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1021:
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331:
20:
935:
917:
32:
295:
The reason of over-constraint is the unique geometry of linkages in these mechanisms, which the mobility formula does not take into account. This unique geometry gives rise to "redundant constraints", i.e. when multiple joints are constraining the same degrees of freedom. These redundant constraints
973:; when their number is more than two, overconstrained mechanisms with negative calculated mobility will result. The companion animated GIFs show overconstrained mechanisms obtained by assembling together four-bar coupler cognates and function cognates of the Watt II type.
692:
321:
The figure on the left shows a two-hinged trunk lid. The calculated mobility for the lid relative to the car body is zero, yet it moves as its hinges (which are pin joints) have colinear axes. In this case, the second hinge is kinematically redundant.
299:
For example, as shown in the figure to the right, consider a hinged door with 3 hinges. The mobility criterion for this door gives the mobility to be −1. Yet, the door moves and has a degree of freedom 1, as all its hinges have colinear axes.
922:
Moreover, the links are assembled in such a way that, for two links that are joined, the common perpendicular to the joint axes of the first link intersects the common perpendicular of the joint axes of the second link.
661:
The dimensional constraints that makes
Bennett's linkage movable are the following. Let us number the links in order that links with consecutive index are joined (first and fourth links are also joined). For the
650:
492:
912:{\displaystyle {\begin{aligned}&d_{1}=d_{3},\quad a_{1}=a_{3}\\&d_{2}=d_{4},\quad a_{2}=a_{4}\\&{\frac {d_{1}^{2}}{\sin ^{2}a_{1}}}={\frac {d_{2}^{2}}{\sin ^{2}a_{2}}}.\end{aligned}}}
697:
271:
159:
1242:
56:
500:
The Sarrus mechanism has one degree of freedom whereas the mobility formula yields M = 0, which means it has a particular set of dimensions that allow movement.
945:
James Watt employed an approximate straight line four-bar linkage to maintain a near rectilinear motion of the piston rod, thus eliminating the need of using a
1205:
Wei, G., Chen, Y. and Dai, J. S., Synthesis, "Mobility and
Multifurcation of Deployable Polyhedral Mechanisms with Radially Reciprocating Motion",
1041:
186:
If the links in the system move planes parallel to a fixed plane, or in concentric spheres about a fixed point, then the mobility formula is
1020:
996:
984:
1222:
529:
374:
64:
1257:
52:
1232:
1132:
Special
Supplement on Spatial Mechanisms and Robot Manipulators, Transactions of the ASME: Journal of Mechanical Design
1092:
192:
80:
1115:
658:
As in the case of the Sarrus linkage, it is a particular set of dimensions that makes the
Bennett linkage movable.
313:
31:
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517:
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40:
1008:
958:
24:
48:
1176:
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Overconstrained
Linkage Geometry by Solution of Closure Equations---Part 1. Method of Study,
1188:
1157:
970:
343:
339:
1236:
1226:
508:
74:
If the links of the system move in three-dimensional space, then the mobility formula is
683:
362:
68:
1192:
1161:
1251:
1130:
Dai, J.S., Huang, Z., Lipkin, H., "Mobility of
Overconstrained Parallel Mechanisms",
523:
A general spatial linkage formed from four links and four hinged joints has mobility
368:
A general spatial linkage formed from six links and six hinged joints has mobility
60:
516:
Another example of an overconstrained mechanism is
Bennett's linkage, invented by
961:, Hoberman mechanisms move because of their particular geometric configurations.
330:
946:
19:
686:
of the link. Bennett's linkage must satisfies the following constraints:
520:
in 1903, which consists of four links connected by four revolute joints.
969:
Overconstrained mechanisms can be also obtained by assembling together
934:
59:. The mobility formula evaluates the degree of freedom of a system of
1177:"Four- and six-bar function cognates and overconstrained mechanisms"
682:
respectively the distance and the oriented angle of the axes of the
933:
926:
Below is an external link to an animation of a
Bennett's linkage.
507:
353:
329:
30:
18:
645:{\displaystyle M=6(N-1-j)+\sum _{i=1}^{j}f_{i}=6(4-1-4)+4=-2,}
487:{\displaystyle M=6(N-1-j)+\sum _{i=1}^{j}f_{i}=6(6-1-6)+6=0,}
338:
A well-known example of an overconstrained mechanism is the
1233:
Page with
Bennett Linkage, above, with explanations, et al
1078:
Mechanism and
Machine Theory, Vol. 8, pp. 94–104, 1973.
1146:"Applications of Watt II function generator cognates"
695:
532:
377:
365:
consists of six bars connected by six hinged joints.
317:
A multi-hinged door is an over-constrained mechanism.
195:
83:
911:
644:
486:
265:
153:
266:{\displaystyle M=3(N-1-j)+\sum _{i=1}^{j}f_{i}.}
154:{\displaystyle M=6(N-1-j)+\sum _{i=1}^{j}f_{i},}
1243:Mobility of Overconstrained Parallel Mechanisms
283:or less, yet still moves, then it is called an
276:If a system of links and joints has mobility
8:
1002:Coupler cognates of a slider-crank linkage.
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35:Trammel of Archimedes with three sliders.
312:
1067:
990:Coupler cognates of a four-bar linkage.
980:
304:Examples of over-constrained mechanisms
296:are the reason of the over-constraint.
1175:Simionescu, P.A.; Smith, M.R. (2001).
1144:Simionescu, P.A.; Smith, M.R. (2000).
655:which is a highly constrained system.
168:is the number of links in the system,
342:with multiple cranks, as seen in the
7:
16:Moveable linkage with zero mobility
1047:3R-P-3R Watt II function cognates.
1026:3R-R-3R Watt II function cognates.
14:
1207:ASME Journal of Mechanical Design
1052:
1040:
1031:
1019:
1007:
995:
983:
334:Overconstrained parallel linkage
179:is the degree of freedom of the
27:is an overconstrained mechanism.
1223:Animation of Bennett's linkage.
782:
727:
309:Multi-hinged doors and the like
1116:J. M. McCarthy and G. S. Soh,
621:
603:
560:
542:
497:and is therefore a structure.
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448:
405:
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223:
205:
111:
93:
1:
1193:10.1016/S0094-114X(01)00031-3
1162:10.1016/S0094-114X(00)00011-2
1118:Geometric Design of Linkages,
172:is the number of joints, and
1181:Mechanism and Machine Theory
1150:Mechanism and Machine Theory
965:Assembly of cognate linkages
666:-th link, let us denote by
67:are imposed in the form of
1274:
1239: (archived 2014-11-23)
1229: (archived 2017-02-20)
1120:2nd Edition, Springer 2010
1209:, 136(9), p.091003, 2014.
957:Same as the crank-driven
291:Reason of over-constraint
285:overconstrained mechanism
55:than is predicted by the
45:overconstrained mechanism
1134:, 128(1): 220–229, 2006.
1089:Bennett, Geoffrey Thomas
518:Geoffrey Thomas Bennett
503:
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346:of steam locomotives.
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41:mechanical engineering
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1258:Linkages (mechanical)
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1091:(December 4, 1903).
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81:
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512:A Bennett's linkage
71:between the links.
953:Hoberman mechanism
943:
909:
907:
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642:
514:
484:
360:
336:
319:
263:
151:
63:that results when
53:degrees of freedom
37:
29:
1156:(11): 1535–1549.
1093:"A New Mechanism"
977:Animation Gallery
939:Watt steam engine
930:Watt steam engine
900:
853:
504:Bennett's linkage
358:A Sarrus linkage.
23:The crank-driven
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340:parallel linkage
326:Parallel linkage
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57:mobility formula
25:elliptic trammel
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1237:Wayback Machine
1227:Wayback Machine
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1217:External links
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1187:(8): 913–924.
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51:that has more
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344:running gear
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185:
163:
73:
61:rigid bodies
44:
38:
1098:Engineering
65:constraints
1105:: 777–778.
1062:References
183:th joint.
947:crosshead
887:
840:
634:−
616:−
610:−
568:∑
555:−
549:−
461:−
455:−
413:∑
400:−
394:−
231:∑
218:−
212:−
119:∑
106:−
100:−
1252:Category
1235:at the
1225:at the
49:linkage
164:where
69:joints
47:is a
43:, an
673:and
1189:doi
1158:doi
878:sin
831:sin
281:= 0
39:In
1254::
1185:36
1183:.
1179:.
1154:35
1152:.
1148:.
1103:76
1101:.
1095:.
949:.
287:.
1195:.
1191::
1164:.
1160::
941:.
903:.
895:2
891:a
882:2
871:2
866:2
862:d
856:=
848:1
844:a
835:2
824:2
819:1
815:d
802:4
798:a
794:=
789:2
785:a
780:,
775:4
771:d
767:=
762:2
758:d
747:3
743:a
739:=
734:1
730:a
725:,
720:3
716:d
712:=
707:1
703:d
679:i
675:a
671:i
668:d
664:i
640:,
637:2
631:=
628:4
625:+
622:)
619:4
613:1
607:4
604:(
601:6
598:=
593:i
589:f
583:j
578:1
575:=
572:i
564:+
561:)
558:j
552:1
546:N
543:(
540:6
537:=
534:M
482:,
479:0
476:=
473:6
470:+
467:)
464:6
458:1
452:6
449:(
446:6
443:=
438:i
434:f
428:j
423:1
420:=
417:i
409:+
406:)
403:j
397:1
391:N
388:(
385:6
382:=
379:M
279:M
261:.
256:i
252:f
246:j
241:1
238:=
235:i
227:+
224:)
221:j
215:1
209:N
206:(
203:3
200:=
197:M
181:i
176:i
174:f
170:j
166:N
149:,
144:i
140:f
134:j
129:1
126:=
123:i
115:+
112:)
109:j
103:1
97:N
94:(
91:6
88:=
85:M
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