REDUCTION OF THE ELEMENTARY BODIES TO SYSTEMS OF MATERIAL POINTS WITH THE SAME INERTIA PROPERTIES. Nicolaie ORASANU 1

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1 U... S. ull., Seres, ol. 7, Iss., ISS - REUTIO OF THE ELEETRY OIES TO SYSTES OF TERIL OITS WITH THE SE IERTI ROERTIES ole ORSU In estă lurre se propun tev odele tete de reduere orpurlor eleentre l sstee de punte terle u eleş propretăţ de nerţe, stfel este vor ve: elş entru de să ş elş tensor de nerţe. leând de l este odele, ore orp pote f redus l un sste de punte terle, r deternre propretăţlor de nerţe devne ult splă. e seene, orpurle rele u fost îpărţte în ptru lse, un spet teoret neesr pentru epl odul în re se pot redue orpurle. estă ordre de deternre propretăţlor de nerţe le orpurlor este un nouă, neîntâlntă în no ltă lurre de speltte. In ts pper te utor proposes soe tetl odels of reduton of te eleentr odes to sstes of terl ponts wt te se nert propertes, w ens: te se entre of ss nd te se tensor of nert. Strtng fro tese odels, e od n e redued to sste of terl ponts nd te posslt to lulte ll tese propertes eoes eser. lso te rel odes were dvded n four lsses to ke etter eplton of te reduton etod. Te reduton operton s n orgnl one. Kewords: entre of ss, reduton of od, tensor of nert, eleentr od, order of od. Introduton Te entre of ss represents te pont were te entrel wegt ss of od n e onentrted wt te se enl effet lke te nturl od, were te dstruton of te wegt s ontnuous funton. Te oents of nert esure ow te ss of od s dstruted wt respet to pont, n s or plne. Te generl epresson for sste of n terl ponts s, [ ]: n Θ λ Were: Θ s te sol of te eleent wt respet to w s defned ts oent of nert, respetvel: pont, n s or plne nd λ - s te dstne etween te pont of ss nd te referene eleent: te pont, te s or te plne. Leturer, eprtent of ens, Unverst olten of urest, Ron, el: norsnu@oo.o.

2 ole Orăşnu Te relton sows tt te oents of nert re strtl postve. If we know te entre of ss nd te oents of nert out te n es of soe oordntes sste, te nert propertes of te od re known. Te produts of nert re defned wt respet to two es of n ortogonl referene sste for eple, f te referene sste s O, te produts of nert wt respet to te es O, O ve te epresson, [ ]: n For ontnuous od, te sus fro reltons nd eoe ntegrls on te volue of te od,, so: Θ λ d d For sste of terl ponts, te lulus of nert propertes ens onl lgerl sus, ut to deterne nert propertes for te ontnuous odes supposes, generll, ver dffult ntegrl lulus. To deterne te poston of te entre of ss or te nert propertes, t ens to dvde te oposte od n te eleentr odes for w reltons of lulus re known. Tus tull, n eleentr od s onsdered od for w te reltons of lulus of te poston of te entre of ss nd te oents of nert or produts of nert re known.. Te order of od We onsder neessr to use soe onepts for lssfton of te odes. So, dependng on te densons, one n lssf te odes n four tegores: -order odes re te odes w n e repled pont or sste of te terl ponts pont s ero densons - order odes re te odes w n e repled segent of lne, urve r or oposed urve urve s sngle denson, w re soetes lled rs - order odes re te odes w n e repled plne surfe, urved surfe or oposed surfe surfe s two densons w re soetes lled pltes - order odes re te rel odes w n ve s eteror fes te plne surfes or urve surfes te volues ve tree densons. Te odes w ve onl te plne eteror surfes - te poledrons - re lled sple odes.

3 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes 7 Te eleentr sple odes re te odes wt known nert propertes, w, troug ddng or etrtng, n for n sple odes fro te se lss wt te.. Te reduton of odes to order odes In ntroduton, we eplned w t s eser to lulte te nert propertes for te sste of terl ponts. In te se ode, f we opre wt te reduton of te fore sstes, we wnt to fnd n es w to deterne te nert propertes on sple odel. Te reduton ens to reple od n nferor order od. Te eleentr od, fro ts pont of vew, s od w n e redued to n order od. ll reduton opertons ust propose od w s te se entre of ss nd te se oents of nert nd produts of nert... Te reduton of strgt r to n order od Teore : n strgt r, n order od, n e redued to n order od fored sste of tree terl ponts, s follows: e end of r s / of te ss of te od nd te entre of ss s te / of te ss of te od. roof: Let us onsder r,, of ss, nd known oordntes of te ends, nd, Fgure, n te plne O. In onfort wt [], te oents of nert nd produts of nert dfferent fro ero re: α O O Fgure Te proposed odel Fgure, n onfort wt reltons nd, s te followng oents of nert nd produts of nert:

4 ole Orăşnu Owng to te setr, te entre of te ss s ovousl dentl wt te entre of ss of te odel. Te deonstrton s ovous... Te reduton of te eleentr order odes to n order od We found two eleentr order odes: te trngle nd te prllelogr. It s well known tt te ltter od, te prllelogrs, nlude lrge fl of odes: te rous, te retngles nd te squres. For n eleentr od we onsdered n, te nuer of te vertes of te od, so tt, n for te trngles nd n for te prllelogrs. Teore : n eleentr order odes n e redued to n order od fored sste of n terl ponts, s follows: e verte of te od s / of te ss of te od nd te entre of ss s te rest of ts ss: / - for te trngles nd, respetvel, / - for te prllelogrs. roof: Te deonstrton s de n O plne, see Fgures nd. Te trngle Let us onsder trngle, stuted n O plne, wt te verte oordntes:,,,,,,,, nd wt te ss. If te entre of ss s n te pont, te known oordntes of te α O O Fgure

5 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes 9 pont re:. If te su s noted wt:, n onfort wt [], te epressons of te oents of nert nd te produts of nert re: Te proposed odel, presented n fgure, s te followng oents of nert nd produts of nert: euse te oent of nert wt respet to te s O s equl to te su, t s not spefed seprtel. Te equltes etween epressons nd 7 deonstrte te frst prt of defnton. Te prllelogr Let us onsder tee prllelogr, fro Fgure, wt te vertes, -, stuted n O plne. Te verte oordntes re:,,,,, nd,. In onfort wt [], te oordntes of te entre of ss re: d d α O Fgure α O O Fgure

6 ole Orăşnu Te oents of nert nd te produts of nert wt respet to te referene sste re deterned usng te ntegrl lulus Fgure, so: ρ ρ d ρ d ρ d ' ρd ρ tgα d ' ρ d ρ tgα d For te proposed odel see fgure, te oents of nert nd te produts of nert re: Te eqult etween epresson, 9 nd deonstrte te seond prt of teore... Te reduton of te eleentr order odes to n order od In te poledron fl, we wll ne te prds nd te prss s sple poledr. It s lred deonstrted tt poledron n e dvded n prs nd prds, dded or etrted. We found two eleentr order odes: n tetredron nd n prs wt prllelogr se. In ft n prd n e dvded n nuer of tetredr nd n prs n e dvded n nuer of prss wt prllelogr se nd tetredr... Te reduton of tetredron Teore.: n tetredron, order od, n e redued to n order od fored sste of terl ponts s follows: e verte of te tetredron s / of te ss of te od nd te entre of ss s te / of te ss, te rest. 9

7 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes w. roof: Frst, we lulte te nert propertes of te tetredron n lss... Te deternton of te entre of ss nd te oents of nert Let us onsder tetredron,, wt te fe stuted n O plne nd te verte on te O s nd te verte on te O s.te verte oordntes of te tetredron re:,,,,,,,, nd,, d Fgure. O To otn te vlues of te nert propertes we ke te ntegrl lulus for tt tetredron. In Fgure s drwn te tetredron on w we ke seton Fgure wt two prllel plns, t egt, we otn te eleentr volue d, proportonl wt te eleentr ss d. To deterne te poston of te entre of te ss we use te reltons: d d d d d d d d euse te trngles Δ nd Δ re slr, nd lso te trngles Δ nd Δ re slr too, t results: In epressons,, re te oordntes of te entre of ss of te eleentr volue, te trngle Δ, tus: Te volue of prd s:

8 ole Orăşnu Δ If we reple te ter fro n, etween te eleentr volue nd te volue we found te relton: d d d Hene, te oordntes of te entre of ss of te prd re: d d d 7 Te reltons 7 sow te forul of te oordntes of te entre of ss of tetredron. In ft, e oordnte s qurter of te su of te verte oordntes wt respet to te se s. For te deternton of te oents of nert nd produts of nert we use te reduton operton for te trngle. So, te oents of nert nd produts of nert of te eleentr od of te volue d nd ss d, re, n ft, te oent of nert of te trngle. So, for oogenous prd of ss ρonst., we get: d d d d ρ ρ In onfort wt reduton operton, te vertes, nd of te trngle ve: d/ nd te entre of ss,, s d/, see Fgure. Te eleentr oent of nert wt respet to O s s: 9 d d d 9 d d d ntegrtng ts, we otn: d d O Fgure

9 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes Slrl, we n lulte te oent of nert wt respet to te O s: 9 d d d fter ntegrtng t s otned: nd wt respet to te O s: 9 9 d d d In epresson ll te ters re lred lulted n reltons nd, so te oent of nert wt respet to te O s s:... Te deternton of te produts of nert Te produt of nert of te eleentr trngle s: 9 d d d [ ]d d 7 fter te ntegrton n relton 7, we otn: [ ] For te oter produts of nert we ke slr lulus, so: d d d 9 Replng: d d fter ntegrton lulus:

10 ole Orăşnu Slr: In onfort wt teore., te oents of nert of te proposed odel, re: Te produts of nert of te proposed odel re: Reltons - prove te vldt of te proposed odel... Te reduton of prs wt prllelogr fes Teore.: n prs wt prllelogr ses, n order od, n e redued to n order od fored sste of 9 terl ponts s follows: e verte of te prs s / of te ss of te prs nd te entre of ss s / of te ss te rest. roof: prs wt prllelogr ses 7, s represented n Fgure O Fgure 7 d d 7 O Fgure

11 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes, s, wt respet to te rtesn oordnte sste O, te ponts O nd te se stuted n O plne, wt te edge stuted on O s. Te oordntes of te prs vertes re:,,,,,,,,,,,,,,,,,,, 7 7, 7,,,,. Te unt vetor of genertor dreton s: e os α os βj osγk Te vetors of te edges ve te for: O 7 λe To splf te nottons, t s eser to wrte te oordntes s funton of slrs. We note te slr: λ osλ, nd te verte oordntes s: λ osα λ os β λ osα λ os β 7 λ osα λ os β λ osα λ os β Te ntegrl lulus of te oents nd te produts of nert Te prs ws setoned wt two prllel plns t te egt nd d dstne etween te. Te eleentr od s te ss nd te volue: d ρ d d d d If we ke te notton: Ou, we n wrte te oordntes epresson: u os γ 9 u osα u os β u osα u os β u osα u os β u osα u os β Te oent of nert of te eleentr od wt respet to O s s: d d d Integrtng: Te oent of nert of te eleentr od wt respet to O s s: d d d Integrtng: osα ρ [ ] d osλ Te oent of nert of te eleentr od wt respet to O s s:

12 ole Orăşnu d d d Integrtng: Te produt of nert of te eleentr od wt respet to O nd O es s: d d 7 Integrtng: ] [ Te proposed odel s presented n Fgure 9. Te oordntes of te entre of ss n e deternte wt te forul: 9 Te oents of nert of ts odel n e lulted wt te followng reltons. Te oent of nert wt respet to O s s: 7 Replng te vlues fro reltons 7 nd 9 n, we otn: Te oent wt respet to te O s: 7 Replng te vlues fro reltons 7 nd 9 n, we otn: 7 O Fgur 9

13 Reduton of te eleentr odes to sste of terl ponts [ ] nert propretes 7 Te oent wt respet to te O s: 7 7 Te produt of nert wt respet to O nd O es: 7 7 [ ] 7 Te produt of nert wt respet to O nd O es: 7 9 Te produt of nert wt respet to O nd O es: 7 Fnll: Te entre of ss of te proposed odel,, n e es verfed usng slr reltons wt 7. If te reltons,,, 7, 9 nd re opred wt te reltons fro pter.., te re slr nd te proposed odel s te se nert propertes tus te teore s proved.. onlusons In ft, ll te ppled senes use odels to stud te nturl penoenon nd te reduton of te odes elp us to work eser nd to lulte te propretes wt nu tetl effort. Ts reduton operton s n orgnl ppro w n open w n ts dreton.

14 ole Orăşnu Frst, we fnd te eleentr odes w n e redued. For te poledron fl order odes nd, lso, for te degenerted poledron solds, order or order odes, we fnd te followng eleentr odes: n tetredron, n prs wt prllelogr ses, n trngle, n prllelogr nd n strgt r. euse n poledron n e fored wt tese eleentr odes, w re presented ove, n ft n poledron n e redued to sste of terl ponts. lso, we know ow to etend ts reduton operton for oter sold fl nd we suppose tt t wll e etended for te prtulr sold fles. Let us rerk tt te reduton operton presents n dvntges f t s opred wt te teoretl stud. lso, t s es to ppl ts operton n oputer progrs. We know tt, t ts te, tere re n oputer progrs w n lulte ll te nert propertes for n sold od tt n e desgned n tenologes, nd our ppro n prove te future progrs. Let us onsder ts stud step n ts dreton. We n not evlute now ll te ppltons of ts etod ut we re sure tt n ppltons would e possle, nd not onl n teoretl ens. s onsequene of te des presented n ts pper, tere s te posslt to redue oter odes to nferor order odes. We wll present ll tese n oter ppers. R E F E R E E S []. rtu, Teoretl ens, Ed. Ipuls, uureşt, n Ronn. []R. one,. oulesu,. euşu, en, Ed. dtă ş edgă, uureşt, 9 n Ronn. [].G. etev, Teoretl ens, Ed. r, osov, 99. []. Rădo, E. eu, en, Ed. dtă ş edgogă, uureşt, 99 n Ronn. []S. Stu, Teoretl ens, Ed. dtă ş edgogă, uureşt, 99 n Ronn. []O. rgne, Geoetr of sses, Ed. dtă ş edgogă, uureşt, 97 n Ronn. [7]S. Stu, ns of plnr prllel root, U, Sentf ulletn, Seres, ol. 7, r.,. []H. ndr, S.K. S, ns nd lnng of ultod Sstes, Hrdover, 9. [9]G. udugn, Strengt of terls, Ed. Tenă, uureşt, 9, n Ronn. [] R. one,. oulesu, Fl.. Son, Introduton n ens of sold wt pplton n engneerng, Ed. dee, uureşt, 99, n Ronn. []. âlov, Şt. ăln, R. one, Teoretl ens, Ed. Tenă, uureşt, 9, n Ronn. []. Guenter, L. Sung-Hee, Sol ns nd Geoetr, K eters, 9.