SaqarTvelos teqnikuri universiteti satransporto da manqanaTmSeneblobis fakulteti departamenti #133

nax. 19. masaTa jgufis erTi masiT Canacvlebis sqema

dauSvaT pirvel masas gaaCnia I₁ inerciis momenti, sididiT mniSvnelovnad gadaWarbebuli sxva yvela masaze I_2, I_3,..... I_n.

masaTa I_2, I_3,..... I_n jgufs vcvliT erTi pirobiTi masiT inerciis momentiT I, romelic tolia danarCen miTiTebul masaTa inerciis momentTa jamis sididisa:

(57)

amasTan pirobiTi damyolobis sidide e – ganisazRvreba formuliT:

(58)

aqedan Cans, rom pirobiTi masa TiTqos modebulia Canacvlebad masaTa simZimis centrSi.

zemoT moyvanili xerxiT, SesaZlebelia vcvaloT masaTa jgufebi erTi pirobiTi masiT, Tu sistemaSi gagvaCnia ori an sami masa, romelTa inerciis momentebic mniSvnelovnad aWarbeben sxva masaTa inerciis momentebs.

nax. 20. sammasiani dayvanili sistema

nax. 21. oTxmasiani dayvanili sistema

nax. 20 a, b-ze warmodgenili sistemisaTvis pirobiTi masebi da damyoloba ganisazRvrbian formuliT:

(59)

(60)

nax. 21 a, b-ze warmodgenili sistemisaTvis pirobiTi masebi da damyoloba ganisazRvrbian formuliT:

(61)

(62)

zemoT ganxilulia masaTa jgufis erTi masiT, miaxloebuli Secvlis meTodi. ufro zust Sedegebs iZleva masaTa jgufis ori masiT Secvlis meTodi.

ganvixiloT nax. 22, a-ze warmodgenili sistema.



nax. 22. masaTa jgufis Canacvlebis sqema ori masiT.

mravalmasiani sistemis (nax. 22, a) oTxmasianiT Canacvlebis sqema, warmodgenilia nax. 22, b-ze. masaTa inerciis momentebi da , romlebic enacvlebian masaTa jgufs I_k (k=3,4,....,n), ganisazRvrebian formuliT:

(63)

monakveTTa da damyoloba ganisazRvreba formuliT:

(64)

sadac e – pirobiTi damyolobaa, saorientacio ganmsazRvreli, rogorc gaormagebuli sidide gamoyvanili formuliT (60).

e.i. rivinisa da l.a. banaxis meTodebi dafuZnebulia rTuli Tanmimdevruli sqemidan metad gamartivebul sqemaze gadasvlis Tanamimdevrulobaze da sakuTar sixSireTa mudmiv analizze.

unda aRiniSnos, rom zemoxsenebuli meTodebi xasiaTdebian Semdegi naklovanebebiT:

1. 
   ZiriTadad gamoiyenebian mxolod mwkriuli mgrexi sistemebisaTvis;
   
2. 
   analizis amocanebis realizeba ver xerxdeba grafikuli gardaqmnebis gareSe, ris gamoc izRudeba analitikuri gardaqmnebi, rac metad mniSvnelovania mizanmimarTuli sinTezis amocanaTa amoxsnisaTvis.
   

3. 
   aproqsimaciuli modelebis agebis axali midgomebi
   

am qveTavSi SemoTavazebulia originaluri Teoriuli midgoma aproqsimaciuli modelebis asagebad.

meqanikuri sistemis SemdgomSi (ms), gantolebaTa sistema gare Zalebis zemoqmedebis dros, SeiZleba Caiweros Semdegi saxiT:

(65)

sadac,

(66)

P – diferencirebis operatori;

I_* – sistemis inerciis matrica;

B_* – blanti winaaRmdegobis matrica;

A_* – gansaxilveli meqanikuri sistemis matrica;

φ(t) – kuTxur koordinatTa veqtori;

q(t) – Semaval zemoqmedebaTa veqtori;

f_H(t) – SemaSfoTebel zemoqmedebaTa veqtori.

formula (66)-is gaTvaliswinebiT ms-is koordinatTaTvis gadamcemi matrica warmoCndeba Semdegi saxiT [3]:

(67)

sadac, aRniSnulia SemaerTebeli matrica, romelSic , matricis elementis algebruli damatebaa, S – laplasis gardamqneli operatoria.

analogiurad isazRvreba gadamcemi matricac SemaSfoTebeli zemoqmedebebiT ms-is gamomavali koordinatTaTvis.

Sesabamisad, gadamcem matricas gaaCnia saxe:

(68)

analogiuri Canaweri gaaCnia gadamcem matricasac .

zogadi saxiT:

(69)

(70)

sadac, i=1,2,...n, k=1,2,..., r=1,2,...m.

momavalSi calkeul koordinatTa dinamikuri maxasiaTeblebis gamokvlevisaTvis gamoviyenebT saxeTa damokidebulebebs.

(71)

sadac Tavis mxriv

(72)

(73)

(74)

xolo da – operatiuli mravalwevri.

Semobrunebuli formiT operatorebi da SesaZloa Caiweros Semdegi saxiT:

(75)

m=1,2,3......

n=1,2,3......

bevr SemTxvevaSi sawyisi saangariSo sqemidan, gamartivebulze gadasvlisaTvis iyeneben arademfirebadi saangariSo sqemebis analizs, SemdegSi (a.s.s) [2,4].

amasTan mravalmasiani saangariSo sqemidan gamartivebul sqemaze gadasvla ganisazRvreba, rogorc sakmarisi piroba kvlevebis sawarmoeblad ara marto gansaxilvel diapazonSi Tavisufal rxevaTa formebis da Sesabamis sakuTar sixSireTa gansazRvra xarisxobrivi miaxloebis gziT, aramed sawyisi da gamartivebuli sqemaTa inerciis jamuri momentebisac.

zogadi saxiT n-sazomi drekad-inerciuli sistema aRiwereba matriculi gantolebiT.

(76)

sadac, I da C – inerciuli da drekadi matricebia.

toloba (76) SeiZleba Caiweros Semdegi saxiT:

(77)

sadac,

(78)

matricis A sakuTari ricxvebi λ_i(i=1,2,...n) warmoadgenen sakuTar sixSireTa ω_i kvadratebs, xolo yoveli λ_i-is Seesabameba sakuTari veqtorebi μ_i – ω_i sixSireze Tavisufal rxevaTa formebi.

analogiuria matricis sakuTari ricxvebic

(79)

aseve warmoadgenen sakuTar sixSireTa kvadratebs, romlebsac Seesabameba zogierTi sakuTari veqtorebi.

cnobilia, rom dinamikuri datvirTvebis struqturaSi yvelaze xSirad gadamwyvetia kerZo amonaxsnis jami da mTavari sixSiris moduli. mTavar sixSired iwodeba is, romelsac gaaCnia yvelaze didi amplituda da araa aucilebeli iyos umdablesi. ormasiani da sammasiani gamartivebuli sqemebi iZlevian sakmaod sarwmuno warmodgenas mTavar sixSirul sidideebze [5].

tolobis pirobis Sesrulebisas gamartivebuli da sawyisi saangariSo sqemebis jamuri inerciis momentebis Sedarebisas, fasdeba gamartivebuli saangariSo sqemis sakuTari λ_i ricxvebi da sakuTar veqtorTa komponentebi μ_i sawyisi saangariSo sqemis Sesabamisi sakuTar ricxvTa da sakuTar veqtorTa komponentebis siaxlove (ricxobrivi) [2].

Tu ganvixilavT matricul sistemas, rodesac q(t) da f_H(t) warmoadgenen regularul funqciebs (safexuriani, siCqaruli, harmoniuli an eqsponencialur zemoqmedebiani), maSin gardamavali procesebis Tavisufali mdgenelebi ganisazRvrebian Tavisufal rxevaTa gantolebebiT [2].

calkeul konkretul koordinatTa dinamikuri maxasiaTeblebis gansazRvrisas, viyenebT damokidebulebis cnobil saxes:

(80)

sadac, - Semavali koordinatia,

- gamomavali koordinati.

arademfirebuli saangariSo sqemisas gveqneba:

(81)

Semobrunebuli formis pirobebSi a.s.s-is dros, operatori (76) miiRebs saxes:

(82)

(83)

m=1,2,...

n=1,2,....

cnobili gamokvlevebis Tanaxmad, rTuli dinamikuri sistemebis miaxloebiTi agebis amoxsnisaTvis erT-erT efeqtur midgomad iTvleba warmosaxviT sixSireTa maxasiaTeblebis gamoyeneba [6].

φ(t) da φ(p) procesis warmosaxviT sixSireTa maxasiaTeblebs uwodeben funqcias, romelic miiReba φ(t) argumentisaTvis rigi arsebiTi mniSvnelobebis p=δ miniWebis Sedegad [6].

amave [6] naSromSi naCvenebia, rom Tu φ(p) gamosaxulebis mixedviT miniWebulia maxasiaTeblis φ(δ) mniSvneloba, arsebiTad dadebiTi naxevarRerZis mcire monakveTis yvela wertilSi, calsaxad ganisazRvreba φ(t) originali, rodesac 0 ≤ t ≤ ∞.

amave SromebSi melinis pirdapiri gardaqmnis gamoyenebis safuZvelze dadgenilia analitikuri damokidebuleba original φ(t) da im mniSvnelobaTa erTobliobas Soris, romlebic Rebuloben φ(δ) gamosaxulebas δ dadebiT naxevarRerZze (ara gansakuTrebul wertilebSi), rac gamomdinareobs mTel marjvena naxevarsibrtyeze, plius xazi p=k+jω (k=const; -ω ≤ ω ≤ ∞), funqciis F(δ) analitikuri gagrZelebis erTianobidan. naSromSi [6] melinis gardaqmnis safuZvelze miRebulia erTmniSvnelovani analitikuri urTierTkavSiri original φ(t)–sa da F(δ) mniSvnelobaTa erTobliobas Soris, romlebic dadebiT naxevarRerZze δ Rebuloben gamosaxulebas. es urTierTkavSiri gamoisaxeba formuliT:

(84)

sadac, s – melinis gardaqmnebSi damxmare kompleqsuri cvladia [6]

saerTo SemTxvevaSi aRwerili meTodi iZleva saSualebas aigos aproqsimaciebi rTuli sistemis amoxsnisaTvis, rogoricaa (75). amave dros aproqsimaciis sizustidan gamomdinare arsebobs sakmaod mniSvnelovani SezRudvebi.

naSromSi [6] miRebulia formulebi, warmosaxviT sixSireTa maxasiaTeblebis cdomilebaTa funqciebSi da amplitudur-fazur maxasiaTeblebSi, originalis cdomilebisaTvis.

miRebuli damokidebulebebis analizma gviCvena, rom im SemTxvevaSi roca (Zlieri rxevaTa sistema), sadac θ – kompleqsur sibrtyeze ganlagebuli naxevarRerZebis F(p) sferos SemzRudveli kuTxea, warmosaxviT sixSireTa aproqsimaciis cdomileba amplitudur-fazuri sixSiris maxasiaTeblebis mixedviT, 1%-iT Seesabameba originalis gansazRvras, modulis mixedviT aproqsimaciis cdomileba miiReba zRvrebSi 10-12%.

Semdgomi kvlevebi ukavSirdeba erTeulovani garegani zemoqmedebebisas meqanikuri sistemis gamomavali koordinatebis gamosaxulebaTa analizis safuZvelze saangariSo sqemis gamartivebas. aqedan gamomdinare momavalSi vioperirebT funqciebiT:

(85)

ganxilvaSi axali operatoris s_*=s² SemotaniT [7], vwerT modificirebul gamosaxulebas:

(86)

romelic Tavis mxriv warmoadgens eqsponencialur milevad mdgenelebiani gardamavali procesebis mqone sistemis gardamavali funqciis analogs. amasTan dakavSirebiT SeiZleba aRiniSnos, rom naSromSi [6] moyvanili kvlevebis Tanaxmad, SezRuduli kuTxis mqone θ ≤ 20⁰, polusebiani gamosaxulebisaTvis, warmosaxviT da amplitudur-fazuri sixSiruli maxasiaTeblebiT originalis sizustis gansazRvra praqtikulad erTi rigisaa [6], xolo polusebiani sistemisaTvis, SezRuduli kuTxiT , amave naSromSi [6] miRebulia damokidebuleba:

(87)

sadac, - warmosaxviT sixSireTa maxasiaTeblebis cdomilebis maqsimaluri mniSvnelobaa.

tolobis (79) gamoyenebiT gadavdivarT warmosaxviT sixSireTa modificirebul maxasiaTeblebze, romelTac vRebulobT S_* argumentisaTvis rigi arsebiTi mniSvnelobebis S_* = δ miniWebiT, romlebic Tavis mxriv warmoadgenen eqsponencialur-milevadi procesebiani sistemis maxasiaTeblebis analogs [7].

aseTi maxasiaTeblebisTvis originalis gansazRvris sizuste sakmaod maRalia, originalisa da maxasiaTeblebis cdomileba TiTqmis tolia.

Tu gansaxilvel aproqsimaciaSi, aproqsimirebis saxiT viyenebT gamosaxulebas miRebuls damokidebulebidan:

maSin, - SesaZloa ganisazRvros sakuTar ricxvTa da sakuTar veqtorTa μ_i siaxlovis safuZvelze, amasTan gansazRvrul koordinatebze gamosaxulebis operirebisas (gadamcemi matricis konkretuli mdgenelebis Φ ganxilvisas) gamosaTvlel aproqsimaciebs virCevT sawyisi sistemis rxevaTa formebidan da sakuTari sixSireebidan, romlebic iqnebian mTavari mocemuli koordinatisaTvis (mocemuli ms kavSirisaTvis).

SegviZlia davaskvnaT, rom modificirebuli warmosaxviTi maxasiaTeblebis gardamavali procesebis integraluri miaxloebis procedurebSi gamoyenebam gviCvena, rom originalisa da aproqsimaciuli maxasiaTeblebis gansazRvris cdomileba, Rebulobs Tanabar mniSvnelobebs, rac miuTiTebs SemoTavazebuli meTodis struqturulad rTuli amZravTa mravalmasiani sistemis aproqsimaciuli modelis agebis sakmaod maRal efeqturobaze.

3.4. aproqsimaciuli modelebis parametrebis gansazRvra

wina qveTavSi formulirebuli iyo mecnieruli safuZvlebi rTuli sistemebis aproqsimaciisaTvis modificirebul warmosaxviT sixSireTa gaTvaliswinebiT.

am qveTavSi ganvixilavT or da sammasian gamartivebuli modelebis gamoyenebiT konkretuli ori da sammasiani gamartivebuli modelebis gamoyenebiT konkretuli parametrebis Zebnis procedurebs.

SromebSi [88,89] ganxilulia warmosaxviT sixSireTa modificirebuli maxasiaTeblebis gamoyenebiT arademfirebuli mravalmasiani dinamikuri sistemebis aproqsimaciis sakiTxebi:

naxazze 23 da 24 mocemulia ormasiani da sammasiani meqanikuri sistemis dinamikuri struqturuli sqema [5].

nax. 23. sammasiani meqanikuri sistemis struqturuli sqema



nax. 24. ormasiani meqanikuri sistemis struqturuli sqema

am naxazebze: I₁, I₂ da I₃ – mbrunavi masaTa inerciis momentebia; C₁₂ da C₂₃ inerciis momentebs Soris uinercio meqanikur drekad elementTa sixisteebia; M Zravis eleqtromagnituri momenti; M_C1, MC₂ da M_C3 – winaRobis momentebia, romlebic Seicaven, rogorc aqtiur Semadgenel datvirTvas, aseve reaqtiul Semadgenelsac mSral da blant xaxunisas da agreTve periodulad Semadgenel datvirTvebsac; M_C1, MC₂ da M_C3 masaTa Soris drekad urTierTzemoqmedebis momenti; P diferencirebis operatori.

arademfirebuli saangariSo sqemis ganxilvisas winaaRmdegobis momentebi ar SeiZleba warmodgenil iqnan reaqtiuli Semadgenlebis saxiT, romlebic warmoadgenen ganzogadoebul koordinatTa da maT warmoebulTa funqciebs. naxazebze 23 da 24 moyvanili struqturuli sqemis gardaqmnisas, vRebulobT sakvlevi sistemis gadamcem funqcias, marTvadi zemoqmedebis M da M_C1, M_C2, M_C3 winaaRmdegobis momentebis mixedviT.

sammasiani sistemisaTvis gveqneba

(88)
(89)

(90)

(91)

(92)

(93)

(94)

(95)

(96)

analogiurad vRebulobT gadamcem funqcias, arademfirebuli ormasiani meqanikuri sistemisTvisac.

(97)

(98)

(99)

aproqsimaciisaTvis viyenebT warmosaxviT sixSireTa modificirebul maxasiaTeblebs [88,89], am mizniT ormasiani modelis aproqsimaciisaTvis unda gamoyenebul iqnas toloba:

(100)

(101)

(102)

sadac: φ₁(P)_a, φ₂(P)_a da M₁₂(P)_a – Sesabamisi koordinatebis maprooqsimirebeli gamosaxuleba;

φ₁(P)_saw., φ₂(P)_saw. da M₁₂(P)_saw. – struqturulad rTuli mravalmasiani meqanikuri sistemis aproqsimaciisaTvis regulirebad koordinatTa sawyisi gamosaxulebebia; P - karsonis operatori.

calkeuli zemoqmedebebisas gamosaxulebis gamoyeneba, warmosaxviT sixSireTa modificirebul maxasiaTeblebze gadasvliT, Tanaxmad tolobebisa (100) da (102) SegviZlia CavweroT:

(103)

(104)

(105)

Sesabamisad gadasawyveti amocanebisa Semdgomi aproqsimaciuli procedurebi SesaZloa agebul iqnas erT-erTi tolobis (103) da (105) safuZvelze. miaxloebis SemTxvevaSi, tolobis (100)-is safuZvelze vwerT Semdeg sawyis gantolebebs:

(106)

(107)

damokidebulebebis gamoyenebiT

(108)

toloba (101)-dan miRebiT, gantolebas (102) gardavqmniT Semdegi saxiT:

(109)

sadac, M₁₂₀ – momentis dadgenili mniSvnelobaa. tolobis (103) safuZvelze vadgenT pirobiT gantolebas:

(110)

sadac: δ_*1- interpolaciur wertilebSi (i=1,2,3...) modificirebul warmosaxviT sixSireTa mniSvnelobaa, wrfiv normalur gantolebaze Semdgomi gadasvliT, saZiebo x,y da c₁₂ –is mimarT. .

aproqsimaciuli amocanis sxvagvarad midgomisaTvis SesaZlebelia gamoyenebul iqnas damokidebulebebi:

(111)

(112)

sadac, φ₁₀, φ₂₀ da M₁₂₀ – Sesabamisi procesebis dadgenili sidideebia.

pirveli ori gantolebidan vsazRvravT:

(113)

(114)

sistemis pirobiTi gantolebebis damuSavebisaTvis

(115)

umciress kvadratTa meTodiT, davdivarT wrfiv normalur gantolebaTa sistemamde saZiebo c₁₂ da I₂ parametrebis mimarT.

sammasiani modelis aproqsimaciis daxmarebiT pirvel rigSi viyenebT tolobas:

(116)

(117)

(118)

(119)

paralelurad (116), (117) da (118) gantolebebis amoxsnisas, gvaqvs

(120)

(121)

damokidebulebebis (119) da (121) gamoyenebiT vwerT pirobiT gantolebaTa sistemas:

(122)

umcires kvadratTa meTodiT miRebuli pirobiT gantolebaTa sistemis damuSavebisas vRebulobT wrfiv normalur gantolebaTa sistemas saZiebo I₁, I₂, I₂c₂₃, c₁₂ da c₁₂c₂₃ parametrebis mimarT. miRebuli sistemis damuSavebisas davdivarT zogierTi saxis damokidebulebebamde:

(123)

amonaxsnis safuZvelze sabolood gveqneba:

aqve avRniSnavT, rom pirobiTi gantolebaTa agebisaTvis gansazRvrul SemTxvevebSi SeiZleba aseve gamoyenebul iqnas miaxloebiTi toloba Semdegi saxiT:

(124)

Catarebuli kvlevebis safuZvelze SemuSavda struqturulad rTuli mravalmasiani meqanikuri sistemebis aproqsimaciuli modelebis parametruli sinTezis inJinruli meTodika.

am SemuSavebuli meTodikis aprobaciisaTvis vixilavT kinematikur sqemas 8 masiani xiduri amwekranis meqanizms, romelic warmodgenilia naxazze 25.



nax. 25. xiduri amwekranis amwe meqanizmis saangariSo sqema

naxazze (25), I₁ – eleqtroamZravis rotoris inerciis momenti; I₂ – muxruWis Skivisa da Zravis naxevarquros inerciis momenti; I₃ - I₆ – reduqtoris kbila Tvlebis inerciis momenti; I₇ – doluras inerciis momenti; m₈ – asawevi tvirTi masa woniT Q; C_i,i+1 – SemaerTebeli elementebis sixistis koeficientebi.

warmodgenili sistema cnobili midgomiT daiyvaneba 6 masian saangariSo drekadkavSirebian sqemaze, SemoTavazebuli meTodis Tanaxmad Semdgomi analizisaTvis viyenebT maTematikur models:

(125)

Semogvaqvs S_* da gadaviyvanoT warmosaxviT gantolebaTa sistemaze;

(126)

es sistema gamsxvilebulad iRebs saxes:

(127)

sadac - ganisazRvrebian sistemidan (126)

am sistemis amoxsniT viRebT

,

sadac:

(128)

(129)

Tu ganvixilavT dinamikas M-is zemoqmedebisas, saqme gveqneba gadamcem funqciasTan

,
sadac:

;

;

. (130)

viRebT saaproqsimacio gamosaxulebas

(131)

Tu gaviTvaliswinebT, rom

,

sadac:

,

miviRebT

.

gamosaxuleba (131) SeiZleba Caiweros Semdegnairad:

. (132)

saZiebel parametrebs warmoadgenen

da .

Tu dauSvebT, rom , maSin dagvrCeba erTi saZiebo parametri

mocemuli tolobebidan gamomdinare unda Caiweros pirobiTi gantolebaTa sistema

. (133)

aqamde cnobili miRebuli damokidebulebebidan gamomdinare, SeiZleba warmovidginoT:

;

;

.
maSin,

.

gardaqmnil aproqsimaciul funqcias eqneba saxe:

.

veZebT da , romlebic gamosaxulebaSi warmodgenili arian warmosaxviTi saxiT.

toloba (133) mimarTebaSi vwerT koordinatTa sawyis tolobas.

an da pirobiT gantolebebs:

SegviZlia CavweroT:

sadac saZiebeli ,

maSin miaxloebiTi toloba miiRebs saxes:

gveqneba pirobiTi gantolebebi:

an

sadac

aviRoT interpolaciis Semdegi kvanZebi:

0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.7; 1.0; 1.5; 2.0; 3.0; 5.0; 7.0; 10

vamuSavebT ra pirobiT gantolebebs umcires kvadratTa meTodiT, davdivarT gantolebaze:

,

sadac:

am gantolebis amonaxsni iqneba:

=246·10³.

saangariSo kvlevebisaTvis SerCeul iqna sawyisi mniSvnelobebi:

I₁ = 0.0576, I₂ = 0.0410, I₃ = 0.0102, I₄ = 0.032, I₅ = 0.029, I₆ = 0.015 kgmwm²; c₁ = 14800, c₂ = 148, c₃ = 0.784, c₄ = 0.629, c₅ = 0.259, c₆ = 0.653.

agebul iqna sami ormasiani aproqsimaciuli modeli, c₁, c₂ da c₃ kavSirebis mimarT da formulis Tanaxmad gaangariSebul iqna gamartivebuli modelis kerZo sixSireTa kvadratebi:

sadac I_1pr da I_2pr - ormasiani sqemis dayvanili momentebia;

c – im kavSiris sixistea, romlis mimarTac xorcieldeba aproqsimaciuli modelis ageba. miRebul iqna, c₁ - Tan mimarTebaSi c₁ = 13995, I₁ = 0.05, I₂ = 0.06, ω² = 529000; c₂ - Tan mimarTebaSi c₂ = 167, I₁ = 0.096, I₂ = 0.013, ω² = 12530; c₃ - Tan mimarTebaSi c₃ = 0.81, I₁ = 0.109, I₂ = 0.036, ω² = 252.

6 masiani sistemis sixSiruli speqtris cifruli gaangariSebebis safuZvelze gansazRvrisas, miRebul iqna: ω₁² = 251,092, ω₂² = 16025, ω₃² = 620085, ω₄² = 1220091, ω₅² = 11223085....

sawyisi da gamartivebuli modelebis mixedviT miRebul sixSireTa Sepirispirebam gviCvena, rom pirveli gamartivebuli ormasiani modelisaTvis sixSiris kvadratis cdomilebam mesame sixSiris kvadratidan, Seadgina 85,3%; meore gamartivebuli modelisaTvis sixSiris kvadrati meore sixSiris kvadratidan Seadgens 78,3%; mesame gamartivebuli modelisaTvis sawyisi mesame sixSiridan Seadgens 100,4%.
4. sinTezis amocana modificirebuli maxasiaTeblebis gamoyenebiT

ganvixiloT rTuli sistemis sinTezis amocanis, aproqsimaciuli modelebis agebis SemuSavebuli Teoriis gamoyenebiT. wina paragrafebis Tanaxmad i da j dayvanili masebia, romelTa Soris mdebareobs sakvlevi kavSiri. P – karsonis gardaqmnis operatoria.

integraluri miaxloebis aparatis saxiT viyenebT warmosaxviT sixSireTa modificirebul maxasiaTeblebs da sakvlev koordinatebs vaniWebT Sesabamis saxes: Μij(P_*) da ij(P_*), sadac P_* = P².

sakvlevi sistemis sinTezis amocanis ganxilvisas, warmosaxviT sixSireTa modificirebuli maxasiaTeblebisa da arademfirebuli saangariSo sqemis gamoyenebiT, optimizaciuri sinTezis amocanis amoxsnisaTvis, SesaZlebelia visargebloT sasurveli procesebiT, romelSic sakuTar sixSireTa rxevebi ganawilebul iqneba garkveuli kanonzomierebiT, geometriuli progresiis mixedviT. amasTan dakavSirebiT naSromSi [6] sasurveli varirebadi procesebis saxiT, SeiZleba SeirCes Semdegi saxis funqcia:

(134)

sadac q – geometriuli progresiis mniSvnelia.

ufro farTo azriT varirebadebis saxiT, SegviZlia avirCioT ori ganzogadoebuli paramtri q da z_m. sadac, z_m - drois maStaburi koeficientia. aseT SemTxvevaSi sasurveli koordinatis gamosaxulebas ij(P_*) ganvixilavT Semdegi saxiT:

(135)

ganxilvadi sistemis sinTezisas, optimizaciis kriteriumebis saxiT, rogorc wesi gvevlinebian dinamiurobis koeficientebi da gardamavali procesebis milevis koeficienti. zogadi saxiT regulirebadi da sasurveli koordinatis saxiT dinamiurobis koeficientis mixedviT parametruli sinTezisas, SegviZlia gamoviyenoT Semdegi saxis funqciebi:

(136)

(137)

sadac, σ_ijk - sakvlevi sistemis sinTezirebadi parametrebia;

M_ijkst. – dadgenili (statikuri) momentebia.

Semdgomi amocana urTierTkavSirSia, sasurveli da integrirebuli procesebis integralur miaxloebebTan uSualod maTi warmosaxviT sixSireTa modificirebuli parametrebis daaxloebiT. am miznisaTvis viyenebT tolobas:

(138)

sadac:

(139)

da - Sesabamis originalTa warmosaxviT sixSireTa modificirebuli maxasiaTeblebia. δ* = ίω – warmosaxviTi sixSireebia; ω - rxevis sixSirea; k = 1,2,3....

toloba (138) saangariSo procedurebis realizebisas davdivarT sakmaod rTul arawrfiv algebrul gantolebaTa sistemamde saZiebo σ_ijk–s mimarT. es erTis mxriv, xolo meores mxriv saangariSo procedurebis procesSi SesaZlebelia SevejaxoT praqtikulad gaangariSebiT miRebuli parametrebis ararealizebad sidideebs, arada praqtikulad inJinruli sinTezis amocanebi Seicaven didi raodenobis teqnikur SezRudvebs.

aRniSnuli sirTuleebis gamosaricxad SemoTavazebulia, integraluri miaxloebis procedurebSi saZiebo σ_ijk–s gamovlenisaTvis, nacvlad gamoviyenoT damokidebulebebi:

(140)

warmodgenilia, pirvel wevrTa dakavebiT, daSla, maklaranis arawrfivi funqciis rigiT, sadac K_ijH funqciis mniSvnelobaa zogierTi saZiebo parametrebis sawyis mniSvnelobaTa dros, magram es saangariSo sqema Tavis mxriv moiTxovs sawyisi sasurveli procesebis miniWebas iseTi xarisxobrivi da raodenobrivi sinTezirebadi procesebis siaxloviT, romelSic arsebobs garkveuli imedi wrfivi daSlis dasaSvebad (140) damokidebulebis saxiT.

miTiTebuli pirobis Sesasruleblad SemoTavazebulia mimdinare saangariSo procedurebis ganxilvadi rTuli sistemisaTvis realizebadi bijuri moZraoba optimizaciur maCveneblebamde:

1. 
   cnobili meTodebis Tanaxmad, statikuri Zaluri parametrebis mixedviT mravalmasiani sistemis parametrebis gaangariSeba.
   
2. 
   aproqsimaciuli gadawyvetis agebis safuZvelze, sistemis drekad kavSirebSi dinamikuri datvirTvebis analizi iangariSeba pirveli punqtis mixedviT, Sesabamisi parametrebis a₄, a₂ da b₂–is pirvel (1) an meore (2) gamosaxulebaSi SerCeviT.
   
3. 
   yovel momdevno procedurebSi, mecnierul aproqsimaciuli gadawyvetilebebis safuZvelze, sinTezirebadi procesebis bijuri moZraobiT optimizaciur maCveneblebamde da sasurvel procesebSi, sawyis gamosaxulebaTa ganzogadoebuli parametrebis z_m da q–s variaciis saxiT sasurveli procesebis ageba.
   
4. 
   sinTezirebadi parametrebis mniSvnelobaTa Zieba sasurveli procesebis ganzogadoebuli varirebad parametrebTan funqcionalur urTierTkavSirSi.
   

zemoT formulirebadi saangariSo proceduraTa kompleqsis realizaciis safuZvelze, maTematikuri kanonzomierebebis gamovlena, gvaZlevs saSualebas avagoT funqcionaluri urTierTkavSiri K_mij (t) da sakvlevi sistemis sinTezirebad parametrTa mniSvnelobebs Soris. xolo es ki Tavis mxriv gvaZlevs saSualebas SevirCioT drekadi sistemis danarCen kavSirebSi misaRebi dinamiuri koeficientebis mniSvnelobaTa pirobidan sinTezirebadi parametrebis mniSvnelobebi.

zemoT moyvanili bijuri saangariSo sqemis xarisxobrivi analizi, garkveuli xarisxiT miuTiTebs mizanmimarTulobas parametruli sinTezis Semobrunebuli sqemis asagebad sasurveli procesebis ganzogadoebuli parametrebis Z_m da q urTierTkavSirisaTvis ori (I da II) saangariSo kompleqsis safuZvelze, Sesabamisad calke miTiTebuli parametrebis variaciiT. I saangariSo kompleqsis realizaciisas, sasurvel procesSi mocemul sakuTar sixSireTa da mocemuli q sididis dros, varirebad parametrebad gvevlineba, mxolod ganzogadoebuli Z_m parametri. amasTan yoveli g-uri bijis variaciisas Z_m= Z_mg integraluri miaxloebiT sasurveli procesis SeCevis saxiT, miRebuli g-1 bijze. aqve unda iTqvas, rom aRniSnuli Tema pirdapiri gagrZelebaa zemoT ganxiluli sakiTxisa, amitom pirdapiri formis saxiT gamoviyenebT pirobiT mniSvnelobebs.

Z_m variaciis pirvel bijze, Z_m = X_m1, sawyisi sasurveli procesis saxiT virCevT funqcias Semdegi saxiT [2,3]

(141)

sadac: b_1H, a_2H da a_1H - sakvlevi sistemis, statikuri gaangariSebidan miRebuli σ_ijkH parametrebiani, aproqsimaciulad miRebuli regulirebadi koordinatTa koeficientebia;

z_mij1 pirveli z_mij parametris varirebadi mniSvnelobaa [1,3].

gamosaxuleba (141)-Tan vanxorcielebT regulirebad koordinatTa integralur miaxloebas, regulirebad koordinatTa K_mij (p_*; σ_ijk) sinTezirebadi σ_ijk parametrebisaTvis K-s SerCevis daxmarebiT, maTi, K_mij gamosaxulebis Secvlis pirobidan, Δσ_ijk1 gardaqmnebis meSveobiT, zogierT sawyis σ_ijH mniSvnelobaTa farglebSi mxolod pirvel daSlil wevrTa dakavebiT. Semdeg vRebulobT parametrTa saZiebo sidideebs, gaangariSebuli Semdegi tolobis mixedviT:

(142)

procesis integraluri miaxloebis pirobidan, vanxorcielebT Semdgomi procesis aproqsimaciuli funqciis pirveli bijis z_m variaciis mniSvnelobaTa dazustebas sasurvel procesTan (141) parametrebis SerCevis xarjze.

(143)

zm variaciis meore bijze, z_m + z_mij2 mniSvnelobisas, sawyisi sasurvel integraluri miaxloebis procesisaTvis virCevT funqcias:

(144)

Semdeg sasurveli funqciis (144) procesis K_mij (p_*; σ_ijk2) integraluri miaxloebis daxmarebiT veZebT Δσ_ijk2 koordinatTa gardaqmnas da Sesabamisad sinTezirebad parametrTa sidideebsac

(145)

Semdeg veZebT funqcias analogiurs (143) funqciisa

(146)

ganzogadoebuli saxiT, yovel momdevno g-iur iteraciul mniSvnelobaze

sadac Δz_mij – ganzogadoebuli z_mij parametris mniSvnelobis bijuri gardaqmnaa. sasurveli saxiT vaniWebT (g-1) dazustebul aproqsimaciul funqcias Semdegi saxiT:

(147)

xolo sinTezirebad regulirebad koordinats Kmij (p_*; σ_ijg) g–iur saangariSo bijisas, veZebT procesis integraluri miaxloebiT g–iur sasurvel procesTan.

Semdeg vsazRvravT g–iur dazustebul aproqsimaciul funqcias:

(148)

Semdeg vsvavT tolobaSi (148) axal mimdinare mniSvnelobas z_mij(g+1), saidanac vRebulobT (g+1) saangariSo bijis sasurvel funqcias.

aRwerili saangariSo procedurebis Catarebis safuZvelze vRebulobT funqcionalur damokidebulebebs:

σ_1ij(z_m), a_2ij(z_m), a_1ij(z_m), σ_ijk = f_ijk(z_m)

Semdgomi kvlevebi dakavSirebulia II saangariSo kompleqsis procedurebTan, romlebic dafuZnebulia q - parametrTa variaciaze – sasurvel procesTa gamosaxulebaSi sakuTar sixSireTa Sesabamis Sefardebis koeficientze. am q variacias vanxorcielebT yovel g-iur saangariSo bijze. es variaciebi xorcieldeba ukve arsebul sawyis q_gH mniSvnelobaTa farglebSi dazustebul aproqsimaciul funqciaSi , q–s axal mniSvnelobaTa nabij-nabij miniWebiT.

saboloo jamSi davdivarT funqcionalur damokidebulebebamde b_1ij (z_m; q), a_2ij (z_m; q), a_1ij (z_m; q) d_a σ_ijk = f_ijk (z_m), romlebic aucilebelia sistemis saboloo parametrTa SerCevisaTvis, sakvlevi sistemis dinamikuri maCveneblebis miRebis mizniT.

5. ganStoebuli sistemis dinamikuri kvleva

ganvixilavT ganStoebuli sistemis modelirebas, romlis kinematikuri sqemac mocemulia naxazze (26).