MEEN Handout 4a ELEMENTS OF ANALYTICAL MECHANICS

Transcription

1 MEEN 67 - Handou 4a ELEMENTS OF ANALYTICAL MECHANICS Newon's laws (Euler's fundamenal prncples of moon) are formulaed for a sngle parcle and easly exended o sysems of parcles and rgd bodes. In descrbng he moons, physcal coordnaes and forces are employed n erms of vecoral quanes (vecoral mechancs). The major drawbac s ha Newon's Laws consder he ndvdual componens of a sysem separaely, hus requrng he calculaon of neracng forces resulng from nemacal consrans. The calculaon of hese forces s many mes of no consequence or neres n he fnal formulaon of he equaons of moon. A dfferen approach nown as ANALYTICAL MECHANICS consders he sysem as a whole and s more general han he smple Newonan formulaon. The moon of a sysem s formulaed n erms of wo scalar quanes, WORK and KINETIC ENERGY. The mahemacal formulaon s ndependen of any specal sysem of coordnaes and reles on he prncple of vrual dsplacemens. When dealng wh mul-degree of freedom sysems, s ofen more expeden o derve he equaons of moon by usng he analycal mechancs approach. Ths mehod s also vald for connuous sysems, and n whch case no only he equaons of moon are obaned bu also he assocaed (naural) boundary condons. MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés

2 WORK AND ENERGY FOR A SINGLE PARTICLE r Pah C r+dr r r dr F dw Consder a parcle (pon mass) movng along he curve C under he acon of a force F. The poson of he parcle a any me s gven by he poson vecor r. If he parcle moves over an elemen of dsance dr, he wor (dw) s gven by he scalar produc: F dr = () If he parcle moves from poson o, along he pah C, he wor performed s W r = F dr r () For a parcle of consan mass, Newon's second law esablshes F d( mr )/ d = mdr / d = (3) Snce dr = r d, wh r as he velocy of he parcle, hen Eq. () s rewren as MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés

3 r dr r r W = F dr = m ( d) r d = m d d d r W = m r m r = T T (4) and T s he nec energy, T = mr (5) Eq. (4) shows ha W s he wor o change he nec energy of he sysem from T o T. In many physcal sysems, he force feld depends on he poson alone and s ndependen on he pah followed,.e. he force feld s a conservave feld, and herefore, can be derved from a poenal funcon V. Inroduce he defnon, dw = F dr = d V ( r) = V dr c where s he graden vecor operaor wh he followng componens n a caresan coordnae sysem (6) j = + + x dy z From (6), s easly nferred ha W = V r V r = V V ( ) ( ) c (7) (8).e., he wor performed by conservave forces s equal o he change n he poenal energy funcon V. Noe ha Eq. (8) MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 3

4 maes evden he conservave naure of he feld,.e. he change n poenal depends only on he begnnng and endng posons and no on he pah followed. In general, here are boh conservave and non-conservave forces acng upon a parcle. The non-conservave forces are energy dsspang forces such as frcon forces, or forces mparng (npung) energy no he sysem,.e. exernal forces. Non-conservave forces usually do no jus depend on poson alone and can no be derved from a poenal funcon. The expresson for wor s hen dvded no conservave and non-conservave pars, ( ) W = W + W = V V + W (9) c nc nc And from Eq. (4), W = T T, hen ( ) ( ) ( ) ( ) W = T T + V V = T+ V T+ V = nc = E E; where E= T+ V (0) E s he mechancal energy of he parcle and equal o he addon of s nec plus poenal energes. Eq. (0) ndcaes ha he wor performed by non-conservave forces s responsble for he change n he mechancal energy of he parcle. For a purely conservave force feld, W nc = 0. Then follows ha he mechancal energy s consan (nvaran) for all mes. d Wnc = 0 E = E, ( T + V) = 0 d MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 4

5 Degrees of freedom for sysems wh consrans Consder a smple pendulum x conssng of a mass m suspended by an L non-exensble srng of lengh L and free o oscllae n he xy plane. The θ coordnaes x and y defne he poson of he mass, and ye hese coordnaes are no ndependen snce y x + y = L denong a consran relaonshp. Thus only one coordnae s needed o express he poson of m. I can be eher x or y, bu s more naural o use he angle θ, snce x = lcos θ; y = lsnθ The mnmum number of ndependen coordnaes needed o descrbe he moon of a sysem s called he degree of freedom of he sysem. Thus, he pendulum s a sngle degree of freedom (SDOF) sysem. Three (3) coordnaes descrbe he poson of a parcle free o move n hree dmensons. If a sysem of N parcles mus sasfy c consran equaons, hen he number of ndependen coordnaes o descrbe he sysem s, n = 3 N c because each consran equaon reduces he degree of freedom of he sysem by one. The sysem s caegorzed as an n-degree of freedom sysem. The consran equaon wren before s reworded as MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 5

6 f ( x, y, ) = f ( x, y) = c A sysem n whch he consrans are funcons of he coordnaes or coordnaes and me (bu no veloces) s called holonomc. GENERALIZED COORDINATES The mnmum number of coordnaes necessary o fully descrbe a sysem s a se of generalzed coordnaes. These are ndependen and each coordnae represens one of he degrees of freedom of he sysem. Generalzed coordnaes, usually denoed as {q, q,... q n }, are no necessarly Caresan coordnaes. The selecon of he se of generalzed coordnaes {q ) =,..,n s evden n some problems. In oher cases where he coordnaes are relaed by consran equaons, coordnae ransformaons are requred o arrve o an ndependen se of generalzed coordnaes. MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 6

7 THE PRINCIPLE OF VIRTUAL WORK: STATICS CASE The prncple of vrual wor s essenally a saemen of an equlbrum sae of a mechancal sysem. Several defnons are needed a he ouse: A dsplacemen coordnae s a quany used o specfy he change of confguraon n a sysem. A consran s a nemac (usually geomercal) resrcon on he possble confguraon he sysem may assume. A vrual dsplacemen s an nfnesmally small and arbrary change of confguraon of a sysem CONSISTENT OR COMPATIBLE wh s consrans. Vrual dsplacemens are no acual dsplacemens snce here s no me change assocaed wh hem. If he acual coordnaes for a sysem are relaed by he consran equaon, (,...,,...,,..., ) f x x y y z z = c () n n n hen, he vrual dsplacemen (δ) mus be such ha ( x δ,..., δ..., δ,..., δ, ) f + x y + y z + z z + z = c () Noe ha me s held consan n Eq. (). The operaons concernng he varaon δ follow he rules of elemenary calculus. Expandng Eq. () as a Taylor seres and neglecng hgher order erms n δx..... δz n leads o n n MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 7

8 (,...,,...,,..., ) f x x y y z z n n n n f f f δx + δy + δz = c x y z = + (3) However, from Eq. (), f =c, and hus n f f f δx + δy + δz = 0 x y z = (4) Ths s he relaon he vrual dsplacemens (δx, δy, δz ) =,..n mus sasfy o be compable wh he sysem consran f=c For he smple pendulum, f = x + y = L, or x = L cos(θ) and y = L sn(θ); hen a varaon n confguraon leads o y + δ y = L sn(θ + δθ) = L ( snθ cosδθ + snδθ cos θ) = L sn(θ)+l cos(θ) δθ snce cos(δθ) and sn(δθ) δθ Thus, for small δθ, y +δy= y + x δθ and smlarly, δy = x δθ, = L cos(θ) δθ δx = -y δθ = -L sn(θ) δθ MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 8

9 Consder a parcle () aced upon by some forces wh resulan vecor R. If he sysem s n STATIC EQUILIBRIUM, he resulan force s zero and herefore, he wor performed over he vrual dsplacemen δ r mus be null,.e. δw = R δr = 0 If here are consrans n he sysem, hen (5) R = F + f (6) F s he resulan vecor of exernal forces appled on he where parcle and f s he resulan of he consran forces. Hence equaon (5) becomes δw = R δr = F δr + f δr = 0 (7) However, consran forces do no perform wor snce (by defnon) he dsplacemens do no have any componens n he drecon of he consran forces. As an example, consder a parcle movng on a smooh surface. The consran force s normal o he surface and he dsplacemens are parallel o he surface. Hence, follows ha, δw = F δr = Fx δ x+ Fy δ y+ Fz δz = 0 (8) In general, for a sysem of N parcles, he sum of he vrual wors over all parcles mus be zero, or MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 9

10 ( ) 0,... δw = δ W = F δ r = = N (9) Ths expresson also ncludes he cancellaon of he vrual wor done by nernal forces on rgd bodes (acon and reacon prncple). Equaon (9) s he expresson of he PRINCIPLE OF VIRTUAL WORK, and saed as: If a sysem of forces s n equlbrum, he wor done by he exernally appled forces hrough vrual dsplacemens compable wh he consrans of he sysem s zero. D'ALEMBERT'S PRINCIPLE: DYNAMICS CASE The prncple of vrual wor can also be exended o he sae of dynamcs (moon),.e. dynamc equlbrum. If here are some unbalanced forces acng upon a parcle m, hen accordng o Newon's nd law, he force resulan vecor mus be equal o he rae of change of he lnear momenum,.e F f p d mr ( ) + = = (0) Thn of an nera force (reversed acceleraon) as a force whose magnude equals he me rae of change of he momenum vecor p, s collnear wh bu acs n he oppose drecon. If such a force s appled o he parcle, hen one can express he dynamc equlbrum condon as d F + f p = 0 () MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 0

11 D'Alember's prncple saes ha he resulan force s n equlbrum wh he nera force. Followng pror reasonng, he vrual wor of he exernal and nera forces mus also be nl,.e. () ( F p ) δ r = 0 and for a sysem of N parcles, N = F p δr = δw = 0 (3) F mx δ x + F m y δ y + F m z δ z = 0 ( ) ( x ) ( y ) ( z ) Thus, he prncple of vrual wor for any sysem of parcles s expressed as δ W δw δw = exernal forces + nera forces = 0 (4) HAMILTON'S PRINCIPLE Hamlon's prncple s (perhaps) he mos advanced varaonal prncple of mechancs. The prncple consders he moon of a whole sysem beween wo nsans of me, and, and s herefore an negral prncple. One remarable advanage of hs formulaon s ha s nvaran o he coordnae sysem used (Prncple of maeral frame ndfference). Consder a sysem of N parcles of consan mass. The sysem may be subjec o nemacal (consran) condons. The vrual wor expresson n conjuncon wh D'Alember's prncple esablshes The suff from whch grea heores are made of. MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés

12 N ( ) δ δw = m r F r = 0 oal = (5) N Le δ W = ( F δ r) = (6) be he vrual wor done by he exernal forces on he sysem. The operaons d/d and δ are nerchangeable (lnear operaors). Then Hence, d ( r δ r) dδ r dr = rδr + r = rδr + r δ = d d d = rδr + r δ ( r ) = r δr + δ ( r r ) d ( r δ r) r δr= δ ( r r ) (7) d Mulply Eq. (7) by m and sum over he whole se of parcles o oban d ( r δ r) mr r = m m r r ( δ ) N N N ( δ ( )) = = d = N d ( r ) N δ r = m δ m r r d = N d ( r δ r) = m δ T = d (8) MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés

13 where T s he nec energy of he sysem (whole se of parcles),.e. N N T= T = m ( r r ) Inserng Eqs. (6) and (8) no Eq. (5) render, N d δt + δw = m ( r δ r) = d (9) The nsananeous confguraon of a sysem s gven by he values of n generalzed coordnaes. These values correspond o a pon n n-dmensonal space nown as he confguraon space. The sysem changes wh me racng a pah nown as "rue" n he confguraon space. A slghly dfferen pah, nown as he vared pah, s obaned f a any gven nsan a small varaon n poson δ r s allowed whou an assocaed change n me,.e. δ =0. δr True pah Vared pah The spulaon s made, however, ha a wo nsans and he rue and vared pahs concde. Tha s, δ = = = r 0 a and a Mulply Eq. (9) by d and negrae beween and o oban Ths means ha vared pah mus sll sasfy he spaal consrans MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 3

14 N ( δt + δw) d = m ( r δ r) snce δ r Thus, N = = = ( ) = 0= δ r ( ) d d d d m ( r δ r) d d = 0 N d = m ( r δ r) = d ( T + W) δ δ d = 0 (3) If he appled forces are dvded no conservave and nonconservave, hen a poenal energy funcon exss such ha, δ W = δw + δw = δv + δw (3) c nc nc Inroduce he Lagrangan funcon, L=T-V (33) Then equaon (3) for a conservave sysem (δw nc = 0) shows ( L) δ d = 0 (34) whle for boh conservave and non-conservave forces we have ( L+ W ) δ δ nc d = 0 (35) MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 4

15 If he sysem has only holonomc consrans,.e. geomercal or dependng only on he coordnaes bu no her me dervaves, hen one can nerchange he δ (varaon) and negral n Eq. (34). Thus, for a conservave sysem, ( ) Ld 0 (36) δ = Ths s he mahemacal saemen of HAMILTON'S PRINCIPLE. In words can be explaned as follows: Only he rue pah renders he value of he negral saonary (a mnmum) wh respec o all possble neghborng pahs ha he sysem may be magned o ae beween wo nsans of me. Noe ha Eq. (35) corresponds o a prncple of leas acon. HAMILTON'S PRINCIPLE IS A FORMULATION AND NOT A SOLUTION OF A DYNAMICS PROBLEM. MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 5

16 LAGRANGE'S EQUATIONS OF MOTION FOR HOLONOMIC SYSTEMS In a sysem of N parcles wh c holonomc consrans, he dependen varables r n erms of n generalzed coordnaes (q ) and me () are expressed as, r = r ( q, q,... q, ); = = n,... N, n N c (36) where n s he number of degrees of freedom of he sysem. Veloces are obaned by dfferenaon of Eq (36) as n dr r r r r r r q... q = = + + n + = q + (37) d q qn = q and he sysem nec energy T s N T = mr r = = N n n r r r r = m qr + qs + r qr = s qs = = = N n n n r r r r r r T = m qr qs + qs + = r= s= qr qs s= qs (38) Noe ha T T( q, q,..., q, q, q,..., q, ) = (39) n.e., he sysem nec energy depends on he generalzed dsplacemens and veloces as well as n me. n MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 6

17 In addon, for conservave force felds, he poenal energy funcon V depends only on poson and me,.e. (,,...,, ) V V q q q = (40) The vrual wor performed by non-conservave forces (boh dsspave and exernal) s he produc of "generalzed forces" Q acng over n generalzed dsplacemens, δq. The drecons of he generalzed forces concde wh he drecons of he generalzed dsplacemens, hus n ( ) δ W = Q δq + Q δq +... Q δq = Q δq (4) nc n n s s s= Noe ha he generalzed force Q may NOT need o acually represen a force or a momen. However, he produc Q δq MUST have uns of wor. Subsue Eqs. (39), (40) and (4) no he generalzed Hamlon's prncple, Eq. (35), o oban ( T V + W ) δ δ δ nc d = 0 (35) n Snce, δv = n n V T T δ q ; δt δ q + δ q = q = = q q And recall ha δ = 0,.e. me does no vary whle obanng he vrual changes n energy. Then Eq. (35) becomes MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 7

18 n T T V 0 = δq + δq δq + Q δq d = q q q (4) The erms nvolvng δq followng manner, (Use δ q = ( δ q) ) may be negraed by pars n he d d ( δ q ) d = = T ] T T δqd d δq d T δqd q q d q d q (43) The frs erm on he RHS vanshes snce (usng Hamlon's Prncple) he nal and fnal confguraon of he sysem (a and, respecvely) are nown,.e., δ Thus, Eq. (4) s rewren as, ( ) δ ( ) q = 0, q = 0 n 0 d T T V = δq δq δq Q δq d d q + + = q q Or n T T V 0 δ q d = Q d d q + + = q q (44) MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 8

19 The varaons {δq } =,...n, are ndependen (correspondng o he n degrees of freedom n he sysem). Hence, Eq. (44) s rue (sasfed a all mes) only when he braceed expresson vanshes for each degree of freedom,.e. d T T V Q 0 d q + + = q q d T T V + = Q = d,... n q q q (45) Eqs. (45) are nown as LAGRANGE'S EQUATIONS OF MOTION. The soluon of hese equaons s equvalen o he saemen ha Hamlon's Prncple s also sasfed. If V = V(q ) only, defne he Lagrangan funcon, L=T-V, V and snce =/ 0 q, hen Eq. (45) becomes d L L = Q = d q,... n q (46) MODIFIED LAGRANGE'S EQUATION FOR SYSTEMS WITH VISCOUS DAMPING If some of he exernal non-conservave forces are of vscous ype,.e. proporonal o he velocy, hen he vscous dsspaed power (P v ) s a general funcon of he veloces,.e. ( q, q,..., q ) = (47) v v n MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 9

20 The n-equaons of moon usng he Lagrangan approach are, d T T V v Q d = q + + = q q q (48),... n REFERENCES Merovch L., ANALYTICAL METHODS IN VIBRATIONS, pp Crag R., STRUCTURAL MECHANICS, pp. 5-6, A THOUGHT: "Those who have medaed on he beauy and uly of he general mehod of Lagrange - who have fel he power and dgny of ha cenral dynamc problem whch he deduced from a combnaon of he prncple of vrual veloces wh he prncple of D'Alember - and who have apprecaed he smplcy and harmony whch he nroduced by he dea of he varaon of parameers, mus feel he unfoldng of a cenral dea. Lagrange has perhaps done more han any oher analys o gve exen and harmony o such deducve researches, by showng he mos vared consequences may be derved from one radcal formula; he beauy of he mehod so sung he dgny of he resuls, as o mae of hs grea wor a nd of scenfc poem." W.R. Hamlon MEEN 67 - Handou 4 Elemens of Analycal Mechancs 008 Lus San Andrés 0