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A mechanical linkage is an assembly of systems 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 In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained (or desired) motion that is the mathematical model for a mechanical system. Reuleaux, F., 187''The Kinematics of Machine ...
. 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. 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
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
(DOF) of the joint. 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 Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
of the linkage, while the ratio of the input speed to the output speed is known as the speed ratio. The speed ratio and mechanical advantage are defined so they yield the same number in an ideal linkage. A kinematic chain, in which one link is fixed or stationary, is called a mechanism, and a linkage designed to be stationary is called a structure.


History

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
applied geometry to the study of the lever. Into the 1500s the work of Archimedes and
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. H ...
were the primary sources of machine theory. It was
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...
who brought an inventive energy to machines and mechanism. In the mid-1700s the
steam engine A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be ...
was of growing importance, and
James Watt James Watt (; 30 January 1736 (19 January 1736 OS) – 25 August 1819) was a Scottish inventor, mechanical engineer, and chemist who improved on Thomas Newcomen's 1712 Newcomen steam engine with his Watt steam engine in 1776, which was ...
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 In kinematics, Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel on a nearly straight line. It was described in ...
. This led to the study of linkages that could generate straight lines, even if only approximately; and inspired the mathematician
J. J. Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...
, who lectured on the Peaucellier linkage, which generates an exact straight line from a rotating crank.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. The work of Sylvester inspired A. B. Kempe, who showed that linkages for addition and multiplication could be assembled into a system that traced a given algebraic curve. Kempe's design procedure has inspired research at the intersection of geometry and computer science. In the late 1800s F. Reuleaux, A. B. W. Kennedy, and L. Burmester formalized the analysis and synthesis of linkage systems using
descriptive geometry Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and ...
, and P. L. Chebyshev introduced analytical techniques for the study and invention of linkages. In the mid-1900s F. Freudenstein and G. N. Sandor 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 and the control of robot manipulators. R. E. Kaufman combined the computer's ability to rapidly compute the roots of polynomial equations with a graphical user interface to unite Freudenstein's techniques with the geometrical methods of Reuleaux and Burmester and form ''KINSYN,'' an interactive computer graphics system for linkage design 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.


Mobility

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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
, of the linkage system. A system of ''n'' rigid bodies moving in space has 6''n'' 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. 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. Thus, the mobility of a linkage system formed from ''n'' moving links and ''j'' joints each with ''f''''i'', ''i'' = 1, ..., ''j'', degrees of freedom can be computed as, :M = 6n - \sum_^j (6 - f_i) = 6(N-1 - j) + \sum_^j\ f_i, where ''N'' includes the fixed link. This is known as Kutzbach–Grübler's equation There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A simple 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 : M = \sum_^j\ f_i . 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 : M = \sum_^j\ f_i - 6. 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. An example of a simple closed chain is the RSSR (revolute-spherical-spherical-revolute) 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.


Planar and spherical movement

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''. In this case, the mobility formula is given by :M = 3(N- 1 - j)+ \sum_^j\ f_i, and we have the special cases, * planar or spherical simple open chain, :: M = \sum_^j\ f_i, * planar or spherical simple closed chain, :: M = \sum_^j\ f_i - 3. 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.


Joints

The most familiar joints for linkage systems are the revolute, or hinged, joint denoted by an R, and the
prismatic An optical prism is a transparent optical element with flat, polished surfaces that are designed to refract light. At least one surface must be angled — elements with two parallel surfaces are ''not'' prisms. The most familiar type of optical ...
, or sliding, joint denoted by a P. Most other joints used for spatial linkages are modeled as combinations of revolute and prismatic joints. For example, * the cylindric joint consists of an RP or PR serial chain constructed so that the axes of the revolute and prismatic joints are parallel, * the
universal joint A universal joint (also called a universal coupling or U-joint) is a joint or coupling connecting rigid shafts whose axes are inclined to each other. It is commonly used in shafts that transmit rotary motion. It consists of a pair of hinges ...
consists of an RR serial chain constructed such that the axes of the revolute joints intersect at a 90° angle; * the
spherical joint In an automobile, ball joints are spherical bearings that connect the control arms to the steering knuckles, and are used on virtually every automobile made. They bionically resemble the ball-and-socket joints found in most tetrapod animal ...
consists of an RRR serial chain for which each of the hinged joint axes intersect in the same point; * the planar joint can be constructed either as a planar RRR, RPR, and PPR serial chain that has three degrees-of-freedom.


Analysis and synthesis of linkages

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. 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 L. Burmester, and is called
Burmester theory In kinematics, Burmester theory comprises geometric techniques for synthesis of linkages. It was introduced in the late 19th century by Ludwig Burmester (1840–1927). His approach was to compute the geometric constraints of the linkage direct ...
.


Planar one degree-of-freedom linkages

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''''i'' = 1, the result is : M = 3(N - 1 - j) + j = 1, \! or : j = \fracN - 2. \! This formula shows that the linkage must have an even number of links, so we have * ''N'' = 2, ''j'' = 1: this is a two-bar linkage known as the
lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
; * ''N'' = 4, ''j'' = 4: this is the
four-bar linkage 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 config ...
; * ''N'' = 6, ''j'' = 7: this is a
six-bar linkage In mechanics, a six-bar linkage is a mechanism with one degree of freedom that is constructed from six links and seven joints. An example is the Klann linkage used to drive the legs of a walking machine. In general, each joint of a 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; * ''N'' = 8, ''j'' = 10: the eight-bar linkage has 16 different topologies; * ''N'' = 10, ''j'' = 13: the 10-bar linkage has 230 different topologies, * ''N'' = 12, ''j'' = 16: the 12-bar has 6856 topologies. See Sunkari and Schmidt 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. The planar
four-bar linkage 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 config ...
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. Examples of four-bar linkages are: * the crank-rocker, in which the input crank fully rotates and the output link rocks back and forth; * the slider-crank, in which the input crank rotates and the output slide moves back and forth; * drag-link mechanisms, in which the input crank fully rotates and drags the output crank in a fully rotational movement.


Other interesting linkages

* Pantograph (four-bar, two DOF) * 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. *
Jansen's linkage Jansen's linkage is a planar leg mechanism designed by the kinetic sculptor Theo Jansen to generate a smooth walking motion. Jansen has used his mechanism in a variety of kinetic sculptures which are known as (Dutch for "beach beasts"). Jans ...
is an eight-bar
leg mechanism A leg mechanism (walking mechanism) is a mechanical system designed to provide a propulsive force by intermittent frictional contact with the ground. This is in contrast with wheels or continuous tracks which are intended to maintain continuous f ...
that was invented by kinetic sculptor
Theo Jansen Theodorus Gerardus Jozef Jansen (; born 14 March 1948) is a Dutch artist. In 1990, he began building large mechanisms out of PVC that are able to move on their own and, collectively, are titled ''Strandbeest''. The kinetic sculptures appear to ...
. *
Klann linkage The Klann linkage is a planar mechanism designed to simulate the gait of legged animal and function as a wheel replacement, a leg mechanism. The linkage consists of the frame, a crank, two grounded rockers, and two couplers all connected by ...
is a six-bar linkage that forms a
leg mechanism A leg mechanism (walking mechanism) is a mechanical system designed to provide a propulsive force by intermittent frictional contact with the ground. This is in contrast with wheels or continuous tracks which are intended to maintain continuous f ...
; * Toggle mechanisms are four-bar linkages that are dimensioned so that they can fold and lock. The toggle positions are determined by the colinearity of two of the moving links. 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.


Straight line mechanisms

* James Watt's parallel motion and
Watt's linkage In kinematics, Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel on a nearly straight line. It was described in ...
*
Peaucellier–Lipkin linkage The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion ...
, the first planar linkage to create a perfect straight line output from rotary input; eight-bar, one DOF. * A
Scott Russell linkage A Scott Russell linkage is a linkage which translates linear motion through a right angle. The linkage is named after John Scott Russell (1808–1882), although watchmaker William Freemantle had already patented it in 1803. A different form o ...
, which converts linear motion, to (almost) linear motion in a line perpendicular to the input. *
Chebyshev linkage In kinematics, Chebyshev's linkage is a four-bar linkage that converts rotational motion to approximate linear motion. It was invented by the 19th-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic mechan ...
, which provides nearly straight motion of a point with a four-bar linkage. * Hoekens linkage, which provides nearly straight motion of a point with a four-bar linkage. *
Sarrus linkage The Sarrus linkage, invented in 1853 by Pierre Frédéric Sarrus, is a mechanical linkage to convert a limited circular motion to a linear motion or vice versa without reference guideways. It is a spatial six-bar linkage (6R) with two grou ...
, which provides motion of one surface in a direction normal to another. *
Hart's inversor Hart's inversors are two planar mechanisms that provide a perfect straight line motion using only rotary joints. They were invented and published by Harry Hart in 1874–5. Hart's first inversor, also known as ''Hart's W-frame'', is based on ...
, which provides a perfect straight line motion without sliding guides.


Biological linkages

Linkage systems are widely distributed in animals. The most thorough overview of the different types of linkages in animals has been provided by Mees Muller, who also designed a new classification system which is especially well suited for biological systems. A well-known example is the
cruciate ligament Cruciate ligaments (also cruciform ligaments) are pairs of ligaments arranged like a letter X. They occur in several joints of the body, such as the knee joint and the atlanto-axial joint. In a fashion similar to the cords in a toy Jacob's la ...
s of the knee. 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 functional constraints (especially the necessity to deliver blood). Biological linkages frequently are 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.
Four-bar linkage 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 config ...
s are by far the most common though. Linkages can be found in joints, such as the
knee In humans and other primates, the knee joins the thigh with the leg and consists of two joints: one between the femur and tibia (tibiofemoral joint), and one between the femur and patella (patellofemoral joint). It is the largest joint in the ...
of
tetrapod Tetrapods (; ) are four-limbed vertebrate animals constituting the superclass Tetrapoda (). It includes extant and extinct amphibians, sauropsids ( reptiles, including dinosaurs and therefore birds) and synapsids ( pelycosaurs, extinct t ...
s, the hock of sheep, and the cranial mechanism of birds and reptiles. The latter is responsible for the upward motion of the upper bill in many birds. Linkage mechanisms are especially frequent and manifold in the head of
bony fishes Osteichthyes (), popularly referred to as the bony fish, is a diverse superclass of fish that have skeletons primarily composed of bone tissue. They can be contrasted with the Chondrichthyes, which have skeletons primarily composed of cartilage ...
, such as
wrasses The wrasses are a family, Labridae, of marine fish, many of which are brightly colored. The family is large and diverse, with over 600 species in 81 genera, which are divided into 9 subgroups or tribes. They are typically small, most of them ...
, which have
evolved Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variati ...
many specialized feeding mechanisms. Especially advanced are the linkage mechanisms of
jaw protrusion Most bony fishes have two sets of jaws made mainly of bone. The primary oral jaws open and close the mouth, and a second set of pharyngeal jaws are positioned at the back of the throat. The oral jaws are used to capture and manipulate prey by b ...
. For
suction feeding Aquatic feeding mechanisms face a special difficulty as compared to feeding on land, because the density of water is about the same as that of the prey, so the prey tends to be pushed away when the mouth is closed. This problem was first identifi ...
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 protrusion of the
premaxilla The premaxilla (or praemaxilla) is one of a pair of small cranial bones at the very tip of the upper jaw of many animals, usually, but not always, bearing teeth. In humans, they are fused with the maxilla. The "premaxilla" of therian mammal has ...
. 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 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.


Image gallery

File:RRRT Func Geen Log(u).gif, Rocker-slider function generator approximating the function Log(u) for 1 < u < 10. File:TRRR Func Geen Tan(u).gif, Slider-rocker function generator approximating the function Tan(u) for 0 < u < 45°. File:Four-bar fixed and moving centrodes.gif, Fixed and moving centrodes of a four-bar linkage File:Rack-and-pinion_4_bar.gif, Rack-and-pinion four-bar linkage File:RTRTR 1.gif, RTRTR mechanism File:RTRTR 2.gif, RTRTR mechanism File:Gear_5-bar_linkage.gif, Gear five-bar mechanisms File:3D slider-crank mechanism.gif, 3D slider-crank mechanism File:Animated Pinochio.gif, Animated Pinocchio character File:Iris mechanism.gif File:Crawford Conicograph.gif, Crawford conicograph File:Deployable Structure1.gif, Outward folding deployable mechanism File:Deployable Structure2.gif, Inward folding deployable mechanism


See also

* Assur Groups * Deployable structure * Dwell mechanism *
Engineering mechanics Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
*
Four-bar linkage 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 config ...
* Mechanical function generator *
Kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
*
Kinematic coupling Kinematic coupling describes fixtures designed to exactly constrain the part in question, providing precision and certainty of location. A canonical example of a kinematic coupling consists of three radial v-grooves in one part that mate with thre ...
*
Kinematic pair In classical mechanics, a kinematic pair is a connection between two physical objects that imposes constraints on their relative movement (kinematics). German engineer Franz Reuleaux introduced the kinematic pair as a new approach to the study ...
* Kinematic synthesis * Kinematic models in
Mathcad Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil engineering. Released in 1986 on DOS, it introduc ...
*
Leg mechanism A leg mechanism (walking mechanism) is a mechanical system designed to provide a propulsive force by intermittent frictional contact with the ground. This is in contrast with wheels or continuous tracks which are intended to maintain continuous f ...
*
Lever A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
*
Machine A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...
*
Outline of machines Machine – mechanical system that provides the useful application of power to achieve movement. A machine consists of a power source, or engine, and a mechanism or transmission for the controlled use of this power. The combination of for ...
*
Overconstrained mechanism In mechanical engineering, an overconstrained mechanism is a linkage that has more degrees of freedom than is predicted by the mobility formula. The mobility formula evaluates the degree of freedom of a system of rigid bodies that results when ...
* Parallel motion *
Reciprocating motion Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single r ...
*
Slider-crank linkage A slider-crank linkage is a four-link mechanism with three revolute joints and one prismatic, or sliding, joint. The rotation of the crank drives the linear movement the slider, or the expansion of gases against a sliding piston in a cylinder ...
*
Three-point hitch The three-point hitch (British English: three-point linkage) is a widely used type of hitch for attaching ploughs and other implements to an agricultural or industrial tractor. The three points resemble either a triangle, or the letter A. Three-p ...


References


Further reading

*  — 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 * * Hartenberg, R.S. & J. Denavit (1964
Kinematic synthesis of linkages
New York: McGraw-Hill — Online link from
Cornell University Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
. *  — "Linkages: a peculiar fascination" (Chapter 14) is a discussion of mechanical linkage usage in American mathematical education, includes extensive references
How to Draw a Straight Line
nbsp;— Historical discussion of linkage design from Cornell University * Parmley, Robert. (2000). "Section 23: Linkage." ''Illustrated Sourcebook of Mechanical Components.'' New York: McGraw Hill. Drawings and discussion of various linkages. * Sclater, Neil. (2011). "Linkages: Drives and Mechanisms." ''Mechanisms and Mechanical Devices Sourcebook.'' 5th ed. New York: McGraw Hill. pp. 89–129. . Drawings and designs of various linkages.


External links


Kinematic Models for Design Digital Library (KMODDL)
nbsp;— 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 Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
, plus 5 other major collections. Includes a
e-book library
of dozens of classic texts on mechanical design and engineering. Includes CAD models and stereolithographic files for selected mechanisms.
Digital Mechanism and Gear Library (DMG-Lib)
(in German: Digitale Mechanismen- und Getriebebibliothek) — Online library about linkages and cams (mostly in German)


Introductory linkage lecture



Linkage-based Drawing Apparatus by Robert Howsare

(ASOM) Analysis, synthesis and optimization of multibar linkages

Linkage animations on mechanicaldesign101.com include planar and spherical four-bar and six-bar linkages.



Animation of Bennett's linkage.

Example of a six-bar function generator that computes the elevation angle for a given range.

Animations of six-bar linkage for a bicycle suspension.

A variety of six-bar linkage designs.

Introduction to Linkages

An open source planar linkage mechanism simulation and mechanical synthesis system.
{{DEFAULTSORT:Linkage (Mechanical) Mechanisms (engineering)