saqarTvelos teqnikuri universiteti

satransporto da manqanaTmSeneblobis fakulteti


departamenti #133

“sawarmoo teqnologiuri manqanebi da meqatronika”

saleqcio kursi saganSi

“მექანიკურ სისტემათა დინამიკური მოდელირება”

სრული პროფესორი თამაზ მჭედლიშვილი

აკადემიური დოქტორი გიორგი გრატიაშვილი

Tbilisi,

2012 w.


Sinaarsi

Sesavali 4

1. 
   manqanaTa meqanikuri amZravebi da maTi sqematizaciis sakiTxebi 4
   
   1. 
      dinamikuri sqemebi 6
      
   2. 
      gare datvirTvebis dayvana 8
      
   3. 
      masebisa da inerciis momentebis dayvana 10
      
   4. 
      sixisteebis dayvana da gansazRvra 21
      
2. 
   meqanikuri sistemebis proeqtirebisa da dinamikuri kvlevis sakiTxebi 26
   
3. 
   maTematikuri modelebi da kvlevis amocanebi 31
   
   1. 
      meqanikuri sistemebis modelirebis sakiTxebi 31
      
   2. 
      gamartivebul modelebze gadasvlis amocanis dasma 39
      
   3. 
      aproqsimaciuli modelebis agebis axali midgomebi 47
      
   4. 
      aproqsimaciuli modelebis parametrebis gansazRvra 54
      
4. 
   sinTezis amocana modificirebuli maxasiaTeblebis gamoyenebiT 67
   
5. 
   ganStoebuli sistemis dinamikuri kvleva 74
   
   1. 
      lilvebis kinetikuri energiis gansazRvra 75
      
   2. 
      dinamikuri gantolebis Sedgena 78
      
   3. 
      saangariSo sqemis inerciuli da sixistobrivi maxasiaTeblebis gansazRvra 81
      
   4. 
      damyarebul samuSao reJimebSi amZravis brunvis kritikuli siCqareebis gansazRvra 82
      
   5. 
      amZravis gamartivebuli sqemis dinamika 85
      

literatura 96

შესავალი

1. manqanaTa meqanikuri amZravebi da maTi sqematizaciis sakiTxebi

manqanaTa elementTa moZraobis siCqaris zrdasTan dakavSirebiT, dinamikuri gaangariSebis roli ganuwyvetliv izrdeba. sakmarisad mtkice da araliTontevadi manqanebis Seqmnis amocana, romelic gauwevs winaaRmdegobas, rogorc statikur aseve dinamikur datvirTvebs gazrdili siCqareebis pirobebSi, sul ufro da ufro rTuldeba, rac Tavis mxriv iwvevs bevri faqtoris gaTvaliswinebas, romlebic adre SezRuduli siCqareebis pirobebSi mxedvelobaSi ar miiReboda. amis gamo dadga aucilebloba statikuri gaangariSebebis nacvlad gamoeyenebinaT dinamikuri gaangariSebebi, romlis drosac manqana ganixileba kompleqnaxad, rogorc erTiani eleqtromeqanikuri agregati. masSi moqmedi gare Zalebi ganisazRvrebian ara marto muSa organoze moqmedi winaaRmdegobebiT, aramed moZravi momentis cvlilebis kanonebiT, rogorc siCqaris an drois funqcia. am praqtikam gansakuTrebuli gamoyeneba hpova eleqtroamZravian manqanebSi, romelTa meqanikuri maxasiaTeblebi sakmaod zustad gamoisaxebian analitikuri damokidebulebebiT.

vinaidan manqanis drekadi sistemis deformacia misi muSa organos gadaadgilebasTan SedarebiT sakmaod mcirea, didi xnis manZilze, gaangariSebebis dros transmisias Tvlidnen absoluturad xists, anu Tvlidnen, rom mTlianobaSi manqanis muSaobis analizisaTvis arsebiT mniSvnelobas ar warmoadgenda. TumcaRa manqanaTa siCqareTa zrdam, rig SemTxvevebSi muSa organoebze statikuri winaaRmdegobebis arastabilurobam, warmoqmna manqanaTa drekad sistemebSi rxeviTi procesebi, romelTa ugulebelyofac Seiqmna SeuZlebeli. amasTanave gamovlinda, rom drekadi sistemis elementebis mcire deformaciebma, gamowveuli rogorc Tavisufali, aseve iZulebiTi rxevebiT, aRarafers vambobT ukve rezonansul procesebze, SeiZleba migviyvanos Zabvebis warmoqmnasTan ara marto statikurTan Tanazomvad, aramed rig SemTxvevebSi am ukanasknelze metad gadaWarbebul zRvrebamdec ki.

amZravebi saWiroa Seswavlil iqnas, rogorc erTiani eleqtromagnituri agregati, amasTan dinamikur sqemaSi unda gaTvaliswinebul iqnas Semdegi faqtorebi:

muSa organoze moqmedi winaaRmdegobis Zalebi, romlebic rig SemTxvevaSi arian arastabilurni, icvlebian gansazRvrul zRvrebSi (magaliTad, sasargeblo wiaRiseulis mompovebel manqanebSi, presebSi da a.S.)

amZravi Zravis eleqtruli maxasiaTeblebi; zogierTi tipis manqanebSi, Zravis statikuri maxasiaTeblebi, Zalian mcire drois procesebisaTvis, romlebic gamosaxaven mbrunavi momentis damokidebulebas kuTxur siCqareze, aRmoCndebian xolme arasakmarisi, ris gamoc icvlebian dinamikuri maxasiaTeblebiT;

elementTaSorisi maxasiaTeblebi: hidravlikuri turboquroebi, drekadi quroebi, zambarebi, Rveduri gadacemebi da a.S.;

manqanis transmisiis elementTa drekadi maxasiaTeblebi (lilvebis, kbila Tvlebis).

manqanaTa gamoyenebiT dinamikis amocanas warmoadgens manqanaTa moZraobis procesebis kvleva, oRond ara saerTod, aramed manqanaTa maxasiaTeblebis (simtkice, kinematikuri) Sedegebis miReba, am ukanasknelTa Seswavlis mizniT. saboloo mizans warmoadgens sawyisi monacemebis gansazRvra Semdgomi gaangariSebebis sawarmoeblad, rogoricaa manqanaTa simtkice, mwarmoebluroba, sicocxlis xangrZlivoba da amZravis simZlavre.

manqanis kinematikuri sqemisa da naxazis mixedviT Znelia msjeloba Tu rogoraa ganawilebuli transmisiaSi masebi da sixisteebi. transmisiis Semadgeneli nawilebi moZraoben gansxvavebuli siCqareebiT, gadascemen gansxvavebul mgrex momentebs, araerTnairia maTi kveTis sixisteebi da Sesabamisad calkeul ubnebze mgrexi kuTxeebic.

amasTan dakavSirebiT, manqanis transmisiis moZraobis gantolebis Sedgenis win, gamosaxaven mas pirobiTi meqanikuri sqemis saxiT, romelsac ewodeba manqanis eqvivalenturi sqemis dayvana. es sqema namdvilad unda warmoadgendes realuri transmisiis eqvivalents, anu sworad asaxavdes mis ZiriTad dinamikur maxasiaTeblebs. dayvanili saangariSo sqemis Sedgena – mniSvnelovani etapia manqanaTa gamoyenebiTi dinamikis amocanaTa amoxsnisas. am etapze Setanili Secdoma moqmedebs amocanis yvela amonaxsnze da mis gamokvlevaze.

1.1. dinamikuri sqemebi

yoveli manqana Sedgeba Zravisagan, gadacemisa da Semsrulebeli organosagan an meqanizmisagan. moqmedi datvirTvebis gansazRvrisaTvis, mizanSewonilia manqanaTa sqemebis moyvana dayvanili saxiT. manqanaTa konstruqciidan da kvlevis miznidan gamomdinare mocemuli gaTvliTi sqemebi Seicaven erT masas an erTmaneTTan drekadi rgolebiT dakavSirebul ramodenime masaTa sistemas (ori, sami, xandaxan oTxic) an nawildebian gansazRvrul monakveTTa zRvrebSi.

nax. 1. Seyursulmasebiani dayvanili saangariSo sqemebi

a) erTmasiani sqema; b) ormasiani sqema; g) sammasiani sqema.



nax. 2. ganawilebulmasiani dayvanili saangariSo sqema

dayvanili masaTa sidide SeiZleba iyos rogorc mudmivi aseve cvladi. saerTod drekadi rgolebis sixiste da Sida Zalebi (mamoZravebeli da winaRobis Zalebi) warmoadgenen cvlad sidideebs, romlebic damokidebuli arian sistemis mdgomareobasa da wamyvani elementis siCqareze. gansxvavebul SemTxvevebSi Sida Zalebi gamoisaxeba drois funqciiT.

dayvanis wertilebs Cveulebisamebr irCeven, meqanizmTa ZiriTadi masebis mdebareobis adgilebSi. dayvanili masebis mniSvnelobebs, romlebic mdebareobs drekadi elementis erT mxares da romlisTvisac ganisazRvreba gaTvliTi datvirTva, krebaven. magaliTad erTi wyvili wamyvani TvlebisaTvis kardanuli gadacemis gaangariSebisaTvis ormasiani sqemis Sedgenisas masebis dayvanas axorcieleben lilvis boloebze.

erT-erTi moyvanili masa SesaZloa Sedgebodes ZravSi moZravi nawilebis masaTa mocemuli mniSvnelobebisagan, mqnevaras masebisagan, lilvebisagan, siCqaris gadacemaTa kolofis moZravi elementebisagan; meore – manqanis Zaris masaze arsebuli tvirTiT moyvanili masebis mniSvnelobebisagan, wamyvani Tvlebisagan, naxevar RerZisagan da a.S. kardanuli lilvis masa SeiZleba miCneul iqnas, rogorc amyoli masa. am SemTxvevaSi momdevno gaangariSebebi gvaZlevs saSualebas ganvsazRvroT SesaZlo udidesi datvirTva ganxilvad elementze.

1.2. gare datvirTvebis dayvana

manqanaTa gare datvirTvebs ganekuTvneba Zalebi an ZalTa momentebi, romlebic winaaRmdegobas uweven manqanaTa moZraobas an elementebs (mag. saxarato Carxis mbrunavi meqanizmis mier dasaZlevi Wris Zalebi; manqanaTa moZraobisas warmoqnili winaRobis Zalebi; sabiZgebelas bunikiT feCSi namzadis CatvirTvisas dasaZlevi xaxunis Zalebi da a.S.), aseve moZravi Zalebi an momentebi (romlebic warmoiqmnebian eleqtroZravebSi magnituri velidan, Tbur ZravebSi gafarToebuli gazis drekadi Tvisebebidan da a.S.)

nax. 3. gare datvirTvebis dayvanis sqema



nax. 4. urikis gadasaadgilebeli meqanizmis sqema

1 – Zravi; 2,3 – savali Tvlebi.
ganvixiloT meqanizmi, romelic Sesdgeba Zravisa da gadacemisagan (nax. 3). sadac, Zravi anviTarebs mbrunav moments M; lilvebi I, II, da III-is mier dasaZlevi datvirTvebi M_1C, M_2C, M_3C; gadacemis gadamcemi fardoba i₁, i₂, i₃ da margi qmedebis koeficientebi η₁, η₂ da η₃ (ix. nax. 3), saWiroa yvela ZalTa momentebi daviyvanoT lilvis III-s boloebze. maSin mocemuli momenti, lilvis III-is marcxena mxares toli iqneba:

M₁ = M_әi₁ i₂ η₁ η₂ - M_1Ci₂ η₂ - M_2C, (1)

xolo marjvnidan

M₂ = M_3C / i₃ η₃. (2)

lilvisTvis II analogiurad miviRebT

M₁ = M_әi₁ η₁ - M_1C , (3)

M₂ = M_3C / i₂i₃η₂η₃ + M_2C / i₂η₂. (4)

Tu gare Zalebi gamoisaxeba Zalebisa da momentebis saxiT, maSin moyvanili mniSvnelobebi unda gamoisaxos an Zalis an momentis saxiT. ZalTa gansazRvra iwarmoeba zemoT moyvanili wesis analogiurad. Zalebi momentebis saxiT an momentebi ZalTa saxiT gamoisaxeba Sesabamisi radiusebis dayvanidan.

magaliTad, Zalebi W meqanikuri urikis gadasaadgilebeli winaRobis Zala (nax. 4) aucilebelia moviyvanoT Zravis lilvTan da gamovsaxoT momentis saxiT. am SemTxvevaSi dayvanis radiusi tolia wamyvani Tvalis R radiusis, xolo dayvanili momenti

MC = WR / i1i2η1η2, (5)

sadac,

i₁ da i₂ - gadamcemis ricxvebia;

η₁ da η₂ - gadacemaTa mqk-s damatebuli danakargebi sayrdenebSi.
1.3. masebisa da inerciis momentebis dayvana

dinamikis amocanaTa amoxsnisaTvis sakvlevi meqanizmebis sqemebi umjobesia warmovidginoT erTmaneTTan drekadi elementebiT SeerTebuli calkeuli elementebis saxiT. avRniSnoT moqmedi meqanizmis elementebis masebi (nax. 5, a) m₁, m₂, m₃ ... m_n-iT, xolo maTi moZraobis siCqare v₁, v₂, v₃ –iT. masaTa dinamikuri dayvanis pirobas warmoadgens dayvanil masaTa da yvela moqmedi meqanizmis masaTa, kinetikur energiaTa toloba¹. Tu masebi dayavT garkveuli v₀ siCqariT moZrav masaTa modebis wertilebSi, maSin SeiZleba daiweros

(6)

saidanac

(7)

sadac, m_n – meqanizmis yvela elementis dayvanil masaTa mniSvnelobaa.

gaTvaliswinebiT imisa, rom

miviRebT

(8)

amrigad, dayvanili masa udris dasayvani masebis jams gadacemis fardobebis kvadratebze.

Tu meqanizmis sqema Seicavs mbrunav masebs (nax. 5, b), maSin calkeuli masaTa inerciis momentebis dayvana xorcieldeba wina SemTxvevis Sesabamisad

(9)

sadac: - meqanizmis yvela elementis masaTa inerciis momentis dayvanis mniSvnelobaa;

, , - meqanizmis realuri sqemis elementTa masaTa inerciis momentebi;

, , - Sesabamisi gadacemaTa damokidebuleba.


nax. 5. masaTa dayvanis pirobiTi sqemebi.

a) masaTa mimdevrobiTi moZraoba; b) masaTa brunviTi moZraoba.

Tu meqanizmi Seicavs moZrav da mbrunav elementebs, masa aucileblad unda gamoisaxos inerciis momentiT, xolo inerciis momenti masiT. magaliTad saWiroa ganvsazRvroT Tokis saSualebiT asawevi tvirTis masa m, romelic axvevia garkveuli R radiusis doluraze da moiZebnos doluras RerZTan mimarTebaSi tvirTisa da doluris inerciis momentebi. doluras inerciis momenti I, amwevi polispatis jeradi – n, nax. 6.

nax. 6. dolurasa da tvirTis masis dayvanis sqema:

1 – dolura; 2 – tvirTi.

tvirTis masa, doluras RerZTan dayvanili da gamosaxuli inerciis momentiT, tolia mR²/n², tvirTisa da doluras dayvanili inerciis momenti iqneba:

. (10)

Tu doluras inerciis moments gamovsaxavT masis saxiT da moviyvanT wonasTan, maSin miviRebT In²/R², xolo dolurasa da tvirTis mTliani dayvanili masa toli iqneba:

. (11)

dinamikis amocanaTaAmiaxloebiTi gamoyvanisTvis, xandaxan mizanSewonilia warmovadginoT Seuyursavi masebi Seyursuli masiT. aseT SemTxvevebSi aseve iyeneben dayvanis meTodebs. metad iolia dayvanil masaTa koeficientebis gamoyeneba, Tu m_n – dayvanilia, m_o – konstruqciis realuri masaa, maSin dayvanili masis koeficients K=m_n /m_o sazRvraven Semdegi saxiT.

dx sididiT SezRuduli sistemis elementaruli nawilis kinetikuri energia, tolia:

,

sadac:

y – elementis masalis kuTri wonaa;

- elementis ganivi kveTi;

ydx – sistemis elementis wona;

g - simZimis ZalTa aCqarebaa;

v - sistemis elementis moZraobis siCqare.

n - monakveTis mqone kinetikuri energiis sistemebi,

(12)

sadac: A da B – sistemis Sesabamisi monakveTTa integrirebis zRvrebia. Tu – v_o wertilis moZraobis siCqare, romelTanac mivyavarT sistemaTa masebs, pirobiTi kinetikuri energia

. (13)

dayvanil masaTa koeficienti

. (14)

formulebidan (8) da (9) Cans, rom gadamcemi damokidebulebisas cvlad mniSvnelobaTa dros, romelTac SeuZliaT qondeT sxvadasxva mniSvnelobebi, sistemis sxvadasxva mdgomareobidan gamomdinare, dayvanili masis mniSvnelobac aseve gveqneba cvladi.

ganvixiloT ufro dawvrilebiT, romelime konkretuli magaliTi masebis dayvanis Sesaxeb.

wonis masa Q da mudmivi kveTis Rero , gawelvisa da SekumSvis zemoqmedebis qveS myofni (nax. 7), unda warmovidginoT Seyursuli masisa da uwonadi drekadi rgolebis saxiT. dx sididiT SezRuduli Reros elementis kinetikuri energia, tolia

,

sadac: v_x – elementis moZraobis siCqarea.

nax. 7. Rerosa da tvirTisagan Semdgari sistemis masaTa dayvanis koeficientis gansazRvris sqema
Tu, l sigrZis Reros bolos moZraobis siCqares avRniSnavT v-Ti, maSin = const. damokidebulebidan miaxloebiT SeiZleba miviRoT, rom v_x = vx/l. amasTan , xolo Reros sruli kinetikuri energia

,

sadac: m – Reros masaa.

Reros masis bolos dayvanili kinetikuri energia, SeiZleba Caiweros saxiT. masaTa dayvanis pirobidan gamomdinare T_n = T, rac aris , saidanac , Tu tvirTis masa aris m₁, maSin tvirTisa da Reros sruli dayvanili masa

roca Reros masa naklebia tvirTis masaze, miRebuli dayvanis formula sakmaod marTebulia [19, 127].

praqtikaSi manqanaTa elementebis gamoTvla xSirad moiTxovs, sistemis masaTa gansazRvras (moyvanas), romelSic tvirTi araa an misi wona naklebia Reros wonaze.

vTqvaT q – Reros grZivi wonaa,

E – Reros masalis modulis drekadoba da Q – tvirTis wona, Reros elementis dagrZeleba dx sigrZiT, tvirTis wonisa da Reros nawilebis zemoqmedebiT iqneba , xolo Reros nawilebis dagrZeleba, SezRuduli x koordinatiT, Caiwereba Semdegi saxiT:

,

sadac:

;

y_x gamosaxvisas y_l-iT vRebulobT

,

Sesabamisad Reros kveTis gadaadgilebis siCqare

;

dx sigrZis Reros elementis kinetikuri energia

.

mTeli sistemis kinetikuri energia SesaZloa Caiweros Semdegi saxiT

.

mniSvnelobaTa T da dT_x CasmiT, gardasaxvis Semdeg vpoulobT

. (15)

me-16 formula SeiZleba gamoyenebul iqnas Reros da tvirTis dayvanili masaTa gansazRvrisaTvis nebismier damokidebulebebSi m₁/m. cxrilSi 1 mocemulia dayvanis koeficientebi sxvadasxva saxis mniSvnelobebisaTvis m₁/m.

cxr. 1. Rero, boloze modebuli tvirTiT da misi dayvanis koeficientebi

	K		K
0	0.533	5.0	0.348
0.5	0.425	10.0	0.342
1.0	0.392	20.0	0.336
2.0	0.368	∞	0.333

cxrili 1-dan Cans, rom masaTa dayvanis koeficienti mxolod damokidebulebidan , axlosaa sididesTan, romelic ganisazRvreba reilis miaxloebiTi formuliT, sxva SemTxvevebSi aRemateba mocemul mniSvnelobas, xolo rodesac xdeba didi 0.5. gamodis, rom Reros masaTa dayvanisTvis, romelic asrulebs grZiv rxevebs, yovelTvis ver gamoviyenebT reilis formulas.

ganvixiloT magaliTi (nax. 8).

nax. 8. erT boloze modebuli tvirTiani konsoluri koWis sistemis masaTa dayvanis koeficientebis gansazRvris sqema.
koWis erTi bolo Camagrebulia, xolo meoreze modebulia tvirTi Q. koWis masaTa dayvanis koeficientebis miaxloebiTi gansazRvrisaTvis, misi Runvis xasiaTs iTvaliswineben ise, rogorc Tavisufal boloze statikuri datvirTvis Zalis Q, SemTxvevaSi, aseT dros koWas Runva x kveTSi gamoisaxeba Semdegnairad:

sadac, J – koWas kveTis inerciis momentia.

SemTxvevisaTvis x = l

.

aqedan

,

xolo koWas Runvis rxevisas, koWas kveTis moZraobis siCqare

,

dx sigrZis koWas elementis kinetikuri energia

,

xolo koWa da tvirTi

.

energiis Senaxvis kanonis Sesabamisad

mniSvnelobaTa CasmiT T da dTx da -is SemotaniT, vipoviT koWas masas, dayvanils mis Tavisufal boloze,

m_n – gamosaxulebis misaRebad, romelic iTvaliswinebs m₁/m, moviyvanoT ufro mkacri daskvna.

koWas x kveTSi Runvis momenti

,

xolo misi drekadi xazis gantoleba

.

orjer integraciisas, miviRebT

.

C da D mudmiv integrirebas ganvsazRvravT sawyisi pirobidan. rodesac x = 0, gvaqvs y = 0 da , aqedan

.

rodesac x = l

Yx –is gamosaxva Yl -iT da gardasaxva, gvapovninebs

dx sigrZis koWas elementis kinetikuri energia

,

xolo mTeli sistema

.

energiis Senaxvis kanonis Sesabamisad

T da dTx mniSvnelobaTa CasmiT da maTi gardasaxviT miviRebT,

(16)

miRebuli formula gamoiyeneba nebismieri m₁/m damokidebulebisas dayvanis koeficientis gansazRvrisaTvis.

cxriliSi 2, moyvanilia am koeficientis mniSvnelobebi zogierTi m₁/m damokidebulebisaTvis, iseve rogorc wona magaliTSi K-s didi mniSvnelobebi Seesabameba m₁/m mcire damokidebulebebs.

cxr. 2. boloze modebuli tvirTiani konsoluri koWas dayvanis koeficientebi

	K		K
0	0.257	5.0	0.238
0.5	0.245	10.0	0.237
1.0	0.241	20.0	0.236
2.0	0.239	∞	0.236

ganvixiloT kidev erTi magaliTi, rodesac tvirTi mdebareobs mudmivi kveTis or Camagrebul koWas Soris, SuaSi (nax. 9).