Linear vibration ke eng?

Ho thothomela ka mola: elasticity ea likarolo tsamaisong e itšetlehile ka molao oa hook, 'me matla a damping a hlahisoang nakong ea motsamao a lekana le equation ea pele ea lebelo le akaretsang (nako e nkiloeng ho li-coordinates tse akaretsang).

mohopolo

Linear system hangata ke mohlala o sa bonahaleng oa ho sisinyeha ha sistimi ea 'nete.Sistimi ea thothomelo ea linear e sebelisa molao-motheo oa superposition, ke hore, haeba karabelo ea sistimi e le y1 tlasa ketso ea ho kenya x1, le y2 tlasa ts'ebetso ea ho kenya x2, joale karabo ea tsamaiso tlas'a ketso ea ho kenya x1 le x2 ke y1 + y2.

Motheong oa molao-motheo oa superposition, ho kenya letsoho ka mokhoa o feteletseng ho ka aroloa ka kakaretso ea letoto la litšusumetso tse sa feleng, 'me joale karabelo e felletseng ea sistimi e ka fumanoa.Kakaretso ea likarolo tsa harmonic tsa phihlello ea nako le nako e ka atolosoa ho ba a letoto la likarolo tsa harmonic ka Fourier fetola, 'me phello ea karolo e' ngoe le e 'ngoe ea harmonic tsamaisong e ka batlisisoa ka thoko.Ka hona, litšoaneleho tsa karabelo tsa litsamaiso tsa linear tse nang le li-parameter tse sa fetoheng li ka hlalosoa ka karabelo ea tšusumetso kapa karabelo ea khafetsa.

Impulse response e bolela karabelo ea tsamaiso ho unit impulse, e khethollang litšobotsi tsa karabelo tsa tsamaiso sebakeng sa nako.Karabelo ea khafetsa e bolela tšobotsi ea karabelo ea tsamaiso ho unit harmonic input.Mangolo pakeng tsa tse peli e ikemiselitse. ka Fourier transform.

karohano

Ho sisinyeha ha linear ho ka aroloa ka mokhoa oa ho sisinyeha o le mong oa sisteme ea degree-of-freedom le ho sisinyeha ha linear tsa sistimi ea degree-of-freedom.

(1) ho sisinyeha ha mola oa tsamaiso ea tekanyo e le 'ngoe ea tokoloho ke ho sisinyeha ha mola oo boemo ba oona bo ka khethoang ke coordinate e akaretsang.Ke thothomelo e bonolo ka ho fetisisa eo ho eona ho ka nkiloeng likhopolo le litšoaneleho tse ngata tsa motheo. ho thothomela ha harmonic, ho thothomela ka bolokolohi, ho thothomela le ho thothomela ka matla.

Ho sisinyeha ha bonolo ha ntho e itseng: ho sisinyeha ha ntho e haufi le sebaka sa eona sa tekano ho latela molao oa sinusoidal tlas'a ketso ea matla a khutlisetsang ho lekana le phalliso ea eona.

Damped vibration/ Ketso ya ho thothomela: ho thothomela hoo bophahamo ba yona bo dulang bo fokotswa ke boteng ba ho hohlana le dielectric resistance kapa tshebediso enngwe ya matla.

Ford vibration/ Ketso ya ho thothomela: ho thothomela hoa sisteme e tlasa pherekano e sa feleng.

(2) the linear vibration of multi-degree-of-freedom system is the vibrate of linear system with n≥2 degrees of freedom.A system of n degrees of freedom e na le maqhubu a tlhaho le mekhoa e meholo.Tlhahiso efe kapa efe ea ho sisinyeha. ea tsamaiso e ka emeloa e le linear motsoako oa mekhoa e meholo.Ka hona, mokhoa o ka sehloohong oa superposition mokhoa o sebelisoa haholo tlhahlobong e matla ea karabo ea litsamaiso tse ngata tsa dof.Ka tsela ena, tekanyo le tlhahlobo ea litšobotsi tsa tlhaho tsa ho thothomela tsamaiso e fetoha mohato o tloaelehileng oa moralo o matla oa tsamaiso.Litšobotsi tse matla tsa mekhoa e mengata ea li-multi-dof li ka boela tsa hlalosoa ka litšobotsi tsa maqhubu ke kamano e tobileng pakeng tsa tšobotsi ea maqhubu le mokhoa o ka sehloohong.Lekhalo la tšobotsi ea amplitude-frequency ea tsamaiso ea tokoloho e mengata e fapane le ea tsamaiso ea bolokolohi bo le bong.

Ho thothomela ha tekanyo ea tekanyo e le 'ngoe ea tsamaiso ea tokoloho

Ho sisinyeha ha mohala moo boemo ba sistimi bo ka khethoang ke khokahanyo e akaretsang. Ke tšisinyeho e bonolo ka ho fetesisa le ea mantlha eo ho eona ho ka nkiloeng mehopolo le litšoaneleho tse ngata tsa ho thothomela. E kenyelletsa ho thothomela ho bonolo, ho sisinyeha ha metsi le ho sisinyeha ho qobelloang. .

Harmonic vibration

Tlas'a ketso ea ho tsosolosa matla a lekanang le ho falla, ntho e khutlela ka mokhoa oa sinusoidal haufi le boemo ba eona ba ho lekana (FIG. 1) .X e emela ho falla 'me t e emela nako.Polelo ea lipalo ea vibrate ena ke:

(1)Moo A e leng boleng bo phahameng ba ho falla x, e bitsoang amplitude, 'me e emela matla a ho thothomela;Omega n ke amplitude Angle increment ea vibrate ka motsotsoana, e bitsoang maqhubu a angular, kapa maqhubu a selikalikoe; e bitsoa mokhahlelo oa pele. Ho ea ka f= n/2, palo ea oscillation ka motsotsoana e bitsoa frequency; Phapang ea sena, T=1/f, ke nako eo e e nkang ho sisinya potoloho e le 'ngoe,' me hoo ho bitsoa. nako.Amplitude A, frequency f (kapa angular frequency n), mohato oa pele, o tsejoang e le ho sisinyeha habonolo ha harmonic likarolo tse tharo.

FEIE.1 e bonolo ea harmonic vibration curve

Joalokaha ho bontšitsoe ho FIG.2, oscillator e bonolo ea harmonic e thehoa ke boima bo tebileng ba m bo kopantsoeng ke seliba se melang.

Ho satalla ha nako ea selemo ho kae

Empa omega n e khethoa feela ke litšoaneleho tsa tsamaiso ka boeona m le k, ntle le maemo a eketsehileng a pele, kahoo omega n e boetse e tsejoa e le maqhubu a tlhaho.

FEIE.2 tekanyo e le 'ngoe ea tsamaiso ea tokoloho

Bakeng sa oscillator e bonolo ea harmonic, kakaretso ea matla a eona a kinetic le matla a ka 'nang a e-ba teng a lula a le teng, ke hore, matla a feletseng a mochine oa tsamaiso a bolokiloe.

The damping vibration

Ho sisinyeha ha amplitude e lulang e fokotsoa ke ho ferekana le ho hanyetsa dielectric kapa tšebeliso e 'ngoe ea matla.Bakeng sa micro vibration, lebelo ka kakaretso ha le leholo haholo,' me khanyetso e mahareng e lekana le lebelo ho matla a pele, a ka ngoloang joalo ka c. The damping equation. Ka hona, equation ea vibrate ea tekanyo e le 'ngoe ea tokoloho e nang le linear damping e ka ngoloa tjena:

(2)Moo, m =c/2m e bitsoang damping parameter, le.Tharollo e akaretsang ea foromo (2) e ka ngoloa:

(3)Kamano ea lipalo lipakeng tsa omega n le PI e ka aroloa ka linyeoe tse tharo tse latelang:

N > (tabeng ea damping e nyane) karoloana e hlahisitsoeng ho sisinyeha ha vibration, equation ea vibration ke:

Bophahamo ba eona bo fokotseha ha nako e ntse e ea ho ea ka molao oa exponential o bontšitsoeng ho equation, joalo ka ha ho bonts'itsoe moleng o nang le matheba ho FIG.3. Ha re bua ka ho hlaka, ho thothomela hona ho etsahala ka nako e telele, empa khafetsa tlhōrō ea eona e ka hlalosoa e le:

E bitsoa sekhahla sa phokotso ea amplitude, moo nako ea ho sisinyeha.Logarithm ea tlhaho ea sekhahla sa phokotso ea amplitude e bitsoa sekhahla sa logarithm minus (amplitude). Ho hlakile hore, =, tabeng ena, e lekana le 2/1. Ka ho toba ka delta ea liteko tsa liteko le, ho sebelisa foromo e kaholimo ho ka baloa c.

Ka nako ena, tharollo ea equation (2) e ka ngoloa:

Hammoho le tataiso ea lebelo la pele, e ka aroloa ka linyeoe tse tharo tse sa sisinyeheng joalokaha ho bontšitsoe ho FIG.4.

N < (tabeng ea damping e kholo), tharollo ea equation (2) e bontšoa ka equation (3) . Hona joale, tsamaiso ha e sa sisinyeha.

Ho thothomela ka matla

Ho sisinyeha ha tsamaiso tlas'a ts'ebetso e sa khaotseng.Tlhahlobo ea ho sisinyeha e batlisisa karabelo ea tsamaiso ho excitation.Periodic excitation is a common excitation.Kaha nako le nako excitation can always decomposed into the sum of multiple harmonic excitation, ho ea ka superposition principle, only. karabelo ea tsamaiso ho e 'ngoe le e' ngoe ea ho tsosoa ha harmonic e ea hlokahala. Tlas'a ketso ea ho tsosoa ha harmonic, phapang e fapaneng ea ho sisinyeha ha tekanyo e le 'ngoe ea tokoloho e kolobisitsoeng tsamaiso e ka ngoloa:

Karabelo ke kakaretso ea likarolo tse peli.Karolo e 'ngoe ke karabelo ea ho thothomela ho nang le mongobo, e senyehang ka potlako ha nako e ntse e ea.Karabelo ea karolo e' ngoe ea ho thothomela e qobelloang e ka ngoloa:

FEIE.3 mothinya oa vibrate o kolobisitsoeng

FEIE.Li-curve tse 4 tsa maemo a mararo a pele a nang le mongobo o boima

Tlanya ho

H /F0= h (), ke karo-karolelano ea amplitude e tsitsitseng ea karabo ho amplitude e tsosang takatso, e khethollang likarolo tsa amplitude-frequency, kapa ho fumana mosebetsi; Bits bakeng sa karabo e tsitsitseng ea boemo le khothatso ea mohato, ho khetholla litšobotsi tsa maqhubu a karolo. maqhubu a excitation a bonts'oa ho FIG.5 le FIG.6.

Joalo ka ha ho ka bonoa ho curve ea amplitude-frequency curve (FIG. 5), ha e le damping e nyane, sekhahla sa amplitude-frequency curve se na le tlhoro e le 'ngoe. e bitsoang "resonant frequency of the system." Tabeng ea damping e nyane, maqhubu a resonance ha a fapane haholo le maqhubu a tlhaho.Ketsahalo ena e bitsoa resonance.At resonance, phaello ea tsamaiso ea eketseha, ke hore, ho sisinyeha ho qobelloang ho matla ka ho fetisisa. thothomelo.

FEIE.5 amplitude frequency curve

E ka bonoa ho tloha mothapong oa maqhubu (setšoantšo sa 6), ho sa tsotelehe boholo ba damping, ho omega zero phase phapano bits = PI / 2, tšobotsi ena e ka sebelisoa ka nepo ho lekanya molumo oa molumo.

Ntle le ho tsosoa ka mokhoa o tsitsitseng, ka linako tse ling litsamaiso li kopana le ts'oaetso e sa tsitsang.E ka aroloa ka mokhoa o batlang o aroloa ka mefuta e 'meli: e' ngoe ke tšusumetso ea tšohanyetso.Ea bobeli ke phello e tšoarellang ea boipelaetso.Ka tlas'a ts'oaetso e sa tsitsang, karabelo ea tsamaiso le eona ha e tsitsa.

Sesebelisoa se matla sa ho hlahloba ho sisinyeha ho sa tsitsang ke mokhoa oa ho arabela ka tšusumetso.E hlalosa litšoaneleho tse matla tsa tsamaiso ka karabelo ea nakoana ea tlhahiso ea unit impulse ea tsamaiso.The unit impulse e ka hlalosoa e le mosebetsi oa delta.Boenjiniere, delta. hangata mosebetsi o hlalosoa e le:

Moo 0- e emelang ntlha e ho t-axis e atamelang zero ho tloha ka ho le letšehali; 0 plus ke ntlha e eang ho 0 ho tloha ka ho le letona.

FEIE.6 phase frequency curve

FEIE.7 tlhahiso efe kapa efe e ka nkuoa e le kakaretso ea letoto la likarolo tsa tšusumetso

Sistimi e tsamaisana le karabelo h(t) e hlahisoang ke unit impulse ho t=0, e bitsoang impulse response function.Ho nka hore tsamaiso e eme pele ho pululo, h(t)=0 bakeng sa t<0.Ho tseba ts'ebetso ea karabelo ea ts'usumetso ea sistimi, re ka fumana karabelo ea sistimi ho tlhahiso efe kapa efe x(t).Nakong ena, o ka nahana ka x(t) e le kakaretso ea letoto la likarolo tsa tšusumetso (FIG. 7) .Karabelo ea sistimi ke:

Ho latela molao-motheo oa superposition, kakaretso ea karabelo ea sistimi e tsamaellanang le x(t) ke:

Kakaretso ena e bitsoa convolution integral kapa superposition integral.

Ho sisinyeha ha mola oa sistimi ea li-degree-of-freedom

Ho sisinyeha ha tsamaiso ea mela e nang le likhato tse n≥2 tsa tokoloho.

Setšoantšo sa 8 se bonts'a litsamaiso tse peli tse bonolo tsa resonant tse hokahantsoeng ke coupling spring.Hobane ke tsamaiso ea li-degree tse peli tsa tokoloho, ho hlokahala lihokahanyi tse peli tse ikemetseng ho fumana hore na boemo ba eona ke bofe.Ho na le maqhubu a mabeli a tlhaho tsamaisong ena:

Leqhubu le leng le le leng le tsamaisana le mokhoa oa ho sisinyeha.The harmonic oscillators phetha harmonic oscillations tsa maqhubu a tšoanang, synchronously feta ka boemo bo leka-lekaneng le synchronously fihla boemong bo feteletseng.Ka thothomelo e ka sehloohong e tsamaisanang le omega e le 'ngoe, x1 e lekana le x2; tšisinyeho e kholo e tsamaellanang le omega omega tse peli, omega omega e le 'ngoe.Tšinyehong e kholo, karo-karolelano ea phalliso ea boima bo bong le bo bong e boloka kamano e itseng 'me e theha mokhoa o itseng, o bitsoang mokhoa o ka sehloohong kapa mokhoa oa tlhaho.The orthogonality of mass and ho satalla ho teng har'a mekhoa e meholo, e bonahatsang boikemelo ba ho thothomela ka 'ngoe.Maqhubu a tlhaho le mokhoa o ka sehloohong li emela litšobotsi tsa tlhaho tsa ho thothomela tsa mekhoa e mengata ea tokoloho.

FEIE.8 tsamaiso e nang le likhato tse ngata tsa tokoloho

Sistimi ea n degree ea tokoloho e na le maqhubu a tlhaho le mekhoa e meholo.Tlhahiso efe kapa efe ea ho sisinyeha ea sistimi e ka emeloa e le motsoako oa linear oa mekhoa e meholo.Ka hona, mokhoa o ka sehloohong oa superposition o sebelisoa haholo tlhahlobong ea karabo e matla ea mefuta e mengata. -dof systems.Ka tsela ena, ho lekanya le ho hlahloba litšobotsi tsa tlhaho tsa ho sisinyeha ha tsamaiso e fetoha mohato o tloaelehileng oa moralo o matla oa tsamaiso.

Litšobotsi tse matla tsa litsamaiso tse ngata tsa li-multi-dof li ka boela tsa hlalosoa ka litšobotsi tsa makhetlo ho tsoa ho ea tsamaiso ea tokoloho e le 'ngoe.

Elastomer ea thothomela

The ka holimo multi - degree of freedom system is an approximate mechanical model of elastomer.Elastomer e na le palo e sa feleng ea likhato tsa tokoloho.Ho na le phapang ea bongata empa ha ho na phapang ea bohlokoa lipakeng tsa tse peli.Elastomer efe kapa efe e na le palo e sa feleng ea maqhubu a tlhaho le palo e sa lekanyetsoang ea mekhoa e lumellanang, 'me ho na le orthogonality pakeng tsa mekhoa ea boima le ho satalla.Tlhahiso efe kapa efe ea vibrational ea elastomer e ka boela ea emeloa e le sebopeho sa linear sa mekhoa e meholo.Ka hona, bakeng sa tlhahlobo e matla ea karabo ea elastomer, mokhoa oa superposition. Mokhoa oa mantlha o ntse o sebetsa (sheba ho sisinyeha ha melastomer).

Nka ho sisinyeha ha khoele. Ha re re khoele e tšesaane ea boima ba m ka bolelele ba yuniti, bolelele ba l, e tsitsitse lipheletsong ka bobeli, 'me tsitsipano ke T. Ka nako ena, maqhubu a tlhaho a khoele a khethoa ke tse latelang. equation:

F =na/2l (n= 1,2,3…).

Hokae, ke lebelo la ho phatlalatsoa ha leqhubu le pota-potileng ka lehlakoreng la khoele.Maqhubu a tlhaho a likhoele a etsahala hore e be li-multiples of the fundamental frequency over 2l.Palo ena e ngata e lebisang ho sebopeho se monate sa harmonic.Ka kakaretso, ha ho na likamano tse felletseng tse ngata har'a maqhubu a tlhaho a elastomer.

Mefuta e meraro ea pele ea khoele e tsitsitsoeng e bontšoa ho FIG.9. Ho na le li-node tse ling mokhoeng oa mokhoa o ka sehloohong.Ka har'a thothomelo e kholo, li-node ha li sisinyehe.FIG.10 e bonts'a mekhoa e 'maloa e tloaelehileng ea poleiti e chitja e tšehelitsoeng ka circumferentially e nang le mela e meng ea nodal e entsoeng ka lidikadikwe le bophara.

Ho hlophisoa hantle ha bothata ba ho sisinyeha ha elastomer ho ka phethoa e le bothata ba boleng ba moeli oa li-equations tse fapaneng. bothata ba ho sisinyeha.Moko oa litharollo tse fapaneng tse hakanyetsoang ke ho fetola se sa feleng ho isa pheletsong, ke hore, ho khetholla tsamaiso ea tokoloho ea maemo a mangata a se nang maoto le matsoho (continuous system) ho kena tsamaisong e lekanyelitsoeng ea mekhahlelo e mengata ea tokoloho (discrete system) .Ho na le mefuta e 'meli ea mekhoa ea discretization e sebelisoang haholo tlhahlobong ea boenjiniere: mokhoa oa finite element le mokhoa oa modal synthesis.

FEIE.9 mokhoa oa khoele

FEIE.10 mokhoa oa poleiti e chitja

Finite Element method is a composite structure that abstracts complex structure into a finite number of elements and connecting them ka palo e lekanyeditsweng ya node.Each unit is an elastomer;The displacement displacement of element e bonts'oa ka interpolation function of node displacement.Ebe mekhahlelo ea kabo ea ntho e 'ngoe le e' ngoe e tsepamisitsoe ho node e 'ngoe le e' ngoe ka mokhoa o itseng, 'me mohlala oa mochine oa tsamaiso ea discrete o fumanoa.

Modal synthesis ke ho bola ha sebopeho se rarahaneng hore e be likaroloana tse 'maloa tse bonolo.Motheong oa ho utloisisa litšobotsi tsa ho sisinyeha tsa kaho e' ngoe le e 'ngoe, karoloana e entsoe ka sebopeho se akaretsang ho latela maemo a khokahano holim'a sebopeho, le morphology ea thothomelo ea kakaretso. sebopeho se fumanwa ka ho sebedisa morphology ya thothomelo ya kaho e nngwe le e nngwe.

Mekhoa ena e 'meli e fapane ebile e amana,' me e ka sebelisoa e le litšupiso.Mokhoa oa modal synthesis o ka boela oa kopanngoa ka katleho le tekanyo ea liteko ho theha mokhoa oa ho hlahloba khopolo-taba le liteko bakeng sa ho sisinyeha ha litsamaiso tse kholo.