Nonideal Constraints and Lagrangian Dynamics

This paper deals with mechanical systems subjected to a general class of non-ideal equality constraints. It provides the explicit equations of motion for such systems when subjected to such nonideal, holonomic and/or nonholonomic, constraints. It bases Lagrangian dynamics on a new and more general principle, of which D’Alembert’s principle then becomes a special case applicable only when the constraints become ideal. By expanding its perview, it allows Lagrangian dynamics to be directly applicable to many situations of practical importance where non-ideal constraints arise, such as when there is sliding Coulomb friction. INTRODUCTION One of the central problems in the field of mechanics is the determination of the equations of motion pertinent to constrained systems. The problem dates at least as far back as Lagrange (1787), who devised the method of Lagrange multipliers specifically to handle constrained motion. Realizing that this approach is suitable to problem-specific situations, the basic problem of constrained motion has since been worked on intensively by numerous scientists, including Volterra, Boltzmann, Hamel, Novozhilov, Whittaker, and Synge, to name a few. About 100 years after Lagrange, Gibbs (1879) and Appell (1899) independently devised what is today known as the Gibbs-Appell method for obtaining the equations of motion for constrained mechanical systems with nonintegrable equality constraints. The method relies on a felicitous choice of quasicoordinates and, like the Lagrange multiplier method, is amenable to problem-specific situations. The Gibbs-Appell approach relies on choosing certain quasicoordinates and eliminating others, thereby falling under the general category of elimination methods (Udwadia and Kalaba 1996). The central idea behind these elimination methods was again first developed by Lagrange when he introduced the concept of generalized coordinates. Yet, despite their discovery more than a century ago, the Gibbs-Appell equations were considered by many, up until very recently, to be at the pinnacle of our understanding of constrained motion; they have been referred to by Pars (1979) in his opus on analytical dynamics as ‘‘probably the simplest and most comprehensive equations of motion so far discovered.’’ Dirac considered Hamiltonian systems with constraints that were not explicitly dependent on time; he once more attacked the problem of determining the Lagrange multipliers of the Hamiltonian corresponding to the constrained dynamical system. By ingeniously extending the concept of Poisson brackets, he developed a method for determining these multipliers in a systematic manner through the repeated use of the consistency conditions (Dirac 1964; Sudarshan and Mukunda 1974). More recently, an explicit equation describing constrained motion of both conservative and nonconservative dynamical systems within the confines of classical mechanics was developed by Udwadia and Kalaba (1992). They used as their starting point Gauss’s principle (1829) and considered general bilateral constraints that could be both nonlinear in the generalized velocities and displacements and explicitly depenProf. of Mech. Engrg., Civ. Engrg., Math., and Decision Sys., 430K Olin Hall, Univ. of Southern California, Los Angeles, CA 90089-1453. Prof. of Biomedical Engrg., Electr. Engrg., and Economics, Univ. of Southern California, Los Angeles, CA. Note. Discussion open until June 1, 2000. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 9, 1999. This paper is part of the Journal of Aerospace Engineering, Vol. 13, No. 1, January, 2000. qASCE, ISSN 0893-1321/00/0001-0017–0022/$8.00 1 $.50 per page. Paper No. 21540. dent on time. Furthermore, their result does not require the constraints to be functionally independent. All the above-mentioned methods for obtaining the equations of motion for constrained systems deal with ideal constraints, wherein the constraint forces do no work under virtual displacements. The motion of an unconstrained system is, in general, altered by the imposition of constraints; this alteration in the motion of the unconstrained system can be viewed at as being caused by the creation of additional ‘‘forces of constraint’’ brought into play through the imposition of these constraints. One view of the main task of analytical dynamics is that it gives a prescription for (uniquely) determining the accelerations of particles at any instant of time, given their masses, positions, and velocities; the kinematic constraints they need to satisfy; and the ‘‘given’’ (impressed) forces acting on them, at that instant. The properties of the constraint forces that are generated depend on the physical situation; these properties need to be provided in order to determine the particle accelerations. Usually, they come from experiments. The principle of D’Alembert, which was first stated in its generality by Lagrange (1787), assumes that the constraints are such that the forces of constraint do no work under virtual displacements. Such constraints are often referred to as ideal constraints and seem to work well in many practical situations. As pointed out by Lagrange, they provide a significant simplification, which enables a relatively easy description of the accelerations of the constrained system. This simplification arises because under this assumption only the ‘‘given forces’’ do work under virtual displacements; the total work done by the constraint forces is zero, and hence no forces of constraint appear in the relation dealing with the total work done on the system under virtual displacements. Additionally, from an algebraic standpoint, the assumption of ideal constraints happens to provide just the right amount of information for the accelerations of the constrained system to be uniquely determined (Udwadia and Kalaba 1996)—that is, the problem of finding the particle accelerations in the presence of ideal constraints is neither overdetermined nor underdetermined. However, the assumption of ideal constraints excludes situations that often arise in practice. Indeed, such occurrences are commonplace in physics and engineering. Typically, the inclusion of nonideal constraint forces that do work under virtual displacements causes considerable difficulties in Lagrangian formulations; consequently, Lagrangian formulations of analytical dynamics exclude these sorts of constraints. Exactly how general, nonideal, equality constraints might be included within the framework of Lagrangian mechanics thus remains an open question today. For example, the empirical sliding friction law suggested by Coulomb has been found to be useful in modeling many mechanical systems; such forces of sliding friction constitute constraint forces that indeed do work under virtual displacements. The special case of Coulomb friction can be handled in the Lagrangian framework, though in a roundabout way that resembles more Newtonian mechanics than Lagrangian mechanJOURNAL OF AEROSPACE ENGINEERING / JANUARY 2000 / 17 ics (Rosenberg 1972). It requires a reformation of the Lagrangian approach by positing that the ‘‘given’’ forces are known functions of the constraint forces. Even after all this, as stated by Rosenberg (1972), ‘‘Lagrangian mechanics is not a convenient vehicle for dealing with [friction forces].’’ Goldstein (1972), in his treatment of Lagrangian dynamics, asserts that ‘‘this [total work done by constraint forces equal to zero] is no longer true if sliding friction forces are present, and we must exclude such systems from our [Lagrangian] formulation.’’ Moreover, as mentioned before, it leaves open the question of how one might handle within the Lagrangian framework more general forces of constraint that indeed do work under virtual displacements. In the 200-year history of analytical dynamics, this problem has resisted a direct assault so far as we know, because, unlike the ideal constraints situation, now the constraint forces must appear in the relations dealing with virtual work. More importantly, a major stumbling block has been the question of what sort of nonideal constraints yield a unique set of accelerations for a given unconstrained system. In this paper, the writers obtain the explicit equations of motion for general, conservative and nonconservative, dynamical systems under the influence of a general class of nonideal bilateral constraints. We show that such nonideal constraints can be brought with simplicity and ease within the general framework of Lagrangian mechanics and prove that, like ideal constraints, they uniquely determine the accelerations of the constrained system of particles. By deriving the explicit equations of motion for nonideal equality constraints, we expand the perview of Lagrangian mechanics to include a much wider variety of situations that often arise in practice, including sliding Coulomb friction. Instead of D’Alembert’s principle, Lagrangian mechanics now becomes rooted in a new principle, of which D’Alembert’s principle becomes a special case. Three simple examples dealing with sliding friction and frictional drag are provided to illustrate the main result.