Introduction to Mechanisms and Kinematics Basic Definitions. Machines are devices used to accomplish work. A mechanism is the heart of a machine. It is the mechanical portion of amachine that has the function of transferring motion and forces from a power source to an output. PDF Mappings are studied of spatial kinematics and the geometry for which a spatial displacement is an element. Study's soma is reviewed and it is shown that Euclidean geometry in three-space. Spall fracture mechanisms and strength evaluation of CFRP against high- speed impact. Daisuke Yoshimizu. Department of Aerospace Engineering, University of Nagoya, Japan. Dust Erosion of the Thermal Protection System for Martian Atmospheric Entry. Takahiro Yamanaka, Jun Koyanagi.

1. Kinematics Spatial Mechanisms Pdf Files Pdf

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.^[1] 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.

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 (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 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,^[2] and a linkage designed to be stationary is called a structure.

• 3Mobility
• 7Other interesting linkages

Uses[edit]

Perhaps the simplest linkage is the lever, which is a link that pivots around a 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 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.

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 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.

Linkages are important components of machines and tools. Examples range from the four-bar linkage used to amplify force in a bolt cutter or to provide independent suspension in an automobile, to complex linkage systems in 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.

Interesting examples of linkages include the windshield wiper, the bicycle suspension, the leg mechanism in a walking machine and 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.

History[edit]

Archimedes^[3] 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.^[4]

In the mid-1700s the 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.^[5]

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.^[6] Kempe's design procedure has inspired research at the intersection of geometry and computer science.^[7][8]

In the late 1800s F. Reuleaux, A. B. W. Kennedy, and L. Burmester formalized the analysis and synthesis of linkage systems using descriptive geometry, and P. L. Chebyshev introduced analytical techniques for the study and invention of linkages.^[5]

In the mid-1900s F. Freudenstein and G. N. Sandor^[9] 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^[10][11] and the control of robot manipulators.^[12]

R. E. Kaufman^[13][14] 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[edit]

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, of the linkage system.

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 = nAkai eie pro drivers mac. + 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,

${\displaystyle M=6n-\sum _{i=1}^j(6-f_i)=6(N-1-j)+\sum _{i=1}^j{f}_i,}$

Kinematics Spatial Mechanisms Pdf Files Pdf

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

${\displaystyle M=\sum _{i=1}^j{f}_i.}$

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

${\displaystyle M=\sum _{i=1}^j{f}_i-6.}$

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 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[edit]

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

${\displaystyle M=3(N-1-j)+\sum _{i=1}^j{f}_i,}$

and we have the special cases,

• planar or spherical simple open chain,

${\displaystyle M=\sum _{i=1}^j{f}_i,}$

• planar or spherical simple closed chain,

${\displaystyle M=\sum _{i=1}^j{f}_i-3.}$

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[edit]

The most familiar joints for linkage systems are the revolute, or hinged, joint denoted by an R, and the prismatic, 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 consists of an RR serial chain constructed such that the axes of the revolute joints intersect at a 90° angle;
• the spherical joint 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[edit]

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.

Planar one degree-of-freedom linkages[edit]

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

${\displaystyle M=3(N-1-j)+j=1,}$

or

${\displaystyle j=\frac{3}{2}N-2.}$

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;
• N = 4, j = 4: this is the four-bar linkage;
• 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;^[15]
• 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^[16] 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 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[edit]

• 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 is an eight-bar leg mechanism that was invented by kinetic sculptor Theo Jansen.
• Klann linkage is a six-bar linkage that forms a Leg mechanism;
• 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.^[17] 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[edit]

• James Watt's parallel motion and Watt's linkage
• Peaucellier–Lipkin linkage, the first planar linkage to create a perfect straight line output from rotary input; eight-bar, one DOF.
• A Scott Russell linkage, which converts linear motion, to (almost) linear motion in a line perpendicular to the input.
• Chebyshev linkage, 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, which provides motion of one surface in a direction normal to another.
• Hart's inversor, which provides a perfect straight line motion without sliding guides.^[18]

Biological linkages[edit]

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,^[19] who also designed a new classification system which is especially well suited for biological systems. A well-known example is the cruciate ligaments 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 mechanical constraints (especially the necessity to deliver blood).^[20] 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.^[19]Four-bar linkages are by far the most common though.

Linkages can be found in joints, such as the knee of tetrapods, 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, such as wrasses, which have evolved many specialized feeding mechanisms. Especially advanced are the linkage mechanisms of 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 protrusion of the premaxilla.

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[edit]

^[21][22][23]

• Rocker-slider function generator of the function Log(u) for 1 < u < 10.

• Slider-rocker function generator of the function Tan(u) for 0 < u < 45°.

• Fixed and moving centrodes of a four-bar linkage

• Rack-and-pinion four-bar linkage

• RTRTR mechanism

• RTRTR mechanism

• Gear five-bar mechanisms

• 3D slider-crank mechanism

• Animated Pinochio character

• Crawford conicograph

• Outward folding deployable mechanism

• Inward folding deployable mechanism

See also[edit]

• Kinematic models in Mathcad^[24]

References[edit]

1. ^P. Moubarak, P. Ben-Tzvi, 'On the Dual-Rod Slider Rocker Mechanism and Its Applications to Tristate Rigid Active Docking,' ASME Journal of Mechanisms and Robotics, 5 (1) (2013) 011010
2. ^OED
3. ^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
4. ^A. P. Usher, 1929, A History of Mechanical Inventions, Harvard University Press, (reprinted by Dover Publications 1968)
5. ^ ^abF. 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
6. ^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
7. ^D. Jordan and M. Steiner, 'Configuration Spaces of Mechanical Linkages,' Discrete and Computational Geometry, 22:297–315, 1999
8. ^R. Connelly and E. D. Demaine, 'Geometry and Topology of Polygonal Linkages,' Chapter 9, Handbook of discrete and computational geometry, (J. E. Goodman and J. O'Rourke, eds.), CRC Press, 2004
9. ^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
10. ^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
11. ^C. H. Suh and C. W. Radcliffe, Kinematics and Mechanism Design, John Wiley, pp:458, 1978
12. ^R. P. Paul, Robot Manipulators: Mathematics, Programming and Control, MIT Press, 1981
13. ^R. E. Kaufman and W. G. Maurer, 'Interactive Linkage Synthesis on a Small Computer', ACM National Conference, Aug.3–5, 1971
14. ^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
15. ^L. W. Tsai, 'Mechanism design: enumeration of kinematic structures according to function', CRC Press, 2000. Books.google.com. Retrieved 2013-06-13.
16. ^Sunkari, R. P.; Schmidt, L. C. (2006). 'Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm'. Mechanism and Machine Theory. 41: 1021–1030. doi:10.1016/j.mechmachtheory.2005.11.007.
17. ^Robert L. Norton; Design of Machinery 5th Edition
18. ^'True straight-line linkages having a rectlinear translating bar'(PDF).
19. ^ ^abMuller, M. (1996). 'A novel classification of planar four-bar linkages and its application to the mechanical analysis of animal systems'. Phil. Trans. R. Soc. Lond. B. 351: 689–720. doi:10.1098/rstb.1996.0065.
20. ^Dawkins, Richard (November 24, 1996). 'Why don't animals have wheels?'. Sunday Times. Archived from the original on February 21, 2007. Retrieved 2008-10-29.Cite uses deprecated parameter deadurl= (help)
21. ^Simionescu, P.A. (2014). Computer Aided Graphing and Simulation Tools for AutoCAD users (1st ed.). Boca Raton, FL: CRC Press. ISBN978-1-4822-5290-3.
22. ^Simionescu, P.A. (21–24 August 2016). MeKin2D: Suite for Planar Mechanism Kinematics(PDF). ASME 2016 Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Charlotte, NC, USA. pp. 1–10. Retrieved 7 January 2017.
23. ^Simionescu, P.A. 'A restatement of the optimum synthesis of function generators with planar four-bar and slider-crank mechanisms examples'(PDF). International Journal of Mechanisms and Robotic Systems. 3 (1): 60–79. doi:10.1504/IJMRS.2016.077038. Retrieved 2 January 2017.
24. ^'PTC Community: Group: Kinematic models in Mathcad'. Communities.ptc.com. Retrieved 2013-06-13.

Further reading[edit]

• Bryant, John; Sangwin, Chris (2008). How round is your circle? : where engineering and mathematics meet. Princeton: Princeton University Press. p. 306. ISBN978-0-691-13118-4. — 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
• Erdman, Arthur G.; Sandor, George N. (1984). Mechanism Design: Analysis and Synthesis. Prentice-Hall. ISBN0-13-572396-5.
• Hartenberg, R.S. & J. Denavit (1964) Kinematic synthesis of linkages, New York: McGraw-Hill — Online link from Cornell University.
• Kidwell, Peggy Aldrich; Amy Ackerberg-Hastings; David Lindsay Roberts (2008). Tools of American mathematics teaching, 1800–2000. Baltimore: Johns Hopkins University Press. pp. 233–242. ISBN978-0-8018-8814-4. — '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 — Historical discussion of linkage design from Cornell University
• Parmley, Robert. (2000). 'Section 23: Linkage.' Illustrated Sourcebook of Mechanical Components. New York: McGraw Hill. ISBN0-07-048617-4 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. ISBN978-0-07-170442-7. Drawings and designs of various linkages.

External links[edit]

• 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 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)

In engineering, a mechanism is a device that transforms input forces and movement into a desired set of output forces and movement. Mechanisms generally consist of moving components that can include:

• Gears and gear trains
• Belt and chain drives
• Cam and followers
• Friction devices, such as brakes and clutches
• Structural components such as a frame, fasteners, bearings, springs, lubricants
• Various machine elements, such as splines, pins, and keys.

The German scientist Reuleaux provides the definition 'a machine is a combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motion.' In this context, his use of machine is generally interpreted to mean mechanism.

The combination of force and movement defines power, and a mechanism manages power to achieve a desired set of forces and movement.

A mechanism is usually a piece of a larger process or mechanical system. Sometimes an entire machine may be referred to as a mechanism. Examples are the steering mechanism in a car, or the winding mechanism of a wristwatch.Multiple mechanisms are machines.

• 1Kinematic Pairs

Kinematic Pairs[edit]

From the time of Archimedes through the Renaissance, mechanisms were viewed as constructed from simple machines, such as the lever, pulley, screw, wheel and axle, wedge, and inclined plane. Reuleaux focused on bodies, called links, and the connections between these bodies called kinematic pairs, or joints.

To use geometry to study the movement of a mechanism, its links are modeled as rigid bodies. This means that distances between points in a link are assumed to not change as the mechanism moves that is, the link does not flex. Thus, the relative movement between points in two connected links is considered to result from the kinematic pair that joins them.

Kinematic pairs, or joints, are considered to provide ideal constraints between two links, such as the constraint of a single point for pure rotation, or the constraint of a line for pure sliding, as well as pure rolling without slipping and point contact with slipping. A mechanism is modeled as an assembly of rigid links and kinematic pairs.

Links and joints[edit]

Reuleaux called the ideal connections between links kinematic pairs. He distinguished between higher pairs with line contact between the two links and lower pairs with area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple model.

Lower pair: A lower pair is an ideal joint that has surface contact between the pair of elements, as in the following cases:

• A revolute pair, or hinged joint, requires a line in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body must maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom.
• A prismatic joint, or slider, requires that a line in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body must maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom.
• A cylindrical joint requires that a line in the moving body remain co-linear with a line in the fixed body. It combines a revolute joint and a sliding joint. This joint has two degrees of freedom.
• A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
• A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.
• A screw joint, or helical joint, has only one degree of freedom because the sliding and rotational motions are related by the helix angle of the thread.

Higher pairs: Generally, a higher pair is a constraint that requires a line or point contact between the elemental surfaces. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.

Kinematic diagram[edit]

A kinematic diagram reduces the machine components to a skeleton diagram that emphasizes the joints and reduces the links to simple geometric elements. This diagram can also be formulated as a graph by representing the links of the mechanism as vertices and the joints as edges of the graph. This version of the kinematic diagram has proven effective in enumerating kinematic structures in the process of machine design.^[1]

An important consideration in this design process is the degree of freedom of the system of links and joints, which is determined using the Chebychev–Grübler–Kutzbach criterion.

Planar mechanisms[edit]

While all mechanisms in a mechanical system are three-dimensional, they can be analyzed using plane geometry, if the movement of the individual components are constrained so all point trajectories are parallel or in a series connection to a plane. In this case the system is called a planar mechanism. The kinematic analysis of planar mechanisms uses the subset of Special Euclidean group SE, consisting of planar rotations and translations, denote SE.

The group SE is three-dimensional, which means that every position of a body in the plane is defined by three parameters. The parameters are often the x and y coordinates of the origin of a coordinate frame in M measured from the origin of a coordinate frame in F, and the angle measured from the x-axis in F to the x-axis in M. This is often described saying a body in the plane has three degrees-of-freedom.

The pure rotation of a hinge and the linear translation of a slider can be identified with subgroups of SE , and define the two joints one degree-of-freedom joints of planar mechanisms. The cam joint formed by two surfaces in sliding and rotating contact is a two degree-of-freedom joint.

See Theo Jansen's Strandbeest walking machine with legs constructed from planar eight-bar linkages

Spherical mechanisms[edit]

It is possible to construct a mechanism such that the point trajectories in all components lie in concentric spherical shells around a fixed point. An example is the gimbaledgyroscope. These devices are called spherical mechanisms.^[2] Spherical mechanisms are constructed by connecting links with hinged joints such that the axes of each hinge passes through the same point. This point becomes center of the concentric spherical shells. The movement of these mechanisms is characterized by the group SO(3) of rotations in three-dimensional space. Other examples of spherical mechanisms are the automotive differential and the robotic wrist.

Select this link for an animation of a Spherical deployable mechanism.

The rotation group SO(3) is three-dimensional. An example of the three parameters that specify a spatial rotation are the roll, pitch and yaw angles used to define the orientation of an aircraft.

Spatial mechanisms[edit]

A mechanism in which a body moves through a general spatial movement is called a spatial mechanism. An example is the RSSR linkage, which can be viewed as a four-bar linkage in which the hinged joints of the coupler link are replaced by rod ends, also called spherical joints or ball joints. The rod ends let the input and output cranks of the RSSR linkage be misaligned to the point that they lie in different planes, which causes the coupler link to move in a general spatial movement. Robot arms, Stewart platforms, and humanoid robotic systems are also examples of spatial mechanisms.

Bennett's linkage is an example of a spatial overconstrained mechanism, which is constructed from four hinged joints.

The group SE(3) is six-dimensional, which means the position of a body in space is defined by six parameters. Three of the parameters define the origin of the moving reference frame relative to the fixed frame. Three other parameters define the orientation of the moving frame relative to the fixed frame.

Linkages[edit]

A linkage is a collection of links connected by joints. Generally, the links are the structural elements and the joints allow movement. Perhaps the single most useful example is the planar four-bar linkage. However, there are many more special linkages:

• Watt's linkage is a four-bar linkage that generates an approximate straight line. It was critical to the operation of his design for the steam engine. This linkage also appears in vehicle suspensions to prevent side-to-side movement of the body relative to the wheels. Also see the article Parallel motion.
• The success of Watt's linkage lead to the design of similar approximate straight-line linkages, such as Hoeken's linkage and Chebyshev's linkage.
• The Peaucellier linkage generates a true straight-line output from a rotary input.
• The Sarrus linkage is a spatial linkage that generates straight-line movement from a rotary input.
• The Klann linkage and the Jansen linkage are recent inventions that provide interesting walking movements. They are respectively a six-bar and an eight-bar linkage.

Compliant mechanisms[edit]

A compliant mechanism is a series of rigid bodies connected by compliant elements. These mechanisms have many advantages, including reduced part-count, reduced 'slop' between joints (no parasitic motion because of gaps between parts), energy storage, low maintenance (they don't require lubrication and there is low mechanical wear), and ease of manufacture ^[3].

Flexure bearings (also known as flexure joints) are a subset of compliant mechanisms that produce a geometrically well-defined motion (rotation) on application of a force.

Cam and follower mechanisms[edit]

A cam and follower is formed by the direct contact of two specially shaped links. The driving link is called the cam (also see cam shaft) and the link that is driven through the direct contact of their surfaces is called the follower. The shape of the contacting surfaces of the cam and follower determines the movement of the mechanism. In general a cam follower mechanism's energy is transferred from cam to follower. The cam shaft is rotated and, according to the cam profile, the follower moves up and down. Now slightly different types of eccentric cam followers are also available in which energy is transferred from the follower to the cam. The main benefit of this type of cam follower mechanism is that the follower moves a little bit and helps to rotate the cam 6 times more circumference length with 70% force.

Gears and gear trains[edit]

The transmission of rotation between contacting toothed wheels can be traced back to the Antikythera mechanism of Greece and the south-pointing chariot of China. Illustrations by the renaissance scientist Georgius Agricola show gear trains with cylindrical teeth. The implementation of the involute tooth yielded a standard gear design that provides a constant speed ratio. Some important features of gears and gear trains are:

• The ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the gear set.
• A planetary gear train provides high gear reduction in a compact package.
• It is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly.
• The speed ratios of chain and belt drives are computed in the same way as gear ratios. (See bicycle gearing.)

Mechanism synthesis[edit]

The design of mechanisms to achieve a particular movement and force transmission is known as the kinematic synthesis of mechanisms.^[4] This is a set of geometric techniques that yield the dimensions of linkages, cam and follower mechanisms, and gears and gear trains to perform a required mechanical movement and power transmission.^[5]

See also[edit]

References[edit]

1. ^Lung-Wen Tsai, 2001, Mechanism design: enumeration of kinematic structures according to function, CRC Press
2. ^J. M. McCarthy and G. S. Soh, Geometric Design of Linkages, 2nd Edition, Springer 2010
3. ^'Compliant Mechanisms About Compliant Mechanisms'. compliantmechanisms. Retrieved 2019-02-08.
4. ^Hartenberg, R.S. and J. Denavit (1964) Kinematic synthesis of linkages, New York: McGraw-Hill — Online link from Cornell University.
5. ^ J. J. Uicker, G. R. Pennock, and J. E. Shigley, Theory of Machines and Mechanisms, Fifth Ed., Oxford University Press, 2016.

External links[edit]

• 507 Mechanical Movements a 1908 publication by Henry T. Brown
• Kinematic Models for Design Digital Library (KMODDL) collections of movies and photos of hundreds of mechanism models.