The meaning of d Alembert s principle for rigid systems and link mechanisms

Transcription

1 De Bedeutung de d Alembertchen Prnzpe für trre Syteme und Gelenkmechnmen (Fortetzung) Arch der Mth und Phy () (90) 98-6 The menng of d Alembert prncple for rgd ytem nd lnk mechnm By KARL HEUN n Berln Trnlted by D H Delphench (Contnuton) 6 Euler equton of moton for rottng bode From equton () nd ( ) n Tble V the bc knetc equton of rottng rgd ytem re: M h M nd M k dm nce the totl moment of the recton nh In the formul: M m x x ɺ one need only to replce xɺ wth the expreon x nd um oer ll m-pont of the body n order to obtn the mpule equton n explct form Howeer one h (cf the ntroducton): x ( x) x ( x ) x Ordnrly one decompoe x long three rectngulr xe tht re rgdly lnked wth the ytem when one et: x + + The correpondng component of the ector n whch we he et: M M M M follow n tht wy nmely: m( + ) m m m ( + ) m m m( + ) m m

2 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) m ( + ) A m ( + ) A m ( + ) A to bbrete nd he et: m D m D m D nd he referred to thoe quntte the moment of nert nd moment of deton The knetc mpule equton then tke on the uul form: (8) M h A D D M h A D D M h A D D In order to contruct the Euler equton we mut now form the dfferentl quotent dm / Inted of dong tht one cn lo dfferentte the elementry ector: wth repect to tme nd get: M ( x x) ( x ) x dm ( x xɺ ) + ( x x) ɺ ( xɺ ) x ( xɺ ) x ( x ) xɺ Howeer one obouly h x xɺ 0 nd xɺ 0 nce the ector n queton re perpendculr to ech other A reult one wll he: dm ( x x) ɺ + ( xɺ ) x ( x ) xɺ ( x x) ɺ + ( xɺ ) x + M nd correpondngly: dm (9) dm + M n whch the brcket round the derte of M ugget tht one mut conder only the quntty to be rble n tht dfferentton In our notton the Euler equton wll then red: dm (0) M k + M They were frt publhed n tht form (nturlly wthout the ymbolm of ector nly) by Lgrnge n h Mécn nl nd ed t pp 9 n whch he dered them

3 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) from the knetc prncple of rtul dplcement (cf no of tht book) In tht book Lgrnge employed the knetc energy E of rottng ytem whch wll he the form: E m ( x x) + + D D D ( A A A ) due to the equton xɺ x From equton (8) one wll then he: M M M A reult (or relly due to the eentl dfference between the derton) for d d d Lgrnge the quntte nd ppered n plced of the dm component of the relte elocty If one refer the ector to the prncple xe then the moment of deton n the expreon for M wll nh nd one wll get the uul Euler equton: d A + ( A A ) M k d () A + ( A A ) M k d A + ( A A ) M k from equton (0) One cn wrte down equton (9) drectly nce t follow mmedtely from the dm prncple of relte moton obouly the ector of the relte chnge n elocty of M wth repect to the rottng ytem whle M the ector of the octed gudng elocty (Führunggechwndgket) Strctly pekng Euler employed the me lne of reonng for the derton of h equton but wthout clothng t defnte nlytcl form 7 Lgrnge trntty equton for rgd ytem D Alembert prncple n the ntegrl form tht Lgrnge nd Hmlton employed: () [ A ] t0 t t δ ( δ E + δ Ak ) t0

4 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 4 ge re to remrkble dffculty when one employ ytem of elocte tht expreed by knemtcl prmeter tht re not t the me tme the tme derte of coordnte nd whch Lgrnge w the frt to clerly recognze nd oercome wth h own kll (for the ce of rottng rgd ytem) Nmely the rton δ n the expreon: () δe δ + δ + δ mut be trnformed n uch wy tht they wll contn only the δθ nd the complete tme derte of thoe quntte Lgrnge (Mécn nl nd ed t pp 9) rred t tht by employng the relton tht ext between the nne drecton cone Inted of tht we hll purue more drect nd conenent pth by trtng mmedtely wth the concept of the ytem of poble elocte A reult of the equton xɺ x we wll he: dx dθ x nd δ x δθ x By ryng nd dfferenttng th one wll get: δ dx δ dθ x + dθ δ x nd dδ x dδθ x + δθ dx Now nce one obouly h δ dx dδ x t wll follow by ubtrctng the foregong equton tht: ( δ dθ dδθ ) x dθ ( δθ x) δθ ( dθ x) ( dθ δθ ) x or nce x completely rbtrry: (4) δ dθ dδθ dθ δθ Tht Lgrnge trntty equton for rottng rgd ytem It n mmedte conequence of the knemtcl expreon for the ytem of elocty We conclude from th tht eery chrctertc form for ytem of elocte functon of eentlly-knemtcl prmeter mut correpond to pecl trntty equton whch lkewe chrctertc of the mterl ytem The relton between the x component: (5) δ dθ dδθ + dθ δθ dθ δθ δ dθ dδθ + dθ δθ dθ δθ δ dθ dδθ + dθ δθ dθ δθ follow from equton (4) Lgrnge communcted n loc ct We ubttute thoe lue n conjuncton wth equton () n the ntegrl expreon:

5 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 5 (6) [ A ] t0 Now one h the relton: δ t t ( δ E + δ Ak ) t0 δe δ + δ + δ M δ n whch from equton (4) one h et: (7) δ d δθ δθ + A reult: d d dm δe M dθ + M δθ ( M d θ) δθ M δθ Equton (6) wll then go to: t [ A ] t0 δ t dm + ( M ) M k δθ ; t0 one wll then he One mut then he: M δθ δ A nd M δθ δ A k nce trnlton re excluded dm + M M k from whch one wll rre t the Lgrngn form of the equton of moton by decompong the ltter equton nto component: (8) d + M d + M d + M k k k 8 Lgrnge knetc equton n generl poton coordnte We next ume n rbtrry ytem of poble elocte whch we ugget by the ymbolc equton: (9) xɺ func ( ε ε ε qɺ qɺ qɺ )

6 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 6 The ε n th re ector n the uul ene whle the q re rel mutully-ndependent poton coordnte The number of the ltter wll be equl to the number of degree of freedom uch tht the moton of the ytem not retrcted by ny condton equton The ector ε re generlly ngle-lued functon of thoe coordnte It follow from the ymbolc equton (9) tht: (0) δ x func ( ε ε ε δ q δ q δ q ) For free mterl pont one lwy h: nd correpondngly: A reult: or f we et: uul to bbrete: xɺ ε qɺ + ε qɺ + ε qɺ δ x ε δ q + ε δ q + ε δ q δ A xɺ δ x S ε xɺ p δ A S p ε xɺ δ q δ q The quntte p re lner functon of the quntte qɺ We hll now mpoe the condton on the functonl relton n equton (9) or equton (0) whch concde wth t tht the clr quntty δ A tht dered from t mut tke the form: () δ A S p δ q nd tht the p mut be lner functon of the qɺ Wth tht umpton one cn lwy et δ A Sh δ q nd n tht wy the bc equton for mpule effect: (I) δ A δ A h wll tke the mple form: () p h ( ) In the equton for tme-ryng force: (II) [ A ] t0 t t δ ( δ E + δ Ak ) t0

7 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 7 the knetc energy E of the totl ytem whch by umpton expreed by equton () qudrtc functon of the qɺ Tht equton mut lo remn ld when δq replced wth dq A reult: () E S q ɺ qɺ Wthout tht one would he: δ E S δ qɺ + S δ q ɺ q q In complete nlogy wth the equton δ A h we further et: S h (4) δ A k k δ x δ q nd lo for tme-ryng force Sh δ q nd from the precedent et by Hertz (Prnzpen pp 8) cll the quntte k k k the component of the Lgrngn force whch we would lke to ymbolclly denote by k A known Hertz clled the ymbol k ector relte to the totl ytem Now nce the q coordnte e the qɺ re complete derte wth repect to tme one wll lwy he the equton: δɺ q nd the bc equton (II) wll go to: dq d q δ δ S p t δ q t0 t t de d S δ q + S + + k δ q ɺ ɺ dq q t t 0 0 q Tht equton cn be fulflled dentclly for rbtrry lue of the δq only when one h: (5) p ɺ q nd dp (6) k q Thee re the Lgrnge equton I would lke to exprely pont out tht the mpule equton tht one get from combnng formul () nd (5) nmely: (7) ɺ q h

8 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 8 go bck to Lgrnge (Méc nl nd ed t pp 8) nd not Nen Routh remrked n h Rgd Dynmc nd ndeed one fnd them n the cted plce n precely the form tht Routh ge to them Nmely the extence of functon Ω umed there tht wll yeld the component: h Ω q Clfford ge derton of the Lgrngn equton n h Element of Dynmc ( of the pothumou publcton pp 8) whoe bc de we hll repet here Let the ytem of elocte be dependent upon only two coordnte q nd q Wth tht umpton one wll he: ε qɺ + ε qɺ Clfford then frt et qɺ nd qɺ 0 nd then qɺ 0 nd qɺ The correpondng d lue of re: nd Now he proed tht dq d The energy of the ytem dq h the lue: E ( ε ε qɺ + ε ε qɺ qɺ + ε ε qɺ ) It then follow from th tht: ɺ q ε ɺ q ε Now one h the followng equton wth no further condton: q dε nd q dε It wll then follow from the energy equton: tht: or Howeer one h: q q q E q d ε q q d ε d ɺ q dε d + ε d ɺ q dε d + ε

9 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 9 One wll get the Lgrnge equton from th drectly: d qɺ q d ε d qɺ q d ε One cn rgorouly dere the Lgrnge equton for free pont knemtclly In tht mple ce wll then he the form: ε qɺ + ε qɺ + ε qɺ It wll then follow tht ε p nd dfferentton of th wth repect to tme wll ge: d ε dp d ε Snce complete derte wth repect to tme due to the ntegrblty condton one wll he the equton: ε ε q q Wth tht: ε ε ε ε ε ε qɺ + qɺ + qɺ qɺ + qɺ + qɺ q q q q q q q or: d ε q One mmedtely obtn the Lgrnge equton from th n the knemtcl form: d ε dp E q The conceptul menng of the Lgrnge equton h lredy been the ubject of repeted netgton Howeer they eem to he yelded no tfyng reult One therefore eentlly ddree the queton of how they emerge from the mpule equton p h If one dfferentte th wth repect to tme then the equton tht one obtn z: Dp Dh mut be dentcl to the equton: dp k q Now one h: p S ε ɺ q

10 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 0 ε dp Sε qɺɺ + SS qɺ qɺ q or f one conder the econd term to be / q : dp S SS ɺɺ ɺ ɺ ( ) ε q + γ λ qλ q λ We cn denote the frt term on the rght-hnd de of th equton by (dp ) n the derton of the Euler equton f the prenthee ugget pure mpule dfferentton by whch the coordnte wll remn unchnged whch content wth the concept of mpule We wll then get: Dp (dp ) + C k The entre dffculty now come down to the nterpretton of the functon: C SS γ ( ) λ q ɺ λ q ɺ λ ( ) In ll lkelhood thee functon C n whch the coeffcent γ re dentcl wth the λ Chrtoffel ymbol re the component (or rther mple combnton of component) of centrfugl ccelerton Howeer up to now I he not ucceeded n prong tht nd n tht wy clrfyng the pecl nture of thng completely Perhp thoe remrk wll ere to tmulte further netgton nto th tuton whch not t ll neentl n knetc λ 9 Explct form of Lgrnge equton A known Hmlton employed not SS SS only the functon E ε q ɺ q ɺ but lo the recprocl functon E η p p whch lnked wth the ltter by the lner relton ε p We hll now employ the equton: ɺ p F F qɺ q p n order to get n explct repreentton of the Lgrnge equton of moton whch we hll ue the b for our tudy of the knetottc of ytem of lnk We next he: q q qɺɺ d F p F dp F S + S qk p p p p ɺ It follow from the uul form of the Lgrnge equton:

11 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) dp k Wth tht the preou equton wll go to: F q (8) dq S F F F F F + S k p q p p p q p p The frt term on the rght-hnd de of th equton homogeneou functon of degree two n the quntte p p p nd thu lo the quntte qɺ qɺ qɺ well The followng term contn only the coordnte q q q n ddton to the generlzed Lgrngn force component k k k For tht reon we cn lo wrte equton (8) n the followng form: (9) dq SS α S qɺ qɺ + η k ( ) λ λ λ The coeffcent quntte α ( ) λ α nd η re known functon of the coordnte n th The ( ) λ cn be expreed mmedtely n term of the Chrtoffel ymbol of the λ econd knd whch re denoted by Howeer t doe not eem necery to go further nto thoe reltonhp t the moment The η re the coeffcent n the recprocl functon F 0 Rodrgue-Cyley poton coordnte for rgd ytem In order to horten the derton much poble I hll now ppel to the Theore de Kreel by F Klen nd A Sommerfeld In tht book (pp nd 4) the component of the ector of rottonl elocty re expreed n term of the four quternon component A B C D n the followng wy: (40) DAɺ ADɺ ( BCɺ CBɺ ) DBɺ BDɺ ( CAɺ ACɺ ) DCɺ CDɺ ( ABɺ BAɺ ) Actully there re complex combnton of the on pp 4 of tht book but they wll mply equton (40) wth no further umpton We hll now ume tht A B C re the component of ector λ Equton (40) cn then be combned nto ngle ector equton nmely: (4) dλ µ λλ µ ɺ

12 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) n whch we he et: λ λ λ µ to bbrete If we ntroduce yet nother ector by wy of the equton: then we wll he µ + µ nd: λ µ (4) + ( ɺ ɺ ) Cyley publhed tht beutful equton [Cmb nd Dubln J (846)] whch expree n term of the necery nd uffcent number of coordnte nd then contructed ery elegnt theory of the rotton of rgd bode Although Somoff knemtc refer to tht book t h tll not found the ttenton tht t deere n our opnon Equton (4) mple the component of the rottonl elocty n the cler nd ymmetrc form: [ ɺ ( ɺ ɺ )] + (4) [ ɺ ( ɺ ɺ )] + [ ɺ ( ɺ ɺ )] + If one ubttute thee quntte nto the lue for the knetc energy E then one cn dere Lgrnge equton of moton from the expreon thu-obtned wth no further dcuon nce the re ndependent poton coordnte The ector B Jut we could repreent the energy E of ytem wth fnte number of degree of freedom n the generl Lgrngn coordnte q q q tht cn lo be done for the ytem ector B Howeer we would lke to retrct ourele here to the elementry ector B for free mterl pont (m ) In the defnng equton: B xɺɺɺ x one then et: xɺ ε qɺ + ε qɺ + ε qɺ nd correpondngly: xɺ ε qɺɺ + ε qɺɺ + ε qɺɺ + ɺ ε qɺ + ɺ ε qɺ + ɺ ε qɺ Crryng out tht ubttuton wll mmedtely yeld:

13 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) B ε ε ( qɺɺɺ q qɺ qɺɺ ) + ε ε ( qɺ qɺɺ qɺ qɺɺ ) + ε ε ( qɺ qɺɺ qɺɺɺ q) + ( ε ɺ ε + ε ɺ ε ) qɺɺɺ q + ε ɺ ε qɺ + ( ε ɺ ε + ε ɺ ε ) qɺ qɺɺ + ε ɺ ε qɺ + ( ε ɺ ε + ε ɺ ε ) qɺ qɺɺ + ε ɺ ε qɺ If one now mgne tht the equton: εɺ q εɺ q εɺ q re ld then one wll ee wth no further umpton tht B cn be put nto the followng form: B ε ε ( qɺ qɺɺ qɺ qɺɺ ) + ε ε ( qɺ qɺɺ qɺɺɺ q) + ε ε ( qɺɺɺ q qɺ qɺɺ ) + H n whch H men homogeneou functon of degree three n the elocty component qɺ qɺ qɺ If one then et: then one wll he: ε ε ε ε ε ε ε ε ε ε p + ε p + ε p uch tht one cn lo expre B n term of the quntte qɺ qɺ qɺ nd q q q The quntte ε nd / q tht enter nto tht expreon re functon of the q We would lke to determne B the ytem ector only for rottng rgd bode In order to do tht we ubttute the lue of xɺ nd xɺɺ nto: nmely: nd obtn: B m x ɺɺɺ x xɺ x nd xɺɺ ɺ x + ( x) x B m ( x ɺ ) x m( x) + m( x ) fter ome reducton Howeer we he: E m ( x ) m( x) m x x Hence we wll he: B E + G

14 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 4 when we et: G m ( ɺ x) x to bbrete The component of tht ector re: n whch: G ( ɺ ɺ ) T + ( ɺ ɺ ) T + ( ɺ ɺ ) T G ( ɺ ɺ ) T + ( ɺ ɺ ) T + ( ɺ ɺ ) T G ( ɺ ɺ ) T + ( ɺ ɺ ) T + ( ɺ ɺ ) T T λµ m xλ xµ If the ngulr ccelerton zero or f t ector le n the me drecton the ngulr elocty then the ector G wll nh nd B wll then contn the drecton of the momentry x B wll then be proportonl to the knetc energy well the ngulr elocty of the ytem The component of B wll then be homogeneou functon of degree three n the component of the ngulr elocty n th pecl ce Euler equton for br chn On pp 54 of the frt olume of Theore de Kreel by F Klen nd A Sommerfeld we fnd the followng ewpont of generl knetc nteret expounded: Euler equton occupy n entrely ngulr poton n the ytem of mechnc nd do not ubordnte themele to the generl type of mechncl dfferentl equton tht Lgrnge preented Nether t poble to exhbt equton for rbtrry mechncl ytem tht would fford dntge tht re mlr to the one tht Euler equton fford for rgd bode For u the queton of whether knetc equton wth the typcl Euler form cn be exhbted for gen ytem cn be nwered n defnte wy n connecton wth the generl rgument tht h been preented up to now Nmely f one ucceed n fnd the nlytcl expreon for the ytem of elocte tht chrcterze the ytem n term of the necery nd uffcent number of purely-knemtcl ector (e prmeter) nd preent the requte trntty equton for the ltter n ddton then d Alembert prncple n the ntegrl form: or n the mpler form: t [ A ] t0 δ t ( δ E + δ Ak ) t0 δ A δ A h for mpule wll lwy yeld the requte number of ector equton whch wll belong to the me genre Euler equton of moton (mpule rep)

15 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 5 In generl one cn exhbt Lgrnge equton (n the nrrow ene of the term) only when the ytem of elocte cn be repreented by the necery nd uffcent number of coordnte nd ther frt derte wth repect to tme If one then rre t both repreentton z the knemtcl nd the geometrcl for certn mterl ytem then nothng wll tnd n the wy of exhbtng the bc knetc equton n both form long one necerly ume tht the trntty equton re known Nturlly the lt requrement uperfluou for the equton of mpule An epeclly mportnt cl of ytem for whch equton of Euler type ext from the outet re the br chn tht re type of engneerng mchne (n the generl ene) The lnk couplng of rgd ubytem re generl ery retrcted n prctce They eentlly reduce to bll jont cylndrcl pn gude nd plnr trght gude In wht follow we hll conder only bll jont becue the other two ce cn be ely reduced to tht ce or led n ny eent to knetc equton tht do not dete from the one tht he been mnly treted here In order to mke the conceptulzton of ytem more prece we mgne our trtng pont to be fxed bll jont (or een eerl of them whch wll not complcte the tretment) A old body of rbtrry hpe wth bll pn mght be n exmple of tht The frt body couple to econd one or other one n the me wy nd o on Tht mult-component br chn cn be open e the lt component not coupled to the frt one or cloed when one contrn t to moe wthn precrbed gude For the ke of mplcty n the followng clculton we would lke to conder br chn tht open t the end nd cont of two rgd component nce tht ce lredy uffcent to mke the chrctertc of the bc knetc equton more ntute The frt lnk n the chn cn exhbt only rotton whch cn be repreented by rotton ector The correpondng ytem of elocte therefore xɺ f we denote the ector tht determne the poton of n rbtrry mterl pont of the ubytem by The reference pont for obouly the center of the fxed bll jont We drw ector c from the me pont to the center of the mong bll jont nd cll the ector the elocty of the econd pont c ɺ One then h c ɺ c the pont of the econd ytem component mght be etblhed n pce by the equton: x c + The relte ector re then referred to the center of the mong bll jont If we then denote the octed rotton ector by then we wll he the equton: x ɺ c + nd we wll get the octed elementry moton n the form: δ x δθ δ x δθ c + δθ

16 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 6 The knetc mpule equton for the combned ytem cn be preented wth the help of thoe relton For mterl pont of the frt ubytem (wth unt m) one h: δ A xɺ δ x δθ or δ ( δθ )( ) ( δθ )( ) A Howeer the moment of the elocty of tht ytem of pont : M ( ) ( ) ( ) A reult one wll he: (45) δ A M δθ For pont (m ) of the econd ubytem we he: δ xɺ δ x [( c ) + ( )][( δθ c ) + ( δθ )] A or when th deeloped: δ c δθ c + c δθ + δθ c + δθ A The frt nd fourth term n th expreon for δ A cn lo be wrtten: c δθ c ( δθ )( c c ) ( c δθ )( c ) c ( c ) δθ M c δθ δθ ( δθ )( ) ( δθ )( ) ( c ) δθ M δθ n whch M nd M re uffcently well-defned elocty moment Interpretng the mddle two term n the expreon for δ A not mple Here we would lke to ntroduce two new ector A nd C whoe mgntude nd drecton cn be deduced from the equton: c δθ A δθ δθ c C δθ In tht wy we wll get the expreon: (46) δ A M c δθ + M δθ + C δθ + A δθ

17 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 7 Now we cn go from equton (45) nd (46) to the correpondng ytem equton by ummng oer the elementry quntte We et: Σ m M M Σ m M M Σ m A A Σ m C C nd get: (47) δ A M + M c + M δθ + M + A δθ Furthermore when one conder the ytem of elocte for the ppled mpule one wll he: δ h δθ h δθ A h δ h δθ + h δθ A h c h δθ + h δθ nd when one goe oer to the ubytem: n whch one h et δ A h A h Σ h δθ M h δθ Σ Σ δ c h δθ + h δθ + δθ c h δθ M h Σ h h If one ubttute thee lue nto the equton: then one wll get the mpule formul: δ A h δ A (48) M h + c h M + M c + C M h M + A The x quntte nd cn be determned from the equton once the mpule tht ct upon the ytem re gen Nturlly the ector M M A nd C depend upon the moment of nert nd the moment of deton of the ubytem The trnton to Euler equton of moton reltely mple now We he only to defne the expreon for the quntty δe n the equton:

18 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 8 t [ A ] b t tb δ [ δ E + δ A] Now for the nddul mterl pont (wth me m nd m ) one h: t well : Hence: E ( )( ) nd δe M δ E [( c ) + ( )][( c ) + ( )] δe M + M c + C δ + M + A δ The trntty equton re: δ d δθ + δθ δ d δθ + δθ If one ubttute thee lue nto the ntegrl equton nd conder the fct tht the reducton of the tme-ryng force k nd k the me t n the ce of mpule then one wll get Euler equton of moton for the two-component ytem of lnk n the ector form: d R + R M k + c k (49) d R + R M k n whch one et: M M C + c + R M + A R k k to bbrete d A dc The ector nd he certn nlogy wth the combned centrpetl ccelerton whch Corol ntroduced n h conderton of the relte moton of the nddul m-pont Nturlly we cn lo preent the Lgrnge equton n generlzed coordnte for our br chn tht cont of rgd component In order to do tht we cn exhbt the Rodrgue-Cyley expreon for ny ubytem defne the knetc energy E of the entre ytem nd rre t the explct equton of moton from the known precrpton In the pecl exmple tht were worked through boe the reult w x Lgrnge equton tht would uffce to determne the moton of the ytem completely

19 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 9 Equton (49) re lwy preferble when the moton reult n the bence of externl force Tht theoretclly-nteretng ce h no menng for engneerng mechnc nce the force of frcton re neer bent n tht relm Howeer n the tretment of knetc mchne problem one wll lo be jut rrely clled upon to conder generl ytem of elocte of the knd tht we umed n the dcuon boe Howeer n ny eent the well-defned conceptulzton of generl procee of moton wll lo he gret gnfcnce when ther relzton n prctce qute remote E Determnng the recton Introducton of the ectonl recton If mple rgd body found n t mot generl form of moton then the coheon of t prt cn be tken dntge of n rou wy For the del tructure tht we cll rgd ytem thoe nternl force cn ech ume rbtrry lue nce we tctly ume tht ther robutne unbounded Moreoer f old body were put nto generl tte of moton (z trnlton nd rotton) then the nternl tree would ttn lue o lrge tht the cohee force t the nddul locton or n certn urfce domn would no longer uffce to preere the connectty of the prt The body would htter nd new tte of moton would re Howeer een when we gnore tht cttrophe for rel ytem (whch re umed to be rgd) tht correpond to well-defned elocty tte n ncree n the tree would occur for ncreng elocty whch would no longer be permble f the hypothe of the rgdty of the ytem were to be mntned nce eltc or pltc deformton of pprecble mgntude would occur The me thng true to greter degree for ytem of lnk tht cont of rgd component (e they re umed to be rgd n the frt pproxmton) In tht ene the knetc of mchne nd n prtculr combuton engne wth prt tht go bck nd forth he lo drected pecl ttenton to the tree long wth concept of moton The qunttte determnton of the recton n mong ytem of me then n mportnt chpter n engneerng mechnc nd uch deere to be treted ytemtclly In order to fx the concept of ytem recton we mgne tht the entre connected mterl ytem decompoed nto two prt by urfce ecton wthout lterng the force-ytem nd the elocty tte tht ext n ny wy If the phycl connecton long the eprtng cut urfce were uddenly cnceled then ech ubytem would generlly he to begn new form of moton t tht moment All of the recton of one pece would combne nto reultnt ytem tht from d Alembert prncple or Newton bc lw of the equlty of cton nd recton would be equlent to the reultnt ytem of the recton of the other pece n the oppote ene The decompoton of the ytem nto two pece correpond to the decompoton of the totl energy E nto two prt E nd E uch tht one would he: E E + E We would lke to further ume tht the entre ytem determned by coordnte uch tht for the mpule effect the Lgrnge equton wll be:

20 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 0 p ɺ q h ( ) We lkewe dere quntte p of knetc chrcter from the energy component E nd E when we et: E p q ɺ p E q ɺ It wll then follow drectly from d Alembert prncple tht: p h r p h r n whch the quntte r r men component of the reultnt ytem of recton when expreed n term of the generl coordnte q Snce one mut he r r one of the foregong ytem of equton y: (50) r h p wll uffce to determne thoe recton component One determne the mpule component h tht re requred to clculte the r from the formul: δ h δ x A h Σ when one expree the δ x n t n term of the q nd δq from whch one wll obtn the equton: δ A h S h δ q If the ytem ubjected to the effect of tme-ryng force (k) then one wll employ the uul Lgrnge equton: pɺ k ( ) q for the determnton of the component nd the forementoned plttng of the ytem wll yeld the recton formul: pɺ k nd q pɺ k + q n whch one mut once gn et:

21 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) p E ɺ q nd p E ɺ q One cn lo employ Euler equton of moton n order to determne the correpondng component of the ectonl recton n n entrely mlr wy In the ce of mple rgd body tht not under the nfluence of externl force tht pth wll dete from the mple one 4 Explct repreentton of the ectonl recton For mpule problem the ectonl recton depend upon only the poton coordnte nd the externl mpule tht ct upon mterl ytem Howeer f tme-ryng force ct upon the ytem then the elocty tte wll lo be crucl for the ectonl recton In the equton for the recton whch we expreed n term of generlzed coordnte boe n the frt ce we cn elmnte the generlzed elocty component qɺ by men of the equton p h nd n the econd ce the ccelerton component qɺɺ cn be elmnted by the ue of the Lgrngn equton of moton nd n tht wy we wll rre t explct repreentton for the component of the ectonl recton Tht exceedngly mple for the mpule recton: r h p If we trnform the knetc energy E by ntroducng the h p n plce of the qɺ n Hmlton recprocl functon F whch now homogeneou functon of degree two n the h then we wll he: qɺ One ubttute thee lue n the equton: p F h S η ɺ q whch lner n the qɺ nd obtn the fnl equton: (5) r h η S F h for the explct repreentton of the ectonl recton for mpule The coeffcent η re known functon of the poton coordnte q q q The nlogou conderton for tme-ryng force cn be mplfed gretly when we ume Lgrnge equton of moton n the explct form whch were repreented

22 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) by equton (8) or (9) n no 9 Tht wll permt u to ubttute the quntte qɺɺ qɺɺ qɺɺ drectly nto the recton formul: (5) We wll then he: nd reult: pɺ k + pɺ q pɺ S η ɺ q SS ( ) λ λ λ S γ qɺ qɺ + η qɺɺ By ubttutng th expreon n equton (5) nd combnng the term of the me type one wll get the fnl formul for the recton component: (5) ( ) ( ) Sη S S ελ λ k k + qɺ qɺ ( ) n whch the coeffcent ( ) η nd ε λ depend upon only the poton coordnte q q q In word tht reult red: The Lgrngn component of the ectonl recton for ytem of lnk upon whch rbtrry externl force ct compoed of two prt: The frt prt depend upon only the drng force nd the poton coordnte whle the econd one wll be repreented by homogeneou functon of degree two n the generlzed elocty component jut lke the knetc energy of the entre ytem We he mde no pecl umpton up to now bout the form of the ytem ecton If the nddul prt n the lnk-ytem re cylndrclly-extended bode (whch frequently the ce for mchne) then one wll motly chooe plnr ecton tht perpendculr to the longtudnl x n order to tudy the cro-ectonl tree tht pper n thoe prt By contrt f one would lke to fnd the preure n the mong lnk whch extremely mportnt n engneerng then one hould mke the ytem ecton long the upportng urfce (Lgerfläche) n queton Here we re concerned wth only gng the generl Anätze tht wll mke t poble to determne the ytem recton on the b of d Alembert prncple nd n tht wy to how tht Lgrnge de bout the oluton of thoe problem wll uffce completely In ny ce the cope of rtonl mechnc cn be extended n frutful wy by ncludng generl problem n knetottc long wth the pecfc ttc nd knetc one nd n tht wy ccommodte ome entrely-jutfed demnd of engneerng 5 The fundmentl recton of mply or multply-coupled lnk-ytem From d Alembert bc equton for mpule:

23 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) r h m xɺ f we conder the entre lnk-ytem then we wll next dere the expreon: nd nd wrte them n the form: (54) r h m x ɺ x r x h m x ɺɺ x r h m xɺ M M M h x ɺ The rgd nd mmoble foundton long wth the lnk-ytem defne lrger mterl complex A reult from d Alembert prncple the reultng recton component r nd M r whch refer to only the mong prt wll not nh n generl They wll be ncluded n the foundton t ret preure nd rtul rottonl moment One mut obere n th tht the ector M r refer to certn ttc reducton pont nd t lue nd drecton wll chnge long tht reference pont hft t poton to the foundton Howeer n ny ttc problem tht concerned wth rgd bode one wll lo be ble to determne the centrl x here nd n tht wy obtn the ector r nd M r n one drecton If the rgd foundton upport the mong ytem wth more thn one retng lnk then one cn poe the queton of how to dtrbute the fundmentl recton oer the nddul upport n tht ce One now plt the common be nto mny pece the number of upport tht re preent ge ech prt the correpondng rtul moton relte to the bolute coordnte ytem nd pple Lgrnge prncple of rtul work for the determnton of the nddul recton For the totl foundton recton under tme-ryng force the followng equton wll enter n plce of equton (54): k mɺɺ x (55) M M h M x ɺɺ Tht ce exmned n detl n the theory of mchne n pecl exmple (prllel crnk mechnm tht ct upon common hft) nd for tht reon we would lke to go nto t omewht deeper n wht follow 6 The problem of djutng the effect of m n lnk ytem The quntte nd M n equton (55) ech cont of two term: The frt one wll be dered from the externl force tht ct upon the ytem nd for tht reon t wll depend upon the m dtrbuton of the entre ytem One mke the econd term equl to zero (up to nhngly-mll redul contrbuton) n the ce of mult-crnk tem engne by utble rrngement of the ytem nd n tht wy elmnte the effect of m on the

24 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 4 foundton n prctce We would now lke to exmne the condton for the nhng of the ector m xɺɺ nd M x ɺɺ for br chn n the generl ce To tht end we hll determne the center of grty n ech rgd ubytem Let the ector of the nddul center of grty be n turn: x x ( ) x ν meured from the bolute reference pont (O) If we chooe them to be the relte reference pont (O O etc) for the ector tht etblh the nddul mterl pont of the ubytem then the bolute ector of thoe pont re: x ( n ) x + ( ) x ν ( ν ) ( ν ) Let the ector of the center of grty of the entre ytem be x We wll then he: nd m x m x m xɺɺ m xɺɺ A reult the recton m xɺɺ wll nh only when the elocty of the common center of grty of ll ubytem remn unchnged durng the moton In order to netgte the moment we form the equton: x ɺɺ x ( ν ) ( ν ) nd upon ummton we wll get: ɺɺ ɺɺ ( ν ) ( ν ) ( ν ) ( ν ) ( x + )( x + ) x ɺɺ x + ɺɺ x ɺɺ + ɺɺ x + ɺɺ ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) nce the quntte: ( ) ( ) ( ) Σ ν m x ν ɺɺ x ν Σ Σ m x ɺɺ ( ) ( ) ( ) Σ m x ɺɺ x + m ɺɺ ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) ( ν ) Σ m x ( ν ) ( ν ) ( ν ) wll nh for rgd bode Therefore we wll he: M x ɺɺ n n ( ν ) ( ν ) ( ν ) ( ν ) S mν x x + S mν ν ν ɺɺ ɺɺ n whch one et Σ (ν) m m (ν) to bbrete For mult-cylndered tem engne the lue of the econd term n th equton wll lwy ty wthn nrrow lmt becue t h the order of mgntude of the rottonl ccelerton In tht pecl ce howeer the condton for the reducton of the m effect to rgd foundton cn be repreented n the form:

25 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 5 dx (56) 0 nd d ( ) ( ) S n ν ν m x xɺ ν 0 ν The further dcuon of thee equton problem for engneerng mechnc H Lorenz ge thorough preentton of the problem of m effect n h Dynmk der Kurbelgetrebe mt beonderer Berückchtgung der Schffmchnen (90) F Knetottc requrement 7 Norml tree nd her tree The theory of ttc tree w frt deeloped for eltc prmtc rod (e bem) In the mplet ce force ct only n the drecton of the longtudnl xe whch one dtnguhe from ech other by cllng them force of tenon nd compreon They generte n nternl tre tte for whch eltc force wll be prooked long tht x In econd ce the externl force reduce to force-couple whoe plne nterect the cro-ecton perpendculrly long one of the two prncpl xe The eltc effect expree telf bendng of the bem Tenon nd bendng tree wll be commonly referred to norml tree If the x of the force-couple le long the longtudnl x of the rod then ngulr deton of the body element whch re umed to be rectngulr prllelepped wll occur long wth the extenon nd one cll thoe ngulr deton her or lp A toronl tre wll re tht ttclly-equlent to the deformng forcecouple Fnlly we cn lo mgne tht the force tht ct on thng le completely wthn the plne of cro-ecton of the rod nd tre to mke the one prt of the body lde long the other one long the ectonl plne A herng tre wll now come bout n the cro-ecton n queton Toronl tree nd her tree wll be collectely referred to her tree We hll now pply th elementry concept from the theory of old to the rgd bode nd the br chn wth rgd br Although deformton wll be excluded n tht wy one cn tll crry out the reducton of the nternl recton force n uch wy tht the component wll correpond to the uul ctegore of nfluence At the me tme by dong tht one wll get n ntute oerew of the reult tht not ntrnc to the generl reducton tht ue Lgrngn coordnte 8 Determnng the tre component It follow from d Alembert bc equton for tme-ryng force: k mɺɺ x + tht (57) Σ Σ m x k Σ m xɺɺ x n whch ll ummton extend oer thoe prt of rgd body tht re rtully eprted by plne ecton Howeer for ech rgd ytem (wthout trnlton) one h: xɺ x

26 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 6 nd t wll follow from th by dfferenttng wth repect to tme tht: or upon expndng the trple ector product: xɺɺ ɺ x + ( x) xɺɺ ɺ x + ( x) ( ) x We decompoe tht ccelerton long three rectngulr xe tht re fxed n the rgd body nd denote the relent projecton of the ector x by : xɺɺ xɺɺ xɺɺ ɺ ɺ + ( + ) + + ɺ ɺ + ( + ) + + ɺ ɺ + ( + ) + + The component ɺ ɺ ɺ n thee expreon mut be elmnted wth the help of Euler equton of rotton (): A ɺ (A A ) + M h A ɺ (A A ) + M h A ɺ (A A ) + M h Tht wll ge: A A xɺɺ A A ( + ) + A (A A + A ) + A (A A + A ) + A M h A M h A A xɺɺ A A ( + ) + A (A A + A ) + A (A A + A ) + A M h A M h If we et: A A xɺɺ A A ( + ) + A (A A + A ) + A (A A + A ) + A M h A M h Σ Σ k k Σ m m to bbrete then equton (57) wll yeld the component of the reultnt force of the nternl tree n the explct form: A A A A k + m (A M k A M k ) + m {A (A A + A ) + A (A A + A ) A A ( + ) }

27 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 7 nd two nlogou expreon for tht wll follow by cyclc permutton of the ndce For the ke of gettng better oerew we wrte: (59) k + m u + m w u then men ector tht depend eentlly upon the totl moment of the rottng force nd w men econd tme-ryng ector tht determned mnly by the elocty tte of the ytem The component of the center of grty ector of the rtully-eprted pece of the body pper mgntude n the component of u nd w well If one goe from one eprtng plne to nother then the poton of tht center of grty wll chnge nd the component of wll be ffected n wy tht ey to ee Wth our notton we cn wrte equton (58) whch determne the moment of the recton force relte to the fxed pont : or M M k Mɺɺ x M M d M k Howeer from no 6 equton (9) one h: Hence one wll he: (60) M or when decompoed nto component: d M d M + M M d M k M ( ) ɺ M M k A A A M M k ( A A ) A ɺ M M k ( A A ) A ɺ A A A re the prncpl moment of nert of the rtully-eprted pece of the body By elmntng the component of the ngulr ccelerton wth the help of Euler equton of moton whch re true for the entre ytem we wll get wth no further dcuon:

28 Heun The menng of d Alembert prncple for rgd ytem nd lnk mechnc (cont) 8 (6) n whch one et: A M A M k A M k + D A M A M k A M k + D A M A M k A M k + D D ( A A A A ) ( A A A A ) D ( A A A A ) ( A A A A ) D ( A A A A ) ( A A A A ) The ector M then h the form: M P + Q P depend eentlly upon the externl force whle Q contrned by the mnly the elocty tte of the ytem The moment of nert of the rtully-eprted prt of the body wll ffect both ector Up to now M h been referred to the fxed pont If we then tke the moment relte to pont of the cro-ecton then the known rule of ttc wll be true for tht trnformton Howeer tht new reference pont cn be choen n the plne of the cro-ecton n uch wy tht the reultnt nd moment of the recton wll le n plne tht perpendculr to the plne of the ecton Snce the locu of thoe reference pont lne we wll tll be free to chooe whch of t pont hould be umed to be the defnte reducton pont Once tht choce h been mde we decompoe the reultnt nd the moment nto component whch wll fll n the plne of the ecton or be perpendculr to t rep nd thu obtn the quntte tht re requred by the rtully-eprted prt of the body n regrd to tenon (or compreon) bendng toron nd herng Should the component of the knetottc requrement be determned for ytem of lnk we would proceed n mnner tht mlr to wht we do for n olted rgd body The ngle dfference cont of the fct tht we mut dd the known recton of the next-lyng lne (or more generlly ll of the lnk tht re on the me de of urfce of the ecton) Snce we he clculted thoe lnk recton completely we cn lo conder th generl problem to be oled Berln Februry 90