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New General Principle of Mechanics and Its Application to General Nonideal Nonholonomic Systems

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New General Principle of Mechanics and Its Application to General Nonideal Nonholonomic Systems New General Principle of Mechanics and Its Application to General Nonideal Nonholonomic Systems Firdaus E. Udwadia1 Abstract: In this paper we develop a general minimum principle of analytical dynamics that is applicable to nonideal constraints. The new principle encompasses Gauss’s Principle of Least Constraint. We use this principle to obtain the general, explicit, equations of motion for holonomically and/or nonholonomically constrained systems with non-ideal constraints. Examples of a nonholonomically constrained system where the constraints are nonideal, and of a system with sliding friction, are presented. DOI: 10.1061/͑ASCE͒0733-9399͑2005͒131:4͑444͒ CE Database subject headings: Constraints; Equations of motion; Mechanical systems; Friction. Introduction ments. Such systems have, to date, been left outside the perview of the Lagrangian framework. As stated by Goldstein ͑1981, p. The motion of complex mechanical systems is often mathemati- 14͒ “This ͓total work done by forces of constraint equal to zero͔ cally modeled by what we call their equations of motion. Several is no longer true if sliding friction is present, and we must exclude formalisms ͓Lagrange’s equations ͑Lagrange 1787͒, Gibbs– such systems from our ͓Lagrangian͔ formulation.” And Pars Appell equations ͑Gibbs 1879, Appell 1899͒, generalized inverse ͑1979͒ in his treatise on analytical dynamics writes, “There are in equations ͑Udwadia and Kalaba 1992͔͒ have been developed for fact systems for which the principle enunciated ͓D’Alembert’s obtaining the equations of motion for such structural and me- principle͔… does not hold. But such systems will not be consid- chanical systems. Though these formalisms do not all afford the ered in this book.” Newtonian approaches are usually used to deal same ease of use in any given practical situation, they are equiva- with the problem of sliding friction ͑Goldstein 1981͒. For general lent to one another. They all rely on D’Alembert’s principle which systems with nonholonomic constraints, the inclusion into the states that, at each instant of time during the motion of the me- framework of Lagrangian dynamics of constraint forces that do chanical system, the sum total of the work done by the forces of work has remained to date an open problem in analytical dynam- constraint under virtual displacements is zero. Such forces of con- ics, because neither D’Alembert’s principle nor Gauss’s principle straint are often referred to as being ideal. D’Alembert’s principle is then applicable. is equivalent to a principle that was first stated by Gauss ͑Gauss In this paper we obtain a general principle of analytical dy- 1829͒ and is referred to nowadays as Gauss’s principle of least namics that encompasses nonideal constraints. It extends Gauss’ constraint. In fact, like D’Alembert’s principle, Gauss’s principle principle to situations where the forces of constraint do do work can be thought of as a starting point from which the machinery of under virtual displacements. It therefore brings nonideal con- analytical dynamics can be developed ͑see Udwadia and Kalaba straints within the scope of Lagrangian mechanics. The power of 1996͒. For example, it has been used in Udwadia and Kalaba the new principle is exhibited by the simple and straightforward ͑1992͒ and Kalaba and Udwadia ͑1993͒, in conjunction with the manner in which we obtain the general, explicit equations of concept of the Penrose inverse of a matrix, to obtain a simple and motion for holonomically and nonholonomically constrained general set of equations for holonomically and nonholonomically mechanical systems where the constraints may not be ideal. We constrained mechanical systems when the forces of constraint are provide two illustrative examples. The first deals with a generali- ideal. zation of a problem first proposed by Appell in which we obtain Though these two fundamental principles of mechanics are the explicit equations of motion for a nonholonomic mechanical often useful to adequately model mechanical system there are, system with nonideal constraints; the second deals with sliding however, numerous situations where they are not applicable since friction. the constraint forces actually do do work under virtual displace- The paper is organized as follows. In the next section we present a statement of the problem and establish our notation. 1Professor, Dept. of Aerospace and Mechanical Engineering, Civil This is followed by the section in which we derive our new gen- Engineering, Mathematics, and Operations and Information Management, eral principle of mechanics applicable to nonideal constraints. 430K Olin Hall, Univ. of Southern California, Los Angeles, CA 90089- The section titled, “General Equations of Motion for Holonomic 1453. E-mail: [email protected] and Nonholonomic Systems with Nonideal Constraints,” illus- Note. Associate Editor: Joel P. Conte. Discussion open until trates the power of this new principle to obtain the general, September 1, 2005. Separate discussions must be submitted for individual explicit equations of motion where the constraints may be non- papers. To extend the closing date by one month, a written request must ideal. In the “Illustrative Examples” section we present two be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November 10, examples showing the simplicity with which the general equation 2003; approved on September 28, 2004. This paper is part of the Journal of motion yields results for nonholonomic, nonideal constraints, of Engineering Mechanics, Vol. 131, No. 4, April 1, 2005. ©ASCE, and for problems with Coulomb friction. The last section gives ISSN 0733-9399/2005/4-444–450/$25.00. the conclusions. 444 / JOURNAL OF ENGINEERING MECHANICS © ASCE / APRIL 2005 Statement of Problem of Constrained Motion with where x¨c is the deviation of the acceleration of the constrained Nonideal Constraints system from what it would have been had there been no con- straints imposed on it at the instant of time t. Alternatively, upon Consider a mechanical system comprised of n particles, of mass premultiplication of Eq. ͑6͒ by M, we see that at the instant of mi, i=1,2,3,...,n. We shall consider an inertial Cartesian coor- time t dinate frame of reference and describe the position of the jth Mx¨ = Ma + Mx¨c = F + Fc ͑7͒ particle in this frame by its three coordinates xj, yj, and zj. Let the “impressed” forces on the jth mass in the X, Y, and Z directions and so a force of constraint Fc is brought into play that causes the be given F , F , and F , respectively. Then the equation of mo- xj yj zj deviation in the acceleration x¨ of the constrained system from tion the “unconstrained” mechanical system can be written as what it might have been ͑i.e., a͒ in the absence of the constraints. Mx¨ = F͑x,x˙,t͒ Thus the constrained mechanical system is described so far by the matrices M and A, and the vectors F and b. As recognized by ͑ ͒ ͑ ͒ ͑ ͒ Lagrange ͑1787͒, the determination from Eqs. ͑7͒ and ͑5͒ of the x 0 = x0, x˙ 0 = x˙0 1 acceleration 3n-vector, x¨, of the constrained system, and of the Here the matrix M isa3n by 3n diagonal matrix with the masses constraint force 3n-vector, Fc, constitutes an underdetermined of the particles in sets of three along the diagonal; x problem and cannot, in general, be solved; for, there are 6n un- ͓ ͔T = x1 ,y1 ,z1 ,...,xn ,yn ,zn ; and the dots refer to differentiation knowns and 3n+m relations. Additional information related to the with respect to time. Similarly, the 3n-vector ͑3n by 1 column nature of the force of constraint Fc is required, and this informa- vector͒ F=͓F ,F ,F ,...,F ,F ,F ͔T. By “unconstrained” x1 y1 z1 xn yn zn tion is situation specific. Thus, to obtain an equation of motion for we mean that the components of the 3n-vector of velocity, x˙͑0͒, a given mechanical system under consideration, additional can be independently prescribed. We note that the acceleration, information—beyond that contained in the four quantities M, A, a͑t͒, of the unconstrained system is then simply given by F, and b—needs to be provided by the mechanician who is mod- eling the motion of that specific system. a͑t͒ = M−1F͑x,x˙,t͒͑2͒ Let us assume that we have this additional information ͑for The impressed force, F͑x,x˙ ,t͒, which has 3n components is a some specific mechanical system͒ regarding the constraint force known vector, i.e., it is a known function of its arguments. The 3n-vector Fc at each instant of time t in the form of the work done matrix M has positive entries along its diagonal; it is therefore by this force under virtual displacements of the mechanical sys- symmetric and positive definite, as is the matrix M−1. tem at time t. A virtual displacement of the system at time t is Let this system now be subjected to a further set of m=h+s defined as any nonzero 3n-vector, v͑t͒, that satisfies the relation constraints of the form ͑see Udwadia et al. 1997͒ ␸͑ ͒ ͑ ͒ x,t =0 3 A͑x,x˙,t͒v͑t͒ =0 ͑8͒ and The mechanician modeling the motion of the given system then c͑ ͒ c ␺͑x,x˙,t͒ =0 ͑4͒ provides the work done, W t , under virtual displacements by F through knowledge of a vector C at each instant of time t, so that where ␸ is an h-vector and ␺ is an s-vector. We shall assume that T c ϵ c͑ ͒ T ͑ ͑ ͒ ͑ ͒ ͒͑͒ the initial conditions x0 and x˙0 satisfy these constraint equations at v F W t = v C x t ,x˙ t ,t 9 time t=0, i.e., ␸͑x ,0͒=0, ␸˙ ͑x ,0͒=0, and ␺͑x ,x˙ ,0͒=0.
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