GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE. Ðorđe Ðukić

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle sére, tome 91(105) (2012), DOI: /PIM D GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE Ðorđe Ðukć Abstract. The major ssues n the analyss of the moton of a constraned dynamc system are to determne ths moton and calculate constrant forces. In the analytcal mechancs, only the frst of the two problems s analysed. Here, the problem s solved smultaneously usng: 1) Prncple of lberaton of constrants; 2) Prncple of generalzed vrtual dsplacement; 3) Idea of deal constrants; 4) Concept of generalzed and supplementary" generalzed coordnates. The Lagrange D Alembert prncple of vrtual work s generalzed ntroducng vrtual dsplacement as vectoral sum of the classcal vrtual dsplacement and vrtual dsplacement n the supplementary" drectons. From such prncple of vrtual work we derved Lagrange equatons of the second knd and equatons of dynamcal equlbrum n the supplementary" drectons. Constraned forces are calculated from the equatons of dynamc equlbrum. At the same tme, ths prncple can be used for consderaton of equlbrum of system of materal partcles. Ths prncple smultaneously gves the connecton between appled forces at equlbrum state and the constraned forces. Fnally, the prncple s appled to a few partcular problems. 1. Introducton There are two problems n analyses of moton of the dynamcal systems under acton of the constrants. The frst problem s to determne the moton of the system, and the second to calculate the reacton forces of the constrants durng the moton. In the analytcal mechancs, see for example [3], [1], [6] and [2], these two tasks are separated by the notons of deal constrants and Lagrange D Alembert prncple of vrtual work (In 1743, Jean Le Rond d Alembert publshed hs mportant Traté de dynamque, a fundamental treatse on dynamcs contanng the famous d Alembert s prncple"). Accordng to the noton of deal constrants, the vrtual work of the reactons forces s equal to zero, and the work vanshes n the Lagrange D Alembert prncple of vrtual work. Therefore, the reacton forces are elmnated from further analyses of moton, whch s based on the Lagrange D Alembert prncple. If there are some reacton forces whose vrtual work s not equal to zero, then 2010 Mathematcs Subject Classfcaton: Prmary 70H45; Secondary 70G10. 49

2 50 ÐUKIĆ those forces are added to the gven set of forces and the correspondng constrants are called non-deal. Reacton forces of deal constrants are absent n the Lagrange equatons of the second knd. In fact, analytcal mechancs consders only the moton of a system, whle for fndng reacton forces the equatons of Newtonan mechancs or Lagrange equatons of the frst knd must be used [6], [5] and [4]. Nevertheless, the Lagrange D Alembert prncple s frequently used for fndng reacton forces of deal constrants. Accordng to that procedure, the mechancal system frees tself of a partcular constrant and performs a vrtual dsplacement n drecton consstent wth the removed constrant. Calculaton of vrtual work on that dsplacement yelds the correspondng reacton force. It s ths author s opnon that any prncple of dynamcs must smultaneously gve answer to both mentoned questons of mechancal moton under constrants. In that sense, the Lagrange D Alembert prncple requres some generalzaton. In ths paper, the basc problem for dynamcal systems s analysed usng the followng deas: (1) Vrtual dsplacements and supplementary vrtual dsplacements, (2) Prncple of lberaton of constrants, (3) Ideal constrants, (4) Generalzed coordnates and supplementary generalzed" coordnates. The supplementary vrtual dsplacements are ntroduced n drectons whch are consstent wth the removed constrants. Because the drectons are n normal drectons to the constrants, the vrtual dsplacements are called normal vrtual dsplacements. Also, supplementary generalzed coordnates are ntroduced, whch are measured along ths normal vrtual dsplacements, and whose number s equal to the number of constrants. In the Lagrange D Alembert prncple of vrtual work, the classcal vrtual dsplacements are replaced by vectoral sum of the classcal and normal vrtual dsplacements. From such generalzed Lagrange D Alembert prncple of vrtual work, the Lagrange equatons of the second knd and the equatons of dynamcal equlbrum n the normal drectons are derved. From the equatons of dynamcal equlbrum the constrants forces are calculated. Hence, usng ths prncple, the equatons of Newtonan mechancs are not necessary for fndng the reacton forces. Ths prncple can be used for consderaton of equlbrum of materal partcles. In that case, ths prncple smultaneously gves the connecton between the appled forces at equlbrum state and the constrant forces. Fnally, the prncple s appled to a few partcular problems. Now, the Lagrange D Alembert prncple s used [7] [30] n analyss of dfferent problems n conservatve, nonconservatve, holonomc and nonholonomc mechancs and dfferent areas of physcs. But, n ths papers there s no generalzaton of the prncple n the sense of the present paper. Here, a small Greek ndex takes values from 1 to n, the captal Greek ndex from 1 to s, small talc ndex from 1 to N and captal talc ndex from 1 to n + s. Every repeated captal talc ndex, small Greek ndex or captal Greek ndex means sum wth respect to that ndex. Bold face letters are vectors.

3 GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE Generalzed Vrtual Dsplacement Let us consder moton of a dynamcal system wth N materal partcles M, whose masses are m, and whch s under the acton of a gven set of forces F, and subject to s holonomc constrants (2.1) f (t, x, y, z ) = 0, = 1,..., s. Here t s tme and x, y and z are the Cartesan coordnates of a partcle M. If partcles are lberated from the constrants, then ther actons on partcles M are replaced by the reacton forces R. The system has n = 3N s degrees of freedom and q α (α = 1,..., n) are the correspondng generalzed coordnates. Let the poston of any partcle M, of the system, whch durng the moton s defned by ts poston vector r n the three dmensonal space be r = x +y j+z k, where, j, and k are the unt vectors of Cartesan coordnate system x, y, and z. Vrtual changes of the constrant equatons (2.1) are (2.2) δf = l δr = 0, where (2.3) l = grad f = f x + f y j + f z k, and δr = δx + δy j + δz k s the vrtual change of the poston vector. It s a well known fact that vectors (2.3) are perpendcular to the constrants. Let us lberate every partcle M n the system of the constrants and permt to every partcle dsplacements l q n+ n the perpendcular drectons l, where q n+ are supplementary generalzed coordnates. In our analyss the ndependent parameters q n+ are consdered as functons of tme. For moton along constrants (2.1), the parameters q n+ and ther dervatves q n+ and q n+ are equal to zero,.e., q n+ = 0, q n+ = 0, q n+ = 0. Any quantty, whch s calculated for the moton along the constrants, s denoted by lower ndex 0. The poston vector of the so lberated partcle M of the constrants s (2.4) ρ (t, q U ) = r (t, q α ) + l (t, q α )q n+, whose vrtual change s δρ = δr + δl q n+ + l δq n+. For the moton of the partcle along the constrants, we have (2.5) (δρ ) 0 = δr + l δq n+. Relatons (2.5) defne generalzed vrtual dsplacement of any partcle M. Let us decompose the generalzed vrtual dsplacement nto the followng two parts (2.6) (δρ ) 0 = (δρ ) 0T + (δρ ) 0N, where (2.7) (δρ ) 0T = δr

4 52 ÐUKIĆ s the classcal vrtual dsplacement n the tangental drectons to the constrants, whle (2.8) (δρ ) 0N = l δq n+, s vrtual dsplacement n perpendculars to the constrants. Calculatng sum (δρ ) 0T (δρ ) 0N, and usng (2.2) we have (δρ ) 0T (δρ ) 0N = δq n+ l δr = 0. In the sense to ths relaton, two parts δr and l δq n+ of the generalzed vrtual dsplacement (δρ ) 0 are mutually orthogonal, and decomposton (2.6) of the generalzed vrtual dsplacement s justfable. 3. Constrants forces Here, the classcal defnton of the deal constrants (see for example [3, p. 218]) s accepted. Ths means that the vrtual work of all reacton forces of the constrants on the tangental vrtual dsplacements must be zero,.e., (3.1) (R ) 0 (δρ ) 0T = 0. The poston vectors of the materal partcles M are functons of tme and the generalzed coordnates r = r (t, q α ), and vrtual dsplacements (2.7) n tangental planes to the constrants are (3.2) (δρ ) 0T = δr = r q α δq α. Substtutng the vrtual dsplacements nto orthogonalty condton (2.2), and because the vrtual changes δq α are mutually ndependent and dfferent from zero, the followng condton s vald (3.3) r q α l = 0. Combnng (3.1) and (3.2), and usng the fact that vrtual changes δq α are mutually ndependent and dfferent from zero, we obtan the followng condtons r (3.4) (R ) 0 = 0, q α whch must be satsfed by the reacton forces of the deal constrants. Comparson of (3.3) and (3.4) suggests the followng form for the reacton forces (3.5) (R ) 0 = (R 1 ) l, where (R 1 ) are mutually ndependent quanttes. The condton (3.4) s dentcally satsfed for arbtrary values of the quanttes (R 1 ). Let us ntroduce a second-order system b Γ by relatons (3.6) l l Γ = b Γ,

5 GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE 53 where b Γ = b Γ, whch means that the system s symmetrc. The correspondng nverse system (b 1 ) Ω s defned by (3.7) (b 1 ) Ω b Γ = δ ΩΓ where δ ΩΓ are the Kronecker symbols, that means δ ΩΓ = 0 for Ω Γ and δ ΩΓ = 1 for Ω = Γ. Usng the second-order system b Γ the nverse vectors R and (R 1 ) Γ are connected wth the relaton (3.8) R = (R 1 ) Γ b Γ. 4. Generalzed Forces The vrtual work of all appled forces F on the generalzed vrtual dsplacements (2.6) s (4.1) δa F = (F ) 0 (δρ ) 0. Usng (2.6), (2.8) and (3.2), the vrtual work becomes (4.2) δa F = Q α δq α + N δq n+, where Q α = ( F r q α, are the classcal generalzed forces, and )0 (4.3) N = (F ) 0 l are generalzed forces n the perpendcular drectons. If all appled forces F have the potental energy Π(t, q U ), then the generalzed forces are (4.4) Q α = Π q α, N = Π q. 5. Knetc Energy of the System Poston n the space of every partcle M of the system, whch s lberated from the correspondng constrants s defned by the vectors ρ (2.4). The knetc energy of the system free of constrants s (5.1) E k = 1 m ρ 2 ρ. The velocty of the partcle free from constrants (2.1) s obtaned as the frst dervatve of the vector ρ n to tme (5.2) ρ = ṙ + l q n+ + l q n+. Here, ṙ s the velocty of the partcle whch s not free from constrants. Usng (5.2), knetc energy (5.1) becomes (5.3) E k = (E k ) 0 + Ẽk + NLT, where (E k ) 0 = 1 2 m ṙ ṙ s knetc energy of the system whch s not free from constrants, and Ẽk = m ṙ ( l q n+ + l q n+ ) s the part of the knetc energy lnear to q n+ and q n+. The term NLT s nonlnear n q n+ and q n+.

6 54 ÐUKIĆ The term NLT s not mportant for analyss and therefore t s omtted n the further consderatons. 6. Generalzed D Alembert Prncple For the moton of a partcle M of the system under acton of a gven force F and a reacton force R, the second Newton law of moton under acton of the constrants means that (F +R m ρ ) 0 = 0. Multplyng the prevous equatons by vrtual dsplacements (δρ ) 0 (2.6) and summng up such equatons for all partcles,.e., over the ndex, we have (F + R m ρ ) 0 (δρ ) 0 = 0, or (6.1) δa I + δa F + δa R = 0, where (6.2) δa I = (m ρ δρ ) 0 s the vrtual work of the nertal forces, δa R = (R δρ ) 0 s the vrtual work of reacton forces and δa F s vrtual work of actve forces gven by (4.1). Equaton (6.1) s a generalzed form of the Lagrange D Alembert prncple of vrtual work. The meanng of the generalzed Lagrange D Alembert prncple of vrtual work (6.1) s that durng the moton of a mechancal system the sum of the vrtual works of nertal, actve and reacton forces s equal to zero. Usng (2.6) and (2.7) the vrtual work of reacton forces s (6.3) δa R = (R ) 0 δr + δq n+ (R ) 0 I. The frst term n (6.3) s equal to zero accordng to (3.1) and the prevous relaton becomes (6.4) δa R = R δq n+, where the generalzed reacton forces are R = (R ) 0 I. 7. Lagrange equatons Accordng to (2.4) ρ s a functon of tme t and of all generalzed coordnates q U, U = 1,..., n + s. Usng the well known relatons for such functons ρ q U = ρ q U, vrtual work of nertal forces (6.2) becomes (7.1) δa I = Z U δq U, where Z U = d ρ = ρ, dt q U q U ( Ek d ) E k. q U dt q U Now, usng (4.2), (6.4), and (7.1), the generalzed form of the Lagrange D Alembert prncple of vrtual work n generalzed coordnates (6.1) becomes (7.2) (Z α + Q α )δq α + (Z n+ + N + R )δq n+ = 0.

7 GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE 55 Because the generalzed coordnates and the supplementary generalzed coordnates are mutually ndependent, and ther vrtual changes are also ndependent and dfferent from zero,.e., δq α 0 and δq n+ 0, vrtual work δa (7.2) s equal to zero for arbtrary vrtual changes δq α and δq n+ only f ( d E k (7.3) E ) k = Q α, dt q α q α and (7.4) ( d E k dt q n+ E k q n+ ) 0 0 = N + R. System of equatons (7.3) are well known Lagrange equatons of moton for the dynamcal system, whch are called the Lagrange equatons of the second knd. Equatons (7.4) form another system of equatons from whch, usng (3.5), (3.8), we calculate the reacton forces on every partcle of a system as [( d E k (7.5) (R ) 0 = E ) ] k N (b 1 ) Ω l Ω. dt q n+ q n+ For ths reason the equatons (7.5) can be called equatons of dynamc equlbrum n normal drectons Examples Example 1. Let us consder the moton of a materal pont wth mass m on a smooth surface whose equaton s F 1 = F(x, y, z), where x, y and z are the Cartesan coordnates of the pont and the axs z s n up drecton. Ths moton has two degrees of freedom. The normal vector (2.3) s (8.1) l 11 = F x + F y j + F z k, where the low ndex means partal dervatve wth respect to that coordnate. Usng (3.6) we have (8.2) b 11 = Fx 2 + F y 2 + F z 2, 1 (b 1 ) 11 = Fx 2 + F y 2 + F z 2. The knetc energy (5.3) of the partcle s (8.3) E k = m 2 (ẋ2 + ẏ 2 + ż 2 ) + (ẋ F x + ẏ F y + ż F z )mq 3, where q 3 s the dsplacement n normal drecton (8.1). There s only one actve force F = mgk, where g s gravty acceleraton. Hence, the generalzed force n normal drecton (4.3) s N 3 = mgf z. Now, applyng (7.4) to (8.3) for the coordnate q 3 we have R 3 = mgf z m(ẋf x +ẏf y +żf z ), and usng (7.5) and (8.2) we have the reacton force R = R 3 Fx 2 + F y 2 + F z 2 (F x + F y j + F z k).

8 56 ÐUKIĆ If the surface s a sphere wth radus r, then F = x 2 +y 2 +z 2 r 2, and the reacton force s R = m[gz (ẋ2 + ẏ 2 + ż 2 )] r 2 r, where r s the poston vector of the pont. Example 2. Let us consder a mathematcal pendulum of the length a and mass m. The correspondng constrant s f 1 = x 2 +y 2 a 2 = 0, where x s horzontal axs and y vertcal orented down. From ths relaton we have the gradent vector (2.3) I 11 = 2x+2yj. The problem has one degree of freedom, the angle α between the pendulum and the vertcal drecton. In that case (8.4) I 11 = 2a sn α + 2a cos αj, and velocty of the partcle ṙ = (a cos α a sn αj) α. Here, there s only gravty appled force F = mgj, where g s the gravty acceleraton. The correspondng potental energy s Π = mga(1 cos α). Therefore, the generalzed forces (4.4) and (4.3) are Q α = mga sn α, N α = 2mga cos α. The knetc energy (5.1) has the form E k = m 2 (a α)2 + 2m(a α) 2 q 2, where q 2 s the addtonal generalzed coordnate. Second-order symmetrc system (3.6), usng (8.4) becomes and ts nverse (3.7) (b 1 ) 11 = 1/4a 2. Fnally, from (7.3) and (7.5) we have the equaton of moton α + g a sn α = 0, and the reacton force to the partcle R = m(a α 2 + g cos α)(sn α + cos αj). Example 3. Let us consder the moton n a vertcal plane of two equal rods OA and AB equal masses m and lengths l connected by a cylndrcal jont at the pont A, whle the rod OA s connected to unmovable cylndrcal jont at the end of rod at the pont O. Ths system has two degrees of freedom and the angles ϕ and ψ between the rod and vertcal drecton are the generalzed coordnates. Let x be the horzontal and y the vertcal axes. The centers of gravty for the rods are the ponts C 1 and C. The gravty forces are (8.5) F c1 = mgj, F c = mgj, where g s the gravty acceleraton. If we want to determne forces at cylndrcal jonts then we must remove them. Let x 1 and y 1 be the moton of the rods and wth respect to jont O, and x 2 and y 2 moton of A of the rod AB wth respect to the jont pont of the rod OA. The correspondng equatons of constrants are (8.6) The knetc energy of the system s F 1x1 = x 1 = 0, F 1y1 = y 1 = 0, F 2x2 = x 2 = 0, F 2y2 = y 2 = 0. E k = 1 2 J c1 ϕ 2 + m 2 v2 c J c ψ 2 + m 2 v2 c,

9 GENERALIZED LAGRANGE D ALEMBERT PRINCIPLE 57 where J c1 and J c are the correspondng moments of nerta and v c1 and v c are veloctes of the ponts C 1 and C. The part of the knetc energy lnear wth respect x 1, y 1 x 2, y 2, ẋ 1, ẏ 1 ẋ 2, ẏ 2 s (8.7) Ẽ k = m 2 l(ẋ 1 ϕ cos ϕ ẏ 1 ϕ sn ϕ) [ ( + ml (ẋ 1 + ẋ 2 ) ϕ cos ϕ + ψ ) 2 cos ψ The vrtual work of actve forces (8.5) s ( (ẏ 1 + ẏ 2 ) ϕ sn ϕ + ψ )] 2 sn ψ. δa F = 2mgδy 1 + mgδy 2 mg 3 l sn ϕδϕ mgl sn ψδψ, 2 and the generalzed forces n perpendcular drectons are (8.8) N x1 = 0, N y1 = 2mg, N x2 = 0, N y2 = mg. Applyng (7.4) to (8.7) and usng (8.8) we have R x1 = d [ ml ( dt 2 ϕ cos ϕ + ml ϕ cos ϕ + ψ )] 2 cos ψ, R y1 = 2mg d [ ml ( (8.9) dt 2 ϕ sn ϕ + ml ϕ sn ϕ + ψ )] 2 sn ψ, R x2 = d [ ( ml ϕ cos ϕ + ψ )] dt 2 cos ψ, R y2 = mg d [ ( ml ϕ sn ϕ + ψ )] dt 2 sn ψ. The vectors n perpendcular drectons to the constrants (8.6) are l 1x1 =, l 1y1 = j, l 2x2 =, l 2y2 = j. Hence the system b Γ and ts nverse system (b 1 ) Γ are 4 4 unt matrxes, and usng (3.8) we have (R 1 ) x1 = R x1, (R 1 ) y1 = R y1, (R 1 ) x2 = R x2, (R 1 ) y2 = R y2. Therefore, the vectors of reacton forces at jonts at the ponts O and A are R O = R x1 + R y1 j, R A = R x2 + R y2 j, where R x1, R y1, R x2 and R y2 are gven by (8.9). References 1. L. A. Pars, Introducton to Dynamcs, Cambrdge Unversty Press, Cambrdge, E. T. Whttaker, A Treatse on the Analytcal Dynamcs of Partcles and Rgd Bodes, Cambrdge Unversty Press, Cambrdge, G. Hamel, Theoretsche Mechank, Sprnger-Verlag, Berln Gottngen Hedelberg, E. I. Vorobev, S. A. Popov, G. I. Scheveleva, Mechancs of Intellgent Robots, Knematcs and Dynamcs, Moscow, Hgher School, 1988 (n Russan). 5. M. Vukobratovć, Appled Dynamcs of Manpulaton Robots, Modellng, Analyss and Examples, Sprnger-Verlag, 1989, reprnted by World Publshng Corporaton, Bejng, A. I. Lurje, Analytcal Mechancs, Phys.-Math. Lt., Moscow, 1961 (n Russan). 7. P. M. Marano, Mechancs of quas-perodc alloys, J. Nonlnear Sc. 16 (2006), A. C. Baptsta, C. Sgaud, P. C. Guaranho de Moraes, Controllng nonholonomc Chaplygn systems, Braz. J. Phys. 40 (2010). 9. Yu. N. Fedorov, B. Jovanovć, Integrable nonholonomc geodesc flows on compact Le groups; n: A. V. Bolsnov, A. T. Fomenko, A. A. Oshemkov (eds), Topologcal Methods n the Theory of Integrable Systems, Cambrge Scentfc Publ., 2006, pp ; arxv: math-ph/

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