NONIDEAL CONSTRAINTS AND LAGRANGIAN DYNAMICS

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1 NONIDEAL CONSTRAINTS AND LAGRANGIAN DYNAMICS By Firdaus E. Udwadia 1 and Robert E. Kalaba ABSTRACT: This paper deals with mehanial systems subjeted to a general lass of non-ideal equality onstraints. It provides the expliit equations of motion for suh systems when subjeted to suh nonideal, holonomi and/or nonholonomi, onstraints. It bases Lagrangian dynamis on a new and more general priniple, of whih D Alembert s priniple then beomes a speial ase appliable only when the onstraints beome ideal. By expanding its perview, it allows Lagrangian dynamis to be diretly appliable to many situations of pratial importane where non-ideal onstraints arise, suh as when there is sliding Coulomb frition. INTRODUCTION One of the entral problems in the field of mehanis is the determination of the equations of motion pertinent to onstrained systems. The problem dates at least as far bak as Lagrange (1787), who devised the method of Lagrange multipliers speifially to handle onstrained motion. Realizing that this approah is suitable to problem-speifi situations, the basi problem of onstrained motion has sine been worked on intensively by numerous sientists, inluding 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 onstrained mehanial systems with nonintegrable equality onstraints. The method relies on a feliitous hoie of quasioordinates and, like the Lagrange multiplier method, is amenable to problem-speifi situations. The Gibbs-Appell approah relies on hoosing ertain quasioordinates and eliminating others, thereby falling under the general ategory of elimination methods (Udwadia and Kalaba 1996). The entral idea behind these elimination methods was again first developed by Lagrange when he introdued the onept of generalized oordinates. Yet, despite their disovery more than a entury ago, the Gibbs-Appell equations were onsidered by many, up until very reently, to be at the pinnale of our understanding of onstrained motion; they have been referred to by Pars (1979) in his opus on analytial dynamis as probably the simplest and most omprehensive equations of motion so far disovered. Dira onsidered Hamiltonian systems with onstraints that were not expliitly dependent on time; he one more attaked the problem of determining the Lagrange multipliers of the Hamiltonian orresponding to the onstrained dynamial system. By ingeniously extending the onept of Poisson brakets, he developed a method for determining these multipliers in a systemati manner through the repeated use of the onsisteny onditions (Dira 1964; Sudarshan and Mukunda 1974). More reently, an expliit equation desribing onstrained motion of both onservative and nononservative dynamial systems within the onfines of lassial mehanis was developed by Udwadia and Kalaba (199). They used as their starting point Gauss s priniple (189) and onsidered general bilateral onstraints that ould be both nonlinear in the generalized veloities and displaements and expliitly depen- 1 Prof. of Meh. Engrg., Civ. Engrg., Math., and Deision Sys., 430K Olin Hall, Univ. of Southern California, Los Angeles, CA Prof. of Biomedial Engrg., Eletr. Engrg., and Eonomis, Univ. of Southern California, Los Angeles, CA. Note. Disussion open until June 1, 000. To extend the losing date one month, a written request must be filed with the ASCE Manager of Journals. The manusript for this paper was submitted for review and possible publiation on August 9, This paper is part of the Journal of Aerospae Engineering, Vol. 13, No. 1, January, 000. ASCE, ISSN /00/ /$8.00 $.50 per page. Paper No dent on time. Furthermore, their result does not require the onstraints to be funtionally independent. All the above-mentioned methods for obtaining the equations of motion for onstrained systems deal with ideal onstraints, wherein the onstraint fores do no work under virtual displaements. The motion of an unonstrained system is, in general, altered by the imposition of onstraints; this alteration in the motion of the unonstrained system an be viewed at as being aused by the reation of additional fores of onstraint brought into play through the imposition of these onstraints. One view of the main task of analytial dynamis is that it gives a presription for (uniquely) determining the aelerations of partiles at any instant of time, given their masses, positions, and veloities; the kinemati onstraints they need to satisfy; and the given (impressed) fores ating on them, at that instant. The properties of the onstraint fores that are generated depend on the physial situation; these properties need to be provided in order to determine the partile aelerations. Usually, they ome from experiments. The priniple of D Alembert, whih was first stated in its generality by Lagrange (1787), assumes that the onstraints are suh that the fores of onstraint do no work under virtual displaements. Suh onstraints are often referred to as ideal onstraints and seem to work well in many pratial situations. As pointed out by Lagrange, they provide a signifiant simplifiation, whih enables a relatively easy desription of the aelerations of the onstrained system. This simplifiation arises beause under this assumption only the given fores do work under virtual displaements; the total work done by the onstraint fores is zero, and hene no fores of onstraint appear in the relation dealing with the total work done on the system under virtual displaements. Additionally, from an algebrai standpoint, the assumption of ideal onstraints happens to provide just the right amount of information for the aelerations of the onstrained system to be uniquely determined (Udwadia and Kalaba 1996) that is, the problem of finding the partile aelerations in the presene of ideal onstraints is neither overdetermined nor underdetermined. However, the assumption of ideal onstraints exludes situations that often arise in pratie. Indeed, suh ourrenes are ommonplae in physis and engineering. Typially, the inlusion of nonideal onstraint fores that do work under virtual displaements auses onsiderable diffiulties in Lagrangian formulations; onsequently, Lagrangian formulations of analytial dynamis exlude these sorts of onstraints. Exatly how general, nonideal, equality onstraints might be inluded within the framework of Lagrangian mehanis thus remains an open question today. For example, the empirial sliding frition law suggested by Coulomb has been found to be useful in modeling many mehanial systems; suh fores of sliding frition onstitute onstraint fores that indeed do work under virtual displaements. The speial ase of Coulomb frition an be handled in the Lagrangian framework, though in a roundabout way that resembles more Newtonian mehanis than Lagrangian mehan- JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000 / 17

2 is (Rosenberg 197). It requires a reformation of the Lagrangian approah by positing that the given fores are known funtions of the onstraint fores. Even after all this, as stated by Rosenberg (197), Lagrangian mehanis is not a onvenient vehile for dealing with [frition fores]. Goldstein (197), in his treatment of Lagrangian dynamis, asserts that this [total work done by onstraint fores equal to zero] is no longer true if sliding frition fores are present, and we must exlude suh 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 fores of onstraint that indeed do work under virtual displaements. In the 00-year history of analytial dynamis, this problem has resisted a diret assault so far as we know, beause, unlike the ideal onstraints situation, now the onstraint fores must appear in the relations dealing with virtual work. More importantly, a major stumbling blok has been the question of what sort of nonideal onstraints yield a unique set of aelerations for a given unonstrained system. In this paper, the writers obtain the expliit equations of motion for general, onservative and nononservative, dynamial systems under the influene of a general lass of nonideal bilateral onstraints. We show that suh nonideal onstraints an be brought with simpliity and ease within the general framework of Lagrangian mehanis and prove that, like ideal onstraints, they uniquely determine the aelerations of the onstrained system of partiles. By deriving the expliit equations of motion for nonideal equality onstraints, we expand the perview of Lagrangian mehanis to inlude a muh wider variety of situations that often arise in pratie, inluding sliding Coulomb frition. Instead of D Alembert s priniple, Lagrangian mehanis now beomes rooted in a new priniple, of whih D Alembert s priniple beomes a speial ase. Three simple examples dealing with sliding frition and fritional drag are provided to illustrate the main result. EQUATIONS OF MOTION Consider first an unonstrained system of partiles, eah partile having a onstant mass. By unonstrained, we mean that the number of generalized oordinates, n, needed to desribe the onfiguration of the system at any time, t, equals the number of degrees of freedom of the system. The equation of motion for suh a system an be written in the form M(q, t)q = Q(q, q, t); q(0) = q, q (0) = q (1) 0 0 where q(t) is the n-vetor (i.e., n by 1 vetor) of generalized oordinates; M is an n by n symmetri, positive-definite matrix; Q is the known n-vetor of impressed fores; and the dots refer to differentiation with respet to time. The aeleration, a, of the unonstrained system at any time t is then given by the relation a(q, q, t) =M 1 Q. We shall assume that both a(t) and the displaement q(t) of the unonstrained system desribed by (1) are loally unique. We shall assume that this system is subjeted to a set of m = h s onsistent onstraints of the form and (q, t) =0 () (q, q, t) =0 (3) where is an h-vetor and is an s-vetor. Furthermore, we shall assume that the initial onditions q 0 and q 0 satisfy these onstraint equations at time t = 0. Assuming that () and (3) are suffiiently smooth, we differentiate () twie with respet to time, and (3) one with respet to time, to obtain the equation A(q, q, t)q = b(q, q, t) (4) where the matrix A is m by n; and b is a suitably defined m- vetor that results from arrying out the differentiations. This set of onstraint equations inludes, among others, the usual holonomi, nonholonomi, sleronomi, rheonomi, atastati, and aatastati varieties of onstraints; ombinations of suh onstraints may also be permitted in (4). In the presene of the onstraints, the number of degrees of freedom of the system is less than n. We shall resist the temptation to eliminate the redundant oordinates (and/or quasi-oordinates), a strategy that has ustomarily been used for the last 00 years. Instead, the underlying theme of our approah will be to determine an expliit equation for the aeleration vetor q (t) of the onstrained system at time t, given the vetors of generalized displaement, q(t); generalized veloity, q (t); the given fore, Q(q(t), q (t), t); and the nature of the onstraints desribed by (4) at time t. Consider any instant of time t. When onstraints are imposed at that instant of time on the unonstrained system, the motion of the unonstrained system is, in general, altered from what it might have been (at that instant of time) in the absene of these onstraints. We view this alteration in the motion of the unonstrained system as being aused by an additional set of fores, alled the fores of onstraint, ating on the system at that instant of time. Sine we shall be dealing with nonideal onstraints, we shall refrain from defining this additional set of fores alled the fores of onstraint as the set of fores that do no work under virtual displaements, as has been the ommon pratie in analytial dynamis (see, for example, Rosenberg 197). The equation of motion of the onstrained system an then be expressed as M(q, t)q = Q(q, q, t) Q (q, q, t), q(0) = q 0, q (0) = q 0 (5) where the additional onstraint fore Q (q, q, t) arises by virtue of the onstraints () and (3) imposed on the unonstrained system, whih is desribed by (1). Our aim is to determine Q expliitly at time t in terms of the known quantities M, Q, A, and b and information about the nonideal nature of the onstraint fore, at time t. Starting from the extended D Alembert s priniple (Udwadia et al. 1997), we shall obtain the onstraint fore Q expliitly, using ideal bilateral onstraints; then, on the basis of a new priniple, an expliit equation for a lass of nonideal bilateral onstraints will be obtained. In what follows, we will usually omit for the sake of brevity the expliit arguments of the various matries and vetors. A generalized virtual displaement at time instant t is defined as any nonzero infinitesimal n-vetor, v, whih satisfies the relation (Udwadia and Kalaba 1996) Av = 0 (6) Defining u = M 1/ v, (6) is equivalent to the equation Bu = 0 (7) where the m by n matrix B = AM 1/. Denoting r = M 1/ q, (4) an be expressed as The general solution of (8) is Br = b (8) r = Bb (I BB)w (9) where the n by m matrix B is the Moore-Penrose (1955) inverse of the matrix B; and w is any arbitrary n-vetor. The first term on the right-hand side of (8) is known beause both the vetor b and the matrix B are known at time t; it therefore remains to determine the vetor (I B B)w, based on the priniples of mehanis. 18 / JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000

3 Ideal Constraints By ideal onstraints we mean that the onstrained system evolves in suh a way that at eah instant of time, the total work done by the onstraint fores under virtual displaements is zero. This is an alternative statement of D Alembert s priniple; it forms the foundation of Lagrangian dynamis as we know it today. Using (4) and (5), this requires that for all nonzero vetors v suh that Av = 0 (Udwadia et al. 1997): T T vq = v [Mq Q] = 0 (10) Expressing (10) in terms of the previously defined vetors u and r, and noting the equivalene between relations (6) and (7), we find that the aeleration r must be suh that for all nonzero vetors u that satisfy the relation Bu = 0, we must have T 1/ T 1/ 1/ T 1/ um Q = um [M r Q] =u [r M Q] = 0 (11) Furthermore, the aeleration r must satisfy the onstraints and must therefore satisfy (8); hene, it must be of the form given by (9). Thus, from (11) it follows that for all nonzero vetors u that satisfy the relation Bu = 0, we require T 1/ u [B b (I BB)wM Q] = 0 (1) However, Bu = 0 implies u B = 0, whih in turn implies u T B = 0, sine u 0. Requirement (1) an then be rephrased as requiring that the vetor w be suh that T 1/ u [w M Q]=0, T T nonzero vetors u that satisfy the relation ub = 0 (13) This implies that w must be given by the relation 1/ T w = M Q Bz (14) where z is an arbitrary m-vetor. Replaing w from (14) into (9) now yields 1/ T r = Bb (I BB)(M Q Bz) (15) But the matrix (I B B) is symmetri; hene T T T T [(I BB)B z] = zb(ibb)=z (B BB B) T = z (B B) = 0 (16) so that (15) redues to 1/ 1/ r = M Q B (b BM Q) (17) Noting that r = M 1/ q and B = AM 1/, (17) yields the equation of motion for the onstrained system as 1/ 1/ Mq = Q M (AM )(baa) (18) where a is the aeleration of the unonstrained system and is defined as a = M 1 Q. We have thus obtained expliitly the onstraint fore at time t when the onstraints are ideal as 1/ 1/ Q i (q, q, t) =M (q, t){a(q, q, t)m (q, t)} {b(q, q, t) A(q, q, t)a(q, q, t)} (19) The subsript i on Q on the left-hand side is used to expliitly indiate that the onstraint fore given by (19) is obtained under the assumption of ideal onstraints. Eq. (19) was first derived by starting from Gauss s priniple of least onstraint (see Udwadia and Kalaba 199). Nonideal Constraints We shall onsider here a lass of nonideal onstraints. For suh onstraints we need to postulate a new priniple that redues to D Alembert s priniple when the onstraints beome ideal. We now state this priniple as follows: The total work done at time t by the onstraint fores Q under virtual displaements at time t is given by T T vq = vc(q, q, t) (0) where C(q, q, t) is a known, presribed, suffiiently smooth n-vetor (it needs only to be 1 ), and v is the virtual displaement n-vetor at time t. When C 0, this priniple redues to D Alembert s priniple, and all the onstraints are ideal. We thus require that for all nonzero vetors v suh that Av =0: T T T vq = v [Mq Q] =vc(q, q, t) (1) As before, this redues to the requirement that for all nonzero vetors u = M 1/ v suh that Bu =0 T 1/ T 1/ u [r M Q]=u [B b (I B B)w M Q] T 1/ = um C(q, q, t) () Again noting that sine u 0, Bu = 0, implies u T B =0,we obtain the requirement that T 1/ 1/ u [w M Q M C(q, q, t)] = 0 (3) for all nonzero vetors u that satisfy the relation u T B T =0. Hene, w is of the form 1/ 1/ T w = M Q M C(q, q, t) Bz (4) Substitution of this w in (9) then gives, along the same lines as before: 1/ 1/ 1/ r = M Q B (b BM Q) (I B B)M C (5) from whih the equation of onstrained motion is obtained as 1/ 1/ Mq = Q M (AM )(baa) 1/ 1/ 1/ 1/ M {I (AM )(AM )}M C (6) Eq. (6) thus provides the expliit equation of motion for a dynamial system subjeted to bilateral, holonomi and/or nonholonomi equality onstraints that are nonideal. When C 0, all the onstraints beome ideal and the third member on the right-hand side of (6) disappears, yielding (18). We have also obtained expliitly the onstraint fore at time t when one or more of the onstraints are nonideal. Noting (19), the total onstraint fore an be expressed as Q = Qi Q ni (7) where the n-vetor Q ni is the ontribution to the total onstraint fore from the presene of nonideal onstraints; it is given by 1/ 1/ 1/ 1/ Q ni = M {I (AM )(AM )}M C 1/ 1/ 1 = C M (AM ) AM C (8) We note that for all nonideal onstraints of the form given by (0), we obtain the partile aelerations uniquely. Eq. (7) shows that the total onstraint fore Q an be deomposed into the sum of two onstraint fore n-vetors, Qi and Q ni. The first of these is the onstraint fore that would have existed had all the onstraints been ideal; the seond may be thought of as a orretion term to aount for the presene of nonideal onstraints. In the presene of onstraints, the first term Q i in (7) is, in general, ever-present; the seond appears only when one or more of the onstraints is nonideal. Example 1 Consider a bead of mass m moving under gravity on a straight wire that is inlined to the horizontal at an angle, 0 < <(/). The unonstrained motion of the bead (in the JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000 / 19

4 absene of the onstraint imposed on its motion by the wire) is given by m 0 ẍ 0 = (9) 0 m ÿ mg where the x-diretion is taken along the horizontal and the y- diretion is taken pointing upwards. The vetor on the righthand side of (9) represents the given fore Q. The wire onstraint an be desribed by the equation y = x tan (30) whih, upon two differentiations with respet to time, yields Sine this an be written as ÿ = ẍ tan (31) ẍ [tan 1] = 0 (3) ÿ the matrix A =[tan 1], and 1/ 1/ tan (AM ) = m os 1 and the salar b =0. Were the onstraint represented by (30) assumed to be ideal, the equation of motion for the system, by (18), would be ẍ 0 sin os m = mg (33) ÿ mg os The seond member on the right-hand side indiates expliitly the onstraint fore Q i generated by the ideal onstraint rep- resented by (30). The magnitude of this onstraint fore is mg os. Were we to inlude Coulomb frition along the inlined wire with a oeffiient of frition, the onstraint will no longer be ideal; the work done by the onstraint fore under any virtual displaement v an then be represented as T T T ẋ vq = vcv mg os (34) ẋ ẏ ẏ Relation (34) states that the fritional fore ats along the onstraint, in a diretion opposing the motion, and has a magnitude of Qi. We note that by virtue of (30), ẏ = ẋ tan, so that the vetor os C = mg os sgn(ẋ) (35) sin The nonideal onstraint given by (34) now provides an additional onstraint fore given by 1/ 1/ 1/ 1/ Q ni = M {I (AM )(AM )}M C sin sin os = I C sin os os mg os = sgn(ẋ) mg os sin (36) so that the onstrained equation of motion is given by os mg sgn(ẋ) ẍ 0 sin os m = mg ÿ mg os os sin (37) where we have expliitly shown on the right-hand side the three different onstituents of the fores ating on the system. The first term orresponds to the given fores Q; the seond orresponds to the fore generated by the presene of the Q i onstraint given by (30), were it an ideal onstraint; and the third orresponds to the additional fore Q ni generated by the presene of the nonideal onstraint given by (30), whose nature is further desribed by (34). Equations of motion using more general desriptions of the fritional fores an be obtained in a similar manner. It should be pointed out that Q ni and the equation of motion for the onstrained system ould have been diretly written down without the simplifiations presented in (35) by using (8) and (6), wherein the vetor C is given by ẋ C = mg os ẋ ẏ ẏ Example Consider a partile of unit mass onstrained to move in a irle in the vertial plane on a irular ring of radius R under the ation of gravity. The unonstrained motion of the partile is given by ẍ 0 = (38) ÿ g and the onstraint is represented by x y = R, whih, upon two differentiations with respet to time, beomes so that A =[x ẍ [x y] = (ẋ ẏ ) (39) ÿ y], and 1 x A = R y Were this onstraint to be ideal, the fore of onstraint would be given by (18) so that (ẋ ẏ gy) Q i x/r Q i = (40) y/r R and the equation of motion of the onstrained system beomes (ẋ ẏ gy) ẍ 0 x/r = (41) ÿ g y/r R The magnitude of this onstraint fore, had the onstraint been ideal, is given by Qi = (ẋ ẏ gy) Let the nature of the nonideal onstraint generated by sliding frition between the ring and the mass be desribed by R Qi T T T ẋ vq = vcv (4) ẏ (ẋ ẏ ) Along the irular trajetory of the partile, xẋ = yẏ, and we get Qi ẋ Qi y sgn(x) C = = sgn(ẏ) (43) ẋ ẏ ẏ R x The ontribution to the onstraint fore provided by this nonideal onstraint is then given, by using (8) (note: M = I )as y sgn(x) y sgn(x)/r i Qi 1 x xy ni Q ={I AA}C = I R R xy y sgn(ẏ) =Q sgn(ẏ) x x/r (44) The equation of motion for the partile then beomes 0 / JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000

5 ẍ 0 x/r (ẋ ẏ gy) = ÿ g y/r R y sgn(x)/r Qi sgn(ẏ) x sgn(x)/r (45) As before, we ould have diretly used the relation for C [given by the first equality in (43)] in (6) to obtain the equation of motion of the nonideally onstrained system. Example 3 We onsider next a partile of unit mass, moving with respet to an inertial frame of referene, subjeted to the given fores f x (x, y, z, t), f y (x, y, z, t), and f z (x, y, z, t) ating on it in the x-, y-, and z-diretions, respetively. The partile is subjeted to the nonholonomi, nonideal, onstraint ẏ = zẋ (46) where the work done by the onstraint fore Q in a virtual displaement v is given by T T v vq = v a0v (47) v Here, v is the veloity of the partile as it exeutes its onstrained motion; and a 0 is a given onstant. Note that the nonideal onstraint fore Q is brought into play only beause the partile is required to satisfy the nonholonomi, kinematial onstraint given by (46). We shall assume that the initial position and veloity of the partile is provided and that it satisfies the onstraint equation (46). Our aim is to determine the equation of motion for the partile in the presene of the nonideal, nonholonomi onstraint desribed by (46) and (47). The unonstrained equation of motion of the partile is then given by (M = I) ẍ f x ÿ = f y = Q (48) z Differentiating (46), (4) yields f z A =[z 1 0], b = ẋż (49) and (19) gives the ontribution to the total onstraint fore, were the nonholonomi onstraint to be ideal, as (ẋż zfx f y) T Q i =[z 1 0] (50) (1 z ) Additionally, the ontribution to the total onstraint fore provided by virtue of the nonholonomi onstraint being nonideal is given by (8) as T Q = a [ẋ zẏ zẋ zẏ ż(1 z )] 1/ (ẋ ẏ ż ) (1 z ) (51) ni 0 Finally, the equation that desribes the motion of the partile with the nonholonomi, nonideal onstraint [as desribed by (46) and (47)] is then simply given by (1 ẍ fx (ẋż zf f ) z x y ÿ = Q Qi Q ni = fy 1 z f (1 z ) 0 a z ẋ zẏ 1/ (ẋ ẏ ż ) zẋ zẏ ż(1 z ) z ) (5) 0 For this nonideal, nonholonomi system, we thus obtain an easy deomposition of the right-hand side in terms of the given fore (vetor), Q; the fore of onstraint that would have been generated were the nonholonomi onstraint ideal, Q i ; and the addition fore, Q ni, engendered beause the onstraint is not ideal. Suh a deomposition of the aeleration of the partile often assists in understanding the physis of the problem. CONCLUSIONS This paper deals with determining the expliit equation of motion for a onstrained dynami system where the fores of onstraint satisfy a more general priniple than that first enuniated by D Alembert and formalized by Lagrange (1787). We state this priniple as follows. Consider an unonstrained system with n degrees of freedom. Let the system be subjeted to general holonomi and nonholonomi onstraints. At eah instant of time t, the virtual work, v T Q, done by the fore-of-onstraint n-vetor, Q, under any virtual displaement n-vetor, v, is given by v T C(q, q, t), 1 where C(q, q, t) is a suffiiently smooth (at least ) n-vetor funtion of its arguments. The following points may be noted relevant to the equation of motion obtained, and to the above-mentioned priniple: 1. The priniple generalizes D Alembert s priniple for bilateral onstraints to situations when the onstraint fores do work under virtual displaements. It enompasses situations suh as sliding Coulomb frition, and many other types of onstraint fores.. When the funtion C(q, q, t) is identially zero, the priniple stated above redues to D Alembert s priniple, and the equation of motion of the onstrained system reverts bak to the equation known earlier (Udwadia and Kalaba 1996). 3. A deeper understanding of the underlying physis is obtained. The total onstraint fore Q [see (6) (8)] is seen to be made up of two additive ontributions. The first ontribution, Q i, to the total onstraint fore omes from onsideration of the onstraints as though they were ideal; the seond term omes from the fat that one or more of the onstraints are not ideal and C(q, q, t) isnot identially zero. 4. We have obtained a simple, expliit equation of motion for a general onservative or nononservative system subjeted to holonomi and/or nonholonomi bilateral onstraints that may be nonideal. 5. For any given suffiiently smooth n-vetor C(q, q, t), the aeleration vetor of the onstrained system is uniquely determined and the trajetory is loally unique. 6. No elimination of oordinates or quasi-oordinates (as required by the Gibbs-Appell approah) is undertaken. The equations of motion pertinent to the onstrained system with nonideal holonomi and/or nonholonomi bilateral onstraints are obtained in the same set of oordinates whih are used to desribe the unonstrained system, thereby showing simply and expliitly the effet of the addition of onstraints on the equations of motion of the unonstrained system. Sine its ineption, Lagrangian mehanis has been built upon the underlying priniple of D Alembert. This priniple makes the onfining assumption that all onstraints are ideal onstraints for whih the sum total of the work done by the fores of onstraint under virtual displaements is zero. Though often appliable, experiments show that this assumption may be invalid in many pratial situations, suh as when sliding frition is important. This paper releases Lagrangian mehanis from this onfinement and obtains the expliit equations of motion allowing for holonomi and/or nonholonomi bilateral onstraints that are nonideal. The expliit equa- JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000 / 1

6 tions of motion obtained here are aordingly based on a more general priniple, whih then inludes D Alembert s priniple as a speial ase when the onstraints are ideal. APPENDIX. REFERENCES Appell, P. (1899). Sur une forme generale des equations de la dynamique. C. R. Aad. Si., Paris, 19, (in Frenh). Dira, P. A. M. (1964). Letures in quantum mehanis. Yashiva University, New York. Gauss, C. (189). Uber ein neues allgemeines grundgesetz der mehanik. J. Reine Angewandte Mathematik, 4, 3 35 (in German). Gibbs, W. (1879). On the fundamental formulae of dynamis. Am. J. Math.,, Goldstein, H. (1981). Classial mehanis. Addison-Wesley, Reading, Mass. Lagrange, J. L. (1787). Meanique analytique. Mme Ve Courier, Paris (in Frenh). Pars, L. A. (1979). A treatise on analytial dynamis. Oxbow Press, Woodbridge, Conn. Penrose, R. (1955). A generalized inverse of matries. Pro., Cambridge Phil. So., Cambridge, U.K., 51, Rosenberg, R. (197). Analytial dynamis of disrete systems. Plenum Press, New York. Sudarshan, E. C. G., and Mukunda, N. (1974). Classial dynamis: a modern perspetive. Wiley, New York. Udwadia, F. E., and Kalaba, R. E. (199). A new perspetive on onstrained motion. Pro., Roy. So. Lon., London, 439, Udwadia, F. E., and Kalaba, R. E. (1996). Analytial dynamis: a new approah. Cambridge University Press, New York. Udwadia, F. E., Kalaba, R. E., and Hee-Chang, E. (1997). Equations of motion for mehanial systems and the extended D Alembert s priniple. Quarterly of Appl. Math., 55(), / JOURNAL OF AEROSPACE ENGINEERING / JANUARY 000