849:
921:
837:
794:
143:
31:
368:
131:
822:
The utilization of the 1:1 gear train is used because of the behavior of parallelogram linkages. Opposite 'sides' of the parallelogram linkages share the same rotational motion function. Because both parallelograms were constructed off of the center coupler link, the new links connected to the ground
888:
One characteristic with cognate linkages is that they often share at least two links in identical configuration. These links are usually oriented 180 degrees of each other, so when pairing, these links can be fused. This creates a 4-bar linkage with two additional links, both of which are defined by
911:
Rather than overlapping the paired links, they can be attached in a fashion which forms a parallelogram, resulting in eight-bar cognate linkages. Triplet cognate linkage pairings and further extensions are also possible, allowing for the number of links of cognate linkages to increase theoretically
876:
The geared five-bar cognate linkage utilizes a 1:1 gear train to ensure two links have the same rotational motion function. However, an alternative to using a gear train to achieve 1:1 rotation is to connect them together using a parallelogram linkage, adding one additional link.
1248:
Similar quadrilaterals will have links which share the same motion function. Because one link is shared between the two quadrilaterals, then as long as the new quadrilaterals are similar to the original, both will still share a link with the same motion function.
111:
For a coupler curve produced by a four-bar linkage with four revolute joints and a prismatic (slider) joint, there exist only two four-bar linkages, as the third would be composed of two sliders, making it a four-bar linkage with two degrees of freedom.
514:
107:
The theorem states, for a given coupler curve produced by a four-bar linkage with four revolute (hinge) joints, there exist three four-bar linkages, three geared five-bar linkages, and more six-bar linkages which will generate the same path.
89:, states that each coupler curve can be generated by three different four-bar linkages. These four-bar linkages can be constructed using similar triangles and parallelograms, and the Cayley diagram (named after
892:
Each of these paired six-bar cognate linkages can also be converted into another cognate linkage by flipping the linkage over, and switching the roles of the rectilinear link and the ground link.
1195:
1141:
1252:
If the new quadrilaterals do not share the same ground joint (as shown with joint C'), they can still be connected together using a parallelogram linkage, forming an 8-bar function cognate.
805:
has two degrees of freedom, and thus there does not exist a five-bar linkage which is capable of acting as a cognate. However, it is possible to generate a 5-bar cognate using gears.
1239:
399:
765:
1011:
972:
1087:
1059:
1034:
931:
linkages are linkages which share the same motion functions for their input and output links. This is performed using a six-bar Watt II linkage arrangement.
885:
An important behavior with four-bar linkages is that once the positions of two links is defined, the other two links in the four-bar linkage can be defined.
889:
the original four-bar linkage. The former ground link of the fusing 4-bar linkage becomes a rectilinear link that travels follows the same coupler curve.
900:
Appending onto the method of the six-bar pairings, the links do not necessarily have to overlap, but rather just need to meet the criterion such that:
1353:
920:
848:
1472:
1288:
82:
1309:
96:
823:
share identical rotational motion functions, allowing for a 1:1 gear train to be used to connect them together.
859:
836:
793:
509:{\displaystyle K_{L}={\frac {\sin(\alpha )}{\sin(\beta )}}\qquad K_{R}={\frac {\sin(\gamma )}{\sin(\beta )}}}
907:
The former ground link of one of the linkages is directly attached to the point producing the coupler curve.
815:
On each parallelogram, find the sides opposite of the connecting link. Apply a 1:1 gear train between them.
797:
Animation showing the construction of a geared five-bar cognate linkages from an initial four-bar cognate.
1312:
configurations that have a DOF of 0, however they are not useful in the construction of cognate linkages.
1146:
1092:
720:
977:
938:
1414:
1389:
1277:
148:
Animation showing the construction of two four-bar cognate linkages from an initial four-bar cognate.
66:
142:
842:
Animation showing the construction of a six-bar cognate linkages from an initial four-bar cognate.
812:
Construct two parallelograms off of the center coupler link and the links connected to the ground.
1374:
1332:
854:
Animation showing the construction of
Chebyshev's table linkage using the two cognate linkages
1349:
1282:
855:
86:
30:
1451:
1426:
1401:
1267:
1203:
802:
241:
74:
70:
73:
geometry or input-output relationship, while being dimensionally dissimilar. In case of
1440:"Coupler cognate mechanisms of certain parallelogram forms of watt's six-link mechanism"
1067:
1039:
1272:
1019:
1439:
1430:
1405:
1466:
1455:
1261:
90:
376:
864:
The pink link indicates the former ground link that has become a rectilinear link.
1370:
367:
17:
58:
781:, one cognate will be a crank-rocker, and the second will be a double-rocker.
130:
784:
If the original is a double-rocker, the cognates will be crank-rockers.
99:
can be obtained by connecting two or more cognate linkages together.
1390:"Four- and six-bar function cognates and overconstrained mechanisms"
366:
1344:
Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2003).
44:
1362:
Samuel
Roberts (1875) "On Three-bar Motion in Plane Space",
935:
Split the six bar linkage into two quadrilaterals (shown as
50:
27:
Linkages of different dimensions with the same output motion
375:
The lengths of the four members can be found by using the
34:
Showcase of nine different coupler-curve cognate linkages.
924:
Animation showing the construction of a function cognate.
904:
There is a pair of two links in identical configuration.
38:
1415:"Applications of Watt II function generator cognates"
1206:
1149:
1095:
1070:
1042:
1022:
980:
941:
723:
402:
136:
Cayley diagram for generating 4-bar coupler cognates.
1233:
1189:
1135:
1081:
1053:
1028:
1005:
966:
759:
508:
896:Construction of coupler cognates with more links
1364:Proceedings of the London Mathematical Society
78:
1373:, p 169, New York: McGraw-Hill, weblink from
8:
1089:, form two similar quadrilaterals (shown as
881:Pairing previously derived cognates together
767:Both 4-bar cognates will be class I chains.
1369:Hartenberg, R.S. & J. Denavit (1964)
1205:
1157:
1148:
1103:
1094:
1069:
1041:
1021:
988:
979:
949:
940:
722:
468:
459:
416:
407:
401:
116:Construction of four-bar coupler cognates
919:
827:Construction of six-bar coupler cognates
792:
789:Construction of geared five-bar cognates
518:
29:
1325:
1301:
872:Alternative to geared five-bar cognates
1413:Simionescu, P.A.; Smith, M.R. (2000).
1388:Simionescu, P.A.; Smith, M.R. (2001).
7:
774:), both cognates will be drag links.
1190:{\displaystyle \square X_{2}C'D'E'}
1136:{\displaystyle \square X_{1}A'B'C'}
809:Select four-bar linkage of choice.
760:{\displaystyle (\ell +s)<(p+q)}
25:
1346:Theory of Machines and Mechanisms
713:If and only if the original is a
358:Put dimensions on Cayley diagram.
355:Separate left and right cognate.
1333:Roberts and Chebyshev (Springer)
1006:{\displaystyle \square X_{2}CDE}
967:{\displaystyle \square X_{1}ABC}
847:
835:
770:If the original is a drag-link (
141:
129:
45:Geared five-bar cognate linkages
1371:Kinematic synthesis of linkages
454:
754:
742:
736:
724:
500:
494:
483:
477:
448:
442:
431:
425:
1:
1431:10.1016/S0094-114X(00)00011-2
1406:10.1016/S0094-114X(01)00031-3
1456:10.1016/0022-2569(70)90023-6
1419:Mechanism and Machine Theory
1394:Mechanism and Machine Theory
299:Using a parallelogram, find
49:Bottom row: Closely related
1348:. Oxford University Press.
1291:- Roberts–Chebyshev Theorem
1285:- Roberts–Chebyshev Theorem
1016:Translate the ground joint
178:Using parallelograms, find
1489:
1064:With the new ground joint
97:Overconstrained mechanisms
53:, derived from middle row.
363:Dimensional relationships
103:Roberts–Chebyshev theorem
79:Roberts–Chebyshev Theorem
39:Four-bar cognate linkages
1200:Reform the shared link
860:Chebyshev lambda linkage
331:Check similar triangles
155:From original triangle,
51:six-bar cognate linkages
47:, derived from top row.
1235:
1234:{\displaystyle C'B'D'}
1191:
1137:
1083:
1055:
1030:
1007:
968:
925:
798:
761:
510:
393:are found as follows.
372:
175:Sketch Cayley diagram.
77:coupler cognates, the
54:
1473:Linkages (mechanical)
1444:Journal of Mechanisms
1236:
1192:
1138:
1084:
1056:
1031:
1008:
969:
923:
796:
777:If the original is a
762:
511:
370:
69:that ensure the same
33:
1335:Retrieved 2012-10-12
1278:Linkage (mechanical)
1204:
1147:
1093:
1068:
1040:
1020:
978:
939:
721:
400:
1438:Soni, A.H. (1970).
1308:There are specific
1036:to a new location,
371:Linkage dimensions.
1375:Cornell University
1244:Separate Cognates.
1231:
1187:
1133:
1082:{\displaystyle C'}
1079:
1054:{\displaystyle C'}
1051:
1026:
1003:
964:
926:
818:Separate cognates.
799:
757:
506:
373:
55:
1425:(11): 1535–1549.
1283:Pafnuty Chebyshev
1029:{\displaystyle C}
916:Function cognates
856:Chebyshev linkage
705:
704:
504:
452:
242:similar triangles
87:Pafnuty Chebyshev
16:(Redirected from
1480:
1459:
1434:
1409:
1359:
1336:
1330:
1313:
1306:
1268:Four-bar linkage
1264:- Cayley Diagram
1240:
1238:
1237:
1232:
1230:
1222:
1214:
1196:
1194:
1193:
1188:
1186:
1178:
1170:
1162:
1161:
1142:
1140:
1139:
1134:
1132:
1124:
1116:
1108:
1107:
1088:
1086:
1085:
1080:
1078:
1060:
1058:
1057:
1052:
1050:
1035:
1033:
1032:
1027:
1012:
1010:
1009:
1004:
993:
992:
973:
971:
970:
965:
954:
953:
929:Function cognate
851:
839:
803:five-bar linkage
766:
764:
763:
758:
701:
687:
673:
659:
640:
626:
612:
598:
579:
569:
559:
549:
519:
515:
513:
512:
507:
505:
503:
486:
469:
464:
463:
453:
451:
434:
417:
412:
411:
392:
385:
351:
327:
307:
295:
279:
261:
252:
236:
215:
195:
186:
170:
145:
133:
75:four-bar linkage
63:cognate linkages
21:
18:Cognate linkages
1488:
1487:
1483:
1482:
1481:
1479:
1478:
1477:
1463:
1462:
1437:
1412:
1387:
1384:
1382:Further reading
1356:
1343:
1340:
1339:
1331:
1327:
1322:
1317:
1316:
1310:overconstrained
1307:
1303:
1298:
1258:
1223:
1215:
1207:
1202:
1201:
1179:
1171:
1163:
1153:
1145:
1144:
1125:
1117:
1109:
1099:
1091:
1090:
1071:
1066:
1065:
1043:
1038:
1037:
1018:
1017:
984:
976:
975:
945:
937:
936:
918:
898:
883:
874:
869:
868:
867:
866:
865:
863:
852:
844:
843:
840:
829:
791:
719:
718:
710:
700:
695:
690:
686:
681:
676:
672:
667:
662:
658:
653:
648:
639:
634:
629:
625:
620:
615:
611:
606:
601:
597:
592:
587:
578:
572:
568:
562:
558:
552:
548:
542:
487:
470:
455:
435:
418:
403:
398:
397:
391:
387:
384:
380:
365:
350:
344:
338:
332:
326:
320:
314:
308:
306:
300:
294:
288:
281:
275:
269:
262:
260:
254:
251:
245:
235:
229:
223:
217:
214:
208:
202:
196:
194:
188:
185:
179:
169:
163:
156:
153:
152:
151:
150:
149:
146:
138:
137:
134:
123:
118:
105:
48:
42:
36:
35:
28:
23:
22:
15:
12:
11:
5:
1486:
1484:
1476:
1475:
1465:
1464:
1461:
1460:
1450:(2): 203–215.
1435:
1410:
1400:(8): 913–924.
1383:
1380:
1379:
1378:
1367:
1360:
1354:
1338:
1337:
1324:
1323:
1321:
1318:
1315:
1314:
1300:
1299:
1297:
1294:
1293:
1292:
1289:Samuel Roberts
1286:
1280:
1275:
1273:Kinematic pair
1270:
1265:
1257:
1254:
1246:
1245:
1242:
1229:
1226:
1221:
1218:
1213:
1210:
1198:
1185:
1182:
1177:
1174:
1169:
1166:
1160:
1156:
1152:
1131:
1128:
1123:
1120:
1115:
1112:
1106:
1102:
1098:
1077:
1074:
1062:
1049:
1046:
1025:
1014:
1002:
999:
996:
991:
987:
983:
963:
960:
957:
952:
948:
944:
917:
914:
912:indefinitely.
909:
908:
905:
897:
894:
882:
879:
873:
870:
853:
846:
845:
841:
834:
833:
832:
831:
830:
828:
825:
820:
819:
816:
813:
810:
790:
787:
786:
785:
782:
775:
768:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
709:
706:
703:
702:
698:
693:
688:
684:
679:
674:
670:
665:
660:
656:
651:
646:
645:Right cognate
642:
641:
637:
632:
627:
623:
618:
613:
609:
604:
599:
595:
590:
585:
581:
580:
576:
570:
566:
560:
556:
550:
546:
540:
536:
535:
532:
529:
526:
523:
517:
516:
502:
499:
496:
493:
490:
485:
482:
479:
476:
473:
467:
462:
458:
450:
447:
444:
441:
438:
433:
430:
427:
424:
421:
415:
410:
406:
389:
382:
364:
361:
360:
359:
356:
353:
346:
340:
334:
329:
324:
318:
310:
302:
297:
292:
286:
273:
267:
258:
249:
238:
233:
227:
219:
212:
206:
198:
192:
183:
176:
167:
161:
147:
140:
139:
135:
128:
127:
126:
125:
124:
122:
121:Cayley diagram
119:
117:
114:
104:
101:
83:Samuel Roberts
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1485:
1474:
1471:
1470:
1468:
1457:
1453:
1449:
1445:
1441:
1436:
1432:
1428:
1424:
1420:
1416:
1411:
1407:
1403:
1399:
1395:
1391:
1386:
1385:
1381:
1376:
1372:
1368:
1365:
1361:
1357:
1355:0-19-515598-X
1351:
1347:
1342:
1341:
1334:
1329:
1326:
1319:
1311:
1305:
1302:
1295:
1290:
1287:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1263:
1262:Arthur Cayley
1260:
1259:
1255:
1253:
1250:
1243:
1227:
1224:
1219:
1216:
1211:
1208:
1199:
1183:
1180:
1175:
1172:
1167:
1164:
1158:
1154:
1150:
1129:
1126:
1121:
1118:
1113:
1110:
1104:
1100:
1096:
1075:
1072:
1063:
1047:
1044:
1023:
1015:
1000:
997:
994:
989:
985:
981:
961:
958:
955:
950:
946:
942:
934:
933:
932:
930:
922:
915:
913:
906:
903:
902:
901:
895:
893:
890:
886:
880:
878:
871:
861:
857:
850:
838:
826:
824:
817:
814:
811:
808:
807:
806:
804:
795:
788:
783:
780:
776:
773:
769:
751:
748:
745:
739:
733:
730:
727:
716:
712:
711:
707:
697:
689:
683:
675:
669:
661:
655:
647:
644:
643:
636:
628:
622:
614:
608:
600:
594:
586:
584:Left cognate
583:
582:
575:
571:
565:
561:
555:
551:
545:
541:
538:
537:
533:
530:
527:
524:
521:
520:
497:
491:
488:
480:
474:
471:
465:
460:
456:
445:
439:
436:
428:
422:
419:
413:
408:
404:
396:
395:
394:
378:
369:
362:
357:
354:
349:
343:
337:
330:
323:
317:
313:
305:
298:
291:
285:
278:
272:
266:
257:
248:
243:
239:
232:
226:
222:
211:
205:
201:
191:
182:
177:
174:
173:
172:
166:
160:
144:
132:
120:
115:
113:
109:
102:
100:
98:
94:
92:
91:Arthur Cayley
88:
84:
80:
76:
72:
71:coupler curve
68:
64:
60:
52:
46:
40:
32:
19:
1447:
1443:
1422:
1418:
1397:
1393:
1363:
1345:
1328:
1304:
1251:
1247:
928:
927:
910:
899:
891:
887:
884:
875:
821:
800:
779:crank-rocker
778:
772:double crank
771:
714:
691:
677:
663:
649:
630:
616:
602:
588:
573:
563:
553:
543:
377:law of sines
374:
347:
341:
335:
321:
315:
311:
303:
289:
283:
276:
270:
264:
255:
246:
230:
224:
220:
209:
203:
199:
189:
180:
164:
158:
154:
110:
106:
95:
62:
56:
43:Middle row:
708:Conclusions
1320:References
59:kinematics
1151:◻
1097:◻
982:◻
943:◻
728:ℓ
539:Original
498:β
492:
481:γ
475:
446:β
440:
429:α
423:
37:Top row:
1467:Category
1366:, vol 7.
1256:See also
1228:′
1220:′
1212:′
1184:′
1176:′
1168:′
1130:′
1122:′
1114:′
1076:′
1048:′
534:Coupler
379:. Both
81:, after
67:linkages
715:Class I
531:Crank 2
528:Crank 1
522:Linkage
244:, find
1352:
717:chain
525:Ground
240:Using
1296:Notes
1350:ISBN
1143:and
974:and
858:and
740:<
386:and
280:and
253:and
216:and
187:and
85:and
65:are
1452:doi
1427:doi
1402:doi
489:sin
472:sin
437:sin
420:sin
93:).
57:In
1469::
1446:.
1442:.
1423:35
1421:.
1417:.
1398:36
1396:.
1392:.
1197:).
1013:).
801:A
333:ΔO
322:DC
309:â–±O
284:DC
231:DB
218:â–±O
210:DA
197:â–±O
171::
165:DB
61:,
1458:.
1454::
1448:5
1433:.
1429::
1408:.
1404::
1377:.
1358:.
1241:.
1225:D
1217:B
1209:C
1181:E
1173:D
1165:C
1159:2
1155:X
1127:C
1119:B
1111:A
1105:1
1101:X
1073:C
1061:.
1045:C
1024:C
1001:E
998:D
995:C
990:2
986:X
962:C
959:B
956:A
951:1
947:X
862:.
755:)
752:q
749:+
746:p
743:(
737:)
734:s
731:+
725:(
699:4
696:R
694:R
692:K
685:3
682:R
680:R
678:K
671:2
668:R
666:R
664:K
657:1
654:R
652:R
650:K
638:2
635:R
633:L
631:K
624:4
621:R
619:L
617:K
610:3
607:R
605:L
603:K
596:1
593:R
591:L
589:K
577:4
574:R
567:3
564:R
557:2
554:R
547:1
544:R
501:)
495:(
484:)
478:(
466:=
461:R
457:K
449:)
443:(
432:)
426:(
414:=
409:L
405:K
390:R
388:K
383:L
381:K
352:.
348:B
345:O
342:C
339:O
336:A
328:.
325:3
319:2
316:C
312:C
304:C
301:O
296:.
293:3
290:B
287:3
282:Δ
277:D
274:2
271:C
268:2
265:A
263:Δ
259:3
256:C
250:2
247:C
237:.
234:3
228:1
225:B
221:B
213:2
207:1
204:A
200:A
193:3
190:B
184:2
181:A
168:1
162:1
159:A
157:Δ
41:.
20:)
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