Kugwedezeka kwa mzere: kusinthasintha kwa zigawo mu dongosolo kumatsatira lamulo la hooke, ndipo mphamvu yochepetsera yomwe imapangidwa panthawi yoyenda imagwirizana ndi equation yoyamba ya liwiro lokhazikika (nthawi yochokera ku ma coordinates okhazikika).
lingaliro
Dongosolo la mzere nthawi zambiri ndi chitsanzo chosamveka bwino cha kugwedezeka kwa dongosolo lenileni. Dongosolo la kugwedezeka kwa mzere limagwiritsa ntchito mfundo ya superposition, ndiko kuti, ngati yankho la dongosolo ndi y1 pansi pa zochita za input x1, ndi y2 pansi pa zochita za input x2, ndiye kuti yankho la dongosolo pansi pa zochita za input x1 ndi x2 ndi y1+y2.
Potengera mfundo ya superposition, cholowetsa chopanda tsankho chingagawidwe kukhala chiwerengero cha mndandanda wa ma impulses osakwanira, kenako yankho lonse la dongosolo lingapezeke. Chiwerengero cha zigawo za harmonic za chisangalalo cha periodic chingakulitsidwe kukhala mndandanda wa zigawo za harmonic ndi Fourier transform, ndipo zotsatira za gawo lililonse la harmonic pa dongosolo zitha kufufuzidwa padera. Chifukwa chake, makhalidwe a mayankho a machitidwe olunjika okhala ndi magawo osasintha amatha kufotokozedwa ndi kuyankha kwa impulse kapena kuyankha kwafupipafupi.
Kuyankha kwa Impulse kumatanthauza kuyankha kwa dongosolo ku unit impulse, komwe kumayimira makhalidwe a yankho la dongosolo mu nthawi. Kuyankha kwa pafupipafupi kumatanthauza khalidwe la yankho la dongosolo ku unit harmonic input. Kugwirizana pakati pa ziwirizi kumatsimikiziridwa ndi Fourier transform.
kugawa magulu
Kugwedezeka kwa mzere kungagawidwe m'magulu awiri: kugwedezeka kwa mzere kwa dongosolo la digiri imodzi ya ufulu ndi kugwedezeka kwa mzere kwa dongosolo la digiri yambiri ya ufulu.
(1) kugwedezeka kolunjika kwa dongosolo la ufulu wa digiri imodzi ndi kugwedezeka kolunjika komwe malo ake amatha kudziwika ndi mgwirizano wodziwika bwino. Ndi kugwedezeka kosavuta komwe malingaliro ambiri oyambira ndi makhalidwe a kugwedezeka angachokere. Kumaphatikizapo kugwedezeka kosavuta kwa harmonic, kugwedezeka kwaulere, kugwedezeka kocheperako ndi kugwedezeka kokakamizidwa.
Kugwedezeka kosavuta kwa harmonic: kuyenda kobwerezabwereza kwa chinthu pafupi ndi malo ake ofanana malinga ndi lamulo la sinusoidal pansi pa mphamvu yobwezeretsa yofanana ndi kusuntha kwake.
Kugwedezeka kofooka: kugwedezeka komwe kukula kwake kumachepa nthawi zonse chifukwa cha kukangana ndi kukana kwa dielectric kapena kugwiritsa ntchito mphamvu zina.
Kugwedezeka kokakamizidwa: kugwedezeka kwa dongosolo lomwe limakhala ndi chisangalalo chosalekeza.
(2) kugwedezeka kwa mzere kwa dongosolo la ufulu wa madigiri ambiri ndi kugwedezeka kwa dongosolo la mzere ndi madigiri a n≥2 a ufulu. Dongosolo la madigiri a n la ufulu lili ndi ma frequency achilengedwe a n ndi ma main modes a n. Kapangidwe kalikonse ka kugwedezeka kwa dongosolo kakhoza kuimiridwa ngati kuphatikiza kwa mzere wa ma major modes. Chifukwa chake, njira yayikulu yopangira mawonekedwe imagwiritsidwa ntchito kwambiri pakusanthula kwamphamvu kwa machitidwe a multi-dof. Mwanjira imeneyi, kuyeza ndi kusanthula kwa makhalidwe achilengedwe a kugwedezeka kwa dongosolo kumakhala gawo lokhazikika pakupanga kwamphamvu kwa dongosolo. Makhalidwe amphamvu a machitidwe a multi-dof amathanso kufotokozedwa ndi makhalidwe a pafupipafupi. Popeza pali ntchito ya frequency characteristic pakati pa cholowera chilichonse ndi chotuluka, matrix ya frequency characteristic imapangidwa. Pali ubale wotsimikizika pakati pa frequency characteristic ndi main mode. Ma amplitude-frequency characteristic curve a dongosolo la ufulu wamitundu yambiri ndi osiyana ndi a single-freedom system.
Kugwedezeka kwa mzere kwa digiri imodzi ya dongosolo la ufulu
Kugwedezeka kwa mzere komwe malo a dongosolo amatha kudziwika ndi cholumikizira chodziwika bwino. Ndi kugwedezeka kosavuta komanso kofunikira kwambiri komwe malingaliro ambiri oyambira ndi makhalidwe a kugwedezeka angachokere. Kumaphatikizapo kugwedezeka kosavuta kwa harmonic, kugwedezeka kochepetsedwa ndi kugwedezeka kokakamizidwa.
Kugwedezeka kwa harmonic
Pogwiritsa ntchito mphamvu yobwezeretsa mogwirizana ndi kusamuka, chinthucho chimabwerera m'mbuyo mwa sinusoidal pafupi ndi malo ake ofanana (Chithunzi 1). X ikuyimira kusamuka ndipo t ikuyimira nthawi. Kufotokozera kwa masamu kwa kugwedezeka uku ndi:
(1)Kumene A ndi mtengo wapamwamba kwambiri wa kusuntha x, komwe kumatchedwa amplitude, ndipo kumayimira mphamvu ya kugwedezeka; Omega n ndi amplitude Kuwonjezeka kwa ngodya ya kugwedezeka pa sekondi, komwe kumatchedwa angular frequency, kapena circular frequency; Izi zimatchedwa initial phase. Ponena za f= n/2, chiwerengero cha oscillations pa sekondi chimatchedwa frequency; Chotsutsana ndi ichi, T=1/f, ndi nthawi yomwe imatenga kuti kuzungulira kamodzi kusunthe, ndipo imatchedwa period. Amplitude A, frequency f (kapena angular frequency n), gawo loyamba, lodziwika kuti simple harmonic vibration three elements.
CHITHUNZI 1 chozungulira chosavuta cha harmonic vibration
Monga momwe zasonyezedwera pa Chithunzi 2, chosinthira chosavuta cha harmonic chimapangidwa ndi kulemera kokhazikika m kolumikizidwa ndi kasupe wolunjika. Pamene kusuntha kwa kugwedezeka kumawerengedwa kuchokera pamalo ofanana, equation ya kugwedezeka ndi:
Kuuma kwa kasupe kuli kuti. Yankho lalikulu la equation yomwe ili pamwambapa ndi (1). A ndipo lingathe kudziwika ndi malo oyamba x0 ndi liwiro loyamba pa t=0:
Koma omega n imangodziwika ndi makhalidwe a dongosolo lokha m ndi k, mosatengera mikhalidwe yowonjezera yoyambirira, kotero omega n imadziwikanso kuti ma frequency achilengedwe.
CHITHUNZI 2 Dongosolo la ufulu wa digiri imodzi
Pa oscillator yosavuta ya harmonic, kuchuluka kwa mphamvu yake ya kinetic ndi mphamvu yake ya potential kumakhala kosasintha, ndiko kuti, mphamvu yonse ya makina ya dongosololi imasungidwa. Mu ndondomeko ya kugwedezeka, mphamvu ya kinetic ndi mphamvu yake ya potential zimasinthidwa nthawi zonse kukhala zina.
Kugwedezeka kwa damping
Kugwedezeka komwe kukula kwake kumachepetsedwa nthawi zonse ndi kukangana ndi kukana kwa dielectric kapena kugwiritsa ntchito mphamvu zina. Pa kugwedezeka kwa micro, liwiro nthawi zambiri silili lalikulu kwambiri, ndipo kukana kwapakati kumakhala kofanana ndi liwiro la mphamvu yoyamba, yomwe ingalembedwe ngati c ndiye damping coefficient. Chifukwa chake, vibration equation ya digiri imodzi ya ufulu ndi linear damping ikhoza kulembedwa motere:
(2)Kumene, m = c/2m amatchedwa damping parameter, ndipo. Yankho lathunthu la fomula (2) lingalembedwe:
(3)Ubale wa manambala pakati pa omega n ndi PI ukhoza kugawidwa m'magawo atatu otsatirawa:
N > (ngati pali damping yaying'ono) tinthu tating'onoting'ono tomwe timapanga kugwedezeka kwa attenuation, equation ya kugwedezeka ndi:
Kutalika kwake kumachepa pakapita nthawi malinga ndi lamulo la exponential lomwe lawonetsedwa mu equation, monga momwe zasonyezedwera pamzere wolembedwa mu Chithunzi 3. Kunena zoona, kugwedezeka kumeneku ndi kosakhazikika, koma kuchuluka kwa nsonga yake kungatanthauzidwe motere:
Amatchedwa amplitude reduction rate, komwe ndi nthawi ya kugwedezeka. Logarithm yachilengedwe ya amplitude reduction rate imatchedwa logarithm minus (amplitude) rate. Mwachionekere, =, pankhaniyi, ndi yofanana ndi 2/1. Mwachindunji kudzera mu delta yoyesera yoyesera ndipo, pogwiritsa ntchito fomula yomwe ili pamwambapa, mutha kuwerengera c.
Pakadali pano, yankho la equation (2) likhoza kulembedwa:
Pamodzi ndi komwe liwiro loyambirira limayambira, lingagawidwe m'magulu atatu osagwedezeka monga momwe zasonyezedwera pa Chithunzi 4.
N < (ngati pali damping yayikulu), yankho la equation (2) likuwonetsedwa mu equation (3). Pakadali pano, dongosolo silikugwedezekanso.
Kugwedezeka kokakamizidwa
Kugwedezeka kwa dongosolo lomwe limakhala ndi kusonkhezera kosalekeza. Kusanthula kugwedezeka kumafufuza makamaka momwe dongosolo limayankhira ku kusonkhezera. Kusonkhezera kwa nthawi ndi kusonkhezera kwachizolowezi. Popeza kusonkhezera kwa nthawi ndi nthawi kumatha kugawidwa kukhala kuchuluka kwa kusonkhezera kosiyanasiyana kwa harmonic, malinga ndi mfundo ya superposition, yankho la dongosololi ku kusonkhezera kulikonse kwa harmonic ndikofunikira. Pansi pa kuchita kwa kusonkhezera kwa harmonic, equation yosiyana ya kayendedwe ka digiri imodzi ya freedom damped system ikhoza kulembedwa:
Yankho lake ndi chiŵerengero cha magawo awiri. Gawo limodzi ndi yankho la kugwedezeka konyowa, komwe kumawola mofulumira pakapita nthawi. Yankho la gawo lina la kugwedezeka kokakamizidwa lingalembedwe:
Chithunzi 3: Mzere wozungulira wozungulira
Chithunzi 4: Ma curve a mikhalidwe itatu yoyambirira yokhala ndi damping yofunika kwambiri
Lembani mu
H /F0= h (), ndi chiŵerengero cha kuyankha kokhazikika kwa kukweza mphamvu, kufotokozera makhalidwe a kukweza mphamvu, kapena ntchito yopeza; Ma bits a kuyankha kokhazikika ndi chilimbikitso cha gawo, kufotokozera makhalidwe a kukweza mphamvu. Ubale pakati pawo ndi kukweza mphamvu ukuonetsedwa mu Chithunzi 5 ndi Chithunzi 6.
Monga momwe tikuonera kuchokera ku amplitude-frequency curve (Chithunzi 5), pankhani ya damping yaying'ono, amplitude-frequency curve ili ndi peak imodzi. Damping ikakhala yaying'ono, peak imakhala yokwera kwambiri; Ma frequency ofanana ndi peak amatchedwa resonant frequency ya system. Pankhani ya damping yaying'ono, ma frequency a resonance sasiyana kwambiri ndi ma frequency achilengedwe. Pamene frequency ya excitation ili pafupi ndi frequency yachilengedwe, amplitude imawonjezeka kwambiri. Chochitikachi chimatchedwa resonance. Pa resonance, phindu la system limakulitsidwa, ndiko kuti, kugwedezeka kokakamizidwa ndiko kwakukulu kwambiri. Chifukwa chake, nthawi zonse, yesetsani kupewa resonance, pokhapokha zida zina ndi zida zina zikugwiritsa ntchito resonance kuti zikwaniritse kugwedezeka kwakukulu.
Chithunzi 5: Ma frequency curve amplitude curve
Zitha kuwoneka kuchokera ku phase frequency curve (chithunzi 6), mosasamala kanthu za kukula kwa damping, mu omega zero phase difference bits = PI / 2, khalidweli lingagwiritsidwe ntchito bwino poyesa resonance.
Kuwonjezera pa kusonkhezera kosalekeza, machitidwe nthawi zina amakumana ndi kusonkhezera kosakhazikika. Kungagawidwe m'mitundu iwiri: imodzi ndi kugwedezeka mwadzidzidzi. Yachiwiri ndi zotsatira zokhalitsa za kukakamiza. Pansi pa kusonkhezera kosakhazikika, yankho la dongosololi silikhazikikanso.
Chida champhamvu chowunikira kugwedezeka kosakhazikika ndi njira yoyankhira ya impulse. Imafotokoza makhalidwe amphamvu a dongosolo ndi yankho losakhalitsa la kulowetsa kwa unit impulse kwa dongosolo. Kuyimitsa kwa unit kumatha kufotokozedwa ngati ntchito ya delta. Mu uinjiniya, ntchito ya delta nthawi zambiri imafotokozedwa ngati:
Kumene 0- ikuyimira mfundo pa t-axis yomwe ikuyandikira zero kuchokera kumanzere; 0 plus ndi mfundo yomwe imapita ku 0 kuchokera kumanja.
Chithunzi 6 cha ma frequency curve
CHITHUNZI 7. Kulowetsa kulikonse kungaganizidwe ngati kuchuluka kwa zinthu zingapo zomwe zimapangitsa kuti munthu ayambe kugwedezeka
Dongosololi likugwirizana ndi yankho h(t) lopangidwa ndi unit impulse pa t=0, lomwe limatchedwa ntchito ya kuyankha kwa kugwedezeka. Poganiza kuti dongosololi silikugwedezeka kugunda kwa mtima, h(t)=0 pa t<0. Podziwa ntchito ya kuyankha kwa kugwedezeka kwa dongosololi, titha kupeza yankho la dongosololi ku chilichonse cholowetsa x(t). Pakadali pano, mutha kuganiza za x(t) ngati chiwonkhetso cha zinthu zingapo za kugwedezeka (Chithunzi 7). Yankho la dongosololi ndi:
Kutengera ndi mfundo ya superposition, yankho lonse la dongosolo lofanana ndi x(t) ndi:
Integral iyi imatchedwa convolution integral kapena superposition integral.
Kugwedezeka kwa mzere kwa dongosolo la ufulu wambiri
Kugwedezeka kwa dongosolo lolunjika ndi madigiri a n≥2 a ufulu.
Chithunzi 8 chikuwonetsa njira ziwiri zosavuta zolumikizirana zomwe zimalumikizidwa ndi kasupe wolumikizira. Popeza ndi njira ya digiri ziwiri zaufulu, ma coordinates awiri odziyimira pawokha amafunika kuti adziwe malo ake. Pali ma frequency awiri achilengedwe mu dongosololi:
Ma frequency aliwonse amafanana ndi njira yogwedera. Ma harmonic oscillators amachita ma harmonic oscillations ofanana, kudutsa motsatizana pamalo ofanana ndikufika pamalo ochulukirapo. Mu vibration yayikulu yofanana ndi omega one, x1 ndi yofanana ndi x2; Mu vibration yayikulu yofanana ndi omega omega two, omega omega one. Mu vibration yayikulu, displacement ratio ya misa iliyonse imasunga ubale winawake ndikupanga njira inayake, yomwe imatchedwa main mode kapena natural mode. Orthogonality ya misa ndi kuuma ilipo pakati pa njira zazikulu, zomwe zimasonyeza kudziyimira pawokha kwa vibration iliyonse. Natural frequency ndi main mode zimayimira makhalidwe ogwedera a multi-degree of freedom system.
Chithunzi 8 dongosolo lokhala ndi madigiri angapo a ufulu
Dongosolo la madigiri n la ufulu lili ndi ma frequency achilengedwe a n ndi ma main modes a n. Kapangidwe kalikonse ka kugwedezeka kwa dongosolo kakhoza kuimiridwa ngati kuphatikiza kolunjika kwa ma major modes. Chifukwa chake, njira yayikulu yopangira mawonekedwe imagwiritsidwa ntchito kwambiri pakusanthula kwamphamvu kwa machitidwe ambiri. Mwanjira imeneyi, kuyeza ndi kusanthula kwa mawonekedwe achilengedwe a kugwedezeka kwa dongosolo kumakhala gawo lokhazikika pakupanga kwamphamvu kwa dongosolo.
Makhalidwe osinthasintha a machitidwe a multi-dof angathenso kufotokozedwa ndi makhalidwe a pafupipafupi. Popeza pali ntchito ya frequency characteristic pakati pa cholowetsa chilichonse ndi chotulutsa, matrix ya frequency characteristic imapangidwa. Mzere wa amplitude-frequency characteristic wa dongosolo la multi-ufulu ndi wosiyana ndi wa dongosolo la single-ufulu.
Elastomer imagwedezeka
Dongosolo la ufulu wamitundu yambiri lomwe lili pamwambapa ndi chitsanzo cha makina cha elastomer. Elastomer ili ndi kuchuluka kosatha kwa madigiri a ufulu. Pali kusiyana kwa kuchuluka koma palibe kusiyana kwakukulu pakati pa ziwirizi. Elastomer iliyonse ili ndi kuchuluka kosatha kwa ma frequency achilengedwe ndi kuchuluka kosatha kwa ma mode ofanana, ndipo pali orthogonality pakati pa ma modes a kulemera ndi kuuma. Kapangidwe kalikonse ka kugwedezeka kwa elastomer kangathenso kuimiridwa ngati linear superposition ya ma major modes. Chifukwa chake, pakusanthula kwamphamvu kwa elastomer, njira ya superposition ya main mode ikugwirabe ntchito (onani linear vibration ya elastomer).
Tengani kugwedezeka kwa chingwe. Tiyerekeze kuti chingwe chopyapyala cha kulemera m pa unit length, long l, chimamangiriridwa mbali zonse ziwiri, ndipo kupsinjika ndi T. Pa nthawiyi, kuchuluka kwachilengedwe kwa chingwe kumatsimikiziridwa ndi equation yotsatirayi:
F = na/2l (n= 1,2,3…).
Kodi liwiro lofalikira la mafunde opingasa likupita kuti? Ma frequency achilengedwe a zingwe amakhala ochulukitsa a ma frequency oyambira pa 2l. Kuchuluka kwa manambala kumeneku kumabweretsa kapangidwe kosangalatsa ka harmonic. Kawirikawiri, palibe ubale woterewu pakati pa ma frequency achilengedwe a elastomer.
Njira zitatu zoyambirira za chingwe chomangiriridwa zikuwonetsedwa pa Chithunzi 9. Pali ma node ena pa curve yayikulu ya mode. Mu kugwedezeka kwakukulu, ma node sagwedezeka. Chithunzi 10 chikuwonetsa njira zingapo za mbale yozungulira yolumikizidwa mozungulira yokhala ndi mizere ina ya nodal yokhala ndi mabwalo ndi mainchesi.
Kapangidwe kake ka vuto la kugwedezeka kwa elastomer kangathe kuganiziridwa ngati vuto la mtengo wa malire a ma equation osiyana pang'ono. Komabe, yankho lenileni lingapezeke kokha m'zochitika zosavuta, kotero tiyenera kugwiritsa ntchito yankho loyerekeza la vuto lovuta la kugwedezeka kwa elastomer. Cholinga cha mayankho osiyanasiyana oyerekeza ndikusintha kosatha kukhala kosatha, kutanthauza, kugawa dongosolo la ufulu wopanda magawo ambiri (dongosolo lokhazikika) kukhala dongosolo lochepera la magawo ambiri la ufulu (dongosolo losasinthika). Pali mitundu iwiri ya njira zosinthira zomwe zimagwiritsidwa ntchito kwambiri pakusanthula kwaukadaulo: njira yochepera ya chinthu ndi njira yopangira modal.
Chithunzi 9 cha chingwe
Chithunzi 10 cha mbale yozungulira
Njira ya Finite element ndi kapangidwe kophatikizana komwe kamaphatikiza kapangidwe kovuta kukhala zinthu zochepa ndikuzilumikiza pa nambala yochepa ya ma node. Chigawo chilichonse ndi elastomer; Kusamuka kwa gawo kumawonetsedwa ndi ntchito yolumikizirana ya kusamuka kwa ma node. Kenako magawo ogawa a gawo lililonse amalumikizidwa ku node iliyonse mumtundu winawake, ndipo chitsanzo cha makina cha dongosolo losiyana chimapezeka.
Kupanga modal ndi kugawikana kwa kapangidwe kovuta kukhala magawo angapo osavuta. Potengera kumvetsetsa makhalidwe a kugwedezeka kwa kapangidwe kalikonse, kapangidwe kake kamapangidwa kukhala kapangidwe kathunthu malinga ndi momwe zinthu zilili pa mawonekedwe ake, ndipo mawonekedwe a kugwedezeka kwa kapangidwe kathunthu amapezeka pogwiritsa ntchito mawonekedwe a kugwedezeka kwa kapangidwe kalikonse.
Njira ziwirizi ndi zosiyana komanso zogwirizana, ndipo zingagwiritsidwe ntchito ngati njira yofotokozera. Njira yopangira modal ingagwiritsidwenso ntchito bwino ndi muyeso woyesera kuti ipange njira yowunikira ya chiphunzitso ndi yoyesera kuti kugwedezeka kwa machitidwe akuluakulu kugwedezeke.
Nthawi yotumizira: Epulo-03-2020
