In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-chain movable linkage. It consists of four bodies, called bars or links, connected in a loop by four joints. Generally, the joints are configured so the links move in parallel planes, and the assembly is called a planar four-bar linkage. Spherical and spatial four-bar linkages also exist and are used in practice.

thumb|right|A [[pumpjack's main mechanism is a four-bar linkage]]

Planar four-bar linkage

Planar four-bar linkages are constructed from four links connected in a loop by four one-degree-of-freedom joints. A joint may be either a revolute joint – also known as a pin joint or hinged joint – denoted by R, or a prismatic joint – also known as a sliding pair – denoted by P.

The figure shows examples of the various cases for a planar quadrilateral linkage.centre|thumb|500x500px|Types of four-bar linkages, s: shortest link, l: longest link.

The configuration of a quadrilateral linkage may be classified into three types: convex, concave, and crossing. In the convex and concave cases no two links cross over each other. In the crossing linkage two links cross over each other. In the convex case all four internal angles are less than 180 degrees, and in the concave configuration one internal angle is greater than 180 degrees. There exists a simple geometrical relationship between the lengths of the two diagonals of the quadrilateral. For convex and crossing linkages, the length of one diagonal increases if and only if the other decreases. On the other hand, for nonconvex non-crossing linkages, the opposite is the case; one diagonal increases if and only if the other also increases.

Design of four-bar mechanisms

The synthesis, or design, of four-bar mechanisms is important when aiming to produce a desired output motion for a specific input motion. In order to minimize cost and maximize efficiency, a designer will choose the simplest mechanism possible to accomplish the desired motion. When selecting a mechanism type to be designed, link lengths must be determined by a process called dimensional synthesis. Dimensional synthesis involves an iterate-and-analyze methodology which in certain circumstances can be an inefficient process; however, in unique scenarios, exact and detailed procedures to design an accurate mechanism may not exist.

Time ratio

The time ratio (Q) of a four-bar mechanism is a measure of its quick return and is defined as follows:

:

:

Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated. Note that the cited conference paper incorrectly conflates Moore-Penrose pseudoinverses with one-sided inverses of matrices, falsely claiming that the latter are unique whenever they exist. This is contradicted by the fact that <math>(1,0)^T</math> admits the set of matrices <math>\{(1,x) \mid x \in \mathbb C\}</math> as all its left inverses.

Bennett's linkage is a spatial four-bar linkage with hinged joints that have their axes angled in a particular way that makes the system movable.

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File:Universal joint.gif|Universal joint.

File:Tractor Bump Steer.GIF|Tractor steering

File:Bennett four-bar linkage.jpg|Bennett four-bar linkage.

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Examples

Other Linkages and Mechanisms

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  • Chebyshev linkage
  • Chebyshev lambda linkage
  • Evans "Grasshopper" linkage
  • Hoecken linkage
  • Horse-head linkage
  • Pantograph
  • Roberts linkage
  • Valve gear
  • Watt's linkage

Applications

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  • Bicycle suspension
  • Biological linkages
  • Double-beam drawbridge
  • Double wishbone suspension
  • Door closer
  • Foldable steps and foldable chairs
  • Foot operated machines (grindstone, lathe, sewing machine, treadle, etc.)
  • Gear shifter
  • Glider (furniture)
  • Oscillating fan
  • Pumpjack
  • Step-on trash can
  • Windshield wiper

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File:Oil well pump jack, NE Colo (7632673158).jpg|Oil well pump-jack.

File:Suspension train watt.svg|Train Suspension using Watt's linkage.

File:Watt's Linkage Rear Suspension.gif|Rear car suspension using Watt's linkage.

File:VPK -3927 Volk 02.jpg|Suspension for VPK-3927 Volk family armored vehicles.

File:Door closer 20170214.jpg|Door closer mechanism.

File:Sewing machine, The Howe Machine Company, Bridgeport CT, c. 1870, cast iron, steel, wood - Bennington Museum - Bennington, VT - DSC08590.JPG|The Howe Machine Co. Sewing machine.

File:Grinding machine with a foot drive.jpg|A pedal-driven grindstone.

File:Screw cutting treadle lathe (Brown bros 1912).jpg|The Brown Bros screwcutting treadle lathe.

File:Ford Design 3-speed OD Transmission w. Hurst Shifter.jpg|Ford 3-speed Overdrive Transmission

File:2003 SV650S Gearshift.jpg|Gearshift on a 2003 SV650S motorcycle.

File:Chair,foldable,Tamil Nadu444.jpeg|A foldable chair.

File:MtbFrameGeometry FSR.png|A diagram of the suspension on a Specialized Stumpjumper FSR mountain bike.

File:Steam locomotive work.gif|Animation of a steam locomotive with Walschaerts valve gear.

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Simulations

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Crank-Rocker 4-bar Linkage.gif|Crank-Rocker 4-Bar Linkage:<br><br>Cyan link completes a full revolution<br>Green link rocks back and forth.

Double-Crank 4-bar Linkage.gif|Double-Crank 4-Bar Linkage<br><br>Cyan link completes a full revolution<br>Green link completes a full revolution.

Double-Rocker 4-bar Linkage.gif|Double-Rocker 4-Bar Linkage:<br><br>Yellow link and Green link rock back and forth.

Parallel 4-bar Linkage 1.gif|Parallelogram and Antiparallelogram 4-Bar Linkages:<br><br>(Demonstration of Point Change Condition in the Middle)

Parallel 4-bar Linkage 2.gif|Parallelogram and Antiparallelogram 4-Bar Linkages (Inversions):<br><br>(Demonstration of Point Change Condition in the Middle)

Deltoid 4-bar Linkage 2.gif|Deltoid/Kite (Galloway) 4-Bar Linkage:<br><br>Both inversions shown:<br>• Crank-Rocker<br>• Double-Crank

Trapezium 4-bar Linkage.gif|Trapezium (Arglin) 4-Bar Linkage:<br><br>All four unique inversions shown:<br>• Crank-Rocker<br>• Crank-Rocker<br>• Double-Crank<br>• Double-Rocker

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File:4-Bar Fixed and Moving Centrodes.gif|Fixed and moving centrodes drawn on a 4-Bar Double-Crank linkage.

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See also

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  • Burmester's theory
  • Cognate linkage
  • Five-bar linkage
  • Kinematic synthesis
  • Linkage (mechanical)
  • Pumpjack
  • Six-bar linkage
  • Slider-crank linkage
  • Spherical trigonometry
  • Straight line mechanism (Approximate straight lines are primarily four-bar linkages)
  • Universal joint

Notes

References

  • The four-bar linkages in the collection of Reuleaux models at Cornell University
  • The four-bar linkages in the collection of Reuleaux models at Cornell University (archive)