Kuyini ukudlidliza okuqondile?

Ukudlidliza okuqondile: ukuqina kwezingxenye ohlelweni kuncike emthethweni ka-hooke, futhi amandla okunciphisa akhiqizwa ngesikhathi sokunyakaza alingana nesibalo sokuqala sejubane elijwayelekile (i-derivative yesikhathi yama-generalized coordinates).

umqondo

Uhlelo oluqondile luvame ukuba yimodeli engaqondakali yokudlidliza kwesistimu yangempela. Uhlelo lokudlidliza oluqondile lusebenzisa isimiso sokuma phezulu, okungukuthi, uma impendulo yesistimu ingu-y1 ngaphansi kwesenzo sokufaka u-x1, kanye no-y2 ngaphansi kwesenzo sokufaka u-x2, khona-ke impendulo yesistimu ngaphansi kwesenzo sokufaka u-x1 kanye no-x2 ingu-y1+y2.

Ngokusekelwe kumgomo we-superposition, okokufaka okungahleliwe kungahlukaniswa kube yisamba sochungechunge lwezimpumuzo ezincane kakhulu, bese impendulo iyonke yesistimu ingatholakala. Isamba sezingxenye ze-harmonic ze-periodic excitation singandiswa sibe uchungechunge lwezingxenye ze-harmonic yi-Fourier transform, futhi umphumela wengxenye ngayinye ye-harmonic ohlelweni ungaphenywa ngokwehlukana. Ngakho-ke, izici zokuphendula zezinhlelo eziqondile ezinemingcele engaguquki zingachazwa ngempendulo ye-impulse noma impendulo yemvamisa.

Impendulo yomfutho ibhekisela empendulweni yesistimu ku-unit impulse, echaza izici zokuphendula zesistimu endaweni yesikhathi. Impendulo yokuphindaphinda ibhekisela empendulweni yesistimu kokufakwayo kwe-unit harmonic. Ukuxhumana phakathi kwalokhu okubili kunqunywa yi-Fourier transform.

ukuhlukaniswa

Ukudlidliza okuqondile kungahlukaniswa kube ukudlidliza okuqondile kwesistimu ye-single-degree-of-freedom kanye nokudlidliza okuqondile kwesistimu ye-multi-degree-of-freedom.

(1) ukudlidliza okuqondile kwesistimu yezinga elilodwa lenkululeko ukudlidliza okuqondile okunendawo okunganqunywa ngayo nge-generalized coordinate. Kungukudlidliza okulula kakhulu lapho imiqondo eminingi eyisisekelo nezici zokudlidliza kungatholakala khona. Kuhlanganisa ukudlidliza okulula kwe-harmonic, ukudlidliza okukhululekile, ukudlidliza okuncishisiwe kanye nokudlidliza okuphoqelelwe.

Ukudlidliza okulula kwe-harmonic: ukunyakaza okuphindaphindayo kwento eduze kwendawo yayo yokulingana ngokomthetho we-sinusoidal ngaphansi kwesenzo samandla okubuyisela ahambisana nokufuduka kwayo.

Ukudlidliza okudambisiwe: ukudlidliza okunamandla ahlala encishiswa ukuba khona kokungqubuzana kanye nokumelana ne-dielectric noma okunye ukusetshenziswa kwamandla.

Ukudlidliza okuphoqelelwe: ukudlidliza kwesistimu ngaphansi kokushukunyiswa okuqhubekayo.

(2) ukudlidliza okuqondile kwesistimu ye-multi-degree-of-freedom kungukudlidliza kwesistimu eqondile enama-degrees enkululeko angu-n≥2. Uhlelo lwama-degrees enkululeko angu-n lunama-frequency emvelo angu-n kanye nama-main mode angu-n. Noma yikuphi ukucushwa kokudlidliza kwesistimu kungamelwa njengenhlanganisela eqondile yama-major modes. Ngakho-ke, indlela ye-main mode superposition isetshenziswa kabanzi ekuhlaziyweni kwempendulo eguquguqukayo kwezinhlelo ze-multi-dof. Ngale ndlela, ukulinganisa nokuhlaziya izici zokudlidliza zemvelo zesistimu kuba yisinyathelo esijwayelekile ekwakhiweni okunamandla kwesistimu. Izici eziguquguqukayo zezinhlelo ze-multi-dof nazo zingachazwa ngezici zemvamisa. Njengoba kukhona umsebenzi wemvamisa phakathi kokufaka ngakunye nokukhiphayo, kwakhiwa i-matrix yemvamisa. Kukhona ubudlelwano obuqondile phakathi kwesici semvamisa kanye nemodi eyinhloko. I-amplitude-frequency characteristic curve yesistimu ye-multi-freedom ihlukile kweyesistimu ye-single-freedom.

Ukudlidliza okuqondile kwesistimu yenkululeko yezinga elilodwa

Ukudlidliza okuqondile lapho indawo yesistimu inganqunywa khona nge-generalized coordinate. Kungukudlidliza okulula nokuyisisekelo lapho imiqondo eminingi eyisisekelo nezici zokudlidliza kungatholakala khona. Kuhlanganisa ukudlidliza okulula kwe-harmonic, ukudlidliza okuncishisiwe kanye nokudlidliza okuphoqelelwe.

Ukudlidliza kwe-Harmonic

Ngaphansi kwesenzo sokubuyisela amandla ngokulingana nokufuduka, into iphinda ngendlela ye-sinusoidal eduze kwesikhundla sayo sokulingana (UMfanekiso 1). X imele ukufuduka kanti u-t umele isikhathi. Ukubonakaliswa kwezibalo kwalokhu kudlidliza yilokhu:

(1)Lapho u-A kuyinani eliphezulu lokufuduka u-x, okubizwa ngokuthi i-amplitude, futhi kumelela ukuqina kokudlidliza; i-Omega n iyi-amplitude Ukwanda kwe-engela kokudlidliza ngomzuzwana, okubizwa ngokuthi i-angular frequency, noma i-circular frequency; Lokhu kubizwa ngokuthi isigaba sokuqala. Ngokuphathelene ne-f= n/2, inani lokudlidliza ngomzuzwana libizwa ngokuthi i-frequency; Okuphambene nalokhu, i-T=1/f, yisikhathi esithathayo ukujikeleza umjikelezo owodwa, futhi lokho kubizwa ngokuthi isikhathi. I-Amplitude A, imvamisa u-f (noma imvamisa ye-angular n), isigaba sokuqala, esaziwa ngokuthi i-simple harmonic vibration elements ezintathu.

UMFANELO 1 ijika lokudlidliza elivumelanayo elilula

Njengoba kuboniswe ku-FIG. 2, i-oscillator elula ye-harmonic yakhiwa yi-concentrated mass m exhunywe yi-linear spring. Lapho ukufuduka kokudlidliza kubalwa kusukela endaweni yokulingana, i-vibration equation yile:

Kuphi ukuqina kwentwasahlobo. Isixazululo esijwayelekile se-equation engenhla yi-(1).A futhi singanqunywa ngesikhundla sokuqala x0 kanye nesivinini sokuqala ku-t=0:

Kodwa i-omega n inqunywa kuphela yizici zesistimu ngokwayo i-m kanye ne-k, ngaphandle kwezimo zokuqala ezengeziwe, ngakho-ke i-omega n yaziwa nangokuthi imvamisa yemvelo.

ISITHOMBE 2 uhlelo lwenkululeko yezinga elilodwa

Kwi-oscillator elula ye-harmonic, isamba samandla ayo e-kinetic kanye namandla angase abe khona ahlala njalo, okungukuthi, amandla aphelele omshini wesistimu ayalondolozwa. Enkambisweni yokudlidliza, amandla e-kinetic kanye namandla angase abe khona aguqulwa njalo abe ngomunye nomunye.

Ukudlidliza okudambisayo

Ukudlidliza okunamandla alo ancishiswa njalo ukungqubuzana nokumelana ne-dielectric noma okunye ukusetshenziswa kwamandla. Ngokudlidliza okuncane, ijubane ngokuvamile alilikhulu kakhulu, futhi ukumelana okuphakathi kuyalingana nejubane lamandla okuqala, okungabhalwa ngokuthi c kuyi-damping coefficient. Ngakho-ke, i-vibration equation yedigri eyodwa yenkululeko ene-linear damping ingabhalwa kanje:

(2)Lapho, i-m = c/2m ibizwa ngokuthi ipharamitha yokudambisa, futhi. Isixazululo esijwayelekile sefomula (2) singabhalwa:

(3)Ubudlelwano bezinombolo phakathi kwe-omega n ne-PI bungahlukaniswa ngamacala amathathu alandelayo:

N > (uma kwenzeka ukudambisa okuncane) izinhlayiya ezikhiqizwe ukudlidliza kokunciphisa, i-vibration equation yile:

Ubukhulu bayo buyancipha ngokuhamba kwesikhathi ngokusho komthetho we-exponential oboniswe ku-equation, njengoba kuboniswe emgqeni onamachashazi ku-FIG. 3. Empeleni, lokhu kudlidliza kuyinto engavamile, kodwa imvamisa yokuphakama kwayo ingachazwa ngokuthi:

Ibizwa ngokuthi izinga lokunciphisa i-amplitude, lapho isikhathi sokudlidliza. I-logarithm yemvelo yezinga lokunciphisa i-amplitude ibizwa ngokuthi izinga le-logarithm minus (amplitude). Ngokusobala, =, kulokhu, ilingana no-2/1. Ngokuqondile ngokusebenzisa i-delta yokuhlola yokuhlola futhi, kusetshenziswa ifomula engenhla kungabalwa u-c.

Ngalesi sikhathi, ikhambi le-equation (2) lingabhalwa kanje:

Kanye nesiqondiso sejubane lokuqala, ingahlukaniswa ngamacala amathathu angadlidlizi njengoba kuboniswe ku-FIG. 4.

N < (esimweni sokunciphisa okukhulu), ikhambi le-equation (2) liboniswa ku-equation (3). Kuleli qophelo, uhlelo alusadlidlizi.

Ukudlidliza okuphoqelelwe

Ukudlidliza kwesistimu ngaphansi kokudlidliza okuqhubekayo. Ukuhlaziywa kokudlidliza kuphenya kakhulu impendulo yesistimu ekudlidlizeni. Ukudlidliza ngezikhathi ezithile kuyindlela evamile yokudlidliza. Njengoba ukudlidliza ngezikhathi ezithile kungahlukaniswa kube yisamba sokudlidliza okuningana kwe-harmonic, ngokusho komgomo we-superposition, impendulo yesistimu kuphela ekudlidlizeni ngakunye kwe-harmonic kuyadingeka. Ngaphansi kwesenzo sokudlidliza kwe-harmonic, i-equation ehlukile yokunyakaza kwesistimu eyodwa yezinga elilodwa lenkululeko engabhalwa:

Impendulo iyisamba sezingxenye ezimbili. Ingxenye eyodwa iyimpendulo yokudlidliza okuncishisiwe, okubola ngokushesha ngokuhamba kwesikhathi. Impendulo yenye ingxenye yokudlidliza okuphoqelelwe ingabhalwa:

UMFANELO 3 ijika lokudlidliza elifiphele

UMFANEKISO 4 amajika ezimo ezintathu zokuqala ezine-critical damping

Thayipha ku-

I-H /F0= h (), isilinganiso se-amplitude yokusabela okuzinzile kuya ku-amplitude yokuvuselela, echaza izici ze-amplitude-frequency, noma umsebenzi wokuzuza; Ama-Bits okuphendula kwesimo esizinzile kanye nesikhuthazo sesigaba, ukuchazwa kwezimpawu zemvamisa yesigaba. Ubudlelwano phakathi kwawo nemvamisa yokuvuselela buboniswe ku-FIG. 5 kanye ne-FIG. 6.

Njengoba kungabonakala ku-amplitude-frequency curve (FIG. 5), uma kwenzeka ukudambisa okuncane, i-amplitude-frequency curve inokuphakama okukodwa. Uma ukudambisa kuncane, kulapho ukuphakama kuba kukhulu khona; Imvamisa ehambisana ne-peak ibizwa ngokuthi imvamisa ye-resonant yesistimu. Uma kwenzeka ukudambisa okuncane, imvamisa ye-resonance ayihlukile kakhulu ku-natural frequency. Uma imvamisa ye-excitation iseduze nemvamisa yemvelo, i-amplitude iyanda kakhulu. Lesi simo sibizwa ngokuthi i-resonance. Uma kwenzeka ukudambisa, inzuzo yesistimu ikhuliswa, okungukuthi, ukudlidliza okuphoqelelwe kunamandla kakhulu. Ngakho-ke, ngokuvamile, hlala uzama ukugwema ukudlidliza, ngaphandle kokuthi ezinye izinsimbi nemishini zisebenzisa i-resonance ukufeza ukudlidliza okukhulu.

Ijika lemvamisa ye-amplitude 5

Kungabonakala kusukela ku-phase frequency curve (isithombe 6), kungakhathaliseki ukuthi usayizi we-damping ungakanani, kuma-bits we-omega zero phase difference = PI / 2, lesi sici singasetshenziswa ngempumelelo ekulinganiseni i-resonance.

Ngaphezu kokuvusa okuqhubekayo, izinhlelo ngezinye izikhathi zibhekana nokuvusa okungazinzile. Kungahlukaniswa cishe zibe izinhlobo ezimbili: eyodwa umthelela ongazelelwe. Owesibili umphumela ohlala njalo wokungakhethi. Ngaphansi kokuvusa okungazinzile, impendulo yesistimu nayo ayizinzile.

Ithuluzi elinamandla lokuhlaziya ukudlidliza okungazinzile yindlela yokuphendula nge-impulse. Ichaza izici eziguquguqukayo zesistimu ngempendulo yesikhashana yokufakwa kwe-unit impulse yesistimu. I-unit impulse ingavezwa njengomsebenzi we-delta. Kobunjiniyela, umsebenzi we-delta uvame ukuchazwa ngokuthi:

Lapho u-0- emele iphuzu ku-t-axis esondela ku-zero ukusuka kwesobunxele; u-0 plus yiphuzu eliya ku-0 ukusuka kwesokudla.

Ijika lemvamisa yesigaba esingu-FIG. 6

UMFANEKISO 7 noma yikuphi ukufaka kungabhekwa njengesamba sochungechunge lwezinto ezishukumisayo

Uhlelo luhambisana nempendulo h(t) ekhiqizwa yi-unit impulse ku-t=0, ebizwa ngokuthi umsebenzi wempendulo ye-impulse. Uma sicabanga ukuthi uhlelo lumile ngaphambi kwe-pulse, h(t)=0 ku-t<0. Njengoba sazi umsebenzi wempendulo ye-impulse wesistimu, singathola impendulo yesistimu kunoma yikuphi ukufaka x(t). Kuleli qophelo, ungacabanga nge-x(t) njengesamba sochungechunge lwezinto ze-impulse (Umfanekiso 7). Impendulo yesistimu yile:

Ngokusekelwe kusimiso se-superposition, impendulo iyonke yesistimu ehambisana no-x(t) ithi:

Lokhu okuhlanganisiwe kubizwa ngokuthi i-convolution integral noma i-superposition integral.

Ukudlidliza okuqondile kwesistimu yenkululeko enamazinga amaningi

Ukudlidliza kwesistimu eqondile ene-n≥2 degrees yenkululeko.

Isithombe 8 sibonisa izinhlelo ezimbili ezilula zokuzwela ezixhunywe yintwasahlobo yokuxhuma. Ngenxa yokuthi iyisistimu yenkululeko enamazinga amabili, kudingeka izixhumanisi ezimbili ezizimele ukuze kunqunywe indawo yayo. Kunezindlela ezimbili zemvelo kulesi simiso:

Imvamisa ngayinye ihambisana nemodi yokudlidliza. Ama-oscillator e-harmonic enza ukudlidliza kwemvamisa efanayo, edlula ngendlela efanayo endaweni yokulingana futhi efinyelela endaweni ephakeme kakhulu. Ekudlidlizeni okuyinhloko okuhambisana ne-omega eyodwa, i-x1 ilingana ne-x2; Ekudlidlizeni okuyinhloko okuhambisana ne-omega omega ezimbili, i-omega omega eyodwa. Ekudlidlizeni okuyinhloko, isilinganiso sokufuduka kwesisindo ngasinye sigcina ubuhlobo obuthile futhi sakha imodi ethile, ebizwa ngokuthi imodi eyinhloko noma imodi yemvelo. Ukulingana kobukhulu nokuqina kukhona phakathi kwezindlela eziyinhloko, okubonisa ukuzimela kokudlidliza ngakunye. Imvamisa yemvelo kanye nemodi eyinhloko kumelela izici zokudlidliza ezingokwemvelo zesistimu yenkululeko enezinga elihlukahlukene.

Uhlelo lwe-FIG. 8 olunezinga eziningi zenkululeko

Uhlelo lwamadigri angu-n lwenkululeko lunamaza emvelo angu-n kanye namamodi ayinhloko angu-n. Noma yikuphi ukucushwa kokudlidliza kwesistimu kungamelwa njengenhlanganisela eqondile yamamodi amakhulu. Ngakho-ke, indlela ye-main mode superposition isetshenziswa kabanzi ekuhlaziyweni kwempendulo enamandla kwezinhlelo eziningi ze-dof. Ngale ndlela, ukulinganisa nokuhlaziya izici zokudlidliza zemvelo zesistimu kuba yisinyathelo esijwayelekile ekwakhiweni kwesistimu okuguquguqukayo.

Izici eziguquguqukayo zezinhlelo ze-multi-dof zingachazwa futhi ngezici zemvamisa. Njengoba kunomsebenzi wesici semvamisa phakathi kokufaka ngakunye nokukhiphayo, kwakhiwa i-matrix yesici semvamisa. Ijika lesici se-amplitude-frequency lesistimu yenkululeko eminingi lihlukile kwelohlelo lwenkululeko eyodwa.

I-elastomer iyadlidliza

Uhlelo lwe-multi-degree of freedom olungenhla luyimodeli ye-elastomer ecishe ibe yimishini. I-elastomer inenani elingenamkhawulo lamazinga enkululeko. Kunomehluko wobuningi kodwa akukho mehluko obalulekile phakathi kwalezi ezimbili. Noma iyiphi i-elastomer inenani elingenamkhawulo lama-frequency emvelo kanye nenani elingenamkhawulo lamazinga ahambisanayo, futhi kukhona ukulingana phakathi kwamazinga obunzima nokuqina. Noma yikuphi ukucushwa kokudlidliza kwe-elastomer nakho kungamelwa njenge-linear superposition yama-major modes. Ngakho-ke, ekuhlaziyweni kwempendulo enamandla kwe-elastomer, indlela ye-superposition yemodi eyinhloko isasebenza (bheka ukudlidliza okuqondile kwe-elastomer).

Thatha ukudlidliza kwentambo. Ake sithi intambo encane enobunzima m ngobude beyunithi, u-l omude, icindezelwe kuzo zombili izinhlangothi, kanti ukucindezelwa kungu-T. Ngalesi sikhathi, imvamisa yemvelo yentambo inqunywa yi-equation elandelayo:

F =na/2l (n= 1,2,3…).

Kuphi, ijubane lokusabalala kwegagasi eliphambeneyo lihambisana nentambo. Amaza okuphindaphinda emvelo ezintambo ayenzeka abe yiziphindaphindo zemvamisa eyisisekelo ngaphezu kuka-2l. Lokhu kuphindaphinda kwenani eliphelele kuholela esakhiweni esimnandi se-harmonic. Ngokuvamile, akukho buhlobo obunjalo benani eliphelele phakathi kwamaza okuphindaphinda emvelo e-elastomer.

Izindlela ezintathu zokuqala zentambo ecindezelwe ziboniswe ku-FIG. 9. Kunezinodi ezithile ku-main mode curve. Ekudlidlizeni okuyinhloko, izinodi azidlidlizi. I-FIG. 10 ibonisa izindlela eziningana ezijwayelekile zepuleti eliyindilinga elisekelwa ngokuzungeza elinemigqa ethile yezinodi eyakhiwe yizindilinga nobubanzi.

Ukwakheka okuqondile kwenkinga yokudlidliza kwe-elastomer kungaphethwa njengenkinga yenani lomngcele wezilinganiso zomehluko ezingaphelele. Kodwa-ke, ikhambi eliqondile lingatholakala kuphela kwezinye zezimo ezilula, ngakho-ke kufanele sisebenzise ikhambi eliqondile lenkinga yokudlidliza eyinkimbinkimbi ye-elastomer. Ingqikithi yezixazululo ezahlukahlukene ezilinganiselwe ukushintsha okungenamkhawulo kube okulinganiselwe, okungukuthi, ukuhlukanisa uhlelo lwenkululeko olungenazo izitho eziningi (uhlelo oluqhubekayo) lube uhlelo lwenkululeko olulinganiselwe olunamazinga amaningi (uhlelo oluhlukile). Kunezinhlobo ezimbili zezindlela zokudlidliza ezisetshenziswa kabanzi ekuhlaziyweni kobunjiniyela: indlela ye-finite element kanye nendlela yokuhlanganisa i-modal.

Imodi yentambo ka-FIG. 9

UMFANELO 10 imodi yepuleti eliyindilinga

Indlela ye-Finite element iyisakhiwo esihlanganisiwe esihlanganisa isakhiwo esiyinkimbinkimbi sibe yinani elilinganiselwe lezinto bese sizixhumanisa nenani elilinganiselwe lama-node. Iyunithi ngayinye iyi-elastomer; Ukusabalala kwe-element kuvezwa umsebenzi wokuhlanganisa ukufuduka kwama-node. Ngemuva kwalokho amapharamitha okusabalalisa e-element ngayinye agxiliswa ku-node ngayinye ngefomethi ethile, bese kutholakala imodeli yemishini yesistimu ehlukile.

Ukuhlanganiswa kwe-Modal kuwukuhlukaniswa kwesakhiwo esiyinkimbinkimbi sibe yizakhiwo eziningana ezilula. Ngokusekelwe ekuqondeni izici zokudlidliza zesakhiwo ngasinye, isakhiwo esingaphansi sihlanganiswa sibe isakhiwo esijwayelekile ngokwezimo zokuxhumanisa esibonakalayo, futhi isimo sokudlidliza sesakhiwo esijwayelekile sitholakala ngokusebenzisa isimo sokudlidliza sesakhiwo ngasinye esingaphansi.

Lezi zindlela ezimbili zihlukile futhi zihlobene, futhi zingasetshenziswa njengereferensi. Indlela yokuhlanganisa i-modal ingahlanganiswa ngempumelelo nesilinganiso sokuhlola ukuze kwakhiwe indlela yokuhlaziya yethiyori neyokuhlola yokudlidliza kwezinhlelo ezinkulu.