Research Articles

Identifying Sets of Constraint Forces by Inspection

[+] Author and Article Information
Carlos M. Roithmayr

NASA Langley Research Center,
Hampton, VA 23681
e-mail: carlos.m.roithmayr@nasa.gov

Dewey H. Hodges

School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: dhodges@gatech.edu

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received February 7, 2012; final manuscript received August 29, 2012; accepted manuscript posted September 10, 2012; published online January 25, 2013. Assoc. Editor: Alexander F. Vakakis.

This material is declared a work of the US Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Appl. Mech 80(2), 021019 (Jan 25, 2013) (13 pages) Paper No: JAM-12-1056; doi: 10.1115/1.4007577 History: Received February 07, 2012; Revised August 29, 2012; Accepted September 10, 2012

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.

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Angeles, J., and Lee, S., 1989, “The Modelling of Holonomic Mechanical Systems Using a Natural Orthogonal Complement,” Trans. Can. Soc. Mech. Eng., 13(4), pp. 81–89.
Géradin, M., and Cardona, A., 2001, Flexible Multibody Dynamics: A Finite Element Approach, Wiley, Chichester, UK.
Nikravesh, P. E., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ.
Saha, S. K., and Angeles, J., 1991, “Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement,” ASME J. Appl. Mech., 58(1), pp. 238–243. [CrossRef]
García de Jalón, J., and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge, Springer-Verlag, New York.
Wampler, C., Buffinton, K., and Shu-hui, J., 1985, “Formulation of Equations of Motion for Systems Subject to Constraints,” ASME J. Appl. Mech., 52(2), pp. 465–470. [CrossRef]
Wang, J. T., and Huston, R. L., 1987, “Kane's Equations With Undetermined Multipliers—Application to Constrained Multibody Systems,” ASME J. Appl. Mech., 54(2), pp. 424–429. [CrossRef]
Anderson, K. S., 1992, “An Order n Formulation for the Motion Simulation of General Multi-Rigid-Body Constrained Systems,” Comput. Struct., 43(3), pp. 565–579. [CrossRef]
Huston, R. L., 1999, “Constraint Forces and Undetermined Multipliers in Constrained Multibody Systems,” Multibody Syst. Dyn., 3(4), pp. 381–389. [CrossRef]
Bajodah, A. H., Hodges, D. H., and Chen, Y.-H., 2003, “New Form of Kane's Equations of Motion for Constrained Systems,” J. Guid. Control Dyn., 26(1), pp. 79–88. [CrossRef]
Kane, T. R., and Levinson, D. A., 1985, Dynamics: Theory and Applications, McGraw-Hill, New York.
Wang, L.-S., and Pao, Y.-H., 2003, “Jourdain's Variational Equation and Appell's Equation of Motion for Nonholonomic Dynamical Systems,” Am. J. Phys., 71(1), pp. 72–82. [CrossRef]
von Flotow, A. H., and Rosenthal, D. E., 1991, “Multi-Body Dynamics: Algorithmic Approaches Based on Kane's Equations” (Course Notes), Massachusetts Institute of Technology, June 10–14, p. 3–58.
Djerassi, S., and Bamberger, H., 2003, “Constraint Forces and the Method of Auxiliary Generalized Speeds,” ASME J. Appl. Mech., 70(4), pp. 568–574. [CrossRef]
Roithmayr, C. M., and Hodges, D. H., 2009, “An Argument Against Augmenting the Lagrangean for Nonholonomic Systems,” ASME J. Appl. Mech., 76(3), p. 034501. [CrossRef]
Roithmayr, C. M., and Hodges, D. H., 2010, “Forces Associated With Nonlinear Nonholonomic Constraint Equations,” Int. J. Non-Linear Mech., 45(4), pp. 357–369. [CrossRef]
Rosenthal, D. E., and Sherman, M. A., 1986, “High Performance Multibody Simulations via Symbolic Manipulation and Kane's Method,” J. Astronaut. Sci., 34(3), pp. 223–239.
Schaechter, D. B., and Levinson, D. A., 1988, “Interactive Computerized Symbolic Dynamics for the Dynamicist,” J. Astronaut. Sci., 36(4), pp. 365–388.
“Motion Genesis—Software and Textbooks for Forces and Motion,” Engineering News and Resources, Retrieved February 2, 2012, http://www.motiongenesis.com/
Roithmayr, C. M., 2007, “Relating Constrained Motion to Force Through Newton's Second Law,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, available at http://hdl.handle.net/1853/14592


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Fig. 2

Two rolling spheres with perpendicular mass center velocities

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Fig. 3

Multipliers for tangential constraint forces

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Fig. 4

Motion variables for A

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Fig. 5

Motion variables for B




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