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Explicit Equations of Motion for Mechanical Systems with Nonideal Constraints Firdaus Udwadia, Robert Kalaba

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Explicit Equations of Motion for Mechanical Systems with Nonideal Constraints Firdaus Udwadia, Robert Kalaba Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints Firdaus Udwadia, Robert Kalaba To cite this version: Firdaus Udwadia, Robert Kalaba. Explicit Equations of Motion for Mechanical Systems With Nonideal Constraints. Journal of Applied Mechanics, American Society of Mechanical Engi- neers, 2001, 68 (3), pp.462 - 462. HAL Id: hal-01399770 https://hal.archives-ouvertes.fr/hal-01399770 Submitted on 20 Nov 2016 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. Distributed under a Creative Commons Attribution 4.0 International License F. E. Udwadia Explicit Equations of Motion Professor of Civil Engineering, Aerospace and Mechanical Engineering, for Mechanical Systems With and Mathematics R. E. Kalaba Nonideal Constraints Professor of Biomedical Engineering, Electrical Engineering, and economics University of Southern California, Los Angeles, CA Abstract Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this con- fining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the con-straints are nonideal. These equations D’Alembert’s principle, and each of these mathematical formal- lead to a simple and new fundamental view of Lagrangian isms is equivalent to the other. Despite the continuous and vigor- mechanics. They provide a geometrical understanding of ous attention that this problem has received, the inclusion of situ- constrained motion, and they highlight the simplicity with which ations where the physically generated forces of constraint in a Nature seems to operate. mechanical system do not satisfy D’Alembert’s principle has so far evaded Lagrangian dynamics. Yet, such forces of constraint are among those quite commonly found in nature. As stated by Goldstein ͓͑9͔,p.17͒, ‘‘This ͓total work done by forces of con- 1 Introduction straint equal to zero͔ is no longer true if sliding friction is present, One of the central problems in classical mechanics is the deter- and we must exclude such systems from our ͓Lagrangian͔ formu- mination of the equations of motion for constrained systems. The lation’’ ͓͑9͔͒. And Pars in his treatise ͓͑5͔͒ on analytical dynamics importance of the problem stems from the fact that what makes a ͑1979, p. 14͒ writes, ‘‘There are in fact systems for which the ͓ ͔ set of point masses and rigid bodies, a ‘‘system,’’ is the presence principle enunciated D’Alembert’s principle . does not hold. of constraints. When physical constraints are imposed on an un- But such systems will not be considered in this book.’’ constrained set of particles, forces of constraint are engendered In this paper we obtain the equations of motion for constrained which ensure the satisfaction of the constraints. The equations of systems where the forces of constraint indeed do not satisfy motion developed to date for such constrained systems are based D’Alembert’s principle, and the sum total of the work done by on a principle first enunciated by D’Alembert, and later elaborated them under virtual displacements no longer need be zero. by Lagrange ͓1͔ in his Mechanique Analytique which dates back The outline of the paper is as follows. In Section 2.1 we gen- to 1787. Today the principle is referred to as D’Alembert’s prin- eralize D’Alembert’s Principle to include constraint forces that do ciple, and it is the centerpiece of classical analytical dynamics. It work. This leads us to a deeper understanding of the specification states, simply, that the total work done by the forces of constraint of constraints in mechanical systems. This we discuss in Section under virtual displacements is always zero. Constraints for which 2.2. Section 3 deals with the mathematical statement of the prob- D’Alembert’s principle is applicable are referred to as ideal lem of constrained motion. Section 4 states and verifies the ex- constraints. plicit equation of motion for constrained systems with nonideal Since its initial formulation by Lagrange more than 200 years equality constraints. This equation leads to a new and fundamental principle of Lagrangian mechanics. The proof we give here is ago, the problem of constrained motion has been vigorously and ͓͑ ͔͒ continuously worked on by numerous scientists including Volt- simpler than the one given in 10 , and it yields an important erra, Boltzmann, Hamel, Whittaker, and Synge, to name a few. In geometrical interpretation that we discuss later. Section 5 gives an 1829, Gauss ͓2͔ provided a new general principle for the motion example of a nonholonomically constrained system for which the of constrained mechanical systems in what is today referred to as constraints are nonideal. We show here the ease of applicability of Gauss’s Principle. About 100 years after Lagrange, Gibbs ͓3͔ and the explicit equation of motion obtained in the previous section Appell ͓4͔ independently discovered what are known today as the and point out the insights it provides into understanding con- Gibbs-Appell equations of motion ͓͑3,4͔͒. Pars ͓͑5͔,p.202͒ refers strained motion where the constraint forces do work. Lastly, Sec- to the Gibbs-Appell equations as ͓͑5͔͒ ‘‘... probably the most tion 6 deals with the geometry of constrained motion and exhibits comprehensive equations of motion so far discovered.’’ Dirac, the simplicity and aesthetics with which Nature seems to operate. because of his interest in constrained systems that arise in quan- tum mechanics, in a series of papers from 1951 to 1969 developed an approach for determining the Lagrange multipliers for con- strained Hamiltonian systems ͓͑6͔͒. More recently, Udwadia and 2 Generalization of D’Alembert’s Principle, Con- Kalaba ͓7͔ presented a simple, explicit, set of equations, appli- straint Forces That Do Work, and Their Specification cable to general mechanical systems, with holonomic and non- holonomic constraints ͓͑7,8͔͒. 2.1 Generalized D’Alembert’s Principle. Consider an un- However, all these alternative descriptions of the motion of constrained system of particles, each particle having a constant constrained systems discovered so far, as well as the numerous mass. By ‘‘unconstrained’’ we mean that the number of general- articles that have subsequently dealt with them, rely on ized coordinates, n, used to describe the configuration of the sys- 1 tem at any time, t, equals the number of degrees-of-freedom of the ample, sliding frictional forces͒ are in fact commonplace, and system. The Lagrangian equation of motion for such a system can have to date defied ͓͑5,9͔͒ inclusion in a simple way within the be written in the form general framework of analytical mechanics. The main reason for M͑q,t͒q¨ ϭQ͑q,q˙ ,t͒, q͑0͒ϭq , q˙ ͑0͒ϭq˙ (1) this difficulty is that three obstacles need to be simultaneously 0 0 surmounted. Firstly, we require the specification of such con- where q(t) is the n-vector ͑i.e., n by 1 vector͒ of generalized straint forces to be general enough so that they encompass prob- coordinates, M is an n by n symmetric, positive-definite matrix, Q lems of practical utility. Secondly, this specification must, in order is the ‘‘known’’ n-vector of impressed forces, and the dots refer to to comply with physical observations, yield the accelerations of differentiation with respect to time. By ‘‘known,’’ we shall mean the constrained system uniquely when using the accepted math- that Q is a known function of its arguments. The acceleration, a, ware of analytical dynamics that has been developed over the last of the unconstrained system at any time t is then given by the 250 years. And lastly, when the constraint forces do no work, we Ϫ relation a(q,q˙ ,t)ϭM 1(q,t)Q(q,q˙ ,t). must obtain the usual formalisms/equations that have thus far We shall assume that this system is subjected to a set of m been obtained ͑e.g., by Lagrange, Gibbs, Appell, and Gauss͒, and ϭhϩs consistent equality constraints of the form are known to be of practical value. Clearly, the work done by such a constraint force under virtual ␸͑q,t͒ϭ0 (2) displacements v at each instant of time needs to be known, and and must therefore be specified using some known n-vector C(q,q˙ ,t), as vTC. Such an additional specification calls for a generalization ␺͑ ˙ ͒ϭ q,q,t 0, (3) of D’Alembert’s principle. We make this generalization in the where ␸ is an h-vector and ␺ an s-vector. Furthermore, we shall following manner: assume that the initial conditions q0 and q˙ 0 satisfy these con- For any virtual displacement v at time t, the constraint force straint equations at time tϭ0. Assuming that Eqs. ͑2͒ and ͑3͒ are n-vector Qc at time t does a prescribed amount of work given by sufficiently smooth, we differentiate Eq. ͑2͒ twice with respect to time, and Eq. ͑3͒ once with respect to time, to obtain the equation vTQc͑t͒ϭvTC͑q,q˙ ,t͒. (6) ͑ ˙ ͒ ¨ ϭ ͑ ˙ ͒ A q,q,t q b q,q,t , (4) Here C(q,q˙ ,t) is a known n-vector ͑i.e., a known function of q, q˙ , where the matrix A is m by n, and b is a suitably defined m-vector and t͒ that needs to be specified and depends on the physics of the that results from carrying out the differentiations.
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