American Institute of Mathematical Sciences

Advanced
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

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 & Optimization, 2013, 3 (3) : 425-443. doi: 10.3934/naco.2013.3.425
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show all references

References:
[1]

C. R. Acad. Sci. III, 129 (1899), 459-460. Google Scholar

[2]

Rend. Circ. Mat. Palermo, 32 (1911), 48-50. doi: 10.1007/BF03014784.  Google Scholar

[3]

Mir Publications, Moscow, 1989. Google Scholar

[4]

Oxford University Press, New York, 1999. Google Scholar

[5]

New York, Yeshiva University Press, 1964. Google Scholar

[6]

J. Reine Angew. Math., 4 (1829), 232-235. doi: 10.1515/crll.1829.4.232.  Google Scholar

[7]

Am. J. Math., 2 (1879), 49-64. doi: 10.2307/2369196.  Google Scholar

[8]

Addison-Wesley, Reading, MA, 1981. Google Scholar

[9]

Wadsworth and Brooks, 1983.  Google Scholar

[10]

Springer-Verlag, Berlin, New York, 1949. Google Scholar

[11]

Paris: Mme Ve Courcier, 1787. Google Scholar

[12]

Woodridge, CT: Oxbow Press, 1979. Google Scholar

[13]

Proc. Cambridge Philos. Soc., 51 (1955), 406-413. doi: 10.1017/S0305004100030401.  Google Scholar

[14]

Journal of Applied Mechanics, 78 (2011), 11 pages. doi: 10.1115/1.4002329.  Google Scholar

[15]

Wiley, New York, 1974.  Google Scholar

[16]

Acta Mechanica., 213 (2010), 111-129. doi: 10.1007/s00707-009-0272-2.  Google Scholar

[17]

Proceedings of the Royal Society of London A, 462 (2006), 2097-2117. doi: 10.1098/rspa.2006.1662.  Google Scholar

[18]

Proceedings of the Royal Society of London A, 439 (1992), 407-410. doi: 10.1098/rspa.1992.0158.  Google Scholar

[19]

Journal of Optimization Theory and Applications, 94 (1997), 23-28.  Google Scholar

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Cambridge University Press, 1996. doi: 10.1017/CBO9780511665479.  Google Scholar

[21]

ASME J. Appl. Mech., 68 (2001), 462-467. doi: 10.1115/1.1364492.  Google Scholar

[22]

Int. J. Nonlin. Mech., 37 (2002), 1079-1090. doi: 10.1016/S0020-7462(01)00033-6.  Google Scholar

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