Pendulum: ixesha kunye ekukhawuleziseni olungumgubo

Inkqubo ngomatshini entlantlu yencopho eziphathekayo (zomzimba), nto leyo oxhonywe filament ikhaphukhaphu inextensible (ubunzima balo ayinamsebenzi kuthelekiswa ubunzima zomzimba) entsimini omxhuzulane efanayo, ebizwa kuqala zemathematika (elinye igama - le oscillator). Kukho ezinye iintlobo ze izixhobo. zingasetyenziswa endaweni filament ngentonga ikhaphukhaphu. Kuqala uyakwazi ukuchaza ngokucacileyo undoqo iziganeko ezininzi umdla. Xa intlokoma awo encinci motion yayo kuthiwa harmonic.

ulwazi ngokubanzi malunga nenkqubo mechanical

Ifomula kwexesha oscillation lwe kuqala yayi vezelwe Huygens Isazinzulu Dutch (1629-1695 gg.). Oku owayephila Isaac Newton wayeyoliswa kakhulu kwinkqubo ngoomatshini. Ngowe-1656 wadala iwotshi yokuqala ngendlela pendulum. Bona Walinganisa ixesha ngobuciko obukhulu ngayo ngaloo maxesha. Olu qambo yaba linyathelo elikhulu kuphuhliso imifuniselo ngokwasemzimbeni kunye nemisebenzi practical.

Ukuba pendulum yi kwindawo lizama (ijinga nkqo), ukuba amandla womxhuzulane ziya silinganiswe yi vulu ngengxabano amandla. pendulum Flat kwi ngemisonto non-stretchable yinkqubo kunye degrees ezimbini inkululeko zonxibelelwano. Xa ukutshintsha icandelo nje enye yokutshintsha iimpawu zonke iinxalenye zayo. Umzekelo, ukuba ngomsonto ithathwa ngentonga, ngoko le nkqubo ngoomatshini yi-1 kuphela isidanga inkululeko. Yintoni, ke, iimpawu pendulum zezibalo? Kule nkqubo elula, phantsi impembelelo perturbation ngamaxesha athile, chaos ibonakala. Xa kunjalo, xa ingongoma ukunqunyanyiswa lo ezihambayo, kwaye oscillates pendulum kukho kwindawo entsha kuvuselele. Ukuba ukuhla olukhawulezileyo phezulu kwaye le nkqubo ngoomatshini kuba isikhundla isitali "ngamahla ndinyuka." Kwakhona unalo igama layo. Ibizwa ngokuba Kapitza kuqala.

Iimpawu kuqala

Kuqala uye iipropati umdla kakhulu. Bonke bephela zixhaswa yimithetho ezaziwayo-kakuhle emzimbeni. Ithuba oscillation ye kuqala nayiphi na enye kuxhomekeke kwiimeko ezahlukeneyo ezifana ubukhulu kunye nokumila komzimba, umgama phakathi ingongoma ukunqunyanyiswa okanye iziko womxhuzulane, ukusasazwa ubunzima ngokuphathelele kule ngongoma. Kungenxa yoko le nto inkcazelo le ijinga ixesha yomzimba kunzima. Ingaba kulula kakhulu ukuba ukubala ixesha pendulum elula, ifomula yayo inikwe apha ngezantsi. Ngenxa yokubona le patheni zingacwangciswa kwi kwiinkqubo ezifana ngomatshini:

• Ukuba, lo gama kugcinwe ubude elifanayo kuqala, anqunyanyiswe ezahlukahlukeneyo zomthwalo, ixesha lo oscillation ukufumana enye, nangona ubunzima zabo ziya kwahluka kakhulu. Ngenxa yoko, ixesha kuqala akuxhomekekanga ubunzima umthwalo.

• Ukuba ngaba urhulumente iqalisa ukuba ukwehla kwi kuqala ayikho inkulu kakhulu, kodwa kwiikona ezahlukeneyo, kuya zinyuka kunye nexesha elinye, kodwa ngezantya ezahlukeneyo. Nangona ezinxaxhileyo ezisuka embindini ukulinganisela amahlandenyuka kakhulu ngendlela yabo iya kuba ikufutshane ngokwaneleyo harmonic. Ixesha pendulum enjalo akuxhomekekanga kwi awo vibrational. Le impahla kwenkqubo ngoomatshini kuthiwa isochronism (in "chronos" Greek - ixesha "Izosov" - kulingana).

Ixesha pendulum elula

Eli nani limele ixesha endalo oscillation. Nangona ekuqulunqeni esimbaxa, inkqubo ngokwayo ilula kakhulu. Ukuba ubude vulu lweMathematika pendulum L, kwaye omxhuzulane lunikezelo g, eli xabiso silingana:

T = 2π√L / g

ixesha elincinane oscillations endalo akukho ndlela komntu akuxhomekekanga ekuzithandeleni ubunzima kuqala kunye namandla awo oscillation. Kulo mzekelo, njengoko pendulum zezibalo ihamba kunye nokuphungulwa ubude.

Oscillations of pendulum zezibalo

pendulum Mathematical oscillates, leyo linokuchazwa yi i'quadratic elula umehluko:

x + ω2 isono x = 0,

apho x (t) - umsebenzi ayaziwa (oku engile aphambuke ukusuka kwindawo esezantsi kuvuselele ngexesha t, ochazwa kwilayini eziqala embindini); ω - njalo esihle okumisiweyo ukususela kuMmandla kuqala (ω = √g / L, apho g - kukukhawuleziswa womxhuzulane, kunye L - ubude pendulum elula (ukumiswa).

Equation oscillations ezincinane kufuphi lizama isikhundla (equation harmonic) ngolu hlobo lulandelayo:

x + ω2 isono x = 0

isiphakamiso Oscillatory of kuqala

Kuqala, nto leyo eyenza oscillations ezincinane, ihamba sinusoid. Yesibini ukuze umahluko equation idibana zonke iimfuno kunye kuMmandla intshukumo enje. Ukuze ubone indlela ufuna ukucwangcisa isantya kunye zilungelelanise, apho kamva wagqiba constants ezizimeleyo:

x = A isono (θ ₀ + ωt),

apho θ ₀ - kwisigaba sokuqala, A - awo of oscillation, ω - rhoqo elibujikele bamiliselwa kwizibalo sentshukumo.

Pendulum (Ifomula ngezantya ezinkulu)

Le nkqubo mechanical, ukwenza oscillations zabo kunye namandla awo amakhulu, oko kuxhomekeke imithetho yendlela ezimbaxa ngakumbi. ukuba zibalwa ngokungqinelana Ifomula pendulum onjalo:

isono x / 2 = u * sn (ωt / u),

apho sn - sine Jacobi, owathi ngenxa u <1 ngumsebenzi ngamaxesha athile, kwaye u omncinane ingqamana sine elula netrigonometri. Ixabiso u bubonwa ibinzana ilandelayo:

u = (ε + ω2) / 2ω2,

apho ε = E / ML2 (ML2 - Amandla pendulum).

Ukuzimisela kwethuba oscillation nonlinear ye kuqala le ndlela ilandelayo:

T = 2π / Ω,

apho Ω = π / 2 * ω / 2K (u), K - okweqanda ebalulekileyo, π - 3,14.

kuqala intshukumo ye separatrix

Oku kubizwa separatrix indlela yenkqubo onamandla, apho isithuba kwisigaba okuntlandlo-mbini. Kuqala ehamba kwi non-njalo. Xa ingongoma ngamabona kude bexesha lehla ukusuka kwindawo kakhulu aphezulu ukuze kwisantya zero, yaye ngoko ke kancinci yande. Ekugqibeleni wayeka, abuyele kwindawo yayo yoqobo.

Ukuba namandla awo of oscillation ye kuqala osondela PI inani, oko kuthiwa isindululo kwisigaba moya likufutshane separatrix. Kulo mzekelo, phantsi ngesenzo lokuqhuba amandla encinane ngamaxesha kwinkqubo ngoomatshini ebonisa ukuziphatha bahlupheke.

Kwimeko pendulum elula ukusuka kwindawo lizama kunye CP engile kwenzeka amandla tangential Fτ = sibi isono -mg φ. uphawu "Thabatha" kuthetha ukuba icandelo tangential ngqo kwi icala evela ngakwicala ukwahluka kuqala. Xa ubhekisela nge pendulum ukubana x ecaleni i arc setyhula kunye L radius ilingana ukubana φ yayo emboxo = x / L. Umthetho yesibini Isaaka Nyutona, eyenzelwe ezayo kumda wesangqa umcephe namandla ukunika ixabiso olifunayo:

mg τ = Fτ = -mg isono x / L

Ngokusekelwe kule ratio, kucacile ukuba kuqala yinkqubo nonlinear, njengokuba amandla ithande ukuba sibuyele endaweni yaso lizama, akusoloko ngokomlinganiselo ukubana x, isono x / L.

Kuphela xa kuqala zemathematika yenza intlokoma ezincinane, yinto oscillator harmonic. Ngamanye amazwi, iba inkqubo ngomatshini ekwaziyo ukwenza oscillations harmonic. Le kelelo isebenza phantse angles 15-20 °. Pendulum kunye ngezantya amakhulu ayikho iyavisisana.

umthetho Newton ngenxa oscillations encinane pendulum

Ukuba inkqubo ngoomatshini yenza oscillations ezincinane, umthetho-2 Newton iya kukhangela ngolu hlobo:

mg τ = Fτ = -m * g / L * x.

Ngenxa yesi sizathu, sinokugqiba ukuba ukukhawulezisa tangential ye pendulum elula na umlinganiselo ukuba ukubana yalo umqondiso "thabatha". Le yimeko apho le nkqubo uba oscillator harmonic. Module ukulingana factor phakathi ukubana kunye lunikezelo ilingana isikweri njalo emboxo:

ω02 = g / L; ω0 = √ g / L.

Le fomyula ibonisa rhoqo zendalo oscillations amancinane olu hlobo pendulum. Ngenxa yesi sizathu,

T = 2π / ω0 = 2π√ g / L.

Ubalo lusekelwe kwi umthetho ukulondolozwa amandla

Izinto zobumnini jinga iintshukumo pendulum kungachazwa ngoncedo umthetho zolondolozo amandla. Kufuneka kukhunjulwe ukuba amandla okunokwenzeka kuqala kwintsimi omxhuzulane ngu:

E = mgΔh = mgL (1 - cos α) = mgL2sin2 α / 2

Full amandla mechanical lilingana amandla entshukumo eliphezulu: Epmax = Ekmsx = E

Emva kokuba ezibhaliweyo emthethweni zolondolozo amandla, ngokuthatha yemvelaphi emacaleni ekhohlo nasekunene lenxaki:

EP + Ek = nengngqi

Ukususela livela i constants ilingana no0, ngoko (EP + Ek) '= 0. I yemvelaphi isixa lilingana sum of yezabelo:

EP '= (mg / L * x2 / 2)' = mg / 2l * 2x * x '= mg / L * V + Ek' = (mv2 / 2) = m / 2 (v2) '= m / 2 * 2v * v '= MV * α,

ke:

Mg / L * XV + mva = V (mg / L * x + m α) = 0.

Ngokusekelwe kwi indlela lokugqibela, sifumana: α = - g / L * x.

Practical isicelo kuqala lweMathematika

Ukukhawulezisa of ukuwa yahluka kunye inkululeko, kuba lexinene kwesikhoko ujikeleze umhlaba hayi ncam. Apho rocks kuhamba ezixineneyo ephakamileyo, kuya kuba ngaphezulu kancinci. Ekukhawuleziseni pendulum lweMathematika isoloko isetyenziswa zokuhlola. Xa bembonakalo yayo uncedo izimbiwa ezahlukeneyo. Nje ngokubala inani oscillations ye pendulum, unako ukubona amalahle okanye ekrwada, ndineemfesane Yomhlaba. Oku kungenxa yokuba ezi zibonelelo zibe lexinene ubunzima ngaphezu elele ngaphantsi rocks evakalala.

pendulum Mathematical esetyenziswa yi abaphengululi owaziwayo ezifana uSocrates, uAristotle, uPlato, Plutarch, Archimedes. Kwakholwa ke ukuba inkqubo koomatshini nefuthe isiphelo nobomi. Archimedes wasebenzisa kuqala zemathematika kunye ubalo bakhe. Kule mihla, occultists ezininzi noosiyazi basebenzisa le nkqubo ngoomatshini ukulungiselela ukuphunyezwa iziprofeto zayo, okanye ukufuna ezilahlekileyo abantu.

Le ngeenkwenkwezi odumileyo French-nzulu, Flammarion ngenxa yophando lwabo wasebenzisa pendulum zezibalo. Wayesithi ngoncedo lwakhe wakwazi ukuqikelela kokufunyanwa iplanethi entsha, ukuvela kwe-Tunguska zimpahla, kunye neminye imicimbi ebalulekileyo. Ngexesha Second World War eJamani (Berlin) wayesebenza ziko ezizodwa lwe kuqala. Kule mihla, uphando loo ayifumaneki Munich Institute of uLawulo അഡ്മിനിസ്ട്രേറ്റര്. Umsebenzi wakhe kuqala abahlohli beli ziko ebizwa ngokuba "radiesteziey".