On general nonlinear constrained mechanical systems

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2013, 3(3): 425-443. doi: 10.3934/naco.2013.3.425

On general nonlinear constrained mechanical systems

Firdaus E. Udwadia ^1, and Thanapat Wanichanon ^2,

1.	Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, University of Southern California, Los Angeles, CA 90089-1453, United States
2.	Department of Mechanical Engineering, Mahidol University, 25/25 Puttamonthon, Nakorn Pathom 73170, Thailand

Received September 2011 Revised February 2013 Published July 2013

Abstract
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This paper develops a new, simple, general, and explicit form of the equations of motion for general constrained mechanical systems that can have holonomic and/or nonholonomic constraints that may or may not be ideal, and that may contain either positive semi-definite or positive definite mass matrices. This is done through the replacement of the actual unconstrained mechanical system, which may have a positive semi-definite mass matrix, with an unconstrained auxiliary system whose mass matrix is positive definite and which is subjected to the same holonomic and/or nonholonomic constraints as those applied to the actual unconstrained mechanical system. A simple, unified fundamental equation that gives in closed-form both the acceleration of the constrained mechanical system and the constraint force is obtained. The results herein provide deeper insights into the behavior of constrained motion and open up new approaches to modeling complex, constrained mechanical systems, such as those encountered in multi-body dynamics.

Keywords: extensions of fundamental equation of mechanics., Nonlinear systems, explicit equations, general nonideal constraints, singular mass matrices.

Mathematics Subject Classification: Primary: 70, 37, 1.

Citation: Firdaus E. Udwadia, Thanapat Wanichanon. On general nonlinear constrained mechanical systems. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 425-443. doi: 10.3934/naco.2013.3.425

References:

[1]	P. Appell, Sur une forme generale des equations de la dynamique, C. R. Acad. Sci. III, 129 (1899), 459-460.
[2]	P. Appell, Example de mouvement d'un point assujeti a une liason exprimee par une relation non lineaire entre les composantes de la vitesse, Rend. Circ. Mat. Palermo, 32 (1911), 48-50. doi: 10.1007/BF03014784.
[3]	N. G. Chataev, "Theoretical Mechanics," Mir Publications, Moscow, 1989.
[4]	C. T. Chen, "Linear System Theory and Design," Oxford University Press, New York, 1999.
[5]	P. A. M. Dirac, "Lectures in Quantum Mechanics," New York, Yeshiva University Press, 1964.
[6]	C. Gauss, Uber ein neues allgemeines grundgesetz der mechanik, J. Reine Angew. Math., 4 (1829), 232-235. doi: 10.1515/crll.1829.4.232.
[7]	W. Gibbs, On the fundamental formulae of dynamics, Am. J. Math., 2 (1879), 49-64. doi: 10.2307/2369196.
[8]	H. Goldstein, "Classical Mechanics," Addison-Wesley, Reading, MA, 1981.
[9]	F. Graybill, "Matrices with Applications in Statistics," Wadsworth and Brooks, 1983.
[10]	G. Hamel, "Theoretische Mechanik: Eine Einheitliche Einfuhrung in die Gesamte Mechanik," Springer-Verlag, Berlin, New York, 1949.
[11]	J. L. Lagrange, "Mechanique Analytique," Paris: Mme Ve Courcier, 1787.
[12]	L. A. Pars, "A Treatise on Analytical Dynamics," Woodridge, CT: Oxbow Press, 1979.
[13]	R. Penrose, A generalized inverse of a matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406-413. doi: 10.1017/S0305004100030401.
[14]	A. Schutte and F. E. Udwadia, New approach to the modeling of complex multi-body dynamical systems, Journal of Applied Mechanics, 78 (2011), 11 pages. doi: 10.1115/1.4002329.
[15]	E. C. G. Sudarshan and N. Mukunda, "Classical Dynamics: A Modern Perspective," Wiley, New York, 1974.
[16]	F. E. Udwadia and A. D. Schutte, Equations of motion for general constrained systems in Lagrangian mechanics, Acta Mechanica., 213 (2010), 111-129. doi: 10.1007/s00707-009-0272-2.
[17]	F. E. Udwadia and P. Phohomsiri, Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics, Proceedings of the Royal Society of London A, 462 (2006), 2097-2117. doi: 10.1098/rspa.2006.1662.
[18]	F. E. Udwadia and R. E. Kalaba, A new perspective on constrained motion, Proceedings of the Royal Society of London A, 439 (1992), 407-410. doi: 10.1098/rspa.1992.0158.
[19]	F. E. Udwadia and R. E. Kalaba, An alternative proof for Greville's formula, Journal of Optimization Theory and Applications, 94 (1997), 23-28.
[20]	F. E. Udwadia and R. E. Kalaba, "Analytical Dynamics: A New Approach," Cambridge University Press, 1996. doi: 10.1017/CBO9780511665479.
[21]	F. E. Udwadia and R. E. Kalaba, Explicit equations of motion for mechanical systems with non-ideal constraints, ASME J. Appl. Mech., 68 (2001), 462-467. doi: 10.1115/1.1364492.
[22]	F. E. Udwadia and R. E. Kalaba, On the foundations of analytical dynamics, Int. J. Nonlin. Mech., 37 (2002), 1079-1090. doi: 10.1016/S0020-7462(01)00033-6.

show all references

References:

[1]	P. Appell, Sur une forme generale des equations de la dynamique, C. R. Acad. Sci. III, 129 (1899), 459-460.
[2]	P. Appell, Example de mouvement d'un point assujeti a une liason exprimee par une relation non lineaire entre les composantes de la vitesse, Rend. Circ. Mat. Palermo, 32 (1911), 48-50. doi: 10.1007/BF03014784.
[3]	N. G. Chataev, "Theoretical Mechanics," Mir Publications, Moscow, 1989.
[4]	C. T. Chen, "Linear System Theory and Design," Oxford University Press, New York, 1999.
[5]	P. A. M. Dirac, "Lectures in Quantum Mechanics," New York, Yeshiva University Press, 1964.
[6]	C. Gauss, Uber ein neues allgemeines grundgesetz der mechanik, J. Reine Angew. Math., 4 (1829), 232-235. doi: 10.1515/crll.1829.4.232.
[7]	W. Gibbs, On the fundamental formulae of dynamics, Am. J. Math., 2 (1879), 49-64. doi: 10.2307/2369196.
[8]	H. Goldstein, "Classical Mechanics," Addison-Wesley, Reading, MA, 1981.
[9]	F. Graybill, "Matrices with Applications in Statistics," Wadsworth and Brooks, 1983.
[10]	G. Hamel, "Theoretische Mechanik: Eine Einheitliche Einfuhrung in die Gesamte Mechanik," Springer-Verlag, Berlin, New York, 1949.
[11]	J. L. Lagrange, "Mechanique Analytique," Paris: Mme Ve Courcier, 1787.
[12]	L. A. Pars, "A Treatise on Analytical Dynamics," Woodridge, CT: Oxbow Press, 1979.
[13]	R. Penrose, A generalized inverse of a matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406-413. doi: 10.1017/S0305004100030401.
[14]	A. Schutte and F. E. Udwadia, New approach to the modeling of complex multi-body dynamical systems, Journal of Applied Mechanics, 78 (2011), 11 pages. doi: 10.1115/1.4002329.
[15]	E. C. G. Sudarshan and N. Mukunda, "Classical Dynamics: A Modern Perspective," Wiley, New York, 1974.
[16]	F. E. Udwadia and A. D. Schutte, Equations of motion for general constrained systems in Lagrangian mechanics, Acta Mechanica., 213 (2010), 111-129. doi: 10.1007/s00707-009-0272-2.
[17]	F. E. Udwadia and P. Phohomsiri, Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics, Proceedings of the Royal Society of London A, 462 (2006), 2097-2117. doi: 10.1098/rspa.2006.1662.
[18]	F. E. Udwadia and R. E. Kalaba, A new perspective on constrained motion, Proceedings of the Royal Society of London A, 439 (1992), 407-410. doi: 10.1098/rspa.1992.0158.
[19]	F. E. Udwadia and R. E. Kalaba, An alternative proof for Greville's formula, Journal of Optimization Theory and Applications, 94 (1997), 23-28.
[20]	F. E. Udwadia and R. E. Kalaba, "Analytical Dynamics: A New Approach," Cambridge University Press, 1996. doi: 10.1017/CBO9780511665479.
[21]	F. E. Udwadia and R. E. Kalaba, Explicit equations of motion for mechanical systems with non-ideal constraints, ASME J. Appl. Mech., 68 (2001), 462-467. doi: 10.1115/1.1364492.
[22]	F. E. Udwadia and R. E. Kalaba, On the foundations of analytical dynamics, Int. J. Nonlin. Mech., 37 (2002), 1079-1090. doi: 10.1016/S0020-7462(01)00033-6.

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