Menene vibration na layi?

Jijjiga layin layi: elasticity na abubuwan da ke cikin tsarin yana ƙarƙashin dokar ƙugiya, kuma ƙarfin damping da aka haifar yayin motsi ya yi daidai da ma'auni na farko na ƙayyadaddun saurin gabaɗaya (wanda ya samo asali daga abubuwan haɗin gwiwar gabaɗaya).

ra'ayi

Tsarin layi na yau da kullun shine samfurin abstract na vibration na ainihin tsarin.Tsarin jijjiga na layi yana amfani da ka'idar superposition, wato, idan amsawar tsarin ya kasance y1 a ƙarƙashin aikin shigarwa x1, da y2 a ƙarƙashin aikin shigarwar x2, sannan martanin tsarin karkashin aikin shigar x1 da x2 shine y1+y2.

A bisa ka'idar superposition, za a iya bazuwar shigarwa ta sabani a cikin jimillar jimillar abubuwan sha'awa marasa iyaka, sa'an nan kuma za a iya samun jimillar amsawar tsarin.Jimillar jimillar abubuwan haɗin kai na tashin hankali na lokaci-lokaci za a iya fadada shi zuwa cikin jerin abubuwan daidaitawa ta hanyar Fourier canzawa, kuma ana iya bincika tasirin kowane ɓangaren jituwa akan tsarin daban-daban.Saboda haka, ana iya bayyana halayen amsawar tsarin layi tare da sigogi akai-akai ta hanyar amsawa ko amsawar mita.

Amsar ƙwanƙwasa tana nufin amsawar tsarin zuwa motsin naúrar, wanda ke nuna halayen amsawar tsarin a cikin yanki na lokaci. ta hanyar Fourier transform.

rarrabawa

Za'a iya raba girgizar layin layi zuwa tsarin girgiza-tsara-ɗaya-na-yanci da kuma rawar jiki na tsarin-digiri-na-yanci da yawa.

(1) girgiza kai tsaye na tsarin yanci-digiri guda ɗaya shine rawar jiki na layi wanda za'a iya ƙayyade matsayinsa ta hanyar haɗin kai. jijjiga masu jituwa, rawar jiki kyauta, girgizar girgizawa da girgizar tilastawa.

Sauƙaƙan jijjiga masu jituwa: motsin wani abu a kusa da matsayin daidaitonsa bisa ga ka'idar sinusoidal ƙarƙashin aikin maido da ƙarfi daidai da ƙaura.

Girgizawar girgiza: girgiza wanda girmansa ke ci gaba da raguwa ta kasancewar gogayya da juriyar dielectric ko sauran amfani da makamashi.

Jijjiga tilas: girgizar tsarin da ke ƙarƙashin tashin hankali akai-akai.

(2) rawar jiki na layi na tsarin tsarin 'yanci da yawa shine rawar jiki na tsarin layi tare da n≥2 digiri na' yanci. na tsarin za a iya wakilta a matsayin haɗin kai tsaye na manyan hanyoyi.Saboda haka, ana amfani da hanyar babban yanayin superposition sosai a cikin nazarin amsawar amsawa na tsarin multi-dof. Ta wannan hanyar, ma'auni da bincike na dabi'un girgizar dabi'a na dabi'a. tsarin ya zama mataki na yau da kullum a cikin tsari mai tsauri na tsarin.Haka kuma za a iya kwatanta halaye masu mahimmanci na tsarin dof da yawa ta hanyar halayen mita.Tun da akwai aikin halayen mitar tsakanin kowane shigarwa da fitarwa, an gina matrix halayyar mita. ƙayyadaddun dangantaka ce tsakanin sifofin mitar da kuma babban yanayin.Halayen siffa mai girman mita na tsarin 'yanci da yawa ya bambanta da na tsarin 'yanci guda ɗaya.

Jijjiga layin layi na digiri ɗaya na tsarin 'yanci

Girgizawar layin layi wanda za'a iya ƙayyade matsayin tsarin ta hanyar haɗin kai na gama gari.Shi ne mafi sauƙi kuma mafi mahimmanci na girgizawa wanda za'a iya samun yawancin ra'ayoyi na asali da halaye na rawar jiki.Ya haɗa da girgiza mai sauƙi mai jituwa, damped vibration da kuma tilasta rawar jiki. .

Jijjiga masu jituwa

A ƙarƙashin aikin maido da ƙarfin da ya dace da ƙaura, abu yana ramawa a cikin hanyar sinusoidal kusa da matsayinsa (FIG. 1) .X yana wakiltar ƙaura kuma t yana wakiltar lokacin.Maganar lissafin wannan girgizarwar ita ce:

(1)Inda A shine matsakaicin ƙimar ƙaura x, wanda ake kira amplitude, kuma yana wakiltar ƙarfin girgiza;Omega n shine haɓakawar Angle Angle na vibration a cikin sakan daya, wanda ake kira mita angular, ko mitar da'ira; ana kiranta da farkon lokaci.A wajen f= n/2 ana kiran adadin motsin motsi a cikin daƙiƙa guda; the period.Amplitude A, mita f (ko angular mita n), farkon lokaci, da aka sani da sauki jituwa vibration abubuwa uku.

FIG.1 sauƙaƙan daidaita jijjiga jigila

Kamar yadda aka nuna a FIG.2, Oscillator mai sauƙi mai jituwa yana samuwa ta hanyar tattarawar taro m da aka haɗa ta hanyar madaidaicin maɓuɓɓugar ruwa. Lokacin da aka ƙididdige ƙaura daga ma'auni, ma'aunin girgiza shine:

A ina ne taurin bazara. Gabaɗayan bayani ga ma'auni na sama shine (1) .A kuma ana iya ƙaddara ta wurin farko x0 da saurin farko a t = 0:

Amma omega n kawai an ƙaddara ta halaye na tsarin kanta m da k, mai zaman kanta daga ƙarin yanayin farko, don haka omega n kuma ana kiranta da mitar yanayi.

FIG.2 digiri ɗaya na tsarin 'yanci

Don oscillator mai sauƙi mai jituwa, jimillar makamashin motsin sa da ƙarfin ƙarfinsa yana dawwama, wato, jimlar makamashin injina na tsarin yana kiyayewa.

Damping vibration

Jijjiga wanda girman girmansa yana ci gaba da ragewa ta hanyar juriya da juriya na dielectric ko sauran amfani da makamashi. Don micro vibration, saurin gudu gabaɗaya baya girma sosai, kuma matsakaicin juriya ya yi daidai da saurin gudu zuwa ƙarfin farko, wanda za'a iya rubuta shi azaman c. damping coefficient.Saboda haka, ana iya rubuta ma'aunin rawar jiki na digiri ɗaya na 'yanci tare da damping na layi kamar:

(2)Inda, m = c/2m ake kira damping parameter, kuma.Game da bayani na dabara (2) za a iya rubuta:

(3)Alakar lamba tsakanin omega n da PI za a iya raba su zuwa lokuta uku masu zuwa:

N> (a cikin yanayin ƙaramar damping) barbashi ya haifar da girgizar girgiza, ma'aunin girgiza shine:

Girman girmansa yana raguwa tare da lokaci bisa ga ka'idar juzu'i da aka nuna a cikin lissafin, kamar yadda aka nuna a cikin layi mai digo a FIG.3. Tsananin magana, wannan jijjiga na lokaci-lokaci ne, amma ana iya siffanta yawan kololuwar sa kamar:

Ana kiran ƙimar raguwar amplitude, inda shine lokacin vibration.Logarithm na dabi'a na raguwar raguwar girman girman ana kiransa logarithm rage (amplitude) rate. gwajin gwaji delta kuma, ta amfani da dabarar da ke sama za'a iya ƙididdige c.

A wannan lokacin, ana iya rubuta maganin daidaitawa (2):

Tare da jagorancin saurin farko, ana iya raba shi zuwa lokuta uku marasa girgiza kamar yadda aka nuna a FIG.4.

N < (a cikin yanayin babban damping), an nuna bayani ga daidaitawa (2) a cikin lissafi (3) . A wannan lokaci, tsarin ba ya yin rawar jiki.

Jijjiga tilas

Vibration na wani tsarin a karkashin m excitation.Vibration bincike yafi bincikar mayar da martani na tsarin zuwa excitation.Lokaci tashin hankali ne na hali na yau da kullum excitation.Tun da lokaci-lokaci tashin hankali iya ko da yaushe a bazu a cikin jimlar da dama jitu excitation, bisa ga superposition ka'idar, kawai. Ana buƙatar amsawar tsarin ga kowane tashin hankali masu jituwa.A ƙarƙashin aikin motsa jiki na jituwa, ana iya rubuta ma'auni na bambancin motsi na digiri ɗaya na tsarin damped 'yanci:

Amsar ita ce jimlar sassa biyu.Wani bangare shine martanin girgizar da aka dasa, wanda ke rubewa da sauri tare da lokaci.

FIG.3 damped vibration kwana

FIG.Hanyoyi 4 na yanayin farko guda uku tare da damping mai mahimmanci

Buga a cikin

H / F0 = h (), shine rabo na tsayin daka na mayar da martani zuwa girman haɓakawa, da ke nuna halayen mitar-girma, ko samun aiki; Bits don amsawar jihar tsayayye da ƙarfafawar lokaci, halayyar halayen mitar lokaci. Dangantakar da ke tsakanin su da Ana nuna mitar tashin hankali a FIG.5 da FIG.6.

Kamar yadda ake iya gani daga madaidaicin mitar mita (FIG. 5), a cikin yanayin ƙaramar damping, madaidaicin mita yana da kololuwa guda ɗaya.Ƙaramin damping, mafi girman kololuwa; Mitar da ta dace da kololuwar ita ce. da ake kira mitar resonant na tsarin.A cikin yanayin ƙananan damping, mitar resonance ba ta bambanta da mitar yanayi ba.Lokacin da mitar motsa jiki ke kusa da mitar yanayi, amplitude yana ƙaruwa sosai.Ana kiran wannan al'amari da ake kira resonance. A resonance, ribar da tsarin ya kara girma, wato, vibration na tilasta shi ne mafi tsanani.Saboda haka, a gaba ɗaya, ko da yaushe yi ƙoƙari don kauce wa resonance, sai dai idan wasu kayan aiki da kayan aiki don amfani da resonance don cimma babban nasara. girgiza.

FIG.5 amplitude mita lankwasa

Ana iya gani daga yanayin mitar lokaci (hoto na 6), ba tare da la'akari da girman damping ba, a cikin omega sifili bambance-bambancen ragi = PI / 2, ana iya amfani da wannan yanayin yadda ya kamata wajen auna sauti.

Bugu da ƙari ga tsayayyen tashin hankali, tsarin wani lokaci yana haɗuwa da tashin hankali mara kyau. Ana iya raba kusan kashi biyu: ɗaya shine tasirin kwatsam. Na biyu shine sakamako mai ɗorewa na arbitrariness. Ƙarƙashin tashin hankali mara kyau, amsawar tsarin kuma ba shi da tsayayye.

Wani kayan aiki mai ƙarfi don nazarin girgizar da ba ta da ƙarfi shine hanyar amsawa ta motsa jiki.Ya bayyana halaye masu ƙarfi na tsarin tare da martani na wucin gadi na shigar da motsin naúrar na tsarin. Ana yawan bayyana aiki da:

Inda 0- ke wakiltar batu akan t-axis wanda ke kusanci sifili daga hagu; 0 da ƙari shine ma'anar da ke zuwa 0 daga dama.

FIG.6 zango mitar lankwasa

FIG.7 kowace shigarwa za a iya la'akari da shi azaman jimillar jerin abubuwan motsa jiki

Tsarin ya dace da amsa h (t) da ke haifar da motsa jiki a t = 0, wanda ake kira aikin amsawa. aikin mayar da martani na tsarin, za mu iya samun martanin tsarin zuwa kowane shigarwar x (t) . A wannan lokaci, za ku iya tunanin x (t) a matsayin jimlar jerin abubuwan motsa jiki (FIG. 7). .Martanin tsarin shine:

Dangane da ka'idar babban matsayi, jimillar martanin tsarin da ya yi daidai da x(t) shine:

Wannan haɗin kai ana kiransa haɗaɗɗen haɗin kai ko babban matsayi.

Jijjiga linzamin kwamfuta na tsarin 'yanci mai yawa-digiri

Jijjiga tsarin layi tare da n≥2 digiri na 'yanci.

Hoto 8 yana nuna sauƙaƙan ƙananan tsarin resonant guda biyu da aka haɗa ta hanyar bazara mai haɗawa.Saboda tsarin tsarin 'yanci ne na digiri biyu, ana buƙatar haɗin kai biyu masu zaman kansu don sanin matsayinsa. Akwai mitoci biyu na halitta a cikin wannan tsarin:

Kowane mita ya dace da yanayin rawar jiki.The masu jituwa oscillators suna aiwatar da oscillations masu jituwa na mitar guda ɗaya, suna wucewa ta hanyar daidaitaccen matsayi kuma tare da kai tsaye zuwa matsakaicin matsayi. Babban girgizar da ta dace da omega omega biyu, omega omega one.A cikin babban rawar jiki, rabon ƙaura na kowane taro yana riƙe da takamaiman alaƙa kuma ya samar da wani yanayi, wanda ake kira babban yanayin ko yanayin yanayi.The orthogonality of mass and taurin yana wanzuwa a cikin manyan hanyoyi, wanda ke nuna 'yancin kai na kowane girgizawa. Mitar yanayi da babban yanayin suna wakiltar halayen rawar jiki na nau'i mai yawa na tsarin 'yanci.

FIG.8 tsarin tare da mahara digiri na 'yanci

A tsarin na n digiri na 'yanci yana da n yanayi mita da kuma n main modes.Duk wani vibration sanyi na tsarin za a iya wakilta a matsayin mikakke hade da manyan halaye.Saboda haka, babban yanayin superposition hanya da ake amfani da ko'ina a tsauri mayar da martani bincike na Multi. -dof tsarin.Ta wannan hanya, ma'auni da bincike na dabi'un girgizar dabi'a na tsarin ya zama mataki na yau da kullum a cikin tsarin tsararren tsarin.

Hakanan za'a iya kwatanta halaye masu ƙarfi na tsarin dof da yawa ta hanyar halayen mitar.Tunda akwai aikin halayen mitar tsakanin kowane shigarwa da fitarwa, ana gina matrix sifa mai mahimmanci. daga tsarin tsarin 'yanci guda ɗaya.

Elastomer na girgiza

The sama Multi-digiri na 'yanci tsarin shi ne wani m inji model na elastomer.An elastomer yana da iyaka adadin digiri na 'yanci.Akwai wani adadi mai yawa bambanci amma babu wani muhimmin bambanci tsakanin su biyu.Kowane elastomer yana da iyaka mara iyaka na halitta mitoci da kuma adadi mara iyaka na yanayin daidaitawa, kuma akwai orthogonality tsakanin nau'ikan nau'ikan taro da taurin kai.Duk wani tsari na vibrational na elastomer kuma ana iya wakilta shi azaman madaidaicin madaidaicin manyan hanyoyin.Saboda haka, don nazarin martani mai ƙarfi na elastomer, hanyar superposition. na babban yanayin har yanzu yana aiki (duba jijjiga na layi na elastomer).

Ɗauki vibration na kirtani. Bari mu ce wani bakin ciki na kirtani na taro m a kowace naúrar tsawon, tsayin l, yana tayar da hankali a duka iyakar, kuma tashin hankali shine T. A wannan lokacin, yanayin yanayi na kirtani yana ƙayyade ta hanyar mai zuwa. daidaito:

F = na/2l (n= 1,2,3…).

Inda, shine saurin yaduwa na madaidaicin igiyar ruwa tare da jagorar kirtani. Matsalolin dabi'a na kirtani sun kasance masu yawa na mitar mahimmanci sama da 2l. Wannan yawan adadin lamba yana kaiwa ga tsarin jituwa mai daɗi. Gaba ɗaya, babu babu. irin wannan alaƙa da yawa a tsakanin mitoci na halitta na elastomer.

Hanyoyi uku na farko na kirtani mai tsauri ana nuna su a FIG.9. Akwai wasu nodes akan babban yanayin yanayin.A cikin babban girgiza, nodes ba sa girgiza.FIG.10 yana nuna nau'i-nau'i da yawa na farantin madauwari mai goyan bayan da'ira tare da wasu layukan nodal da suka ƙunshi da'ira da diamita.

Za'a iya ƙaddamar da ainihin tsari na matsalar girgizawar elastomer a matsayin matsalar ƙimar iyaka na ma'auni daban-daban. Duk da haka, ana iya samun ainihin maganin kawai a wasu lokuta mafi sauƙi, don haka dole ne mu yi amfani da madaidaicin bayani don hadaddun elastomer. Matsalolin jijjiga.Dalili na daban-daban kimanin hanyoyin magance shi shine canza mara iyaka zuwa iyaka, wato, don kawar da tsarin yanci da yawa mara iyaka (tsarin ci gaba) zuwa tsarin tsarin 'yanci mai iyaka (tsari mai hankali) .Akwai nau'ikan hanyoyi guda biyu na ɓacin rai da ake amfani da su sosai a cikin binciken injiniya: Hanyar ƙayyadaddun ƙayyadaddun abubuwa da hanyar haɗaɗɗiyar modal.

FIG.9 yanayin kirtani

FIG.10 yanayin farantin madauwari

Hanyar ƙayyadaddun ƙayyadaddun ƙayyadaddun abubuwa tsari ne mai haɗaka wanda ke ɗaukar hadadden tsari zuwa ƙayyadaddun abubuwa masu iyaka kuma ya haɗa su a iyakacin adadin nodes.Kowace naúrar elastomer ne; Rarraba matsugunin kashi yana bayyana ta hanyar interpolation aiki na ƙaura daga kumburi. sigogi na rarraba kowane nau'i suna mayar da hankali ga kowane kumburi a cikin wani tsari, kuma ana samun samfurin injiniya na tsarin mai hankali.

Modal kira shi ne bazuwar wani hadadden tsari a cikin wasu sassa masu sauƙi masu sauƙi. A kan tushen fahimtar halayen girgiza kowane tsarin, an haɗa tsarin da aka haɗa a cikin wani tsari na gaba ɗaya bisa ga yanayin daidaitawa a kan mahaɗin, da kuma yanayin yanayin girgizawa na gaba ɗaya. Ana samun tsari ta hanyar amfani da yanayin yanayin girgiza kowane tsarin ƙasa.

Hanyoyi guda biyu sun bambanta kuma suna da alaƙa, kuma ana iya amfani da su azaman tunani. Hakanan za'a iya haɗawa da hanyar haɗin kai tare da ma'auni na gwaji don samar da hanyar bincike na ka'idar da gwaji don girgiza manyan tsarin.