Geometric relationships between the velocity screw and momentum screw are presented, and the dual angle between these two screws is shown to provide important insight into the kinetics of a rigid body. Then the centripetal screw is defined, and the significance of this screw in a study of the dynamics of a rigid body is explained. The dual-Euler equation, which is the dual form of the Newton-Euler equations of motion, is shown to be a spatial triangle. The vertices of the triangle are the centripetal screw, the time rate of change of momentum screw, and the force screw. The sides of the triangles are three dual angles between the three vertices. The spatial triangle provides valuable geometrical insight into the dynamics of a rigid body and is believed to be a meaningful alternative to existing analytical techniques. The authors believe that the work presented in this paper will prove useful in a dynamic analysis of closed-loop spatial mechanisms and multi-rigid body open-chain systems.
Issue Section:
Technical Papers
1.
Poinsot
, L.
, 1806
, “Sur la Composition des Moments et la Composition des Aires
,” Journal de L’Ecole Polytechnique
, 6
(13
), pp. 182
–205
.2.
Chasles
, M.
, 1830
, “Note sur les Properties Generales du Systeme de Deux Corps Semblables Entr’eux et Places d’une Maniere Quelconque dans l’Espace; et sur le Deplacement Fini ou Infiniment Petit d’une Corps Solide Libre
,” Bulletin des Sciences Mathematiques, Ferussac
, 14
, p. 321
321
.3.
Ball, R. S., 1990, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, England. (Reprinted in 1999).
4.
Yang, A. T., 1974, “Calculus of Screws,” Basic Questions of Design Theory, North Holland Publishing Company, Amsterdam, pp. 266–281.
5.
Kotelnikov, A. P., 1895, “Screw Calculus and Some of its Applications to Geometry and Mechanics,” Annals of the Imperial University of Kazan.
1.
von Mises
, R.
, 1924
, “Motorrechnung, ein neues Hilfsmittel der Mechanik
,” Zeitshrift fur Angewandte Mathemati und Mechanik
,4
(2
), pp. 155
–181
. 2.
“Motor Calculus: A New Theoretical Device for Mechanics,” English Translation by J. E. Baker and K. Wohlhart, Published by the Institute for Mechanics, University of Technology, Graz, 1996.
1.
Dimentberg, F. M., 1968, “The Screw Calculus and its Application in Mechanics,” Izdat. Nauka, Moscow, U.S.S.R., 1965. English Translation: AD 680993, Clearinghouse for Federal and Scientific Technical Information, Virginia.
2.
Hunt, K. H., 1970, “Screw Systems in Spatial Kinematics,” MMERS Report No. 3, Department of Mechanical Engineering, Monash University, Melbourne, Australia.
3.
Keler
, M. L.
, 1973
, “Kinematics and Statics Including Friction in Single-Loop Mechanisms by Screw Calculus and Dual Vectors
,” ASME J. Eng. Ind.
, 95
(2
), pp. 471
–480
.4.
Ohwovoriole
, M. S.
, and Roth
, B.
, 1981
, “An Extension of Screw Theory
,” ASME J. Mech. Des.
, 103
(4
), pp. 725
–735
.5.
Roth
, B.
, 1967
, “On the Screw Axes and Other Special Lines Associated with Spatial Displacements of a Rigid Body
,” ASME J. Eng. Ind.
, 89
(1
), pp. 102
–110
.6.
Yang
, A. T.
, 1969
, “Displacement Analysis of Spatial Five-Link Mechanisms Using (3×3) Matrices with Dual Number Elements
,” ASME J. Eng. Ind.
, 91
(1
), pp. 152
–157
.7.
Yang
, A. T.
, 1969
, “Analysis of an Offset Unsymmetric Gyroscope with Oblique Rotor Using (3×3) Matrices with Dual-Number Elements
,” ASME J. Eng. Ind.
, 91
(3
), pp. 535
–542
.8.
Woo
, L. S.
, and Freudenstein
, F.
, 1970
, “Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems
,” ASME J. Mech., Transm., Autom. Des.
, 5
, pp. 417
–460
.9.
Yuan
, M. S. C.
, and Freudenstein
, F.
, 1971
, “Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 1—Screw Coordinates
,” ASME J. Eng. Ind.
, 93
(1
), pp. 61
–66
.10.
Kohli
, D.
, and Soni
, A. H.
, 1975
, “Kinematic Analysis of Spatial Mechanisms Via Successive Screw Displacements
,” ASME J. Eng. Ind.
, 97
(2
), pp. 739
–747
.11.
Veldkamp
, G. R.
, 1976
, “On the Use of Dual Numbers, Vectors, and Matrices in Instantaneous, Spatial Kinematics
,” Mech. Mach. Theory
, 11
(2
), pp. 141
–156
.12.
Lipkin, H., and Duffy, J., 1982, “Analysis of Industrial Robots Via the Theory of Screws,” Proceedings of the 12th International Symposium on Industrial Robots, Paris, France, pp. 359–370.
13.
Pennock
, G. R.
, and Yang
, A. T.
, 1983
, “Dynamic Analysis of a Multi-Rigid-Body Open-Chain System
,” ASME J. Mech., Transm., Autom. Des.
, 105
(1
), pp. 28
–34
.14.
Waldron, K. J., and Hunt, K. H., 1987, “Series-Parallel Dualities in Actively Coordinated Mechanisms,” Proceedings of the 4th International Symposium on Robotics Research, Santa Cruz, California, pp. 175–181.
15.
Pennock, G. R., and Oncu, B. A., 1992, “Static Force Analysis of a Three-Cylindric Robot Using the Theory of Screws,” Journal of Robotic and Autonomous Systems, Elsevier Science Publishing Co., Inc., Special Issue: Trends in Robot Kinematics, Dynamics, Control, Sensing, Programming, and Simulation, 9, (4), pp. 201–211.
16.
Pennock
, G. R.
, and Oncu
, B. A.
, 1992
, “Application of Screw Theory to Rigid Body Dynamics
,” ASME J. Dyn. Syst., Meas., Control
, 114
(2
), pp. 262
–269
.17.
Meehan, P. J., 1999, “Geometric Insight into the Dynamics of a Rigid Body Using the Theory of Screws,” M. Eng. Sc., School of Mechanical Engineering, Purdue University, West Lafayette, Indiana.
18.
Pennock, G. R., and Meehan, P. J. 2000, “Geometric Insight into the Dynamics of a Rigid Body Using the Theory of Screws,” Proceedings of the Ball Symposium 2000, CD-ROM Format, Trinity College, Cambridge University, England, July 9–11.
19.
Roth, B., 1984, “Screws, Motors, and Wrenches That Cannot Be Bought in a Hardware Store,” Robotics Research: The First International Symposium, Brady, M., and Paul, R., eds., Chapter 8, pp. 679–693; The MIT Press, Cambridge, Massachusetts.
20.
Brand, L., 1947, Vector and Tensor Analysis, John Wiley and Sons, Inc., Chapter 2, pp. 63–83.
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