MEG6007: Advanced Dynamics -Principles and Computational Methods (Fall, 2017) Lecture 4: Euler-Lagrange Equations via Hamilton s Principle

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1 MEG6007: Avnce Dynmics -Principles n Computtionl Methos (Fll, 2017) Lecture 4: Euler-Lgrnge Equtions vi Hmilton s Principle This lecture covers: Begin with Alembert s Principle. Cst the virtul work emnting from Alembert s principle into the time-efinite integrl to obtin Hmilton s Principle. Crry out the vrition of the time-efinite integrl to obtin the Euler-Lgrnge equtions of motion. Apply the Euler-Lgrnge equtions of motion to simple exmple problems. 4.1 A Précis of Euler s n Lgrnge s contributions to mechnics (s well s science n mthemtics) Leonhr Euler ( ) Nturl logrithmic constnt: e = lim n (1+ 1 n )n = Introuction of mny mthemticl nottions: f(x) s function f pplie to the rgument x. Series expnsion of e x = n=0 x n n! Complex exponentil function e iθ = cos θ + i sin θ 1

2 Euler-Bernoulli bem theory: EI 4 w x 4 = q(x) Euler s rottionl equtions for rigi boy(theori motus corporum soliorum seu rigiorum, 1760): J ω + ω (J ω) = M Euler ngles to represent the orienttion of soli boy in the 3-imensionl spce Joseph-Louis Lgrnge ( ) His book, Mécnique Anlytique, ppere in In tht book, Lgrnge orgnize the principles of ynmics into the four principles the conservtion of living forces (first trete by Glileo n perfecte by Huyghens); the conservtion of the motion of the center of grvity(newton); the conservtion of moments or the principle of res (Euler, Bernoulli, Arcy); n the principle of the lest quntity of ction (Mupertuis). Metho of multipliers, now referre to s Lgrnge s metho of multipliers. The principle of virtul work (or sometimes referre to of virtul velocities). He solve the three-boy(erth, Sun n Moon) problem(1764) n ientifie the so-clle Lgrngin points in the Erth- Moon-stellite system ( points). His nme is inscribe on the Eiffel Tower long with 71 other nmes. 2

3 4.2 Extene Hmilton s Principle Let s begin with Alembert s principle: (F i m i r i ) δr i = 0 δt + δ W t (m iṙ i δr i ) = 0, δw = F i δr i, δt = δt i, δt i = δ( 1m 2 iṙ i ṙ i ) i (4.1) Let s integrte (4.1) in time: {δt + δ W {δt + δ W } t = t (m iṙ i δr i )} t = 0 t (m iṙ i δr i ) t (4.2) Noting tht we hve t (m iṙ i δr i ) t = so tht, if r i ( ) n r i (t 2 ) re specifie, we hve (m i ṙ i δr i ) t 2 t1 (4.3) δr i ( ) = δr i (t 2 ) = 0 (4.4) Hence, eqution (4.1) becomes {δt + δ W } t = 0 (4.5) which is known s extene Hmilton s principle. In generl the work one, δ W, consists of two prts: 3

4 δ W = δ W cons + δ W noncons (4.6) where subscripts cons n noncons esignte conservtive n nonconservtive systems, respectively. 4.3 Hmilton s Principle for Conservtive Systems For conservtive systems, we hve hve from (4.5) n (4.6) with δ W noncons = 0: {δt + δ W cons } t = 0 (4.7) which is known s Hmilton s principle for conservtive systems. 4.4 Action Integrl for Conservtive Systems Observe tht the work one on the conservtive systems cn be expresse in terms of the corresponing potentil energy: δ W rref cons = δv, V = ( r F i r i ) (4.8) Substituting (4.8) into (4.7), one obtins: {δt δv } t = 0 {δt δv } t = 0 (4.9) δs = {δl} t = 0, L = T V where L is clle system Lgrngin. 4

5 4.5 Euler-Lgrnge Eqution s of Motion Let s substitute the generl work expression (4.6) into the extene Hmilton s principle (4.5) to obtin: {δt + δ W cons + δ W noncons } t = 0 (4.10) which, with (4.8) n the efinition of the system Lgrngin L = T V, becomes {δl + δ W noncons } t = 0 (4.11) Since δl cn be expresse in terms of the generlize coorintes s δl = one obtins { L δ q k + L δq k } (4.12) q k q k δl t = { L δ q k + L δq k } t (4.13) q k q k Integrting by prt the first term of the preceing integrl, we obtin { L δ q k } t = { L δq k } t 2 q k q k t1 { t ( L )δq k } t q k (4.14) Since δq i ( ) = δq(t 2 ) = 0 s we iscusse in eriving (4.4), we hve { L δ q k } t = q k { t ( L )δq k } t (4.15) q k Substituting (4.15) into (4.13), then introucing the resulting 5

6 expression into (4.11), we finlly obtin: { t ( L ) + L + Q k }δq k t = 0, δ q k q W noncons = Q k δq k k (4.16) Since δq k re rbitrry, we obtin: t ( L q k ) L q k = Q k (4.17) which is clle Euler-Lgrnge s equtions of motion. 4.6 Appliction of Euler-Lgrge s Equtions of Motion: A Spring-Mss-Br System for Euler-Lgrnge s Equtions Notice below how simple the problem escription n its erivtion become now! Step 1. The position vectors for the exmple problem: Position vectors of the sliing mss M, the penulum br t C, n B where the nonconservtive force F is pplie re given by: r A = xi r C = xi + L (sin θi cos θj) 2 r B = xi + L(sin θi cos θj) (4.18) r C = L 2 j Step 2. The kinetic energy of the exmple problem: 6

7 Fig. 1. An exmple problem for pplying Euler-Lgrnge s equtions of motion T = 1 2 m(ṙ A ṙ A ) M(ṙ C ṙ C ) I C θ 2 (4.19) = 1 2 [(M + m)ẋ2 + MLẋ θ cos θ ML2 θ2 ] Step 3. Potentil energy: V totl = V spring + V grvity V spring = V grvity = 0 x rc r C ( kxi) r A = 1 2 kx2 ( Mgj) r C (4.20) = Mg L (1 cos θ) 2 Step 4. Nonconservtive work δ W noncons : δ W noncons = F B δr B = F i [δxi + L(cos θi + sin θj)δθ] = F (δx + L cos θ δθ) Q x = F, Q θ = F L cos θ (4.21) Note tht there re only two stte vribles: x n θ in the 7

8 system kinetic energy, potentil energy n the nonconservtive work. Step 5. Euler-Lgrnge s Equtions of Motion for Exmple Problem t ( L ẋ ) L x = Q x = F t ( L L ) θ θ = Q θ = F L cos θ t [(M + m)ẋ ML θ cos θ] + kx = F ML 6 t (2L θ + 3ẋ cos θ) + 1ML(ẋ θ + g) sin θ = F L cos θ 2 (4.22) 4.7 Mthemticl Derivtion of the Euler-Lgrnge Equtions Mthemticl erivtion of the Euler-Lgrnge equtions cn be crrie out by utilizing the clculus of vritions of efinite integrl. To this en, let us now ress how one obtins sttionry vlue of function vi the clculus of vritions. First, the concept of virtul isplcement shoul not be confuse with the concept of the vrition of function. For we hve the virtul isplcement t our isposl, but the vrition of the function is not. Suppose we re to obtin sttionry vlue of Π = G(y, y, x)x (4.23) with the bounry conitions y() = α, y(b) = β (4.24) 8

9 Assume tht y = f(x) by hypothesis gives sttionry vlue to Π. One wy to prove this to be true is to evlute the sme integrl for slightly moifie function ȳ n estblish tht the rte of chnge of Π ue to the chnge in y is zero (why?). We cn thus write ȳ = f(x) + ɛg(x) = y(x) + ɛg(x) (4.25) where g(x) is n rbitrry function tht must be continuous n ifferentible s y. Since g(x) is n rbitrry function, this ifference is clle the vrition of the function y n is enote by δy: δy = ȳ y = ɛg(x) (4.26) This is seemingly trivil expression but there is n importnt property: The vrition of δy refers to n rbitrry infinitesiml chnge of the vlue of the epenent vrible of y, t the point x. The inepenent vrible, x, oes not prticipte in the process of vrition. A consequence of the bove sttement is δx = 0, (4.27) thus resulting in δy() = δy(b) = 0 (4.28) Before we procee to minimize Π in (4.23), we nee to estblish two itionl properties of the δ-process. Since G(y, y, x) involves y, we nee to know how to express δy. To this en, we note from (4.26) x δy = (ȳ y) = x x (ɛg(x)) = ɛg (4.29) which is the erivtive of the vrition δy. On the other hn, 9

10 for the vrition of the erivtive, we hve δy = ȳ y = (y + ɛg) y = ɛg (4.30) Equtions (4.29) n (4.30) give y δy = δ x x (4.31) Hence, the erivtive of the vrition is the sme s the vrition of the erivtive. Similrly, one cn show δ G(y, y, x)x = δg(y, y, x)x (4.32) In other wors, vrition n ifferentition re commuttive. Similrly, one cn show tht vrition n integrtion re lso commuttive. We re now rey to minimize Π in (4.23). First, we hve δg(y, y, x) = G(y + ɛg, y + ɛg, x) G(y, y, x) = ɛ( G y g + G y g ) (4.33) Now for the vrition of the efinite integrl (4.23), we pply (4.31) n (4.33): δπ = δ = ɛ G(y, y, x)x = ( G y g + G y g )x δg(y, y, x)x (4.34) Upon utilizing the rule of integrtion by prts, we hve G y g x = [ G y g]b x ( G )gx (4.35) y 10

11 Since the vrition of y t x = n x = b is zero from (4.28), we hve the following vritionl quntities: {δy() = ɛg() = 0, δy(b) = ɛg(b) = 0} => g() = g(b) = 0 (4.36) so tht we hve [ G g(b)] [ Gg()] = 0 (4.37) y y Substituting (4.36) n (4.37) into (4.35), we obtin δπ = ɛ ( G y G )gx (4.38) x y As ɛ is ssocite with n rbitrry vrition of y, the sttionry vlue of Π is δπ ɛ = ( G y G )g(x)x = 0 (4.39) x y Since g(x) is n rbitrry function tht represents the vrition of y, we must hve for the sttionry vlue of Π: G y G = 0 (4.40) x y This is the celebrte Euler-Lgrnge eqution in mechnics when G(y, y, x) is replce by the Lgrngin function, L, with substitutions of y by q n x by t: G(y, y, x) = L( q, q, t) = {T ( q, q) V (q)} (4.41) so tht we hve from (4.40) the equtions of motion s given by T t q T q + V q = 0 (4.42) 11

12 Remrk: When G is of the form G = G(y, y, y, x) (4.43) the resulting Euler-Lgrnge eqution is given by G y G x y + 2 G = 0 (4.44) x 2 y with the ssocite bounry conitions given by [( G y x G G )δy + y y δy ] b = 0 (4.45) The preceing equtions re pplicble for bem uner grvity lo for which G becomes G = EI( 2 w(x) x 2 ) 2 mgw(x) (4.46) where m is the mss per unit bem length. 12