1884:
1920:
1908:
1816:
2095:
2061:
2014:
891:
2120:
1997:
2046:
2031:
1844:
1896:
2076:
2151:
1950:
2163:
1962:
2136:
1974:
1828:
1669:
1934:
111:
1856:
1872:
898:
1657:
914:
degrees, and in the concave configuration one internal angle is greater than 180 degrees. There exists a simple geometrical relationship between the lengths of the two diagonals of the quadrilateral. For convex and crossing linkages, the length of one diagonal increases if and only if the other decreases. On the other hand, for nonconvex non-crossing linkages, the opposite is the case; one diagonal increases if and only if the other also increases.
1645:
1470:
74:
99:
91:
1147:
Other applications require that the mechanism-to-be-designed has a faster average speed in one direction than the other. This category of mechanism is most desired for design when work is only required to operate in one direction. The speed at which this one stroke operates is also very important in
913:
The configuration of a quadrilateral linkage may be classified into three types: convex, concave, and crossing. In the convex and concave cases no two links cross over each other. In the crossing linkage two links cross over each other. In the convex case all four internal angles are less than 180
926:
is important when aiming to produce a desired output motion for a specific input motion. In order to minimize cost and maximize efficiency, a designer will choose the simplest mechanism possible to accomplish the desired motion. When selecting a mechanism type to be designed, link lengths must be
429:
The movement of a quadrilateral linkage can be classified into eight cases based on the dimensions of its four links. Let a, b, g and h denote the lengths of the input crank, the output crank, the ground link and floating link, respectively. Then, we can construct the three terms:
992:, the forward and return, which when added together create a cycle. Each stroke may be identical or have different average speeds. The time ratio numerically defines how fast the forward stroke is compared to the quicker return stroke. The total cycle time (
1125:
Some mechanisms that produce reciprocating, or repeating, motion are designed to produce symmetrical motion. That is, the forward stroke of the machine moves at the same pace as the return stroke. These mechanisms, which are often referred to as
1048:
1329:
between two or more mechanisms. They graphically display information showing where and when each mechanism is stationary or performing its forward and return strokes. Timing charts allow designers to qualitatively describe the required
983:
1883:
1148:
certain machine applications. In general, the return and work-non-intensive stroke should be accomplished as fast as possible. This is so the majority of time in each cycle is allotted for the work-intensive stroke. These
2353:
Cvetkovic, Ivana & Stojicevic, Misa & Popkonstantinović, Branislav & Cvetković, Dragan. (2018). Classification, geometrical and kinematic analysis of four-bar linkages. 261-266. 10.15308/Sinteza-2018-261-266.
1495:
An in-line slider-crank has its slider positioned so the line of travel of the hinged joint of the slider passes through the base joint of the crank. This creates a symmetric slider movement back and forth as the crank
155:
A link connecting to a ground line by a prismatic joint is called a slider. Sliders are sometimes considered to be cranks that have a hinged pivot at an infinitely long distance away perpendicular to the travel of the
1120:
2094:
1502:
If the line of travel of the hinged joint of the slider does not pass through the base pivot of the crank, the slider movement is not symmetric. It moves faster in one direction than the other. This is called a
1243:
1170:
With offset mechanisms, it is very important to understand how and to what degree the offset affects the time ratio. To relate the geometry of a specific linkage to the timing of the stroke, an imbalance angle
386:
If the sum of the shortest and longest link of a planar quadrilateral linkage is less than or equal to the sum of the remaining two links, then the shortest link can rotate fully with respect to a neighboring
2060:
1315:
2045:
2013:
1623:
1996:
931:
methodology which in certain circumstances can be an inefficient process; however, in unique scenarios, exact and detailed procedures to design an accurate mechanism may not exist.
2075:
1007:
588:
535:
482:
2568:
Phoengsongkhro, S., Tangpornprasert, P., Yotnuengnit, P. et al. Development of four-bar polycentric knee joint with stance-phase knee flexion. Sci Rep 13, 22809 (2023).
1571:
2198:
2162:
1361:
for velocity as a function of time. Constant acceleration allows for the velocity vs. time graph to appear as straight lines, thus designating a relationship between
692:
664:
636:
2030:
949:
1516:
If the linkage has four hinged joints with axes angled to intersect in a single point, then the links move on concentric spheres and the assembly is called a
2119:
1339:
The velocity of a link is the time rate at which its position is changing, while the link's acceleration is the time rate at which its velocity is changing.
881:
1961:
2150:
1919:
1907:
1973:
594:
The movement of a quadrilateral linkage can be classified into eight types based on the positive and negative values for these three terms, T
1520:. The input-output equations of a spherical four-bar linkage can be applied to spatial four-bar linkages when the variables are replaced by
1066:
1185:
1843:
218:
It is denoted as RRRP, PRRR, RPRR, or RRPR, constructed from four links connected by three revolute joints and one prismatic joint. The
320:
Planar quadrilateral linkage, RRRR or 4R linkages have four rotating joints. One link of the chain is usually fixed, and is called the
1053:
Most four-bar mechanisms are driven by a rotational actuator, or crank, that requires a specific constant speed. This required speed (
2256:
2135:
1855:
1477:
A slider-crank linkage is a four-bar linkage with three revolute joints and one prismatic, or sliding, joint. The rotation of the
1258:
1455:
Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated.
1638:
is a spatial four-bar linkage with hinged joints that have their axes angled in a particular way that makes the system movable.
2589:
208:
Parallelogram (Parallel Motion) and
Antiparallelogram (Contraparallelogram, Inverse Parallelogram, Butterfly, Bow-tie) linkages
222:(RRRP) is one type of arrangement such that one link is a crank, which is then connected to a slider by a connecting rod. The
2459:
1827:
1895:
2616:
2484:
1933:
1815:
2536:
2517:
2368:
1576:
2342:
1534:
of matrices, falsely claiming that the latter are unique whenever they exist. This is contradicted by the fact that
1527:
1353:; however, only their magnitudes are used in timing charts. When used with two mechanisms, timing charts assume
2792:
2731:
1747:
880:= 0 are interesting because the linkages fold. If we distinguish folding quadrilateral linkage, then there are
123:
176:
There are three basic types of planar four-bar linkage, depending on the use of revolute or prismatic joints:
1668:
17:
1949:
150:
A link connecting to the ground by a revolute joint that cannot perform a complete revolution is called a
66:. Generally, the joints are configured so the links move in parallel planes, and the assembly is called a
2665:
1925:
1710:
202:
1752:
232:
142:
A link connecting to the ground by a revolute joint that can perform a complete revolution is called a
2500:
Angeles, Jorge (2012), "The Dual
Generalized Inverses and Their Applications in Kinematic Synthesis",
1871:
1043:{\displaystyle \Delta t_{\text{cycle}}={\text{Time of slower stroke}}+{\text{Time of quicker stroke}}}
292:
Planar four-bar linkages can be designed to guide a wide variety of movements, and are often the base
2649:
1695:
1337:
These charts are also used to estimate the velocities and accelerations of certain four-bar links.
542:
489:
436:
2384:
226:
is another type of arrangement such that there are two cranks with a slider acting as the coupler.
2767:
2655:
2532:
2295:
1700:
1346:
923:
293:
31:
2609:
2541:
2251:
1980:
1537:
890:
309:
166:
A coupler connecting a crank and a slider in a single slider crank mechanism is often called a
2416:
1656:
359:
Assuming the frame is horizontal there are four possibilities for the input and output links:
2762:
2752:
2211:
2171:
1362:
1354:
1130:
design, usually do work in both directions, as they exert the same force in both directions.
978:{\displaystyle Q={\frac {\text{Time of slower stroke}}{\text{Time of quicker stroke}}}\geq 1}
268:
1803:
259:
It is denoted as PRRP, and is constructed by connected two sliders with a coupler link. The
2625:
2246:
2231:
1644:
1634:
1626:
1464:
1358:
1350:
670:
642:
614:
305:
297:
47:
2590:
The four-bar linkages in the collection of
Reuleaux models at Cornell University (archive)
182:
It is denoted as RRRR, constructed from four links connected by four revolute joints. The
8:
2726:
2226:
122:
Planar four-bar linkages are constructed from four links connected in a loop by four one-
927:
determined by a process called dimensional synthesis. Dimensional synthesis involves an
366:
A rocker: can rotate through a limited range of angles which does not include 0° or 180°
2736:
2708:
2407:
2380:
2343:
J. M. McCarthy and G. S. Soh, Geometric Design of
Linkages, 2nd Edition, Springer, 2010
2323:
2285:
Sometimes, it is also denoted as 'S' to refer to slider, rather than a prismatic joint.
1787:
1742:
372:
A π-rocker: can rotate through a limited range of angles which includes 180° but not 0°
369:
A 0-rocker: can rotate through a limited range of angles which includes 0° but not 180°
2559:
Hunt, K. H., Kinematic
Geometry of Mechanisms, Oxford Engineering Science Series, 1979
943:) of a four-bar mechanism is a measure of its quick return and is defined as follows:
2718:
2680:
2645:
2602:
2513:
2480:
2455:
2364:
1834:
1730:
1690:
1531:
1478:
144:
2584:
2411:
1248:
Through simple algebraic rearrangement, this equation can be rewritten to solve for
2660:
2505:
2403:
2399:
2221:
1481:
drives the linear movement the slider, or the expansion of gases against a sliding
1342:
1326:
887:
The figure shows examples of the various cases for a planar quadrilateral linkage.
2703:
2690:
2675:
2319:
2266:
2241:
2216:
2141:
1720:
1705:
989:
336:
and are generally the input and output links of the system, sometimes called the
278:
131:
43:
2585:
The four-bar linkages in the collection of
Reuleaux models at Cornell University
2509:
94:
Coupler curves of a crank-rocker four-bar linkage. Simulation done with MeKin2D.
1774:
168:
127:
110:
70:. Spherical and spatial four-bar linkages also exist and are used in practice.
63:
2786:
2757:
2698:
2569:
1862:
1967:
A diagram of the suspension on a
Specialized Stumpjumper FSR mountain bike.
1940:
284:
1757:
1521:
897:
274:
27:
Mechanical linkage consisting of four links connected by joints in a loop
2670:
1766:
1725:
1715:
301:
51:
73:
2156:
Ackermann steering linkage, which uses a 4-Bar Double-Rocker linkage.
1331:
1115:{\displaystyle \omega _{\text{crank}}=(\Delta t_{\text{cycle}})^{-1}}
249:
2363:
Design of
Machinery 3/e, Robert L. Norton, 2 May 2003, McGraw Hill.
1238:{\displaystyle Q={\frac {180^{\circ }+\beta }{180^{\circ }-\beta }}}
2236:
2126:
1795:
195:
78:
2709:
Hart's First
Inversor / Hart's Antiparallelogram / Hart's W-frame
2477:
Engineering Design
Synthesis: Understanding, Approaches and Tools
1778:
137:
A link that is fixed in place relative to the viewer is called a
130:– also known as a pin joint or hinged joint – denoted by R, or a
2066:
Parallelogram and Antiparallelogram 4-Bar Linkages (Inversions):
1469:
245:
Crank and slotted lever Quick Return mechanism (used in shapers)
173:
however, it has also been used to refer to any type of coupler.
1482:
239:
98:
90:
1770:
376:
Some authors do not distinguish between the types of rocker.
2385:"Simple proofs of a geometric property of four-bar linkages"
1310:{\displaystyle \beta =180^{\circ }\times {\frac {Q-1}{Q+1}}}
2594:
1526:
Note that the cited conference paper incorrectly conflates
1511:
1488:
There are two types of slider-cranks: in-line and offset.
308:
of planar four-bar linkages are also important topics in
238:
Whitworth Quick Return mechanism (used in early types of
2069:(Demonstration of Point Change Condition in the Middle)
2054:(Demonstration of Point Change Condition in the Middle)
332:. The two links connected to the frame are called the
2174:
1579:
1540:
1473:
Crank slider mechanisms with 0 and 1.25 eccentricity.
1261:
1188:
1069:
1010:
952:
673:
645:
617:
545:
492:
439:
384:
The Grashof condition for a four-bar linkage states:
1175:) is used. This angle is related to the time ratio,
2452:
Machines and Mechanisms: Applied Kinematic Analysis
2051:
Parallelogram and Antiparallelogram 4-Bar Linkages:
1485:in a cylinder can drive the rotation of the crank.
1133:Examples of symmetrical motion mechanisms include:
2192:
1617:
1565:
1309:
1237:
1114:
1042:
977:
686:
658:
630:
582:
529:
476:
211:Deltoid (Galloway) and Trapezium (Arglin) linkages
2479:. Great Britain: Springer-Verlag London Limited.
158:A link connecting to two other links is called a
2784:
1684:
1325:Timing charts are often used to synchronize the
917:
2039:Yellow link and Green link rock back and forth.
1618:{\displaystyle \{(1,x)\mid x\in \mathbb {C} \}}
315:
134:– also known as a sliding pair – denoted by P.
389:In other words, the condition is satisfied if
2610:
257:Two revolute joints and two prismatic joints:
1612:
1580:
356:because it connects an input to the output.
2474:
2617:
2603:
2570:https://doi.org/10.1038/s41598-023-49879-4
2338:
2336:
2334:
2332:
2322:, New York: McGraw-Hill, online link from
2259:(Approximate straight lines are primarily
1913:The Brown Bros screwcutting treadle lathe.
1060:)is related to the cycle time as follows:
85:
2445:
2379:
2318:Hartenberg, R.S. & J. Denavit (1964)
1849:Rear car suspension using Watt's linkage.
1608:
2443:
2441:
2439:
2437:
2435:
2433:
2431:
2429:
2427:
2425:
1468:
896:
889:
263:refers to all arrangements in this type.
194:Crank-rocker (Treadle) linkage (used in
186:refers to all arrangements in this type.
109:
97:
89:
72:
2737:Hart's Second Inversor / Hart's A-frame
2531:
2504:, Springer Netherlands, pp. 1–10,
2499:
2329:
2024:Green link completes a full revolution.
1512:Spherical and spatial four-bar linkages
1458:
1159:Examples of offset mechanisms include:
988:With four-bar mechanisms there are two
231:Single-slider crank mechanism (used in
81:'s main mechanism is a four-bar linkage
14:
2785:
2449:
2129:drawn on a 4-Bar Double-Crank linkage.
2081:Deltoid/Kite (Galloway) 4-Bar Linkage:
1384:). The following equations show this.
922:The synthesis, or design, of four-bar
363:A crank: can rotate a full 360 degrees
2598:
2422:
2288:
2279:
2022:Cyan link completes a full revolution
2005:Cyan link completes a full revolution
1979:Animation of a steam locomotive with
1889:The Howe Machine Co. Sewing machine.
1152:mechanisms are often referred to as
379:
2502:Latest Advances in Robot Kinematics
2468:
2168:Four-bar linkage for approximating
1926:Ford 3-speed Overdrive Transmission
1345:quantities, in that they have both
1341:Both velocity and acceleration are
265:Examples of 2R2P linkages include:
228:Examples of 3R1P linkages include:
24:
1762:Foldable steps and foldable chairs
1086:
1011:
277:(slotted link) mechanism (used in
25:
2804:
2578:
2454:. New Jersey: Pearson Education.
2103:All four unique inversions shown:
2100:Trapezium (Arglin) 4-Bar Linkage:
424:
188:Examples of 4R linkages include:
2294:This is not to be confused with
2161:
2149:
2144:of a 4-Bar Crank-Rocker linkage.
2134:
2118:
2093:
2074:
2059:
2044:
2029:
2012:
2007:Green link rocks back and forth.
1995:
1972:
1960:
1948:
1932:
1918:
1906:
1894:
1882:
1870:
1854:
1842:
1826:
1814:
1667:
1655:
1643:
1320:
248:Fixed piston mechanism (used in
126:joints. A joint may be either a
2562:
2553:
2525:
2493:
2320:Kinematic synthesis of linkages
1736:
201:Double-rocker linkage (used in
2404:10.1080/00029890.2003.11919986
2373:
2357:
2347:
2312:
2296:Watt's Parallel Motion linkage
2187:
2181:
1988:
1595:
1583:
1554:
1541:
1100:
1083:
62:, connected in a loop by four
13:
1:
2475:Chakrabarti, Amaresh (2002).
2392:American Mathematical Monthly
2305:
1685:Other Linkages and Mechanisms
934:
918:Design of four-bar mechanisms
583:{\displaystyle T_{3}=b+h-a-g}
530:{\displaystyle T_{2}=b+g-a-h}
477:{\displaystyle T_{1}=g+h-a-b}
2624:
2036:Double-Rocker 4-Bar Linkage:
1528:Moore-Penrose pseudoinverses
1140:Engine mechanisms or pistons
901:Types of four-bar linkages,
316:Planar quadrilateral linkage
184:planar quadrilateral linkage
7:
2656:Evans "Grasshopper" linkage
2510:10.1007/978-94-007-4620-6_1
2205:
2002:Crank-Rocker 4-Bar Linkage:
1701:Evans "Grasshopper" linkage
1679:
1573:admits the set of matrices
1357:. This assumption produces
233:internal combustion engines
10:
2809:
2732:Peaucellier–Lipkin linkage
2019:Double-Crank 4-Bar Linkage
1901:A pedal-driven grindstone.
1753:Double wishbone suspension
1518:spherical four-bar linkage
1462:
114:An example of an RRPR, or
102:An example of an RRRP, or
2745:
2717:
2689:
2636:
2632:
1674:Bennett four-bar linkage.
1566:{\displaystyle (1,0)^{T}}
1334:behavior of a mechanism.
348:, which is also called a
42:, is the simplest closed-
2650:Chebyshev lambda linkage
2533:Bennett, Geoffrey Thomas
2272:
1865:family armored vehicles.
1765:Foot operated machines (
1696:Chebyshev lambda linkage
2549:(Dec 4, 1903): 777–778.
2257:Straight line mechanism
2193:{\displaystyle \log(u)}
1833:Train Suspension using
1143:Automobile window crank
344:. The last link is the
300:. Because of this, the
86:Planar four-bar linkage
68:planar four-bar linkage
2450:Myszka, David (2012).
2252:Spherical trigonometry
2194:
2084:Both inversions shown:
1981:Walschaerts valve gear
1877:Door closer mechanism.
1619:
1567:
1505:quick-return mechanism
1474:
1311:
1239:
1166:Package-moving devices
1116:
1044:
1037:Time of quicker stroke
1001:) for a mechanism is:
979:
965:Time of quicker stroke
910:
894:
688:
660:
632:
584:
531:
478:
409:is the shortest link,
310:mechanical engineering
216:Three revolute joints:
119:
107:
95:
82:
50:. It consists of four
2793:Linkages (mechanical)
2763:Scott Russell linkage
2195:
1620:
1568:
1472:
1369:), maximum velocity (
1355:constant acceleration
1312:
1240:
1117:
1045:
1029:Time of slower stroke
980:
962:Time of slower stroke
900:
893:
689:
687:{\displaystyle T_{3}}
661:
659:{\displaystyle T_{2}}
633:
631:{\displaystyle T_{1}}
585:
532:
479:
421:are the other links.
269:Trammel of Archimedes
224:inverted slider-crank
180:Four revolute joints:
116:inverted slider-crank
113:
101:
93:
76:
2247:Slider-crank linkage
2232:Linkage (mechanical)
2172:
1939:Gearshift on a 2003
1577:
1538:
1465:slider-crank linkage
1459:Slider-crank linkage
1359:polynomial equations
1259:
1186:
1067:
1008:
950:
671:
643:
615:
543:
490:
437:
413:is the longest, and
271:(Elliptical trammel)
220:slider-crank linkage
191:Double-crank linkage
2381:Toussaint, Godfried
2227:Kinematic synthesis
1821:Oil well pump-jack.
1748:Biological linkages
929:iterate-and-analyze
2753:Burmester's theory
2666:Horse-head Linkage
2324:Cornell University
2212:Burmester's theory
2190:
1788:Glider (furniture)
1743:Bicycle suspension
1711:Horse-head linkage
1615:
1563:
1532:one-sided inverses
1475:
1307:
1235:
1112:
1040:
975:
911:
895:
882:27 different cases
696:Grashof condition
684:
656:
628:
580:
527:
474:
203:Ackermann steering
120:
108:
96:
83:
2780:
2779:
2776:
2775:
2646:Chebyshev linkage
2537:"A New Mechanism"
2461:978-0-13-215780-3
2417:Additional Proofs
2261:four-bar linkages
2125:Fixed and moving
1955:A foldable chair.
1941:SV650S motorcycle
1800:Step-on trash can
1691:Chebyshev linkage
1635:Bennett's linkage
1376:), acceleration (
1305:
1233:
1137:Windshield wipers
1096:
1077:
1038:
1030:
1021:
967:
966:
963:
905:: shortest link,
866:
865:
380:Grashof condition
285:Oldham's coupling
124:degree-of-freedom
16:(Redirected from
2800:
2727:Jansen's linkage
2661:Hoeckens linkage
2634:
2633:
2619:
2612:
2605:
2596:
2595:
2572:
2566:
2560:
2557:
2551:
2550:
2529:
2523:
2522:
2497:
2491:
2490:
2472:
2466:
2465:
2447:
2420:
2415:
2389:
2377:
2371:
2361:
2355:
2351:
2345:
2340:
2327:
2316:
2299:
2292:
2286:
2283:
2222:Five-bar linkage
2199:
2197:
2196:
2191:
2165:
2153:
2142:Coupler cognates
2138:
2122:
2097:
2078:
2063:
2048:
2033:
2016:
1999:
1976:
1964:
1952:
1936:
1922:
1910:
1898:
1886:
1874:
1858:
1846:
1830:
1818:
1808:Prosthetic knee
1804:Windshield wiper
1671:
1662:Tractor steering
1659:
1650:Universal joint.
1647:
1624:
1622:
1621:
1616:
1611:
1572:
1570:
1569:
1564:
1562:
1561:
1451:
1442:
1440:
1439:
1436:
1433:
1419:
1408:
1406:
1405:
1402:
1399:
1316:
1314:
1313:
1308:
1306:
1304:
1293:
1282:
1277:
1276:
1244:
1242:
1241:
1236:
1234:
1232:
1225:
1224:
1214:
1207:
1206:
1196:
1163:Cutting machines
1121:
1119:
1118:
1113:
1111:
1110:
1098:
1097:
1094:
1079:
1078:
1075:
1049:
1047:
1046:
1041:
1039:
1036:
1031:
1028:
1023:
1022:
1019:
1000:
984:
982:
981:
976:
968:
964:
961:
960:
939:The time ratio (
693:
691:
690:
685:
683:
682:
665:
663:
662:
657:
655:
654:
637:
635:
634:
629:
627:
626:
609:
608:
589:
587:
586:
581:
555:
554:
536:
534:
533:
528:
502:
501:
483:
481:
480:
475:
449:
448:
38:, also called a
36:four-bar linkage
30:In the study of
21:
2808:
2807:
2803:
2802:
2801:
2799:
2798:
2797:
2783:
2782:
2781:
2772:
2741:
2713:
2704:Parallel motion
2685:
2676:Roberts linkage
2628:
2623:
2581:
2576:
2575:
2567:
2563:
2558:
2554:
2530:
2526:
2520:
2498:
2494:
2487:
2473:
2469:
2462:
2448:
2423:
2387:
2378:
2374:
2362:
2358:
2352:
2348:
2341:
2330:
2317:
2313:
2308:
2303:
2302:
2293:
2289:
2284:
2280:
2275:
2267:Universal joint
2242:Six-bar linkage
2217:Cognate linkage
2208:
2201:
2173:
2170:
2169:
2166:
2157:
2154:
2145:
2139:
2130:
2123:
2112:
2111:• Double-Rocker
2110:
2108:
2106:
2104:
2102:
2101:
2098:
2089:
2087:
2085:
2083:
2082:
2079:
2070:
2068:
2067:
2064:
2055:
2053:
2052:
2049:
2040:
2038:
2037:
2034:
2025:
2023:
2021:
2020:
2017:
2008:
2006:
2004:
2003:
2000:
1991:
1984:
1977:
1968:
1965:
1956:
1953:
1944:
1937:
1928:
1923:
1914:
1911:
1902:
1899:
1890:
1887:
1878:
1875:
1866:
1861:Suspension for
1859:
1850:
1847:
1838:
1831:
1822:
1819:
1792:Oscillating fan
1739:
1721:Roberts linkage
1706:Hoecken linkage
1687:
1682:
1675:
1672:
1663:
1660:
1651:
1648:
1607:
1578:
1575:
1574:
1557:
1553:
1539:
1536:
1535:
1514:
1467:
1461:
1437:
1434:
1431:
1430:
1428:
1422:
1414:
1403:
1400:
1397:
1396:
1394:
1388:
1374:
1323:
1294:
1283:
1281:
1272:
1268:
1260:
1257:
1256:
1220:
1216:
1215:
1202:
1198:
1197:
1195:
1187:
1184:
1183:
1103:
1099:
1093:
1089:
1074:
1070:
1068:
1065:
1064:
1059:
1035:
1027:
1018:
1014:
1009:
1006:
1005:
999:
996:
993:
959:
951:
948:
947:
937:
920:
909:: longest link.
879:
875:
871:
678:
674:
672:
669:
668:
650:
646:
644:
641:
640:
622:
618:
616:
613:
612:
605:
601:
597:
550:
546:
544:
541:
540:
497:
493:
491:
488:
487:
444:
440:
438:
435:
434:
427:
382:
318:
279:valve actuators
264:
258:
227:
217:
187:
181:
157:
154:
149:
141:
132:prismatic joint
88:
28:
23:
22:
15:
12:
11:
5:
2806:
2796:
2795:
2778:
2777:
2774:
2773:
2771:
2770:
2765:
2760:
2755:
2749:
2747:
2743:
2742:
2740:
2739:
2734:
2729:
2723:
2721:
2715:
2714:
2712:
2711:
2706:
2701:
2695:
2693:
2687:
2686:
2684:
2683:
2681:Watt's linkage
2678:
2673:
2668:
2663:
2658:
2653:
2642:
2640:
2630:
2629:
2622:
2621:
2614:
2607:
2599:
2593:
2592:
2587:
2580:
2579:External links
2577:
2574:
2573:
2561:
2552:
2524:
2518:
2492:
2486:978-1852334925
2485:
2467:
2460:
2421:
2398:(6): 482–494.
2372:
2356:
2346:
2328:
2310:
2309:
2307:
2304:
2301:
2300:
2287:
2277:
2276:
2274:
2271:
2270:
2269:
2264:
2254:
2249:
2244:
2239:
2234:
2229:
2224:
2219:
2214:
2207:
2204:
2203:
2202:
2189:
2186:
2183:
2180:
2177:
2167:
2160:
2158:
2155:
2148:
2146:
2140:
2133:
2131:
2124:
2117:
2114:
2113:
2109:• Double-Crank
2107:• Crank-Rocker
2105:• Crank-Rocker
2099:
2092:
2090:
2088:• Double-Crank
2086:• Crank-Rocker
2080:
2073:
2071:
2065:
2058:
2056:
2050:
2043:
2041:
2035:
2028:
2026:
2018:
2011:
2009:
2001:
1994:
1990:
1987:
1986:
1985:
1978:
1971:
1969:
1966:
1959:
1957:
1954:
1947:
1945:
1938:
1931:
1929:
1924:
1917:
1915:
1912:
1905:
1903:
1900:
1893:
1891:
1888:
1881:
1879:
1876:
1869:
1867:
1860:
1853:
1851:
1848:
1841:
1839:
1835:Watt's linkage
1832:
1825:
1823:
1820:
1813:
1810:
1809:
1806:
1801:
1798:
1793:
1790:
1785:
1782:
1775:sewing machine
1763:
1760:
1755:
1750:
1745:
1738:
1735:
1734:
1733:
1731:Watt's linkage
1728:
1723:
1718:
1713:
1708:
1703:
1698:
1693:
1686:
1683:
1681:
1678:
1677:
1676:
1673:
1666:
1664:
1661:
1654:
1652:
1649:
1642:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1560:
1556:
1552:
1549:
1546:
1543:
1513:
1510:
1509:
1508:
1500:
1497:
1493:
1463:Main article:
1460:
1457:
1453:
1452:
1420:
1412:
1372:
1322:
1319:
1318:
1317:
1303:
1300:
1297:
1292:
1289:
1286:
1280:
1275:
1271:
1267:
1264:
1246:
1245:
1231:
1228:
1223:
1219:
1213:
1210:
1205:
1201:
1194:
1191:
1179:, as follows:
1168:
1167:
1164:
1145:
1144:
1141:
1138:
1123:
1122:
1109:
1106:
1102:
1092:
1088:
1085:
1082:
1073:
1057:
1051:
1050:
1034:
1026:
1017:
1013:
997:
994:
986:
985:
974:
971:
958:
955:
936:
933:
919:
916:
877:
873:
869:
868:The cases of T
864:
863:
860:
857:
854:
851:
848:
844:
843:
840:
837:
834:
831:
828:
824:
823:
820:
817:
814:
811:
808:
804:
803:
800:
797:
794:
791:
788:
784:
783:
780:
777:
774:
771:
768:
764:
763:
760:
757:
754:
751:
748:
744:
743:
740:
737:
734:
731:
728:
724:
723:
720:
717:
714:
711:
708:
704:
703:
700:
697:
694:
681:
677:
666:
653:
649:
638:
625:
621:
603:
599:
595:
592:
591:
579:
576:
573:
570:
567:
564:
561:
558:
553:
549:
538:
526:
523:
520:
517:
514:
511:
508:
505:
500:
496:
485:
473:
470:
467:
464:
461:
458:
455:
452:
447:
443:
426:
425:Classification
423:
381:
378:
374:
373:
370:
367:
364:
354:connecting rod
334:grounded links
317:
314:
296:found in many
290:
289:
288:
287:
282:
272:
261:doubler slider
255:
254:
253:
246:
243:
236:
214:
213:
212:
209:
206:
199:
192:
169:connecting rod
128:revolute joint
87:
84:
26:
9:
6:
4:
3:
2:
2805:
2794:
2791:
2790:
2788:
2769:
2766:
2764:
2761:
2759:
2758:Leg mechanism
2756:
2754:
2751:
2750:
2748:
2744:
2738:
2735:
2733:
2730:
2728:
2725:
2724:
2722:
2720:
2716:
2710:
2707:
2705:
2702:
2700:
2699:Klann linkage
2697:
2696:
2694:
2692:
2688:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2651:
2647:
2644:
2643:
2641:
2639:
2635:
2631:
2627:
2620:
2615:
2613:
2608:
2606:
2601:
2600:
2597:
2591:
2588:
2586:
2583:
2582:
2571:
2565:
2556:
2548:
2544:
2543:
2538:
2534:
2528:
2521:
2519:9789400746190
2515:
2511:
2507:
2503:
2496:
2488:
2482:
2478:
2471:
2463:
2457:
2453:
2446:
2444:
2442:
2440:
2438:
2436:
2434:
2432:
2430:
2428:
2426:
2418:
2413:
2409:
2405:
2401:
2397:
2393:
2386:
2382:
2376:
2370:
2369:0-07-247046-1
2366:
2360:
2350:
2344:
2339:
2337:
2335:
2333:
2325:
2321:
2315:
2311:
2297:
2291:
2282:
2278:
2268:
2265:
2262:
2258:
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2209:
2184:
2178:
2175:
2164:
2159:
2152:
2147:
2143:
2137:
2132:
2128:
2121:
2116:
2115:
2096:
2091:
2077:
2072:
2062:
2057:
2047:
2042:
2032:
2027:
2015:
2010:
1998:
1993:
1992:
1982:
1975:
1970:
1963:
1958:
1951:
1946:
1942:
1935:
1930:
1927:
1921:
1916:
1909:
1904:
1897:
1892:
1885:
1880:
1873:
1868:
1864:
1863:VPK-3927 Volk
1857:
1852:
1845:
1840:
1836:
1829:
1824:
1817:
1812:
1811:
1807:
1805:
1802:
1799:
1797:
1794:
1791:
1789:
1786:
1783:
1780:
1776:
1772:
1768:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1740:
1732:
1729:
1727:
1724:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1688:
1670:
1665:
1658:
1653:
1646:
1641:
1640:
1639:
1637:
1636:
1631:
1630:
1628:
1627:left inverses
1604:
1601:
1598:
1592:
1589:
1586:
1558:
1550:
1547:
1544:
1533:
1529:
1523:
1519:
1506:
1501:
1498:
1494:
1491:
1490:
1489:
1486:
1484:
1480:
1471:
1466:
1456:
1449:
1445:
1426:
1421:
1418:
1411:
1392:
1387:
1386:
1385:
1383:
1379:
1375:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1335:
1333:
1328:
1321:Timing charts
1301:
1298:
1295:
1290:
1287:
1284:
1278:
1273:
1269:
1265:
1262:
1255:
1254:
1253:
1251:
1229:
1226:
1221:
1217:
1211:
1208:
1203:
1199:
1192:
1189:
1182:
1181:
1180:
1178:
1174:
1165:
1162:
1161:
1160:
1157:
1155:
1151:
1142:
1139:
1136:
1135:
1134:
1131:
1129:
1107:
1104:
1090:
1080:
1071:
1063:
1062:
1061:
1056:
1032:
1024:
1015:
1004:
1003:
1002:
991:
972:
969:
956:
953:
946:
945:
944:
942:
932:
930:
925:
915:
908:
904:
899:
892:
888:
885:
883:
861:
858:
855:
852:
849:
846:
845:
841:
838:
835:
832:
829:
826:
825:
821:
818:
815:
812:
809:
806:
805:
801:
798:
795:
792:
789:
786:
785:
781:
778:
775:
772:
769:
766:
765:
761:
758:
755:
752:
749:
746:
745:
741:
738:
735:
732:
729:
726:
725:
721:
718:
715:
712:
709:
706:
705:
701:
698:
695:
679:
675:
667:
651:
647:
639:
623:
619:
611:
610:
607:
577:
574:
571:
568:
565:
562:
559:
556:
551:
547:
539:
524:
521:
518:
515:
512:
509:
506:
503:
498:
494:
486:
471:
468:
465:
462:
459:
456:
453:
450:
445:
441:
433:
432:
431:
422:
420:
416:
412:
408:
404:
400:
396:
392:
388:
377:
371:
368:
365:
362:
361:
360:
357:
355:
351:
347:
346:floating link
343:
339:
335:
331:
327:
323:
313:
311:
307:
303:
299:
295:
286:
283:
280:
276:
273:
270:
267:
266:
262:
256:
251:
247:
244:
241:
237:
234:
230:
229:
225:
221:
215:
210:
207:
204:
200:
197:
193:
190:
189:
185:
179:
178:
177:
174:
172:
170:
165:
161:
160:floating link
153:
148:
146:
140:
135:
133:
129:
125:
117:
112:
105:
100:
92:
80:
75:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
18:4-bar linkage
2637:
2564:
2555:
2546:
2540:
2527:
2501:
2495:
2476:
2470:
2451:
2395:
2391:
2375:
2359:
2349:
2314:
2290:
2281:
2260:
1784:Gear shifter
1737:Applications
1633:
1632:
1525:
1522:dual numbers
1517:
1515:
1504:
1487:
1476:
1454:
1447:
1443:
1424:
1416:
1409:
1390:
1381:
1380:), and time(
1377:
1370:
1366:
1363:displacement
1338:
1336:
1324:
1249:
1247:
1176:
1172:
1169:
1158:
1153:
1150:quick-return
1149:
1146:
1132:
1127:
1124:
1054:
1052:
987:
940:
938:
928:
921:
912:
906:
902:
886:
867:
702:Output link
593:
428:
418:
414:
410:
406:
402:
398:
394:
390:
385:
383:
375:
358:
353:
349:
345:
341:
337:
333:
329:
325:
321:
319:
291:
260:
223:
219:
183:
175:
167:
163:
159:
152:rocker link.
151:
143:
139:ground link.
138:
136:
121:
115:
104:slider-crank
103:
67:
59:
55:
39:
35:
29:
2768:Whippletree
2542:Engineering
1989:Simulations
1758:Door closer
1625:as all its
856:Non-Grashof
836:Non-Grashof
816:Non-Grashof
796:Non-Grashof
699:Input link
342:output link
322:ground link
275:Scotch yoke
2671:Pantograph
2306:References
1767:grindstone
1726:Valve gear
1716:Pantograph
935:Time ratio
924:mechanisms
876:= 0, and T
338:input link
326:fixed link
302:kinematics
294:mechanisms
250:hand pumps
145:crank link
118:, linkage.
106:, linkage.
32:mechanisms
2719:Eight-bar
2179:
2127:centrodes
1605:∈
1599:∣
1351:direction
1347:magnitude
1332:kinematic
1288:−
1279:×
1274:∘
1263:β
1230:β
1227:−
1222:∘
1212:β
1204:∘
1105:−
1087:Δ
1072:ω
1012:Δ
970:≥
862:π-Rocker
842:0-Rocker
822:π-Rocker
802:0-Rocker
575:−
569:−
522:−
516:−
469:−
463:−
328:, or the
196:pumpjacks
54:, called
2787:Category
2638:Four-bar
2626:Linkages
2535:(1903).
2412:16625019
2383:(2003).
2237:Pumpjack
2206:See also
1796:Pumpjack
1680:Examples
1496:rotates.
859:0-Rocker
839:π-Rocker
819:π-Rocker
799:0-Rocker
405:, where
306:dynamics
298:machines
164:coupler.
79:pumpjack
46:movable
40:four-bar
2691:Six-bar
1781:, etc.)
1779:treadle
1492:In-line
1441:
1429:
1407:
1395:
1128:in-line
995:Δt
990:strokes
782:Rocker
776:Grashof
756:Grashof
742:Rocker
736:Grashof
716:Grashof
602:, and T
350:coupler
240:shapers
156:slider.
48:linkage
2516:
2483:
2458:
2410:
2367:
1499:Offset
1483:piston
1343:vector
1327:motion
1154:offset
872:= 0, T
779:Rocker
762:Crank
759:Rocker
722:Crank
64:joints
52:bodies
2746:Other
2408:S2CID
2388:(PDF)
2273:Notes
1771:lathe
1530:with
1479:crank
1095:cycle
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