample rather than a catalogue: the four-bar's usefulness is precisely that its list of applications has no natural end. ### Other linkages and mechanisms Several four-bars are named after the person who proportioned them, nearly always to make a coupler point travel in an approximate straight line before there was any way to machine a flat guide. [[Watt's_linkage|Watt's linkage]] traces a figure eight whose central portion is nearly straight, and was devised to guide the piston rod of a [[Beam_engine|beam engine]]; [[Chebyshev_linkage|Chebyshev's linkage]] and the closely related [[Hoecken_linkage|Hoecken linkage]] achieve a straight portion with an approximately constant speed along it, which suits a walking or conveying motion; [[Roberts_linkage|Roberts's linkage]] is the symmetric solution of the same problem. The family is collected under [[Straight-line_mechanism|straight-line mechanisms]], and its exact members, which need more than four bars, sit beside it. The [[Pantograph|pantograph]] belongs here too, as a parallelogram four-bar whose geometry scales a traced path by a fixed factor. ### Applications Working four-bars are chosen for their strokes rather than their curves. The [[Pumpjack|pumpjack]] over an oil well is a crank-rocker whose rocker is the walking beam; a [[Windscreen_wiper|windscreen wiper]] pair couples its two arms through a four-bar so that they rock together, and a [[Door_closer|door closer]] is a four-bar with a damper in one joint. Vehicle suspensions supply the best-known examples: the [[Double_wishbone_suspension|double wishbone]] is a four-bar in the transverse plane whose coupler carries the wheel, so the camber and track changes through the stroke are simply the coupler's motion, and many [[Bicycle_suspension|bicycle rear suspensions]] are four-bars designed to place an instant centre where the designer wants the axle path. Foot-operated machines — the [[Treadle|treadle]] of a [[Lathe|lathe]] or a [[Sewing_machine|sewing machine]] — are crank-rockers driven backwards, from the rocker, which is why they need a flywheel to carry them through the toggle positions. ## Simulations Simulating a four-bar means solving the loop-closure equation repeatedly as the input turns. Writing the links as vectors in the [[Complex_number|complex plane]] gives the constraint in one line, `a e^(j θ₂) + b e^(j θ₃) − c e^(j θ₄) − d = 0`, two real equations in the unknown angles θ₃ and θ₄ for each input θ₂. The equations can be solved in closed form — geometrically, as the intersection of a circle of radius b about the crank pin with a circle of radius c about the rocker pivot — or iteratively with a Newton solver of the kind [[Multibody_system|multibody]] codes use. The closed form is exact and fast, and what a teaching simulation should use; its one subtlety is that two intersections exist, so the solver must choose a branch and hold it or the animation will flip mechanisms mid-turn. The framework microsim does exactly that. Position comes from the library's `fourBar` routine, the vector-loop closed form evaluated as a circle intersection, with the second drawn as a ghost so both assemblies are visible; the Grashof class from the link lengths; the transmission angle from the law of cosines on the loop's diagonal, folded to 90 degrees or less so the reported angle is the one a designer checks; and the rocker's travel limits from the extremes of the same diagonal.[^linkspec] The geometry it opens on — `a = 15`, `b = 35`, `c = 30`, `d = 40` mm — is the design sub-manual's shared illustrative set rather than a machine's dimensions, and this article's worked numbers are computed from it, not measured.[^linkspec] The library's checks are the design text's statements about Grashof classes, transmission angles and toggle positions, since that text's equations survive only as prose.[^jensen209][^linkspec] What a simulation adds over the algebra is the coupler curve. Once positions are solved for a full turn, the path of any point fixed in the coupler can be plotted, and moving it around the coupler sweeps a family of curves no formula makes obvious — the practical face of the classical result that the coupler curve of a four-bar is a sextic, and the reason [[Kinematic_synthesis|synthesis]] for path generation was a graphical art long before it was a numerical one. A simulation also exposes what the algebra hides: the branch structure, the approach to a toggle, and how far a small change in one length moves the transmission angle while barely touching the coupler curve. ## See also - [[Linkage_(mechanical)]] - [[Slider-crank_linkage]] - [[Six-bar_linkage]] - [[Five-bar_linkage]] - [[Cognate_linkage]] - [[Burmester's_theory]] - [[Kinematic_synthesis]] - [[Straight-line_mechanism]] - [[Pumpjack]] - [[Universal_joint]] - [[Cam_(mechanism)]] - [[Gear]] - [[Simple_machine]] ## References [^jensen201]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 201–206: the Grashof condition and the four-bar classes. [^jensen209]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 209–214: transmission angle, toggle positions, and position analysis by the vector loop. [^jensen215]: Jensen, N. *Introduction to Mechanical Design and Manufacturing*. Portal Book 109, pp. 215–218: mechanical advantage of a linkage and its instant centres. [^linkspec]: Engineering portal sim spec `specs/sims/Four-bar_linkage.json`, framework module `design.linkage`: `fourBar` (circle-intersection form of the standard vector loop, both branches), `grashof` (class from the link lengths), `transmissionAngle` (law of cosines, folded to ≤ 90°) and `rockerLimits`. The default geometry `a = 15`, `b = 35`, `c = 30`, `d = 40` mm is the design sub-manual's shared **illustrative** set, not a machine's dimensions; the article's transmission angles (44.4° minimum, 64.6° maximum) and Grashof and classification terms are computed from it. The underlying design text's equation images were lost, so the library was verified against that text's statements rather than its formulas. [^mccarthy2010]: McCarthy, J. Michael; Soh, Gim Song (2010). *Geometric Design of Linkages*, 2nd edition. New York: Springer. (Chapter and page numbers not pinned for this run; the crank / rocker / 0-rocker / π-rocker taxonomy, the eight-case sign classification and the dual-number extension to spatial linkages are that text's standard treatment.) [^bennett1903]: Bennett, Geoffrey Thomas (1903). "A new mechanism." *Engineering* 76. (Page numbers not pinned for this run.)  **Microsim — three.js (Wikitube framework):** *Four-bar linkage* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Four-bar_linkage.html" data-title="Four-bar linkage"></div> *Built from `MICROSIM_GUIDE/specs/sims/Four-bar_linkage.json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Four-bar_linkage) : [Wikitube](https://en.wikitube.io/wiki/Four-bar_linkage) - skeleton pinned to revision 1372304672 (2026-09-18).