International Journal of Non-Linear 37 (2002) 1079–1090

On the foundations of analytical F.E. Udwadiaa; ∗, R.E. Kalabab

aAerospace and , Civil Engineering, Mathematics, and Information and Operations Management, 430K Olin Hall, University of Southern California, Los Angeles, CA 90089-1453, USA bBiomedical Engineering and Economics, University of Southern California, Los Angeles, CA 90089, USA

Abstract In this paper, we present the general structure for the explicit equations of for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D’Alembert’s principle, and our derivation is not based on the principle of virtual . Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint acting on the systemcan always be viewed as madeup of the sumof two components.The explicit formfor each of the two components is provided. The ÿrst of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs speciÿcation by the mechanician and depends on the particular situation at hand, this component nonetheless has a speciÿc form. The paper also provides a generalized formof D’Alembert’sprinciple which is then used to obtain the explicit for constrained mechanical systems where the constraints may be non-ideal. We show an example where the new general, explicit equations of motion obtained in this paper are used to directly write the equations of motion for describing a non-holonomically constrained system with non-ideal constraints. Lastly, we provide a geometrical description of constrained motion and thereby exhibit the simplicity with which Nature seems to operate. ? 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction referred to as ideal constraints, and the assumption that they do no work under virtual displacements is Ever since the initial description of constrained referred to as D’Alembert’s principle [8]. However, motion given by Lagrange [1] the problem of the inclusion of general constraint (which obtaining the equations of motion for constrained may do work under virtual displacements) within mechanical systems has been of considerable inter- the framework of has posed est to both mathematicians and engineers. Various a considerable problemto date. Such forces indeed ways have been devised [2–7] for determining these do exist and are in fact commonplace in Nature, like equations under the assumption that the work done the force of . No general equation of by the forces of constraint under virtual displace- motion within the Lagrangian framework for such ments is always zero. Such constraints are often systems has so far been developed. As stated by Goldstein (1981), “This [total work done by forces ∗ Corresponding author. Tel.: +1-213-740-0484; of constraint equal to zero] is no longer true if slid- fax: +1-213-740-8071. ing friction is present, and we must exclude such E-mail address: [email protected] (F.E. Udwadia). systems from our [Lagrangian] formulation” [9].

0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0020-7462(01)00033-6 1080 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090

And Pars (1979, p. 14) in his treatise [5] on ana- where q(t)isthen-vector (i.e. n by 1 vector) of lytical dynamics writes, “There are in fact systems , M is an n × n symmetric, for which the principle enunciated [D’Alembert’s positive-deÿnite matrix, Q is the ‘known’ n-vector Principle] ::: does not hold. But such systems will of impressed (also, called ‘given’) forces, and the not be considered in this book.” dots refer to diQerentiation with respect to . In this paper we obtain the general formof By unconstrained, we mean that the components the equation of motion for holonomically and of the n-vectorq ˙0 can be arbitrarily speciÿed. By non-holonomically constrained systems that is ‘known’, we mean that Q is a known function of its valid independent of whether or not the constraints arguments. The , a, of the unconstrained satisfy D’Alembert’s principle. Unlike our previ- systemat any time t is then given by the relation ous results [10,11], we derive the general formof a(q; q;˙ t)=M −1(q; t)Q(q; q;˙ t). the equation of motion without considerations of We shall assume that this system is subjected to . We ÿnd that the total constraint force a set of m = h + s consistent equality constraints of can always be thought of as made up of two com- the form ponents. The ÿrst component is ever-present and can be uniquely determined from information about ’(q; t)=0 (2) the unconstrained systemand the kinematicequa- tions of constraint that this unconstrained system and is required to satisfy. It is the force of constraint that would have arisen had all the constraints been (q; q;˙ t)=0; (3) ideal. The second component is caused by the presence of the non-ideal nature of the constraints, where ’ is an h-vector and an s-vector. Further- if such non-ideal constraints exist. We show that more, we shall assume that the initial conditions q andq ˙ satisfy these constraint equations at time this second component must appear in the equation 0 0 t = 0, i.e., ’(q ; 0) = 0, and (q ; q˙ ; 0)=0. of motion of a constrained mechanical system in 0 0 0 a speciÿc form, though. Using this formwe then Assuming that Eqs. (2) and (3) are suRciently 1 particularize these non-ideal constraint forces to smooth, we diQerentiate Eq. (2) twice with respect situations where the work that they do under vir- to time, and Eq. (3) once with respect to time, to tual displacements is known and prescribed by the obtain an equation of the form: mechanician. This leads us to state a generaliza- A(q; q;˙ t)Mq = b(q; q;˙ t); (4) tion of D’Alembert’s principle, and we show that for constrained systems that satisfy this general- where the matrix A is m × n, and b is the m-vector ized principle one obtains an explicit and unique that results fromcarrying out the diQerentiations. equation of motion. This equation is then used to We place no restrictions on the rank of the matrix A. further understand the general formof the equation This set of constraint equations includes among previously obtained. An example (a generalization others, the usual holonomic, non-holonomic, of Appell’s problem[12]) showing the application scleronomic, rheonomic, catastatic and acatastatic of our results to a non-holonomically constrained varieties of constraints; combinations of such systemwith non-ideal constraints is provided. The constraints may also be permitted in Eq. (4). It is last section of the paper exposes the geometry of important to note that Eq. (4), together with the constrained motion with non-ideal constraints. initial conditions, is equivalent to Eqs. (2) and (3). Consider now any instant of time t. When the 2. Statement of problem equality constraints (Eqs. (2) and (3)) are imposed Consider an unconstrained mechanical system described by the Lagrange equations 1 We assume throughout this paper that the presence of constraints does not change the rank of the matrix M. This is M(q; t)Mq = Q(q; q;˙ t);q(0) = q0; q˙(0) =q ˙0; (1) almost always true in mechanical systems. F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 1081 at that instant of time on the unconstrained system, In what follows, instead of the we the motion of the unconstrained system is, in gen- shall use, for convenience, the ‘scaled’ accelera- eral, altered fromwhat it would have been (at that tions deÿned by instant of time) in the absence of these constraints. qM = M 1=2q;M (8) We view this alteration in the motion of the uncon- s strained systemas being caused by an additional a = M −1=2Q = M 1=2a; (9) set of forces, called the ‘forces of constraint’, act- s ing on the systemat that instant of time.The equa- and tion of motion of the constrained system can then qMc = M −1=2Qc = M 1=2qMc: (10) be expressed as s Thus ‘scaling’ consists of simply premultiplication M(q; t)Mq = Q(q; q;˙ t)+Qc(q; q;˙ t); of the acceleration n-vectors at any time t,bythe matrix M 1=2 at that time. Eq. (6) can now be written q(0) = q ; q˙(0) =q ˙ ; (5) 0 0 in terms of the scaled accelerations. These scaled where the additional ‘constraint force’ n-vector, accelerations then satisfy, at each time t, the relation Qc(q; q;˙ t), arises by virtue of the constraints (2) c qM = as +Mq ; (11) and (3) imposed on the unconstrained system, s s which is described by Eq. (1). Our aimis to and, by Eq. (4), also the relation determine a general explicit formfor Qc at any BqMs = b; (12) time t. In what follows, for brevity, we shall sup- where press the arguments of the various quantities, unless necessary for purposes of clariÿcation. B = AM −1=2: (13) Following Gauss [2], another way of viewing We note that since, by assumption we know q and constrained motion is as follows. Let us premultiply q˙ at time t, the quantities M; A; B; Q, and b are Eq. (5) by M −1. We then obtain the acceleration all known at time t. Eq. (11), therefore, informs equation us that at time t, the (scaled) acceleration of the c qM = a +Mq ; (6) constrained system,q Ms, deviates fromthe known (scaled) acceleration of the unconstrained system, where a = M −1Q, andq Mc = M −1Qc. Now imagine c as, by the quantityq Ms. (As before, the unconstrained that the constrained mechanical system has evolved acceleration as is known at time t, because as = up until some time, say t. We assume that we know M −1=2Q.) This (scaled) deviation is caused by the the of the constrained system,q ˙(t) , and presence of the constraints (12) and their nature, its conÿguration, q(t), at that time. Our aim is to c and it is the general form of this deviation,q Ms, that ÿnd the subsequent motion of the constrained sys- we shall now determine. By virtue of Eq. (10) the temat the next instant of time.This would naturally knowledge of this deviation is tantamount to the be accomplished if we knew the acceleration,q M(t), knowledge of the constraint force Qc. of the constrained systemat time t. Now since the acceleration of the unconstrained system, a,isa function of q; q˙, and t, it is, by assumption, known 3. General form for the equation of motion for at time t. Hence we see that the problemof con- constrained mechanical systems strained motion can be viewed as requiring us to ÿnd the deviation, Sq M, at each instant of time, of We begin by considering the matrices T = B+B the acceleration of the constrained systemfromthat and N =(I − B+B), where the matrix B+ is it would have had, were there no constraints (i.e., the Moore–Penrose (MP) inverse of the ma- fromthat of the unconstrained system).This devi- trix B [13]. 2 The matrix T is an orthogonal ation can be expressed, using Eq. (6), as 2 Some of the basic properties of the Moore–Penrose inverse c SMq(t)=q M− a =Mq : (7) used in this paper may be found in Chapter 2 of Reference 8. 1082 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 projection operator since (B+B)T = B+B, and T 2 = where we have used the relations (9) and (13) to + + + 1=2 (B B)(B B)=B B = T. Both these results follow yield Bas = BM a = Aa. This then is the most fromthe deÿnition of the MP inverse of the ma- general form in which the force of constraint can trix B. Also, N is an orthogonal projection opera- appear in a mechanical system constrained by tor since (I − B+B)T = I − (B+B)T = I − B+B, and Eqs. (2) and (3). N 2 = N. We note that to obtain the unique constraint force Furthermore, given an n-vector, w, the vector Nw acting on a given constrained mechanical system, is such that B(Nw)=B(I − B+B)w = 0; hence Nw we need a further speciÿcation of the constraints belongs to the null of the matrix B. Also the (beyond that provided through the knowledge of vector Tw belongs to the range space of BT [8]. the matrix A and the vector b); that is, we need a Since Rn = R(BT) ⊕ N(B), any n-vector w has a speciÿcation of the vector z at each instant of time. unique orthogonal decomposition w =B+Bw +(I − In the next section we shall consider this in detail. + B B)w; and so also our n-vectorq Ms. We can thus We now consider the ÿrst member on the write the identity right-hand side of Eq. (19). We note that were the acceleration, a = M −1Q, of the unconstrained + + qMs = B BqMs +(I − B B)Mqs; (14) systemat time t to be inserted into the equation of constraint (4), this equation would not, in general, where the right-hand side simpliÿes toq M . Since s be satisÿed at that time. The extent to which the the (scaled) acceleration of the constrained system constraint (Eq. (4)) would not be satisÿed by this must satisfy Eqs. (11) and (12), we can replace, acceleration, a, of the unconstrained systemat time on the right-hand side of Eq. (14),q Ms in the second c t would then be given by member by (as +Mqs), and BqMs in the ÿrst member by the m-vector b. This yields the relation e = b − Aa: (20)

+ + c qMs = B b +(I − B B)(as +Mqs) Eq. (19) can now be rewritten as = B+b +(I − B+B)a +(I − B+B)Mqc; (15) s s Qc = M 1=2B+e + M 1=2(I − B+B)z (21) which can also be expressed as and, using Eqs. (18), (7) and (10), the deviation, + + c SMq, becomes qMs = as + B (b − Bas)+(I − B B)Mqs: (16) Comparing the right-hand sides of Eqs. (11) and SMq =Mq − a = M −1=2B+e + M −1=2(I − B+B)z: (16) we see that (22)

+ c + B BqMs = B (b − Bas) (17) Eqs. (21) and (22) encapsulate a new fundamen- tal principle of mechanics which we state in two c which, upon solving forq Ms, gives equivalent statements.

c + + + + + qMs = B BB (b − Bas)+{I − (B B) (B B)}z 1. A constrained mechanical system evolves in = B+(b − Ba )+(I − B+B)z (18) such a way that, at each instant of time, the s deviation, Sq M, of its acceleration fromwhat it for any n-vector z. We have thus obtained the most would have been at that instant had there been c no constraints on it, is given by a sumof two general form that the (scaled) acceleration,q Ms,of a mechanical system subjected to the constraints components: the ÿrst component is proportional (2) and (3) can have. FromEq. (10), we note that to the extent, e, to which the unconstrained ac- c 1=2 c celeration does not satisfy the constraints at that the force of constraint Q = M qMs, and Eq. (18) becomes instant of time, the matrix of proportionality being the matrix M −1=2B+; the second is pro- Qc = M 1=2B+(b − Aa)+M 1=2(I − B+B)z; (19) portional to an n-vector z that needs, in general, F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 1083

to be speciÿed at each instant of time, the matrix non-ideal constraints in any given mechanical sys- of proportionality being M −1=2(I − B+B), where temwhose equation of motionwe want to obtain. B = AM −1=2. As we shall soon see, it is useful to express the 2. At each instant of time, the force of constraint total force of constraint, Qc, as being made up of acting on a constrained mechanical system is the two additive components: made up of two components: the ÿrst compo- Qc = M 1=2B+(b − Aa) (24) nent is proportional to the extent, e, to which the i unconstrained acceleration of the systemdoes and not satisfy the constraints at that instant of time, Qc = M 1=2(I − B+B)z; (25) and the matrix of proportionality is M 1=2B+; the ni c second is proportional to an n-vector z that, in so that the total constraint force n-vector, Q ,is general, needs further speciÿcation at each in- given by c c c stant of time, the matrix of proportionality being Q = Qi + Qni (26) 1=2 + −1=2 M (I − B B), where B = AM . This vector and Eq. (23) takes the less formidable form z is speciÿc to a given mechanical system and c c needs to be prescribed by the mechanician who MqM = Q + Qi + Qni: (27) is modeling the system. The reason for using the subscripts ‘i’ and ‘ni’ will become clear in the following section. We note that these statements are natural extensions of those previously obtained only for the case of ideal constraints [7]. Using Eq. (21) in Eq. (5) we c 4. Understanding the two components Qi and are now ready to state the following fundamental Qc of the force of constraint Qc result. ni To understand the two components of the total Result 1. The equation of motion of any mechani- constraint force n-vector, Qc, given by Eqs. (24)– cal systemsubjected to the constraints (2) and (3) c (26) we note ÿrstly that the component, Qi ,isex- has, at each instant of time t, the explicit general plicitly known and is dependent on the description form of: the unconstrained systemgiven by Eq. (1); and,

1=2 + 1=2 + the constraints as given by Eq. (4). It thus only de- MqM = Q + M B (b − Aa)+M (I − B B)z pends on the four known entities A; M; Q, and b. = Q + Qc; (23) By ‘known’ we mean, as usual, known functions of their arguments. −1=2 −1 c c where B = AM , and a = M Q. The n-vector z The second component, Qni,ofQ is dependent needs to be speciÿed at each instant of time t, and on a vector z (which may be, in general, a function it depends on the nature of the speciÿc mechanical of q; q˙ and t) that needs to be suitably speciÿed systemunder consideration. In general, A; B; a; b, for a given mechanical system, beyond the speciÿ- and z are each functions of q; q˙ and t; the matrix cation of the four quantities just mentioned before. M is a function of q and t. We shall now attempt to understand the physical meaning of the speciÿcation of this vector z, that We remark that the generality of this result stems appears in Eq. (23) and remains to be speciÿed so fromthe fact that we have been led to it, somewhat that the appropriate equations of motion relevant to surprisingly, simply through the use of Eqs. (11)– a speciÿc constrained mechanical system may be (14). No appeal to the principle of virtual work obtained. was made in obtaining it, and therefore the result is We begin by showing that if the constraints valid, irrespective of whether the constraint forces satisfy D’Alemebert’s principle, then the n-vector c are ideal or not. Qni ≡ 0 for all time. For, D’Alembert’s principle We next explore the speciÿcation of the vector states that at each instant of time t, and for all vir- z, which we shall show reTects the nature of the tual displacements, v, at that time t, the work done 1084 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 by the force of constraint, Qc, under these virtual Let us further decompose the n-vector z at time T c displacements, v, must be zero; that is, v Q =0at t into its orthogonal components z = zn + zr, where each instant of time t. A virtual [8] zn belongs to the null space of the matrix B, and T at time t is any non-zero n-vector, v, such that at zr = B w (for some m-vector w) belongs to the that time, Av =0. range space of BT. Then the last equality in Eq. (29) Let us consider a speciÿc instant of time t, and requires that at time t, for all n-vectors u, assume that vTQc =0, but only at that instant of time T + T + t. We shall refer to this assumption as D’Alembert’s u (I − B B)z = u (I − B B)(zn + zr) prescription. When D’Alembert’s prescription is = uTz + uT(I − B+B)BTw satisÿed at every instant of time, we obviously ob- n T tain D’Alembert’s Principle, and the constraints are = u zn =0: (30) then called ideal. If D’Alembert’s Principle is not satisÿed, the constraint forces (and, for short, the For the last equality in Eq. (30) to be valid for all constraints) will be referred to as non-ideal. n-vectors u, we must have zn = 0, and therefore Setting v = M −1=2, we note that Av = 0 implies we see that at each instant of time t at which the AM −1=2(M 1=2v)=B=0. D’Alembert’s prescription constraint force n-vector Qc satisÿes D’Alembert’s T then requires that at the instant of time t, prescription, z = zr = B w, for some m-vector w. Furthermore, at each such instant of time we have { | B =0;=0}} ⇒ TM −1=2Qc c 1=2 + Qni = M (I − B B)z T −1=2 c T −1=2 c =  M Qi +  M Qni = M 1=2(I − B+B)BTw =0: (28) = M 1=2{I − (B+B)T}BTw The ÿrst termon the left-hand side of the last equal- ity in Eq. (28) gives the work done at time t under = M 1=2{BT − BT(BT)+BT} =0: (31) virtual displacements by the component Qc, and the i Hence at those instants of time at which the con- second termthe work done by the component Qc . ni straint forces satisfy D’Alembert’s prescription, we But B = 0 implies  =(I − B+B)u, for any must have Qc = Qc, and Qc = 0! When the con- n-vector u at that time [8]. Thus relation (28) im- i ni straints are ideal and therefore satisfy D’Alembert’s plies that for all n-vectors u; uT(I−B+B)M −1=2Qc= prescription for every instant of time, (i.e., they sat- 0 at time t. By Eqs. (24)–(28) we then require that isfy D’Alembert’s Principle), Qc = Qc is then al- at time t, for all n-vectors u, i ways true. We have shown the following important T + −1=2 c T + −1=2 c result. u (I − B B)M Q = u (I − B B)M Qi T + −1=2 c + u (I − B B)M Qni Result 2. At each instant of time at which the forces of constraint obey D’Alembert’s prescrip- = uT(I − B+B)B+(b − Aa) tion, the equation of motion is uniquely determined + uT(I − B+B)(I − B+B)z solely through knowledge of: (a) the unconstrained systemas contained in the known matrix M and T + =0+u (I − B B)z =0: the known n-vector Q, and, (b) the constraints as (29) contained in Eqs. (2) and (3) (or alternatively by Eq. (4)), and described by the m × n matrix A and Notice that we have explicitly pointed out that the c the m-vector b. The contribution, Qni, to the total ÿrst member on the right in the second equation constraint force n-vector, Qc, is zero at that instant, above is always zero, proving that the work done and the explicit equation of motion at that instant c by the component, Qi , of the total constraint force, is then given by Qc, under virtual displacements is always (for all c 1=2 + time) zero. MqM = Q + Qi = Q + M B (b − Aa): (32) F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 1085

When the constraints satisfy D’Alembert’s Princi- and, in general, at each instant of time t, the work ple, Eq. (32) is valid for all instants of time. The done may be positive, negative or zero. (At instants equation of motion of the constrained system be- of time when this work done is zero, the constraints c comes (32). The second member on the right-hand satisfy D’Alembert’s prescription, and hence Qni = side in Eq. (32) explicitly gives the unique con- 0, at those instants, by our previous result.) c straint force that the mechanical system is subjected The component Qni depends on the speciÿcation to under the assumption that all the constraints are of the vector z for any given mechanical system. ideal. Using Eq. (34), for a given constrained mechanical system, the vector z (which, in general, may be a We point out that Eq. (32) agrees with the ex- function of q; q˙ and t) may be viewed as providing plicit equation of motion given by Udwadia and the needed additional speciÿcation related to the Kalaba [7,8] for constrained systems that satisfy constraints in the system(beyond that contained in D’Alembert’s principle. Comparing Eq. (32) with the matrix A and the vector b) that informs us of the general form(23) stated in Result 1, we obtain the extent of work done by the constraint forces at the following fundamental result. each instant of time t. During such a speciÿcation, it should be remembered that no work is done under Result 3. In any constrained motion of a mechan- virtual displacements by the component zr of z that ical systemsubjected to the constraints (2) and lies in the range space of the matrix BT. (3), whether or not the constraints are ideal, the c c c total force of constraint, Q = Qi + Qni,ismade We now provide further insight into the physical up of two additive components. The component meaning of the vector z. Consider again the instant c 1=2 + Qi = M B (b − Aa)isalways present in general, of time t, and a given virtual displacement v of and it is the force of constraint that would have been the constrained systemat that instant. Furthermore, generated had all the constraints been ideal. The let us say that we could prescribe the work done, c c c work done by this component Qi of the constraint W (t), by the total constraint force n-vector, Q , force under virtual displacements is always zero. under any given inÿnitesimal virtual displacement c The second component Qni of the total constraint v at the instant t, by the quantity force is caused by the non-ideal nature of the con- W c(t)=vT(t)C(q; q;˙ t) (35) straints. which may be positive, negative or zero at that The work done by the constraint force, Qc, under instant of time. The n-vector C is, in general, any virtual displacement v at any time t is, therefore, time-dependent. always given by For any given mechanical system at hand, C comes from a proper examination of the speciÿc T c T c T c T c v Q = v Qi + v Qni = v Qni; (33) systemat hand; it depends on our understanding of because, as pointed out in the comment following the nature of the constraint forces speciÿc to the T c given system, as best discernable to the mechani- Eq. (29), at each instant of time v Qi = 0 for all virtual displacements, v, at that instant. We then cian. In particular, it would be diQerent for a body have the next result. undergoing sliding friction (subjected to a set of impressed forces) on a rough surface as opposed to c a smooth surface (and to the same set of impressed Result 4. Only the component, Qni, of the total constraint force, Qc, can do work under virtual forces). The n-vector C has the dimensions of force displacements. The amount of work done by this and can be thought of as a kind of ‘generalized component of the total constraint force at any in- force’ for describing the non-ideal nature of the con- stant of time t under a virtual displacement, v,at straints in any given, speciÿc, mechanical system. c that time, always has the explicit form In general, as shown below it does not equal Qni. Eq. (35), which prescribes, for each instant of T c T c T 1=2 + v Q = v Qni = v M (I − B B)z (34) time t, the work done under any inÿnitesimal virtual 1086 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 displacement v at time t by the forces of constraint n-vector C that describes the non-ideal constraints acting at that time, constitutes a natural generaliza- does not itself appear directly in the equation of tion of D’Alembert’s prescription, and reduces to motion (38). What appears instead is the projection D’Alembert’s Principle when C ≡ 0 for all instants of M −1=2C on the null space of the matrix B.As of time. We shall refer to Eq. (35) as the General- seen fromEq. (38), only when M −1=2C belongs c ized D’Alembert’s Principle. to the null space of B does Qni = C. Also, if C T c We now show that the prescription (at each in- belongs to the range space of B ;Qni =0. stant of time) of this vector C, allows us to uniquely The last result and the remarks preparatory to c determine the contribution, Qni, which is caused by it have been aimed at understanding how one the presence of non-ideal constraints, to the total might prescribe the vector z that arises in Eq. constraint force, Qc. In short, such a prescription in (23) for any given practical situation. We have eQect, informs us of the relevant nature of z. found that the use of the generalized D’Alembert’s Our generalized D’Alembert’s Principle then Principle (which requires a speciÿcation of the T c T c c states that v Q =v Qni =W at each instant of time vector C at each instant of time) yields a unique c c c t, where W (t) is speciÿed by the n-vector C. We characterization of Qni, and therefore of Q .We again set v=M −1=2=M −1=2(I −B+B)u,aswedid have thus obtained, within the framework of La- before in Eqs. (28) and (29). This would require grangian dynamics, the explicit equation of motion that at each instant of time, for all n-vectors u, for systems with non-ideal constraints. But were we to simply set z = M −1=2C (and this uT(I − B+B)z = uT(I − B+B)M −1=2C; (36) we can always do, since M is positive deÿnite) then Eqs. (23) and (38) would be identical! This points fromwhich it follows that to the generality of Eq. (38), and gives us our last c 1=2 + 1=2 + −1=2 result. Qni = M (I − B B)z = M (I − B B)M C; (37) Result 6. The equation

c and Qni is, therefore, uniquely determined. Using MqM = Q + M 1=2B+(b − Aa) this in Result 1, we obtain our next result. +M 1=2(I − B+B)M −1=2C Result 5. If for a given constrained mechanical sys- c c tem, the work done, W c, by the total force of con- = Q + Qi + Qni (39) straint, Qc, under virtual displacements is prescribed is the general equation of motion for any me- by the mechanician through a speciÿcation of the chanical systemthat is holonomicallyand =or vector C(q; q;˙ t) (see Eq. (35)), then the unique non-holonomically constrained. The entities equation describing the constrained motion of the A; B; Q; b, and C are, in general, functions of q; systemis given by q˙, and t. MqM = Q + M 1=2B+(b − Aa) The n-vector C(q; q;˙ t) needs to be speciÿed at each instant of time t, and it depends on the nature + M 1=2(I − B+B)M −1=2C: (38) of the non-ideal constraints in the speciÿc mechan- ical systemunder consideration. This n-vector can It is indeed satisfying that the generalized always be viewed as that vector that speciÿes the D’Alembert’s principle through a speciÿcation of work done by the force of constraint, Qc, at time the ‘generalized’ constraint force, C, yields just t under virtual displacements v (at that instant of enough information to uniquely determine (the time) via the relation W c(t) ≡ vTQc = vTC. contribution of the non-ideal nature of the con- c straints to) the constraint force component, Qni, that Eq. (39) informs us that to model a given speciÿc arises in the equation of motion of the constrained mechanical system and obtain its equation of mo- mechanical system. It should be noted that this tion, once the coordinates, qi, are chosen, and the F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 1087 constraint equation (4) determined by the mechani- On diQerentiating the constraint equation with re- cian, (s)he needs to further prescribe, in general, spect to time, we obtain the vector C(t). Furthermore, no matter how (s)he   arrives at (perhaps by inspection, or otherwise) the x˙   requisite vector C(t) (or, equivalently z(t)) that [˙x y˙ − z˙] y˙ = $g;˙ (40) characterizes the nature of the non-ideal constraints z˙ in the system, this vector can always be viewed in where terms of the virtual work done by the forces of con- @g @g @g @g g˙ = x˙ + y˙ + z˙ + : straint. Also, when C ≡ 0, and the constraints are @x @y @z @t ideal then Eq. (39) reduces to the equation given in Ref. [7]. Thus we have A=[˙x y˙ −z˙], and the scalar b=$g˙. + 2 T Since M = I3;B= A. Hence B =1=u [˙x y˙ − z˙] , 2 T and the vector C = −(a0u =|u|)[x ˙ y˙ z˙] . The equation of motion for this constrained sys- 5. Illustrative example temcan now be written down directly by using Eq. (38). It is given by       To illustrate the simplicity with which we can xM F x˙ x ($g˙ − xF˙ − yF˙ +˙zF ) write out the equations of motion for non-holonomic  yM  =  F  + x y z  y˙  y u2 systems with non-ideal constraints we consider zM F −z˙ here a generalization of a well-known problemthat z     was ÿrst introduced by Appell [12]. (˙y 2 +˙z2) −x˙y˙ x˙z˙ x˙ a Consider a particle of unit moving in a − 0  −y˙x˙ (˙x2 +˙z2)˙yz˙   y˙  ; |u| Cartesian inertial frame subjected to the known im- x˙z˙ y˙z˙ (˙x2 +˙y 2) z˙ pressed (given) forces F (x;y;z;t);F (x;y;z;t), and x y (41) Fz(x;y;z;t) acting in the X -, Y - and Z-directions. Let the particle be subjected to the non-holonomic which simpliÿes to constraintx ˙2 +˙y 2 − z˙2 =2$g(x;y;z;t), where $ is a       given scalar constant and g is a given, known func- xM F x˙ x ($g˙ − xF˙ − yF˙ +˙zF ) tion of its arguments. (Appell [12] took $ = 0.) Let  yM  =  F  + x y z  y˙  y u2 us say that the mechanician (who has supposedly zM Fz −z˙ examined the physical mechanism which is being   2˙xz˙2 modeled here) ascertains that this constraint sub- a jects the particle to a force proportional to the square − 0  2˙yz˙2  : (42) |u| of its velocity and opposing its motion, so that the 2˙z(˙x2 +˙y 2) virtual work done by the force of constraint on the particle is prescribed (by the mechanician) as The ÿrst termon the right-hand side is the impressed   force. The second termon the right-hand side is the x˙ c u2 constraint force Qi that would prevail were all the W c(t)=−a vT(t)  y˙  ; (39) constraints ideal so that they did no work under vir- 0 |u| z˙ tual displacements. The third term on the right-hand c side of (42) is the contribution, Qni, to the total con- where u(t) is the of the particle. Thus the con- straint force generated by virtue of the fact that the straint is non-holonomic and non-ideal. We point constraint force is not ideal, and its nature in the out that this force is caused solely because of the given physical situation is speciÿed by the vector C, presence of the constraint. In the absence of the which gives the work done by this constraint force c non-holonomic constraint, it would disappear. under virtual displacements. Note that Qni = C. We shall obtain the equations of motion of this We observe that it is because we do not eliminate system. any of the q’s or theq ˙’s (as is customarily done in 1088 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 the development of the equations of motion for con- strained systems) that we can explicitly assess the eQect of the ‘given’ force, and of the components c c Qi and Qni on the motion of the constrained system. Problems that arise with sliding friction can be handled in a similar manner [10]. We note that holo- nomic constraints that do work could at be handled by Newtonian mechanics. However, to date we know of no general formulations of mechan- ics that provide the explicit equations of motion for non-holonomically constrained mechanical systems where the non-holonomic constraints do work.

Fig. 1. The geometry of constrained motion is depicted using T 6. The geometry of constrained motion with projections on N(B) and R(B ). The projection ofq Ms on non-ideal constraints N(B) is the same as that of (as +cs) because NqMs =N(as +cs). The vector B+b is orthogonal to this projection. We present here a geometrical description of con- strained motion. Using Eq. (18) in Eq. (1) the scaled Using the last equality, B+b, the component of the acceleration of the constrained systemcan be writ- (scaled) acceleration of the constrained systemthat ten as lies in R(BT) is given by (see Fig. 1)

+ + + qMs = N(as + cs)+B b; (43) B b = B e + Tas: (46) Fig. 1 depicts the relations (43)–(46) pictori- where we have denoted here the n-vector z by cs. Recalling that z can be expressed as z = M −1=2C, ally, and reveals the geometrical elegance with which Nature appears to operate. The n-vectors the acceleration cs can be viewed as the ‘scaled’ 1=2 −1 B+e; Ta B+b are seen to belong to R(BT) which acceleration [cs = M (M C)] created by the s ‘generalized’ force C that speciÿes the virtual work is orthogonal to N(B) to which the n-vectors Ncs done by the non-ideal constraint forces. As shown (or Nz) and Nas belong. in Fig. 1, Eq. (43) thus informs us that the scaled At any time t, the component, B+b,ofthe acceleration of the constrained systemis simply (scaled) acceleration of the constrained system the sumof two orthogonal vectors, one belonging only depends on the three entities A; M, and b to the null space of B—denoted N(B), and the (at time t) that occur in Eqs. (1) and (4); it is other belonging to the range space of BT—denoted not aQected by whether or not the constraints are R(BT). ideal. The presence of constraint forces that do Our geometrical understanding is also enhanced not satisfy D’Alembert’s principle (what we have by considering the deviation of the (scaled) acceler- referred to in this paper as non-ideal constraint forces) only aQects the component of the (scaled) ation, Sq Ms(t), of the constrained systemfromwhat it would have been, had there been no constraints acceleration that lies in N(B) (see Fig. 1). Such on it at time t, given by (see Eqs. (11) and (18)) forces engender the n-vector cs, which then needs to be suitably speciÿed in order to obtain the rel- + + SMqs =Mqs − as = B (b − Bas)+(I − B B)cs: (44) evant equation of motion for a given constrained mechanical system. Their only eQect is on the com- Recalling Eq. (20), Eq. (44) becomes ponent of the acceleration that lies in N(B), and is seen to be Ncs. At those instants of time at which SMq =Mq − a = B+b − Ta + Nc = B+e + Nc : s s s s s s D’Alembert’s prescription is satisÿed, cs = 0, and (45) Fig. 1 reverts to the one that was obtained earlier, F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090 1089 but only for constrained mechanical systems that in which the constraints are not ideal, and the obeyed D’Alembert’s Principle [14]. forces of constraint may do positive, nega- tive, or zero work under virtual displacements. This generalized principle reduces to the usual 7. Conclusions D’Alembert’s Principle when the constraints are ideal. In this paper we present a comprehensive descrip- 4. The framework of Lagrangian mechanics is used tion of constrained motion where the forces of con- to show that this generalized D’Alembert’s Prin- straint need not satisfy D’Alembert’s principle. We ciple provides a deeper insight into the nature summarize the main results of this paper as follows: of the vector z. In addition, this principle is shown to provide just the right extent of infor- 1. We obtain the general formof the explicit mation to uniquely yield the accelerations of the equations of motion for constrained mechani- constrained system(subjected to non-ideal con- cal systems. This general form is applicable to straints), as demanded by practical observation. all holonomically and non-holonomically con- In the situation that the constraints are ideal, strained systems irrespective of whether they these accelerations agree with those determined satisfy D’Alembert’s Principle or not.Itis using formalisms (like those of Gibbs, Appell, obtained without appealing to the principle of and Gauss) that have been developed earlier [8], virtual work. The general formshows that the and that are only applicable to the case of ideal total force of constraint, Qc, is made up of the constraints. sumof two contributions. The ÿrst contribution, 5. The general equations that describe constrained c Qi , is what would have been caused were all motion (Eqs. (23) and (39)) obtained here ap- the constraints ideal and had the unconstrained pear to be the simplest and most comprehensive systembeen required to satisfy the prescribed so far discovered. constraint equations during its motion. This con- 6. The paper shows how we can include gen- tribution is uniquely determined once M, and eral constraint forces that do work within the Q (which describe the unconstrained system), scope of the Lagrangian formulation of an- and A and b (which characterize the alytical dynamics. Speciÿcally, we provide of the constraints), are known. The second con- the explicit general equations of motion for c tribution, Qni, to the total constraint force deals non-holonomically constrained systems where with the non-ideal nature of the constraints. the constraints do work. So far, this has been It needs to be speciÿed by the mechanician beyond the reach of Lagrangian formulations after an inspection of the given mechanical sys- of mechanics (see Refs. [5,9]). We thus sur- temat hand through speciÿcation of the vector mount one of the long-standing diRculties in z(q; q;˙ t). This speciÿcation depends, in general, mechanics. on an understanding of the underlying 7. The geometry of constrained motion described that is involved in generating the forces of con- here reveals the simplicity and elegance with straint; one needs to rely here on the judgement which Nature seems to operate. and discernment of the mechanician. Yet, it is c shown that the contribution Qni takes a speciÿc form in the equation of motion that governs the References constrained system, as shown in Eq. (25). 2. We provide two fundamental principles that gov- [1] J.L. Lagrange, Mecanique Analytique, Mme Ve Courcier, ern the motion of general constrained mechani- Paris, 1787. cal systems. [2] C.F. Gauss, Uber Ein Neues Allgemeines Grundgesetz der Mechanik, J. Reine Angew. Math. 4 (1829) 3. In an eQort to understand the nature of 232–235. the n-vector z, we are led to generalize [3] J.W. Gibbs, On the fundamental formulae of dynamics, D’Alembert’s Principle to include situations Amer. J. Math. 2 (1879) 49–64. 1090 F.E. Udwadia, R.E. Kalaba / International Journal of Non-Linear Mechanics 37 (2002) 1079–1090

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