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| {{About|assemblies of links designed to manage forces and movement|other uses|Linkage (disambiguation){{!}}Linkage}}
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| [[File:Variable stroke engine (Autocar Handbook, Ninth edition).jpg|thumb|Variable stroke engine (Autocar Handbook, Ninth edition)]]
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| A '''mechanical linkage''' is an assembly of bodies connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for example, and are called joints. A linkage modeled as a network of rigid links and ideal joints is called a [[kinematic chain]].
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| Linkages may be constructed from open chains, closed chains, or a combination of open and closed chains. Each link in a chain is connected by a joint to one or more other links. Thus, a kinematic chain can be modeled as a graph in which the links are paths and the joints are vertices, which is called a linkage graph.
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| [[File:Pantograph Mirror.gif|thumb|The deployable mirror linkage is constructed from a series of rhombus or scissor linkages.]]
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| [[Image:Hebebuehne Scissorlift.jpg|thumb|An extended [[scissor lift]]]]
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| The movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the [[degree of freedom (mechanics)|degrees of freedom]] (DOF) of the joint.
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| Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement. The ratio of the output force to the input force is known as the [[mechanical advantage]] of the linkage, while the ratio of the input speed to the output speed is known as the [[Gear ratio#Speed ratio|speed ratio]]. The speed ratio and mechanical advantage are defined so they yield the same number in an ideal linkage.
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| A kinematic chain, in which one link is fixed or stationary, is called a mechanism,<ref>OED</ref> and a linkage designed to be stationary is called a '''structure'''.
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| ==Uses==
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| [[Image:Linkage spacial 3DOFL.gif|thumb|A spatial 3 DOF linkage for joystick applications.]]
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| Perhaps the simplest linkage is the [[lever]], which is a link that pivots around a [[wikt:fulcrum|fulcrum]] attached to ground, or a fixed point. As a force rotates the lever, points far from the fulcrum have a greater velocity than points near the fulcrum. Because [[Power (physics)|power]] into the lever equals the power out, a small force applied at a point far from the fulcrum (with greater velocity) equals a larger force applied at a point near the fulcrum (with less velocity). The amount the force is amplified is called [[mechanical advantage]]. This is the law of the lever.
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| Two levers connected by a rod so that a force applied to one is transmitted to the second is known as a [[four-bar linkage]]. The levers are called [[crank (mechanism)|cranks]], and the fulcrums are called pivots. The connecting rod is also called the coupler. The fourth bar in this assembly is the ground, or frame, on which the cranks are mounted.
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| Linkages are important components of [[machine (mechanical)|machines]] and [[tool]]s. Examples range from the four-bar linkage used to amplify force in a [[bolt cutter]] or to provide independent [[suspension (vehicle)|suspension]] in an automobile, to complex linkage systems in [[robot|robotic arms]] and walking machines. The [[internal combustion engine]] uses a slider-crank four-bar linkage formed from its [[piston]], [[connecting rod]], and [[crankshaft]] to transform power from expanding burning gases into rotary power. Relatively simple linkages are often used to perform complicated tasks.
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| Interesting examples of linkages include the [[windshield wiper]], the [[bicycle suspension]], and [[hydraulic cylinder|hydraulic actuators]] for [[heavy equipment]]. In these examples the components in the linkage move in parallel planes and are called "planar linkages." A linkage with at least one link that moves in three dimensional space is called a "spatial linkage." The skeletons of robotic systems are examples of spatial linkages. The geometric design of these systems relies on modern [[computer aided design]] software.
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| The [[4-bar linkage]] is an adapted mechanical linkage used on bicycles. With a normal full-suspension bike the back wheel moves in a very tight arc shape. This means that more power is lost when going uphill.{{clarify|date=December 2012|reason=Why}} With a bike fitted with a 4-bar linkage, the wheel moves in such a large arc that it is moving almost vertically. This way the power loss is reduced by up to 30%.
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| ==History==
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| [[Archimedes]]<ref>T. Koetsier, "From Kinematically Generated Curves to Instantaneous Invariants: Episodes in the History of Instantaneous Planar Kinematics," Mechanism and Machine Theory, 21(6):489- 498, 1986</ref> applied geometry to the study of the lever. Into the 1500s the work of Archimedes and [[Hero of Alexandria]] were the primary sources of machine theory. It was [[Leonardo da Vinci]] who brought an inventive energy to machines and mechanism.<ref>A. P. Usher, 1929, A History of Mechanical Inventions, Harvard University Press, (reprinted by Dover Publications 1968)</ref>
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| In the mid-1700s the [[Watt steam engine|steam engine]] was of growing importance, and [[James Watt]] realized that efficiency could be increased by using different cylinders for expansion and condensation of the steam. This drove his search for a linkage that could transform rotation of a crank into a linear slide, and resulted in his discovery of what is called [[Watt's linkage]]. This led to the study of linkages that could generate straight lines, even if only approximately; and inspired the mathematician [[J. J. Sylvester]], who lectured on the [[Peaucellier linkage]], which generates an exact straight line from a rotating crank.<ref name="C. Moon, 2009">F. C. Moon, "History of the Dynamics of Machines and Mechanisms from Leonardo to Timoshenko," International Symposium on History of Machines and Mechanisms, (H. S. Yan and M. Ceccarelli, eds.), 2009. {{doi|10.1007/978-1-4020-9485-9-1}}</ref>
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| The work of Sylvester inspired [[Alfred Kempe|A. B. Kempe]], who showed that linkages for addition and multiplication could be assembled into a system that traced a given algebraic curve.<ref>A. B. Kempe, "On a general method of describing plane curves of the nth degree by linkwork," Proceedings of the London Mathematical Society, VII:213-216, 1876</ref> Kempe's design procedure has inspired research at the intersection of geometry and computer science.<ref>D. Jordan and M. Steiner, "Configuration Spaces of Mechanical Linkages," Discrete and Computational Geometry, 22:297-315, 1999</ref><ref>R. Connelly and E. D. Demaine, "Geometry and Topology of Polygonal Linkages," Chapter 9, Handbook of discrete and computational geometry, ([[Jacob E. Goodman|J. E. Goodman]] and J. O’Rourke, eds.), CRC Press, 2004</ref>
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| In the late 1800s [[Franz Reuleaux|F. Reuleaux]], A. B. W. Kennedy, and [[Ludwig Burmester|L. Burmester]] formalized the analysis and synthesis of linkage systems using [[descriptive geometry]], and [[Pafnuty Chebyshev|P.L.Chebyshev]] introduced analytical techniques for the study and invention of linkages.<ref name="C. Moon, 2009"/>
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| In the mid-1900s [[Ferdinand Freudenstein|F. Freudenstein]] and G. N. Sandor<ref>F. Freudenstein and G. N. Sandor, "Synthesis of Path Generating Mechanisms by Means of a Programmed Digital Computer," ASME Journal of Engineering for Industry, 81:159-168, 1959</ref> used the newly developed digital computer to solve the loop equations of a linkage and determine its dimensions for a desired function, initiating the computer-aided design of linkages. Within two decades these computer techniques were integral to the analysis of complex machine systems<ref>P. N. Sheth and J. J. Uicker, "IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis system for Mechanisms and Linkages," ASME Journal of Engineering for Industry, 94:454-464, 1972</ref><ref>C. H. Suh and C. W. Radcliffe, Kinematics and Mechanism Design, John Wiley, pp:458, 1978</ref> and the control of robot manipulators.<ref>R. P. Paul, Robot Manipulators: Mathematics, Programming and Control, MIT Press, 1981</ref>
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| R.E.Kaufman<ref>R. E. Kaufman and W. G. Maurer, "Interactive Linkage Synthesis on a Small Computer", ACM National Conference, Aug.3-5, 1971</ref><ref>A. J. Rubel and R. E. Kaufman, 1977, "KINSYN III: A New Human-Engineered System for Interactive Computer-aided Design of Planar Linkages," ASME Transactions, Journal of Engineering for Industry, May</ref> combined the computer’s ability to rapidly compute the roots of polynomial equations with a graphical user interface to unite [[Ferdinand Freudenstein|Freudenstein’s]] techniques with the geometrical methods of Reuleaux and [[Burmester's theory|Burmester]] and form ''KINSYN,'' an interactive computer graphics system for linkage design
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| The modern study of linkages includes the analysis and design of articulated systems that appear in robots, machine tools, and cable driven and tensegrity systems. These techniques are also being applied to biological systems and even the study of proteins.
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| == Mobility ==
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| [[File:4 bar linkage animated.gif|thumb|Simple linkages are capable of producing complicated motion.]] | |
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| The configuration of a system of rigid links connected by ideal joints is defined by a set of configuration parameters, such as the angles around a revolute joint and the slides along prismatic joints measured between adjacent links. The geometric constraints of the linkage allow calculation of all of the configuration parameters in terms of a minimum set, which are the ''input parameters''. The number of input parameters is called the ''mobility'', or [[degree of freedom (mechanics)|degree of freedom]], of the linkage system.
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| A system of ''n'' rigid bodies moving in space has ''6n'' degrees of freedom measured relative to a fixed frame. Include this frame in the count of bodies, so that mobility is independent of the choice of the fixed frame, then we have M=6(N-1), where N=n+1 is the number of moving bodies plus the fixed body.
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| Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints ''c'' that a joint imposes in terms of the joint's freedom ''f'', where ''c=6-f''. In the case of a hinge or slider, which are one degree of freedom joints, we have ''f=1'' and therefore ''c=6-1=5''.
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| Thus, the mobility of a linkage system formed from ''n'' moving links and ''j'' joints each with ''f<sub>i</sub>'', ''i=1, ..., j,'' degrees of freedom can be computed as,
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| :<math>M = 6n - \sum_{i=1}^j (6 - f_i) = 6(N-1 - j) + \sum_{i=1}^j\ f_i,</math>
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| where ''N'' includes the fixed link. This is known as [[Chebychev–Grübler–Kutzbach criterion|Kutzbach-Gruebler's equation]]
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| There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A single open chain consists of ''n'' moving links connected end to end by ''j'' joints, with one end connected to a ground link. Thus, in this case ''N=j+1'' and the mobility of the chain is
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| :<math> M = \sum_{i=1}^j\ f_i .</math>
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| For a simple closed chain, ''n'' moving links are connected end-to-end by ''n+1'' joints such that the two ends are connected to the ground link forming a loop. In this case, we have ''N=j'' and the mobility of the chain is
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| :<math> M = \sum_{i=1}^j\ f_i - 6.</math>
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| An example of a simple open chain is a serial robot manipulator. These robotic systems are constructed from a series of links connected by six one degree-of-freedom revolute or prismatic joints, so the system has six degrees of freedom.
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| An example of a simple closed chain is the RSSR spatial four-bar linkage. The sum of the freedom of these joints is eight, so the mobility of the linkage is two, where one of the degrees of freedom is the rotation of the coupler around the line joining the two S joints.
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| === Planar and spherical movement ===
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| [[Image:Linkage mobility.png|thumb|Linkage mobility]]
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| [[Image:Locking pliers.jpg|thumb|Locking [[Pliers (tool)|pliers]] exemplify a four-bar, one [[degrees of freedom (mechanics)|degree of freedom]] mechanical linkage. The adjustable base pivot makes this a two degree-of-freedom five-bar linkage.]]
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| It is common practice to design the linkage system so that the movement of all of the bodies are constrained to lie on parallel planes, to form what is known as a ''planar linkage''. It is also possible to construct the linkage system so that all of the bodies move on concentric spheres, forming a ''spherical linkage''. In both cases, the degrees of freedom of the link is now three rather than six, and the constraints imposed by joints are now ''c=3-f''.
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| In this case, the mobility formula is given by
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| :<math>M = 3(N- 1 - j)+ \sum_{i=1}^j\ f_i, </math>
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| and we have the special cases,
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| * planar or spherical simple open chain,
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| :<math> M = \sum_{i=1}^j\ f_i, </math>
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| * planar or spherical simple closed chain,
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| :<math> M = \sum_{i=1}^j\ f_i - 3. </math>
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| An example of a planar simple closed chain is the planar four-bar linkage, which is a four-bar loop with four one degree-of-freedom joints and therefore has mobility M=1.
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| == Joints ==
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| The most familiar joints for linkage systems are the [[revolute joint|revolute]], or hinged, joint denoted by an R, and the [[prismatic joint|prismatic]], or sliding, joint denoted by a P. Most all other joints used for spatial linkages are modeled as combinations of revolute and prismatic joints. For example,
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| * the cylindric joint consists of an RP or PR serial chain constructed so that the axes of the revolute and prismatic joints are parallel,
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| * the [[spherical joint]] consists of an RRR serial chain for which each of the hinged joint axes intersect in the same point;
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| * the planar joint can be constructed either as a planar RRR, RPR, and PPR serial chain that has three degrees-of-freedom.
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| == Analysis and synthesis of linkages ==
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| The primary mathematical tool for the analysis of a linkage is known as the kinematics equations of the system. This is a sequence of rigid body transformation along a serial chain within the linkage that locates a floating link relative to the ground frame. Each serial chain within the linkage that connects this floating link to ground provides a set of equations that must be satisfied by the configuration parameters of the system. The result is a set of non-linear equations that define the configuration parameters of the system for a set of values for the input parameters.
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| [[Ferdinand Freudenstein|Freudenstein]] introduced a method to use these equations for the design of a planar four-bar linkage to achieve a specified relation between the input parameters and the configuration of the linkage. Another approach to planar four-bar linkage design was introduced by [[Ludwig Burmester|L. Burmester]], and is called [[Burmester theory]].
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| == Planar one degree-of-freedom linkages ==
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| The mobility formula provides a way to determine the number of links and joints in a planar linkage that yields a one degree-of-freedom linkage. If we require the mobility of a planar linkage to be M=1 and f<sub>i</sub>=1, the result is
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| :<math> M = 3(N-1-j) + j = 1, \!</math>
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| or
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| :<math> j = (3/2)N - 2. \!</math>
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| This formula shows that the linkage must have an even number of links, so we have
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| * N=2, j=1: this is a two-bar linkage known as the [[lever]];
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| * N=4, j=4: this is the [[four-bar linkage]];
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| * N=6, j=7: this is a [[six-bar linkage]] [ it has two links that have three joints, called ternary links, and there are two topologies of this linkage depending how these links are connected. In the Watt topology, the two ternary links are connected by a joint. In the Stephenson topology the two ternary links are connected by binary links;<ref>{{cite web|url=http://books.google.com/books?id=X0AHKxwWTsYC&pg=PA107&lpg=PA107&dq=tsai,+enumeration&source=bl&ots=KsaVQo-N-Z&sig=xUPIkQcpC8ldoxjAJeVifSK2MRY&hl=en&ei=j4oSTpO2FcfYiAKu1ejdDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA#v=onepage&q&f=false |title=L. W. Tsai, ''Mechanism design: enumeration of kinematic structures according to function'', CRC Press, 2000 |publisher=Books.google.com |date= |accessdate=2013-06-13}}</ref>
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| * N=8, j=10: the eight-bar linkage has 16 different topologies;
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| * N=10, j=13: the 10-bar linkage has 230 different topologies,
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| * N=12, j=16: the 12-bar has 6856 topologies.
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| See Sunkari and Schmidt<ref>{{cite journal | last1 = Sunkari | first1 = R. P. | last2 = Schmidt | first2 = L. C. | year = 2006 | title = Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm | url = | journal = Mechanism and Machine Theory | volume = 41 | issue = | pages = 1021–1030 }}</ref> for the number of 14- and 16-bar topologies, as well as the number of linkages that have two, three and four degrees-of-freedom.
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| The planar [[four-bar linkage]] is probably the simplest and most common linkage. It is a one degree-of-freedom system that transforms an input crank rotation or slider displacement into an output rotation or slide.
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| [[Image:Linkage four bar.png|thumb|Types of four-bar linkages, s = shortest link, l = longest link]]
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| Examples of four-bar linkages are:
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| * the crank-rocker, in which the input crank fully rotates and the output link rocks back and forth;
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| * the slider-crank, in which the input crank rotates and the output slide moves back and forth;
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| * drag-link mechanisms, in which the input crank fully rotates and drags the output crank in a fully rotational movement.
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| ==Other interesting linkages==
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| [[Image:Linkage function generator.png|thumb|A function generator linkage that approximates a [[Parabola|parabolic]] output.]]
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| * [[Pantograph]] (four-bar, two DOF)
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| * Five bar linkages often have meshing gears for two of the links, creating a one DOF linkage. They can provide greater power transmission with more design flexibility than four-bar linkages.
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| * [[Klann linkage]] is a six-bar linkage that forms the leg of a walking mechanism;
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| * Toggle mechanisms are four-bar linkages that are dimensioned so that they can fold and lock. The linkage is dimensioned so that the linkage reaches a toggle position just before it folds. The high mechanical advantage allows the input crank to deform the linkage just enough to push it beyond the toggle position. This locks the input in place. Toggle mechanisms are used as clamps.
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| === Straight line mechanisms ===
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| {{main|Straight line mechanism}}
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| * James Watt's [[parallel motion]] and [[Watt's linkage]]
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| * [[Peaucellier–Lipkin linkage]], the first planar linkage to create a perfect straight line output from rotary input; eight-bar, one DOF.
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| * A [[Scott Russell linkage]], which converts linear motion, to (almost) linear motion in a line perpendicular to the input.
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| * [[Chebyshev linkage]], which provides nearly straight motion of a point with a four-bar linkage.
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| * [[Hoekens linkage]], which provides nearly straight motion of a point with a four-bar linkage.
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| * [[Sarrus linkage]], which provides motion of one surface in a direction normal to another.
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| * [[Hart's inversor]], which provides a perfect straight line motion without sliding guides.<ref name=Dijksman>{{cite web |url=http://alexandria.tue.nl/repository/freearticles/603890.pdf |title=True straight-line linkages having a rectlinear translating bar}}</ref>
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| ===Biological linkages===
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| Linkage systems are widely distributed in animals. The most thorough overview of the different types of linkages in animals has been provided by M. Muller,<ref name = muller>{{cite journal |doi=10.1098/rstb.1996.0065 |author=Muller, M. |title=A novel classification of planar four-bar linkages and its application to the mechanical analysis of animal systems |journal=Phil. Trans. R. Soc. Lond. B |volume=351 |pages=689–720 |year=1996 }}</ref> who also designed a new classification system which is especially well suited for biological systems. A well-known example is the [[cruciate ligament]]s of the knee.
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| An important difference between biological and engineering linkages is that revolving bars are rare in biology and that usually only a small range of the theoretically possible is possible due to additional mechanical constraints (especially the necessity to deliver blood).<ref name = dawkins>{{cite news | last = Dawkins | first = Richard | authorlink = Richard Dawkins | title = Why don't animals have wheels? | work = [[Sunday Times]] | date = November 24, 1996 | url = http://www.simonyi.ox.ac.uk/dawkins/WorldOfDawkins-archive/Dawkins/Work/Articles/1996-11-24wheels.shtml | accessdate = 2008-10-29 |archiveurl = http://web.archive.org/web/20070221073440/www.simonyi.ox.ac.uk/dawkins/WorldOfDawkins-archive/Dawkins/Work/Articles/1996-11-24wheels.shtml |archivedate = February 21, 2007}}</ref> Biological linkages frequently are [[compliant mechanism|compliant]]. Often one or more bars are formed by ligaments, and often the linkages are three-dimensional. Coupled linkage systems are known, as well as five-, six-, and even seven-bar linkages.<ref name = muller/> [[Four-bar linkage]]s are by far the most common though.
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| Linkages can be found in joints, such as the [[cruciate ligament|knee]] of [[tetrapod]]s, the hock of [[sheep]], and the cranial mechanism of [[bird]]s and [[reptile]]s. The latter is responsible for the upward motion of the upper bill in many birds.
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| Linkage mechanisms are especially frequent and manifold in the head of [[bony fishes]], such as [[wrasses]], which have [[evolution|evolved]] many specialized [[Aquatic predation|feeding mechanisms]]. Especially advanced are the linkage mechanisms of [[Aquatic predation#Protrusion|jaw protrusion]]. For [[suction feeding]] a system of linked four-bar linkages is responsible for the coordinated opening of the mouth and 3-D expansion of the buccal cavity. Other linkages are responsible for [[Aquatic predation#Protrusion|protrusion]] of the [[premaxilla]].
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| Linkages are also present as locking mechanisms, such as in the knee of the horse, which enables the animal to sleep standing, without active muscle contraction. In [[Aquatic predation#Pivot feeding|pivot feeding]], used by certain bony fishes, a four-bar linkage at first locks the head in a ventrally bent position by the alignment of two bars. The release of the locking mechanism jets the head up and moves the mouth toward the prey within 5-10 ms.
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| ==See also==
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| <div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
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| * [[Burmester theory]] of the design of linkages to reach multiple specified poses
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| * [[Cam]]
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| * [[dwell mechanism]]
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| * [[Engineering mechanics]]
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| * [[Four-bar linkage]]
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| * [[Jansen's linkage]]
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| * [[Kinematic coupling]]
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| * [[Kinematic pair]]s
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| * [[Kinematics]]
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| * Kinematic models in [[Mathcad]]<ref>{{cite web|author={{subject}} |url=http://communities.ptc.com/groups/kinematic-models-in-mathcad |title=PTC Community: Group: Kinematic models in Mathcad |publisher=Communities.ptc.com |date= |accessdate=2013-06-13}}</ref>
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| * [[Lever]]
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| * [[Machine (mechanical)|Machine]]
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| * [[Outline of machines]]
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| * [[Overconstrained mechanism]]
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| * [[Parallel motion]]
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| * [[Reciprocating motion]]
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| * [[Three-point hitch]]
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| * [[Whippletree (mechanism)|Whippletree]], a multi-bar linkage to evenly distribute force.
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| </div>
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| ==References==
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| {{Reflist|2}}
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| ==Further reading==
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| * {{cite book|last=Bryant|first=John|title=How round is your circle? : where engineering and mathematics meet|year=2008|publisher=Princeton University Press|location=Princeton|isbn=978-0-691-13118-4|page=306|coauthors=Sangwin, Chris}} — Connections between mathematical and real-world mechanical models, historical development of precision machining, some practical advice on fabricating physical models, with ample illustrations and photographs
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| * {{cite book | author=Erdman, Arthur G.; Sandor, George N.| title=Mechanism Design: Analysis and Synthesis | publisher=Prentice-Hall | year=1984 | isbn=0-13-572396-5}}
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| * Hartenberg, R.S. & J. Denavit (1964) [http://kmoddl.library.cornell.edu/bib.php?m=23 Kinematic synthesis of linkages], New York: McGraw-Hill — Online link from [[Cornell University]].
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| * {{cite book|last=Kidwell|first=Peggy Aldrich|title=Tools of American mathematics teaching, 1800–2000|year=2008|publisher=Johns Hopkins University Press|location=Baltimore|isbn=978-0-8018-8814-4|coauthors=Amy Ackerberg-Hastings, David Lindsay Roberts|pages=233–242}} — "Linkages: a peculiar fascination" (Chapter 14) is a discussion of mechanical linkage usage in American mathematical education, includes extensive references
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| *[http://kmoddl.library.cornell.edu/tutorials/04/ How to Draw a Straight Line] — Historical discussion of linkage design from Cornell University
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| * Parmley, Robert. (2000). "Section 23: Linkage." ''Illustrated Sourcebook of Mechanical Components.'' New York: McGraw Hill. ISBN 0070486174 Drawings and discussion of various linkages.
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| * Sclater, Neil. (2011). "Linkages: Drives and Mechanisms." ''Mechanisms and Mechanical Devices Sourcebook.'' 5th ed. New York: McGraw Hill. pp. 89-129. ISBN 9780071704427. Drawings and designs of various linkages.
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| ==External links==
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| {{commons|Linkage (mechanical)|Linkage (mechanical)}}
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| * [http://kmoddl.library.cornell.edu/index.php Kinematic Models for Design Digital Library (KMODDL)] — Major web resource for kinematics. Movies and photos of hundreds of working mechanical-systems models in the Reuleaux Collection of Mechanisms and Machines at [[Cornell University]], plus 5 other major collections. Includes an [http://kmoddl.library.cornell.edu/e-books.php e-book library] of dozens of classic texts on mechanical design and engineering. Includes CAD models and stereolithographic files for selected mechanisms.
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| * [http://www.howround.com/ How round is your circle?] — Supplementary online information to support the book, ''How round is your circle? : where engineering and mathematics meet'', by John Bryant and Chris Sangwin. Includes sections about Watt's and other linkages, many photographs and videos of linkages, errata and elaborations to the published book
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| * [http://www.dmg-lib.de Digital Mechanism and Gear Library (DMG-Lib)] (in German: Digitale Mechanismen- und Getriebebibliothek) — Online library about linkages and cams (mostly in German)
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| * [http://www.roymech.co.uk/Useful_Tables/Mechanics/Linkages.html Linkage calculations]
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| * [http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/index.html Java animated linkages]
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| * [http://www.math.ntnu.edu.tw/~jcchuan/demo/gear/machine.html GIF-format animated linkages]{{dead link|date=March 2012}}
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| * [http://web.archive.org/web/20070926103500/http://pergatory.mit.edu/2.007/lectures/2002/Lectures/Topic_04_Linkages.pdf Introductory linkage lecture]
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| * [http://www.brockeng.com/mechanism/index.htm Virtual Mechanisms Animated by Java]
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| * [http://vimeo.com/31933085 Linkage-based Drawing Apparatus by Robert Howsare]
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| * [http://mechanicaldesign101.com/category/linkage-animation/ Linkage animations on Mechanical Design 101]
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| * [https://www.info-key.de/en/asom (ASOM) Analysis, synthesis and optimization of multibar linkages]
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| {{DEFAULTSORT:Linkage (Mechanical)}}
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| [[Category:Machines]]
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| [[Category:Mechanisms]]
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| [[Category:Linkages]]
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