Yintoni ukungcangcazela okuthe ngqo?

Ukungcangcazela okuthe tye: ukuthamba kwezinto ezikwinkqubo kuxhomekeke kumthetho ka-hooke, kwaye amandla okudambisa aveliswa ngexesha lokushukuma alingana ne-equation yokuqala yesantya esijikelelekileyo (i-time derivative ye-generalized coordinates).

ingcamango

Inkqubo yomgca idla ngokuba yimodeli engabonakaliyo yokungcangcazela kwenkqubo yokwenyani. Inkqubo yokungcangcazela emgca isebenzisa umgaqo we-superposition, oko kukuthi, ukuba impendulo yenkqubo ingu-y1 phantsi kwesenzo sokufaka u-x1, kunye no-y2 phantsi kwesenzo sokufaka u-x2, ngoko impendulo yenkqubo phantsi kwesenzo sokufaka u-x1 kunye no-x2 ngu-y1+y2.

Ngokusekelwe kumgaqo we-superposition, igalelo elingacwangciswanga linokwahlulwa libe yi-sum of a infinitesimal impulses, kwaye emva koko impendulo iyonke yenkqubo ingafunyanwa. Isimbuku see-harmonic components ze-periodic excitation sinokwandiswa sibe yi-series ye-harmonic components yi-Fourier transform, kwaye isiphumo se-harmonic component nganye kwinkqubo sinokuphandwa ngokwahlukeneyo. Ke ngoko, iimpawu zempendulo zeenkqubo ezithe ngqo ezineeparameters ezingaguqukiyo zinokuchazwa yi-impulse response okanye i-frequency response.

Impendulo yempembelelo ibhekisa kwimpendulo yenkqubo kwi-unit impulse, echaza iimpawu zempendulo yenkqubo kwicandelo lexesha. Impendulo yefrikhwensi ibhekisa kwimpawu yempendulo yenkqubo kwi-unit harmonic input. Ukuhambelana phakathi kwezi zimbini kumiselwa yi-Fourier transform.

ukuhlelwa

Ukungcangcazela okuthe tye kunokwahlulwahlulwa kube kukungcangcazela okuthe tye kwenkqubo ye-single-degree-of-freedom kunye nokungcangcazela okuthe tye kwenkqubo ye-multi-degree-of-freedom.

(1) ukungcangcazela okuthe ngqo kwenkqubo ye-single-degree-of-freedom kukungcangcazela okuthe ngqo apho indawo yayo inokugqitywa yi-generalized coordinate. Kukungcangcazela okulula apho iingcamango ezininzi ezisisiseko kunye neempawu zokungcangcazela zinokufunyanwa khona. Kubandakanya ukungcangcazela okulula kwe-harmonic, ukungcangcazela okukhululekileyo, ukungcangcazela okuncitshisiweyo kunye nokungcangcazela okunyanzelekileyo.

Ukungcangcazela okulula kwe-harmonic: intshukumo ephindaphindayo yento ekufutshane nendawo yayo yokulingana ngokomthetho we-sinusoidal phantsi kwesenzo samandla okubuyisela ahambelana nokufuduka kwayo.

Ukungcangcazela okudambileyo: ukungcangcazela okuhlala kuncitshiswa bubukho bokungqubana kunye nokumelana ne-dielectric okanye ukusetyenziswa kwamandla okunye.

Ukungcangcazela okunyanzelekileyo: ukungcangcazela kwenkqubo phantsi koxinzelelo oluqhubekayo.

(2) ukungcangcazela okuthe ngqo kwenkqubo ye-multi-degree-of-freedom kukungcangcazela kwenkqubo ethe ngqo ene-n≥2 degrees yenkululeko. Inkqubo ye-n degrees yenkululeko ine-n natural frequency kunye ne-n main modes. Naluphi na ulungiselelo lokungcangcazela kwenkqubo lungamelwa njengomdibaniso othe ngqo weendlela eziphambili. Ke ngoko, indlela ye-main mode superposition isetyenziswa kakhulu kuhlalutyo lwempendulo eguquguqukayo lweenkqubo ze-multi-dof. Ngale ndlela, ukulinganisa kunye nohlalutyo lweempawu zendalo zokungcangcazela zenkqubo ziba linyathelo eliqhelekileyo kuyilo oluguquguqukayo lwenkqubo. Iimpawu ze-dynamic zeenkqubo ze-multi-dof zinokuchazwa ngeempawu ze-frequency. Ekubeni kukho umsebenzi we-frequency characteristic phakathi kwe-input nganye kunye ne-output, kwakhiwa i-frequency characteristic matrix. Kukho ulwalamano olucacileyo phakathi kwe-frequency characteristic kunye ne-main mode. I-amplitude-frequency characteristic curve yenkqubo ye-multi-freedom yahlukile kweyenkqubo ye-single-freedom.

Ukungcangcazela okuthe tye kwenkqubo yenkululeko yenqanaba elinye

Ukungcangcazela okuthe ngqo apho indawo yenkqubo inokugqitywa khona yi-generalized coordinate. Yeyona ngcangcazela ilula neyona isisiseko apho iingcamango ezininzi ezisisiseko kunye neempawu zokungcangcazela zinokufunyanwa khona. Ibandakanya ukungcangcazela okulula kwe-harmonic, ukungcangcazela okudampisiweyo kunye nokungcangcazela okunyanzelekileyo.

Ukungcangcazela kweHarmonic

Phantsi kwesenzo sokubuyisela amandla ngokulingana nokufuduka, into iphinda-phinda ngendlela ye-sinusoidal kufutshane nendawo yayo yokulingana (UMfanekiso 1). X imele ukufuduka kwaye u-t umele ixesha. Inkcazo yemathematika yale ntshukumo yile:

(1)Apho u-A lixabiso eliphezulu lokufuduka u-x, elibizwa ngokuba yi-amplitude, kwaye limele ubunzulu bokungcangcazela; i-Omega n yi-amplitude Ukunyuka kwe-Angle kokungcangcazela ngomzuzwana, okubizwa ngokuba yi-angular frequency, okanye i-circular frequency; Oku kubizwa ngokuba yi-initial phase. Ngokwe-f= n/2, inani le-oscillations ngomzuzwana libizwa ngokuba yi-frequency; I-inverse yale, i-T=1/f, lixesha elithathayo ukungcangcazela umjikelo omnye, kwaye oko kubizwa ngokuba yi-period. I-Amplitude A, i-frequency f (okanye i-angular frequency n), isigaba sokuqala, esaziwa ngokuba yi-simple harmonic vibration elements ezintathu.

UMZOBO 1 ijika elilula le-harmonic vibration

Njengoko kubonisiwe kuMfanekiso 2, i-oscillator elula ye-harmonic yenziwe yi-concentrated mass m edityaniswe yi-linear spring. Xa i-vibration displacement ibalwa ukusuka kwindawo ye-equilibrium, i-vibration equation yile:

Iphi na i-spiring hardness. Isisombululo esiqhelekileyo kwi-equation engasentla yi-(1).A kwaye sinokumiselwa ngendawo yokuqala x0 kunye ne-initial velocity kwi-t=0:

Kodwa i-omega n imiselwa kuphela ziimpawu zenkqubo ngokwayo i-m kunye ne-k, ngaphandle kweemeko zokuqala ezongezelelweyo, ngoko ke i-omega n ikwaziwa njenge-natural frequency.

UMZOBO 2 Inkqubo yenkululeko yenqanaba elinye

Kwi-harmonic oscillator elula, isimbuku samandla ayo e-kinetic kunye namandla anokubakho ahlala ehleli, oko kukuthi, amandla e-mechanical ewonke enkqubo ayagcinwa. Kwinkqubo yokungcangcazela, amandla e-kinetic kunye namandla anokubakho zihlala ziguqulwa zibe zezinye.

Ukungcangcazela kokunciphisa

Ukungcangcazela okumandla alo ahlala encitshiswa kukungqubana kunye nokumelana ne-dielectric okanye ukusetyenziswa kwamandla okunye. Kwi-micro vibration, isantya ngokubanzi asisikhulu kakhulu, kwaye ukumelana okuphakathi kuhambelana nesantya samandla okuqala, anokubhalwa ngolu hlobo c yi-damping coefficient. Ke ngoko, i-vibration equation ye-degree enye yenkululeko kunye ne-linear damping ingabhalwa ngolu hlobo:

(2)Apho, i-m = c/2m ibizwa ngokuba yi-damping parameter, kwaye. Isisombululo esiqhelekileyo sefomula (2) singabhalwa:

(3)Ubudlelwane phakathi kwe-omega n kunye ne-PI bunokwahlulwahlulwa zibe zezi meko zintathu zilandelayo:

N > (kwimeko yokudambisa okuncinci) isuntswana eliveliswe kukungcangcazela kokuthomalalisa, i-vibration equation yile:

Ubungakanani bayo buyancipha ngokuhamba kwexesha ngokomthetho we-exponential oboniswe kwi-equation, njengoko kubonisiwe kumgca onamachaphaza kwi-FIG. 3. Ngokuthe ngqo, olu ngcangcazelo luyinto engaqhelekanga, kodwa ubuninzi bencopho yalo bungachazwa ngolu hlobo:

Ibizwa ngokuba yi-amplitude reduction rate, apho lixesha lokungcangcazela. I-logarithm yendalo ye-amplitude reduction rate ibizwa ngokuba yi-logarithm minus (amplitude) rate. Ngokucacileyo, =, kule meko, ilingana no-2/1. Ngokuthe ngqo kwi-delta yovavanyo lovavanyo kwaye, kusetyenziswa le fomyula ingasentla ingabalwa u-c.

Ngeli xesha, isisombululo se-equation (2) singabhalwa:

Kunye necala lesantya sokuqala, inokwahlulwa ibe ziimeko ezintathu ezingangcangcazeliyo njengoko kubonisiwe kuMfanekiso 4.

N < (kwimeko yokudambisa okukhulu), isisombululo se-equation (2) siboniswe kwi-equation (3). Kweli nqanaba, inkqubo ayisashukumi.

Ukungcangcazela okunyanzelekileyo

Ukungcangcazela kwenkqubo phantsi koxinzelelo oluqhubekayo. Uhlalutyo lokungcangcazela luphanda ikakhulu impendulo yenkqubo kuxinzelelo. Uxinzelelo oluqhubekayo luxinzelelo oluqhelekileyo oluqhelekileyo. Ekubeni uxinzelelo oluqhubekayo lunokuhlukaniswa lube sisiphumo seenkqubo ezininzi zoxinzelelo oluhambelanayo, ngokomgaqo we-superposition, kuphela impendulo yenkqubo kuxinzelelo ngalunye oluhambelanayo olufunekayo. Phantsi kwesenzo soxinzelelo oluhambelanayo, i-equation eyahlukileyo yentshukumo yenqanaba elinye lenkqubo exineneyo yenkululeko inokubhalwa:

Impendulo yindibaniselwano yamacandelo amabini. Inxalenye enye yimpendulo yokungcangcazela okudampisiweyo, okubola ngokukhawuleza ngokuhamba kwexesha. Impendulo yenxalenye enye yokungcangcazela okunyanzelekileyo ingabhalwa:

UMZOBO 3 ijika lokungcangcazela elixineneyo

UMZOBO 4 iigophe zeemeko ezintathu zokuqala ezine-critical damping

Thayipha kwi

I-H /F0= h (), ngumlinganiselo we-amplitude yempendulo ezinzileyo ukuya kwi-amplitude ye-excitation, ebonisa iimpawu ze-amplitude-frequency, okanye umsebenzi we-gain; iiBits zempendulo yesimo esizinzileyo kunye nenkuthazo yesigaba, ukuchaza iimpawu ze-phase frequency. Ulwalamano phakathi kwazo kunye ne-excitation frequency luboniswe kuMfanekiso 5 kunye noMfanekiso 6.

Njengoko kunokubonwa kwi-amplitude-frequency curve (UMfanekiso 5), kwimeko ye-amplitude-frequency curve encinci, i-amplitude-frequency curve ine-peak enye. Okukhona i-damping incinci, kokukhona i-peak inyuka; I-frequency ehambelana ne-peak ibizwa ngokuba yi-resonant frequency yenkqubo. Kwimeko ye-damping encinci, i-resonance frequency ayihlukanga kakhulu kwi-natural frequency. Xa i-excitation frequency ikufutshane ne-natural frequency, i-amplitude iyanda kakhulu. Le nto ibizwa ngokuba yi-resonance. Kwi-resonance, inzuzo yenkqubo iyanda, oko kukuthi, i-forced vibration yeyona inamandla kakhulu. Ke ngoko, ngokubanzi, soloko uzama ukuphepha i-resonance, ngaphandle kokuba ezinye izixhobo kunye nezixhobo zisebenzisa i-resonance ukufezekisa i-vibration enkulu.

UMZOBO 5: Ijika lefrikhwensi ye-amplitude

Ingabonakala kwi-phase frequency curve (umfanekiso 6), nokuba ubungakanani bokuxinana bungakanani na, kwi-omega zero phase difference bits = PI / 2, olu phawu lungasetyenziswa ngempumelelo ekulinganiseni i-resonance.

Ukongeza kwi-activation engaguqukiyo, iinkqubo ngamanye amaxesha zihlangabezana ne-activation engaguqukiyo. Ingahlulwahlulwa ibe ziintlobo ezimbini: enye yi-impact ye-automatic. Eyesibini yi-activation ehlala ixesha elide. Phantsi kwe-activation engaguqukiyo, impendulo yenkqubo nayo ayiguquki.

Isixhobo esinamandla sokuhlalutya ukungcangcazela okungazinzanga yindlela yokuphendula nge-impulse. Ichaza iimpawu ezitshintshayo zenkqubo ngempendulo yesikhashana yokufakwa kwe-unit impulse yenkqubo. I-unit impulse inokubonakaliswa njengomsebenzi we-delta. Kubunjineli, umsebenzi we-delta udla ngokuchazwa ngolu hlobo:

Apho u-0- emele inqaku elikwi-t-axis esondela ku-zero ukusuka ekhohlo; u-0 plus yinqaku eliya ku-0 ukusuka ekunene.

UMZOBO 6 ijika lesakhelo sesigaba

UMZOBO 7 naluphi na igalelo lunokuthathwa njengesimbuku sothotho lwezinto ezibangela impulse

Le nkqubo ihambelana nempendulo h(t) eveliswa yiyunithi ye-impulse kwi-t=0, ebizwa ngokuba ngumsebenzi wempendulo ye-impulse. Xa sicinga ukuba inkqubo ayimi phambi kwe-pulse, h(t)=0 ye-t<0. Ukwazi umsebenzi wempendulo ye-impulse yenkqubo, singafumana impendulo yenkqubo kuyo nayiphi na into engeniswayo x(t). Kweli nqanaba, ungacinga nge-x(t) njengesimbuku sothotho lwezinto ze-impulse (Umfanekiso 7). Impendulo yenkqubo yile:

Ngokusekelwe kumgaqo we-superposition, impendulo iyonke yenkqubo ehambelana no-x(t) yile:

Le integral ibizwa ngokuba yi-convolution integral okanye i-superposition integral.

Ukungcangcazela okuthe tye kwenkqubo yenkululeko enamanqanaba amaninzi

Ukungcangcazela kwenkqubo ethe tye ene-n≥2 degrees yenkululeko.

Umfanekiso 8 ubonisa iinkqubo ezimbini ezilula ezisebenzisanayo ezidityaniswe yintwasahlobo yokudibanisa. Ngenxa yokuba yinkqubo yenkululeko enamanqanaba amabini, kufuneka ii-coordinates ezimbini ezizimeleyo ukuze kuchongwe indawo yayo. Kukho ii-frequency ezimbini zendalo kule nkqubo:

I-frequency nganye ihambelana nendlela yokungcangcazela. Ii-harmonic oscillators zenza i-harmonic oscillations ye-frequency efanayo, zidlula ngendlela efanayo kwindawo yokulingana kwaye zifikelele kwindawo egqithisileyo. Kwi-vibration ephambili ehambelana ne-omega one, i-x1 ilingana ne-x2; Kwi-vibration ephambili ehambelana ne-omega omega two, i-omega omega one. Kwi-vibration ephambili, umlinganiselo wokuhamba we-mass nganye ugcina ulwalamano oluthile kwaye wenza i-mode ethile, ebizwa ngokuba yi-main mode okanye i-natural mode. I-orthogonality yobunzima kunye nokuqina ikhona phakathi kwe-main modes, ebonisa ukuzimela kwe-vibration nganye. I-frequency yendalo kunye ne-main mode zimele iimpawu ze-vibration ezingaphakathi ze-multi-degree of freedom system.

Inkqubo yeFIG. 8 enamanqanaba amaninzi enkululeko

Inkqubo ye-n degrees yenkululeko inee-n natural frequencies kunye nee-n main modes. Naluphi na ulungiselelo lwe-vibration lwenkqubo lungamelwa njengomdibaniso othe ngqo we-major modes. Ke ngoko, indlela ye-main mode superposition isetyenziswa kakhulu kuhlalutyo lwe-dynamic response lweenkqubo ze-multi-dof. Ngale ndlela, ukulinganisa kunye nohlalutyo lweempawu ze-natural vibration zenkqubo ziba linyathelo eliqhelekileyo kuyilo lwe-dynamic lwenkqubo.

Iimpawu ezitshintshayo zeenkqubo ze-multi-dof zinokuchazwa ngeempawu ze-frequency. Ekubeni kukho umsebenzi we-frequency characteristic phakathi kwe-input nganye kunye ne-output, kwakhiwa i-frequency characteristic matrix. I-amplitude-frequency characteristic curve yenkqubo ye-multi-freedom yahlukile kweyenkqubo ye-single-freedom.

I-elastomer iyangcangcazela

Inkqubo yenkululeko ye-multi-degree engentla yimodeli yomatshini ye-elastomer eqikelelweyo. I-elastomer inenani elingenamkhawulo leedigri zenkululeko. Kukho umahluko wobuninzi kodwa akukho mahluko ubalulekileyo phakathi kwezi zimbini. Nayiphi na i-elastomer inenani elingenamkhawulo lee-frequency zendalo kunye nenani elingenamkhawulo lee-modes ezihambelanayo, kwaye kukho i-orthogonality phakathi kweendlela zobunzima kunye nokuqina. Naluphi na ulwakhiwo lokungcangcazela kwe-elastomer lunokubonakaliswa njenge-linear superposition yee-modes eziphambili. Ke ngoko, kuhlalutyo lwempendulo eguquguqukayo ye-elastomer, indlela ye-superposition ye-main mode isasebenza (jonga i-linear vibration ye-elastomer).

Thatha ukungcangcazela komtya. Masithi umtya omncinci wobunzima m ngobude beyunithi, u-l omde, uxinezelekile kuzo zombini iziphelo, kwaye uxinzelelo ngu-T. Ngeli xesha, imvamisa yendalo yomtya imiselwa yile equation ilandelayo:

F =na/2l (n= 1,2,3…).

Iphi isantya sokusasazeka kwewave enqamlezileyo kwicala lomtya. Iifrequencies zendalo zeestrings ziphindaphinda zefrequencies ezisisiseko ngaphezulu kwe-2l. Olu phindo lwe-integer lukhokelela kwisakhiwo esimnandi se-harmonic. Ngokubanzi, akukho nxu lumano olunjalo lwe-integer multiple phakathi kweefrequencies zendalo ze-elastomer.

Iindlela ezintathu zokuqala zentambo exineneyo ziboniswe kuMfanekiso 9. Kukho amaqhuqhuva athile kwigophe lendlela ephambili. Kwintshukumo ephambili, amaqhuqhuva awashukumi. UMfanekiso 10 ubonisa iindlela ezahlukeneyo eziqhelekileyo zepleyiti ejikelezayo exhaswa ngokujikelezileyo enemigca ethile yamaqhuqhuva eyenziwe zizangqa kunye nobubanzi.

Indlela echanekileyo yokwenziwa kwengxaki yokungcangcazela kwe-elastomer ingagqitywa njengengxaki yexabiso lomda wee-equation ezingafaniyo. Nangona kunjalo, isisombululo esichanekileyo sinokufunyanwa kuphela kwezinye zezona meko zilula, ngoko ke kufuneka sisebenzise isisombululo esiqikelelweyo sengxaki yokungcangcazela kwe-elastomer enzima. Ingongoma yezisombululo ezahlukeneyo eziqikelelweyo kukutshintsha i-infinite ibe yi-finite, oko kukuthi, ukwahlulahlula inkqubo yenkululeko engenazo ii-limb ezininzi (inkqubo eqhubekayo) ibe yinkqubo yenkululeko ene-finite multi-degree (inkqubo edityanisiweyo). Kukho iintlobo ezimbini zeendlela zokuqhekeza ezisetyenziswa kakhulu kuhlalutyo lobunjineli: indlela ye-finite element kunye nendlela yokudibanisa i-modal.

UMZOBO 9 imo yomtya

UMZOBO 10 imo yepleyiti ejikelezayo

Indlela ye-Finite element sisakhiwo esidibeneyo esidibanisa isakhiwo esintsonkothileyo sibe linani elilinganiselweyo lezinto size sizidibanise kwinani elilinganiselweyo lee-nodes. Iyunithi nganye yi-elastomer; Ukuhanjiswa kwe-element kubonakaliswa ngumsebenzi wokudibanisa we-node displacement. Emva koko iiparameter zokusasazwa kwe-element nganye zixutywe kwi-node nganye kwifomathi ethile, kwaye imodeli ye-mechanical yenkqubo ehlukeneyo ifunyenwe.

Ukwenziwa kweModal kukuqhekeka kwesakhiwo esintsonkothileyo sibe zizakhiwo ezimbalwa ezilula. Ngokusekelwe ekuqondeni iimpawu zokungcangcazela kwesakhiwo ngasinye, isakhiwo esingaphantsi senziwa sibe yisakhiwo ngokubanzi ngokweemeko zokudibanisa kwi-interface, kwaye imo yokungcangcazela yesakhiwo ngokubanzi ifunyanwa ngokusebenzisa imo yokungcangcazela yesakhiwo ngasinye.

Ezi ndlela zimbini zahlukile kwaye zinxulumene, kwaye zingasetyenziswa njengesalathiso. Indlela yokwenziwa kwe-modal ingadibaniswa ngempumelelo nomlinganiselo wovavanyo ukwenza indlela yohlalutyo lwethiyori kunye novavanyo lokungcangcazela kweenkqubo ezinkulu.