SaqarTvelos teqnikuri universiteti satransporto da manqanaTmSeneblobis fakulteti departamenti #133

nax. 9. orsayrdeniani SuaSi modebuli tvirTiani koWas masaTa dayvanis koeficientis gansazRvris sqema
am SemTxvevaSi [132]

;

;

.

sawyisi pirobidan, sadac: x = 0, y_x = 0; x = ½, dy_x/dx = 0.

sawyisi pirobidan C da D-s mudmivas integrirebis gansazRvrisas da gardasaxvisas, miviRebT:

;

.

gamovsaxoT -i -iT

.

koWas elementis kinetikuri energia

xolo koWas SuaSi dayvanili masebi

dayvanis pirobidan

T da dT_x mniSvnelobaTa CasmiT, vipoviT

saidanac,

(17)

miRebuli formuliT SegviZlia visargebloT koWaze tvirTis nebismieri damokidebulebisas dayvanis koeficientis gansazRvrisaTvis. kerZo SemTxvevaSi, rodesac koWas wona mcirea tvirTis wonaze (Q >> ql) an roca koWas deformaciis xasiaTi daaxloebiT SeiZleba miviRoT iseTad, rogorc misi erTi ZaliT Runvisas, koncetrirebuls SuaSi, miviRebT K=17/35 ≈ 0.486, rodesac Q = 0, K = 3968/7875 ≈ 0.505.

1.4. sixisteebis dayvana da gansazRvra

meqanikuri sistemis an misi calkeuli elementis sixistis qveS gulisxmoben im datvirTvis damokidebulebas, romelic iwvevs mis deformacias. sixiste SesaZloa iyos xazobrivi (Zala xazobriv deformaciasTan) da kuTxuri (Zaluri momenti kuTxur deformaciasTan). martivi sistemis elementebisaTvis mas gansazRvraven Semdegi saxiT.

mudmivi kveTis Reros deformacia, gawelvadi an kumSvadi ZaliT P [3,5]

,

saidanac Reros xazobrivi sixiste

. (18)

boloebiT Tavisuflad dayrdnobili mudmivi kveTis koWas deformacia, romelic datvirTulia SuaSi P ZaliT, datvirTvis modebis nacvlad tolia [3,5]

,

xolo koWas SuaSi CaRunvis wrfivi sixiste

(19)

mudmivi kveTis grZivi Reros grexisas, kuTxuri deformacia [5]

,

sadac:

M – mbrunavi momentia;

G – meore saxeobis sixistis moduli;

J_p – inerciis polaruli momenti.

Sesabamisad Reros kuTxuri sixiste

. (20)

ufro rTuli elementebisaTvis (cvladi kveTis RerZebi da lilvebi, zambarebi, resorebi da a.S.) drekadobis Teoriis meTodebiT pouloben elementTa deformacias datvirTvis funqciaSi da gamoyofen datvirTvebis damokidebulebas deformaciebis mimarT, rac aris sixiste.

ramodenime Seyursulmasebiani drekadi rgolebiT SeerTebuli sistemebi, mizanSewonilia gamovsaxoT dayvanili sqemebis, erTi, ori an sami dayvanili masebis saxiT, romlis drosac dayvanili masebis SemaerTebeli rgolebis sixisteebic unda iqnas dayvanili. magaliTad, drekadi rgolebiT SeerTebuli masaTa gadataniTi moZraobisgan Sedgenili sistema (nax. 10), romelsac gaaCnia sixisteebi C₁, C₂, C₃… datvirTuli ZalebiT P₁,P₂,P₃…., SeiZleba dayvanil iqnas m₁ masaze, romlis drosac dayvanili sistemis sixisteebis dayvanas ganvaxorcielebT Semdegi saxiT.



nax. 10. wrfivi sixisteebis dayvanis sqema
statikuri Zala P₁, moqmedi m₁ masaze, gamoiwvevs gadaadgilebas, P₁/C₁ rgolis C₁ drekadi deformaciis xarjze. m₂ masis gadaadgileba Sesabamisad toli iqneba , xolo Semdegi masis gadaadgileba da a.S. Tu margi qmedebis koeficientebiT gaviTvaliswinebT gadacemebSi xaxunze danakargebs, maSin m₁ masis gadaadgileba, P₁ Zalis zemoqmedebiT iqneba

imave m₁ masis gadaadgileba P₂ Zalis zemoqmedebiT

da a.S. sistemis sruli deformacia, m₁ masis gadaadgileba, Seadgens f = f₁ + f₂ + f₃ + f₄ +.....

sistemis datvirTva, dayvanili m₁ masaze

(21)

sistemis dayvanili sixiste

. (22)

umetesoba manqanebisa gare datvirTvebs aRiqvamen mxolod ukiduresi masebiT. am SemTxvevisaTvis formulaSi (23) yvela Zala, garda P₁, unda iyos nulis toli. amasTanave

(23)



nax. 11. kuTxuri sixisteebis dayvanis sqema
mbrunavi masebis mqone sistemebisaTvis (nax.11) dayvanili kuTxuri sixiste ganisazRvreba, rogorc damokidebuleba kuTxuri deformaciis mimarT dayvanili masisa

(24)

rodesac, M₂ = M₃ = M₄ = ……= 0

(25)

Tu sistemaSi aris mimdevrobiT moZravi da mbrunavi masebi, maSin sixiste SeiZleba warmodgenil iqnas, rogorc xazobrivi an kuTxuri. magaliTad, Tu Sereul sistemaSi dayvanili xazobrivi sixiste tolia C-si, xolo kuTxuri C_y-is, maSin saerTo dayvanili sixiste, R dayvanili radiusis dros iqneba

an

2. meqanikuri sistemebis proeqtirebisa da dinamikuri kvlevis sakiTxebi.

meqanizmebis struqturasTan dakavSirebuli sakiTxebi pirvelad iqna ganxiluli l.v. anaxis da a.g. maliSevis naSromebSi. meqanizmebis klasifikaciis Semdgomi ganviTareba asaxulia i.i. arTobacevskis, v.v. dobrovolckis [10-25] da r. frankes [26] naSromebSi.

brtyeli meqanizmebis sinTezis sakiTxebs da mravalrgolian saxsrovani meqanizmebis kvlevebs ganixilaven Tavis SromebSi n.i. levitski [27,28] da s.a. Cerkudinovi [29].

swored d.s. TavxeliZis, g.a. jabuas da o.s. ezikaSvilis [30,41] Sromebi miZRvnilia brtyeli meqanizmebis kvlevis sakiTxebisadmi.

g.p. baranovis naSromebi [32] miZRvnilia winaswar dasaxuli pirobebis mixedviT meqanizmebis kvlevisa da proeqtirebis sakiTxebisadmi, agreTve mravalsafexuriani kbilanuri da rTuli planetaruli meqanizmebis kinematikuri kvlevebisadmi.

s.n. koJevnikovis wignSi [33] Seiswavleba struqturuli da analizebis sakiTxebi; dgindeba meqanizmebis struqtura da ganisazRvreba rgolebis calkeuli wertilebis traeqtoriebi, gadaadgilebebi, siCqareebi da aCqarebebi, agreTve rgolebis zomebi winaswar dasaxuli pirobebis mixedviT. ganixileba gareSe mamoZravebeli da winaaRmdegobis Zalebis zemoqmedebiT gamowveul rgolebis moZraobebis gansazRvris meTodebi, moZraobis regulirebis, inerciis Zalebis gawonasworebis sakiTxebi da sxv.

rxevebis Teoriis safuZvlebi, Tavisuflebis erT da mravali xarisxis mqone meqanikuri sistemebis rxevebis Teoria da sistemis sakuTari sixSiris gansazRvris meTodebi ganixileba s.p. timoSenkos, d.x. iangis, u. uiveris, f.m. czas, i.e. morzes, p.t. xinkpas SromebSi [34,36].

v.a. iudinisa da l.v. petrokasas naSromebSi agreTve SemoTavazebulia meqanizmebis Zalovani gaTvlis meTodebi, manqanebis agregatebis dinamikuri analizis da sinTezis zogierTi sakiTxebi, romelTac miekuTvneba perioduli rxevebis regulireba da manqanebis gawonasworebis amocanebi.

v.a. kudinovis naSromSi [37] SemoTavazebulia Carxebis dinamikuri xarisxis maCveneblebis sistema da mocemulia am maCveneblebis mixedviT Carxebis Sefasebis sakiTxebi, Teoriuli da eqsperimentuli analizis Catarebis saerTo meTodika.

miwodebis racionaluri siCqareebiT aucilebelia simZlavris, teqnologiuri wnevebis gansazRvra, maRali xarisxis da damuSavebis sizustis uzrunvelyofis pirobebis uzrunvelyofiT, optimaluri warmadobis, gawyobis meTodebi da marTvis sakiTxebi ganxilulia f.m. manJosis naSromebSi [38].

v.l. bidermanis wignSi [39] gadmocemulia wrfivi da arawrfivi meqanikuri sistemebis rxevebis Teoriis safuZvlebi da manqanaTmSenebeli konstruqciebis dinamikuri gaTvlisTvis saerTo meTodebis gamoyeneba.

m.f. dimentbergis naSromSi [40] ganxilulia moZraobis ramodenime SesaZlebel reJimSi myofi arawrfivi sistemebis rxevebis kvlevis sakiTxebi, moyvanilia sistemebis xarisxuri da araxarisxuri identifikaciis meTodebi rxeviTi procesebis statistikuri analizis safuZvelze. ganxiluli sakiTxebis gadawyveta xdeba analizurad, eleqtronul gamomTvlel manqanaze modelirebis gziT.

v.l. veicis naSromSi [41] meqanizmis kvanZebis aradartymiTi urTierTqmedebis amsaxvel dinamikur modelebTan erTad, ganxilulia dartymiTi da vibridartymiTi tipis modelebi. maT safuZvelze Catarebulia iZulebiTi rxevebis kompleqnaxi kvlevebi, romelic saSualebas gvaZlevs aRmovaCinoT TviTdamuxruWebadi sistemebisaTvis damaxasiaTebeli rigi axali movlenebisa. aqve mocemulia arawrfivi disipaciuri Zalebis kvlevebi drekad rgolebiani meqanizmebis dinamikis amocanebSi. ganxilulia disipaciuri Zalebis koreqtuli eqvivalenturi gawrfivebis sakiTxebi, agreTve damuSavebuli meTodebis gamoyeneba Tavisuflebis erTi da mravali xarisxis mqone sistemebSi rxevebis gaangariSebisas.

s.i. sergeevis naSromSi [42] ZiriTadi yuradReba eqceva hidravlikur dempferebs da erTi an ramodenime adgilSi Tavmoyrili blanti xaxunis mqone wrfivi meqanikuri sistemebis rxevebis Teoriul da eqsperimentul monacemebs.

drekad-plastikuri sxeulebis SemTxveviTi rxevebi Seiswavleba v.a. palmovis naSromebSi [43]. sasazRvro amocanis amoxsna igeba galerkinis meTodis amonaxsniT sxeulis Tavisufali drekadi rxevebis formebis mixedviT rigSi daSlis gamoyenebiT. aRniSnulia rxevebis sxvadasxva sixSiruli Semdgenebis urTierTzemoqmedeba.

dinamikis mravali amocanis amoxsnisas mivdivarT efeqturi miaxloebiT ricxviT-analitikuri da ricxviTi meTodebis SemuSavebis aucileblobasTan yvelaze ufro perspeqtiuls warmoadgens Tanamedrove egm-ze praqtikulad realizebadi farTo amocanebis Semcveli ricxviTi analizuri meTodebi.

a.p. kartaSovisa da b.l. roJdenstvenskis naSromi [44] eZRvneba Cveulebrivi diferencialuri gantolebebis Teoriasa da ZiriTad ganmartebebs, aseve variaciuli gamoTvlebis martiv amocanebs. mocemulia agreTve pirveli rigis kerZo warmoebulis Semcveli gantolebebis amoxsnis maxasiaTeblebis meTodebi, diferencialuri gantolebebis amoxsnis miaxloebiTi meTodebis periodul koeficientebiani diferencialuri gantolebebis wrfivi sistemebi.

amerikeli specialistebis j. forsotisa da k. moleris gamoyenebiT maTematikasTan dakavSirebul SromebSi [45] aRwerilia wrfivi algebruli sistemebis egm-ze amoxsnis Tanamedrove meTodebi. naSromebi [46,48] miZRvnilia manqanebSi dinamikuri procesebis Seswavlisadmi.

dinamikuri procesebis modelirebis da kvlevis sakiTxebisadmi miZRvnilia profesorebis d.d. TavxeliZisa da v.n. gogilaSvilis naSromebi [49,53].

Tanamedrove manqanebis amZravTa jaWvebSi gamoyenebuli realuri mimyoli sistemebi, ZiriTadad miekuTvnebian rTul arawrfivs, garkvel SemTxvevebSi ki arastacionarul sistemebs, rac mniSvnelovnad arTulebs maTi gaangariSebisa da daproeqtebis sakiTxebs [48,54,60].

Tavisufali da iZulebiTi rxevebis analizis meTodebi, dafuZnebuli a.m. liapunovis moZraobis mdgradobis Sesaxeb fundametur Teoriebze, fazuri sivrceebis struqturis geometriul agebasTan dakavSirebuli tipologiuri meTodebi, diferencialuri gantolebebis xarisxobrivi Teoriis meTodebi, morgebisa, gadamcemi funqciis gansazRvrebaze da sistemebis sixSiriT maxasiaTeblebze dayrdnobili sxvaobiTi meTodebi da sxva [61,62], romlebic saSualebas gvaZleven miviRoT mkacrad dasabuTebuli Sedegebi, Cveulebrivad Zalian rTulni arian dasaproeqtebeli sistemebis struqturisa da parametrebis winaswari SerCevis procesSi da inJinruli gaTvlebis praqtikaSi gamoyenebisaTvis. amitom analizis zust meTodebTan erTad did praqtikul gamoyenebas iZenen miaxloebiTi meTodebi, romlebic xasiaTdebian parametrebis winaswari SerCevis procesSi praqtikuli gamoyenebis simartiviT da calkeuli struqturuli elementebis sistemebis saerTo struqturul mTlianobaSi Serwymisas gamovlenili midamoebis SigniT parametrebis da struqturis zusti meTodebis gamoyenebisas maTi Semdgomi dazustebiT. amasTan Zalian xSirad meTodis praqtikuli gamoyenebis simartives ufro didi mniSvneloba aqvs, vidre maRal sizustes. es aixsneba imiT, rom nebismieri xarisxis arawrfiv sistemebSi dinamikuri procesebis sakmarisad zusti da detaluri kvleva (sxvadasxva sawyis pirobebSi da sxvadasxva gare zemoqmedebisas) axlandel droSi SeiZleba ganxorcieldes eleqtronuli samodelo mowyobilobebiT da gamomTvleli manqanebiT.

arawrfiv sistemebSi procesebis miaxloebiTi kvlevisTvis farTo gamoyeneba hpoves harmoniuli gawrfivebis da masTan monaTesave mcire parametrebis, harmoniuli balansis da sxva meTodebma [56,62,64].

arawrfivi sistemebis miaxloebiTi gaTvlis meTodebi, ganxilulia i.a. orurkis, v.i. stankeviCis, i.i. krineckis da sxvaTa SromebSi [65,67]. SeiZleba iqnas gamoyenebuli monotonur procesebTan axlos garkveulad SezRudul klasSi myofi arawrfivi avtomaturi sistemebis kvlevaSi, romelTa arawrfivi funqciebi moicaven erT-or erTmniSvnelovan uban-uban wrfiv maxasiaTeblian arawrfiv funqciebs. amasTan erTad, xsenebuli meTodebi saSualebas gvaZleven vawarmooT sistemis gaangariSebebi, romelime erTi sistemis dinamikuri Tvisebis maxasiaTebeli konkretuli kriteriumisaTvis.

ufro farTo klasis dinamikuri sistemebis kvlevis sakiTxebi ganxilulia naSromebSi [68-78].

Tavis bolos SeiZleba aRiniSnos, rom miuxedavad, Tanamedrove manqanebis amZravTa meqanikuri da eleqtromeqanikuri sistemebis dinamikasTan dakavSirebuli mniSvnelovani raodenobis samecniero kvlevebisa, dResdReobiT, manqanaTa dinamikis amocanebi isev inarCuneben aqtualobas.

amasTan erTad unda aRiniSnos, rom Tanamedrove manqanebis amZravebi zogadad warmoadgenen sakmaod rTul mravalmasian meqanikur sistemebs, romelTa struqturac SesaZloa iyos, rogorc mwkriuli, aseve ganStoebuli an rgoluri, rac mniSvnelovnad arTulebs kvlevis meTodebis sirTules. aseTi sistemebis proeqtirebisas dinamikuri kvlevis erT-erT ZiriTad amocanas warmoadgens dinamikuri datvirTvebis minimizacia, rac dakavSirebulia gardamavali procesebis analizisa da sinTezis sakiTxebTan.

3. maTematikuri modelebi da kvlevis amocanebi

3.1. meqanikuri sistemebis modelirebis sakiTxebi

amZravTa meqanikur sistemebSi, warmoqmnili dinamikuri movlenebis kvlevaTa gansakuTrebulobis TvalsazrisiT, SesaZloa maTi, sxvadasxva movlenaTa saxiT klasificireba.

pirvel SemTxvevaSi, sistemis Semadgeneli elementebi absoluturad xisti da ara deformirebadi Zalebisa da momentebis datvirTvisas, ramdenadac elementTa deformacia manqanaTa zomebTan SedarebiT mcirea, rigi dinamikis amocanebisa SeiZleba gamoyvanil iqnas elementTa drekadobis gauTvaliswineblad. magaliTad, manqanaTa moZravi elementebis siCqarisa da aCqarebis gansazRvrisas, aseve manqanaTa gamSvebi simZlavrisa, samuxruWe momentisa, romelic saWiroa manqanis gasaCereblad Sesabamis droSi an gansazRvrul samuxruWe manZilze, dinamikuri datvirTvebis, romlis drosac gansazRvrulia moZrav elementTa siCqare, aCqarebis dro, manqanis gaCereba da sxv. xisti sistemebi erTmaneTTan xisti rgolebiT gadabmuli SeiZleba warmodgenil iqnas Seyursuli masebis saxiT.

drekad sistemebSi manqanis yvela elementi, romlebic gadascemen moZraobas da iyolieben manqanis nawilebs, miiCnevian drekadad; amasTanave elementTa deformacia ar aRemateba sixistis zRvars, asec aris narCeni deformaciebi ar gaiTvaliswineba. erTi drekadi kavSiris SemTxvevaSi sistemas uwodeben erTkavSirians, ori kavSiris SemTxvevaSi – orkavSirians da a.S. drekad kavSirebs SeiZleba gaaCndeT mudmivi da cvladi sixisteebi. pirvel SemTxvevaSi sistemis rxeviTi procesebisa da drekadi kavSirebis gansazRvris kvlevisas, davdivarT rogorc wesi mudmiv koeficientiani wrfivi gantolebis amoxsnamde. meore SemTxvevaSi igive movlenebis kvlevisas davdivarT arawfrivi gantolebebis amoxsnamde. wrfivi da arawrfivi Tvisebebis mqone sqemebis magaliTebi naCvenebia nax. 12-ze.

nax. 12. Seyursuli da ganawilebul masiani sistemebi

1. 
   wrfivi Tvisebebis mqone drekadi sistema;
   
2. 
   arawrfivi Tvisebebis mqone drekadi sistema.
   

moyvanili sistemebi SeiZleba warmodgenil iqnas Seyursuli masebis saxiT dakavSirebuli xisti an drekadi rgolebiT, an kidev elementTa ganawilebuli masebis saxiT. pirvel SemTxvevaSi dinamikis amocanaTa gadawyveta SeiZleba gamosaxuli iqnas niutonis diferencialuri gantolebiT (xisti kavSirebi) an langraJis gantolebiT (drekadi kavSirebi). gamomdinare Seyursul masaTa raodenobidan sistema iwodeba erTmasianad, ormasianad, sammasianad da a.S.

ganawilebuli masebis SemTxvevaSi drekadi sistemis dinamikis ganxilvisas davdivarT gadawyvetilebamde e.w. talRur gantolebebamde.

konservatiul sistemebs uwodeben iseT sistemebs, romlebSic moqmed Zalebs an Zalur momentebs gaaCniaT potenciali. sxva sityvebiT, rom iTqvas konservatiul sistemebSi praqtikulad ar arsebobs energiis Semodena da gadena.

raime gansazRvruli niSnis mqone disipaciuri sistemebi (wamyvani rgolis siCqaris SenarCuneba, drois funqciaSi datvirTvebis cvlilebis kanonis SenarCuneba da sxv.) xasiaTdebian imiT, rom moZraobis procesSi arsebuli CamxSobis an wyaros xarjze, rogoricaa magaliTad moZravi cvladi Zalebis mniSvnelobebi, romelic dakavSirebulia Zravis TvisebebTan, adgili aqvs ukucemas an energiis Semonakads.

konkretuli sistemebis dinamikuri amocanebis gamoyvana advilia, oRond manqanebisa da meqanizmebis muSaobis konkretuli pirobebisaTvis iSviaTad gamoiyenebian.

wonasworuls uwodeben iseT sistemebs, romlebic nebismier mdgomareobaSi, gare datvirTvebis ar arsebobisas inarCuneben wonasworobas. magaliTisaTvis SegviZlia moviyvanoT meqanizmebi an manqanebi, Sedgenili mxolod mbrunavi elementebisagan (ventilatori, saxarato Carxis Spindelis mabrunebeli meqanizmi da sxv.).

arawonasworuli sistemebi mxolod zogierT mdgomareobaSi wonaswordebian sakuTari wonis ZalebiT (eqscentruli meqanizmi, mrudxara barbaca da sxv.).

xisti sistemebis dinamikuri amocanebi mdgomareobs imaSi, rom mocemuli ZalebiT an momentebiT ganisazRvros sistemis moZraobis kanoni (mdgomareoba, nebismieri drois momentSi sistemis yvela wertilis siCqare da aCqareba) an mocemuli moZraobis kanoniT ganisazRvros Zalebi, romelTa moqmedebiTac igi xorcieldeba.

xisti sistemebi moZravi dayvanili Zalis zemoqmedebis qveS (momenti), SeiZleba warmodgenil iqnas erTi dayvanili masis saxiT (inerciis momenti). amave dros SeiZleba mkacrad ganisazRvros mdebareoba (koordinatebi), dayvanili masis siCqare da aCqareba, mocemul koordinatebis zRvrebSi dayvanili masis moZraobis dro, sistemis dinamikuri datvirTvebis saSualo dayvanili mniSvnelobebi (rxevebis gauTvaliswineblad).

dayvanili Zalebi SeiZleba damokidebuli iyos kooedinatebze x, siCqareze v da droze t. dayvanili masis sidide aseve SesaZloa iyos cvladi da damokidebuli mdebareobaze (koordinati x). avRniSnoT cvladi dayvanili Zala P (x,v,t) da dayvanili masa m(x). Tanaxmad energiis Senaxvis kanonisa, sistemis kinetikuri energiis namati tolia moqmedi Zalebis elementaruli muSaobisa.

an

diferencirebisas vipoviT

an Canacvlebisas da gardaqmnisas,

(26)

miRebuli gantoleba saerTo saxiT maTematikur formaSi gamosaxavs niutonis meore kanons. sadac m(x) = constant da P(x,v,t) = constant, igi martivdeba da Rebulobs saxes

(27)

Tu dayvanili sistema warmodgenilia mbrunavi masis inerciis momentis J(φ) saxiT, damokidebuli bolo mdebareobaze, xolo dayvanili ZalTa momenti M damokidebulia koordinatze φ, kuTxur siCqareze da droze t, maSin energiis Senaxvis kanonis diferencialuri gantoleba miiRebs saxes:

(28)

es gantoleba analogiuria gantolebisa (26) sadac J(φ) = constant da M(φ,ω,t) = constant miviRebT

(29)

garda analitikuri meTodisa, arsebobs rigi saSualebebisa, xisti sistemebisaTvis dinamikuri amocanebis amoxsnisaTvis, rogoricaa grafikuli da grafo-analitikuri meTodebi [6.10.36 da sxv.].

drekadi sistemebis dinamikis amocanebi mdgomareobs TvisebaTa cvlilebebis gansazRvraSi, rgolebis dinamikuri datvirTvebis maqsimalur mniSvnelobebSi, periodebsa da sixSirul rxevebSi da sistemis rezonansuli mdgomareobis pirobebSi. iseve rogorc xisti sistemebisas, manqanaTa realur sqemebs cvlian dayvanilebiT. rig SemTxvevebSi (Seyursul masaTa aSkara gamovlinebisas) dayvanili sqema gamoisaxeba, erTi an ramdenime Tavisuflebis xarisxis mqone sistemis saxiT. dayvanili masebi, aseve drekadi rgolebis sixisteebi da moqmedi gare Zalebi (aseve moqmedi), SeiZleba iyvnen cvalebadni, damokidebuli mdebareobaze, moZraobis siCqareze an droze.

dinamikuri datvirTvebi yovelTvis ar arian pirdapir kavSirSi sistemis drekad rgolebis sixistesTan. yvela meqanikuri sistema flobs drekadobas, amitom arasworia imis mtkiceba, rom xisti sistemis rgolebze dinamikuri datvirTvebis gansazRvrisaTvis saWiroa myari sxeulebis dinamikis formulaTa gamoyeneba da mxolod drekadi rgolebisaTvis drekadi sistemis dinamikis formulebisa.

ori an meti Tavisuflebis xarisxis mqone drekadi sistemis dinamikis amocanaTa amoxsnisas, moxerxebulia miRebuli koordinatTa sistemis gamoyeneba. ganvrcobil koordinatTa qveS igulisxmeba rigi damoukidebeli sidideebisa (mzomi xazobrivi an kuTxuri sidideebiT), romelic gansazRvravs sistemis mdebareobas.

ganvrcobil koordinatTa meSveobiT Tanmimdevruli mcire gardaqmnebis gziT, TiToeuli maTganidan SeiZleba miviRoT sistemis yvela damoukidebeli SesaZlo gadaadgilebis tipebi. dinamikuri amocanebis amoxsna ganvrcobili koordinatebiT, varaudobs aseve sidideebis ganvrcobas, romlebic moqmedeben ganvrcobil koordinatTa mimarTulebiT.

dauSvaT Φ,Ψ da Θ – ganvrcobili Zaluri faqtorebia, xolo φ,ψ da θ – Sesabamisi koordinatebi. maSin gare ZalTa elementaruli muSaoba SeiZleba Caiweros jamis saxiT.

Φδφ + Ψδψ + Θδθ,

sadac δφ, δψ, δθ da φ,ψ da θ koordinatebi ganusazRvrelad mcire gardaqmnebia. materialuri wertili iqneba wonasworobaSi, Tu moqmed ZalTa muSaoba yovel gadaadgilebaze nulis tolia. drekad sistemaTa deformaciisas gare da Sida (drekadi) Zalebi axdenen niSnis mixedviT sapirispiro muSaobas. Sesabamisad, yoveli SesaZlo gadaadgilebisaTvis arsebobs toloba:

Φδφ + Ψδψ + Θδθ = δΠ, (30)

sadac Π – Sesabamis gadaadgilebebze Sida Zalebis jamuri muSaobaa (deformirebuli sistemis potenciuri energia)

Tu, gamosaxulebis (30) diferencirebiT, gamovsaxavT

(31)

gantoleba miiRebs saxes

(32)

tolobis (31) ganxilvisas, SeiZleba gamotanil iqnas daskvna, rom potenciuri energiidan pirveli warmoebuli, koordinatebze warmoadgens gare Zalebidan pirveli xarisxis funqcias da Sesabamisad potenciuri energiis gamosaxuleba warmoadgens koordinatTa meore xarisxis funqcias. ukanasknelis damtkiceba SesaZlebelia Semdegis miRebiT:

(33)

(33) toloba samarTliania deformaciasa da Zalebs Soris wrfivi damokidebulebis dros (deformaciebi warmoiqmneba drekadi datvirTvebis zRvrebSi) da damoukideblad moqmed ZalTa pirobebSi, maTi (31) formulaSi CasmiT da integrirebiT, miviRebT:

(34)

rac saWiro iyo dasamtkiceblad.

(30) formulis mixedviT drekadi sistemis potenciuri energiis saerTo gamosaxuleba iqneba:

an (34) gamosaxulebis gaTvaliswinebiT

(35)

materialuri wertilis T kinetikuri energia (wertilovani masis m) SeiZleba gamosaxul iqnas siCqareTa da kvadratebze, wertilis warmoebuli masaTa wertilovani jamis naxevris saxiT mimarTuli Sesabamis koordinatebze da :

(36)

ganzogadoebul koordinatebze gadasvlisas, gamovsaxoT

x = f₁(φ,ψ,θ); y = f₂(φ,ψ,θ); z = f₃(φ,ψ,θ);

SesaZlo wertilTa gadaadgilebebi δx, δy, da δz, ganzogadoebul koordinatTa cvlilebisas φ-s, δφ–ze miviRebT:

niutonis kanonisa da SesaZlo gadaadgilebaTa principis Tanaxmad:

(37)

sadac, Φ – ganzogadoebuli gare Zalaa.

vinaidan

maSin gantoleba (37) SeiZleba Caiweros Semdegnairad:

(38)

(38) gantolebis pirveli wevri warmoadgens diferencials t-s kerZo warmoebuliT kinetikuri energiidan ganzogadoebuli siCqaris φ-s mixedviT. asec aris . meore wevri gamoiyeneba –Tan. Tanaxmad (31) formulisa ; Tu Φ ekuTvnis konservatiul ZalTa Semadgenlobas, maSin misi niSani iqneba minusi. ase rom:

(39)

da aseve analogiurad gamoisaxeba sxva ganzogadoebuli koordinatebisaTvis:

(40)

(41)

da a.S. swored esaa langranJis materialuri wertilis moZraobis gantoleba.

wertilTa sistemis gantolebisaTvis inarCuneben Tavis formas, xolo T da Π qveS unda vigulisxmoT mTliani sistemis kinetikuri da potenciuri energia. Tu sistemaze, garda konservatiuli Zalebisa moqmedeben Zalebi F₁, F₂, F₃ maSin:

(42)

(43)

. (44)

da a. S.

3.2. gamartivebul modelebze gadasvlis amocanis dasma

rogorc wina TavSi ganxiluli meqanikuri sistemebis struqturis ganxilvidan Cans (meqanikuri transmisiebis) Seyursul masebian da arainerciul kavSirebad dayofis safuZvelze vRebulobT arsebiTad ara mniSvnelovan uSvelebel saangariSo sqemebs. Tavisuflebis xarisxTa dinamikuri datvirTvebis formirebisaTvis. aseTi sistemebis sakuTar sixSireTa maqsimaluri mniSvnelobebi mniSvnelovnad aWarbeben eqsperimentul monacemebis dinamikuri datvirTvebis zeda sixSiruli diapazonis zRvars.

amasTan dakavSirebiT, meqanikuri transmisiis analizisa da sinTezis procedurebSi farTod gamoiyeneba dinamikuri sistemebis gamartivebuli modelebi, anu modelebi Semcirebuli Tavisuflebis xarisxebiT. amasTan, arsebuli mravalmasiani modelidan gamartivebul modelebze gadasvlis koreqtuloba warmoadgens sakmaris pirobas kvlevebis Casatareblad imisaTvis, rom gansaxilvel sixSireTa diapazonSi ganxorcieldes Sesabamis sixSireTa da rxevaTa formebis xarisxobrivi miaxloeba.

Sesabamisad miRebuli dinamikuri kriteriumis msgavsad, sawyisi monacemebis saxiT transmisiis gamartivebisaTvis sawyisi sqemidan SeiZleba miviRoT inerciis jamuri momentis mniSvneloba, sakuTari sixSireebi da Tavisufali rxevaTa formebi. saxe (mwkriuli an ganStoebuli, ganStoebebis raodenoba, maTi adgili) da rigi (Seyursuli masebis raodenoba), gamartivebuli saangariSo sqemis, ganisazRvreba sawyisi sqemis saxiT, Casatarebeli kvlevebis miznebiT da sakvlev sixSireTa diapazonSi [79] moxvedrili dinamikuri sistemis sakuTar sixSireTa rxevaTa ricxviT.

ganvixiloT gamartivebuli saangariSo sqemis matricul formaSi Cawerili Tavisufali rxevebis gantoleba:

(45)

sadac J – sistemis inerciis diagonaluri (nxn) matricaa; C – sistemis sixistis simetriuli (nxn) matricaa; Φ - kuTxur koordinatTa n–zomuri veqtori, romlis yoveli komponenti warmoadgens moxvevis kuTxis i – iuri masis mimdinare mniSvnelobas; n – gamartivebuli saangariSo sqemis rigi.

gantolebis (45) marcxnidan gamravlebiT Semobrunebul matricaze J^-1, miviRebT:

(46)

sadac: A – gansaxilveli dinamikuri sistemis matricaa, romelic Seesabameba gamartivebul saangariSo sqemas [8]:

(47)

amasTan, A matricis sakuTari ricxvebi λ(i = 1,2,....n) warmoadgenen sakuTar sixSireTa ω_i kvadratebs, xolo yovel sakuTar λ_i ricxvs Seesabameba sakuTari veqtorebi μ_i - ω_i sixSireze gamartivebuli saangariSo sqemis Tavisufali rxevaTa formebi. Sesabamisad, sawyisi saangariSo sqemis gamartivebis amocana daiyvaneba iseTi A matricis povnis amocanamde, saidanac SemdegSi SesaZlebeli iqneba gamartivebuli saangariSo sqemis parametrebis gansazRvra (Seyursuli masebis, inerciis momentebi da drekadi kavSirebis damyoloba).

gamartivebuli da sawyisi saangariSo sqemebis inerciis jamuri momentebis tolobis pirobis Sesrulebisas, maTi daaxloebis xarisxi fasdeba maTi sakuTar λ_i ricxvTa da sakuTar μ_i veqtorTa kmponentebis siaxloviT.

(48)

maSin, matricaTa Teoriis “saukunis” gantolebis (48) gamoyenebisas da A matricisaTvis (nxn) ganzomilebis miniWebiT, xolo sakuTar ricxvTa saxiT λ_i da sakuTar veqtorTa μ_i komponentTa saxiT, sakuTari ricxvebi λ_i(saw.) da sakuTari veqtoris komponentebi μ_i(saw.), sadac i = 1,2,....n, miviRebT gantolebas A matricis elementTa mimarT:

(49)

gamartivebuli saangariSo sqemis n rigi yovelTvis mcirea sawyis N rigze, amitom veqtorTa komponentTa saxiT μ_i(saw.) gamoiyeneba sawyisi saangariSo sqemis i-iur sakuTar veqtorTa n komponentebi. umeteswilad zustad aproqsimirebadi sawyisi saangariSo sqemis i-iur formis rxevaTa saxe.

dReisaTvis, mravalmasiani modelebis gamartivebis meTodebidan, SeiZleba gamovyoT meTodebi SemuSavebuli a.p. Cerevkovis, e.i. rivinis, s.a. kazakis da l.a. banaxis [5,80-87].

profesor a.p. Cerevkovisa da s.a. kazakis meTodebi efuZneba mravalmasiani sistemis mcire masaTa jgufis miaxloebiT Canacvlebas erTi an ori masiT.

ara umetes sami Seyursuli masis mqone martivi sistemis sakuTar sixSireTa gansazRvra xorcieldeba mza formulebiT, romlebic miRebulia Sesabamisi sixSiruli gantolebebis amoxsniT.

erTmasiani sistema (nax. 13)

(50)

(51)

erTmasiani sistema (nax. 14):

(52)

ormasiani sistema (nax. 15):

(53)

ormasiani sistema CamagrebiT (nax. 16):

(54)

ormasiani sistema ori CamagrebiT (nax. 17):

(55)

sammasiani sistema (nax. 18):

(56)

dayvanili formulebi aseve SeiZleba gamoviyenoT mravalmasian sistemebSi sakuTar sixSireTa winaswari gansazRvrisaTvis.

am dros mravalmasiani saangariSo sistema icvleba gamartivebuli ori an sammasiani sistemiT.

Tu sistemis kidura erT an or masas gaaCnia mniSvnelovnad didi inerciis momentebi (miRebuli da meti) vidre Sualedur masebs, maSin saorientacio gaangariSebisaTvis SegviZlia aviRoT erTi an ori amonaxsniani sqema.

sakuTar sixSireTa ganszRvris gamartivebuli meTodi SeimuSava profesorma a.p. Cerevkovma. es meTodi efuZneba mravalmasiani sistemis mcire masaTa jgufis miaxloebul Canacvlebas erTi an ori masiT [38].

ganvixiloT nax. 19 a, b – ze warmodgenili sistema: