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Equations of motion for general constrained systems in Lagrangian mechanics

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Abstract

This paper develops a new, simple, explicit equation of motion for general constrained mechanical systems that may have positive semi-definite mass matrices. This is done through the creation of an auxiliary mechanical system (derived from the actual system) that has a positive definite mass matrix and is subjected to the same set of constraints as the actual system. The acceleration of the actual system and the constraint force acting on it are then directly provided in closed form by the acceleration and the constraint force acting on the auxiliary system, which thus gives the equation of motion of the actual system. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. The use of this new equation is illustrated through its application to the important and practical problem of finding the equation of motion for the rotational dynamics of a rigid body in terms of quaternions. This leads to a form for the equation describing rotational dynamics that has hereto been unavailable.

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, Systems Architecture Engineering, and Information and Operations Management, University of Southern California, 430K Olin Hall, Los Angeles, CA, 90089-1453, USA

    Firdaus E. Udwadia

  2. Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 90089-1453, USA

    Aaron D. Schutte

Authors
  1. Firdaus E. Udwadia
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  2. Aaron D. Schutte
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Correspondence to Firdaus E. Udwadia.

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Udwadia, F.E., Schutte, A.D. Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech 213, 111–129 (2010). https://doi.org/10.1007/s00707-009-0272-2

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  • DOI: https://doi.org/10.1007/s00707-009-0272-2

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