Dab tsi yog linear vibration?

Linear vibration: Lub elasticity ntawm cov khoom nyob rau hauv lub system yog raws li txoj cai ntawm hooke, thiab lub damping quab yuam generated thaum lub sij hawm lub zog yog proportional rau thawj kab zauv ntawm lub generalized tshaj tawm (lub sij hawm derivative ntawm lub generalized coordinates).

tswvyim

Cov kab hluav taws xob feem ntau yog cov qauv paub daws teeb meem ntawm kev co ntawm qhov system tiag tiag.Lub linear vibration system siv txoj cai superposition, uas yog, yog tias cov lus teb ntawm qhov system yog y1 nyob rau hauv qhov kev txiav txim ntawm cov tswv yim x1, thiab y2 nyob rau hauv qhov kev txiav txim ntawm cov tswv yim x2, Tom qab ntawd cov lus teb ntawm lub kaw lus nyob rau hauv qhov kev txiav txim ntawm cov tswv yim x1 thiab x2 yog y1 + y2.

Nyob rau hauv lub hauv paus ntawm superposition txoj cai, ib tug arbitrary tswv yim yuav decomposed mus rau hauv lub sum ntawm ib tug series ntawm infinitesimal impulses, thiab ces tag nrho cov lus teb ntawm lub system yuav tau txais.Cov sum ntawm cov harmonic Cheebtsam ntawm ib tug periodic excitation yuav nthuav mus rau hauv ib tug. series ntawm harmonic Cheebtsam los ntawm Fourier transform, thiab cov nyhuv ntawm txhua harmonic tivthaiv ntawm lub system yuav raug soj ntsuam cais. Yog li ntawd, cov lus teb cov yam ntxwv ntawm linear systems nrog tas li tsis tau piav qhia los ntawm impulse teb los yog zaus teb.

Impulse teb yog hais txog cov lus teb ntawm lub system mus rau lub zog impulse, uas characterizes cov lus teb cov yam ntxwv ntawm lub system nyob rau hauv lub sij hawm domain.Frequency teb yog hais txog cov lus teb yam ntxwv ntawm lub system mus rau lub unit harmonic input.Cov ntawv xov xwm ntawm ob yog txiav txim. los ntawm Fourier transform.

kev faib tawm

Linear vibration tuaj yeem muab faib ua linear vibration ntawm ib-degree-of-freedom system thiab linear vibration ntawm multi-degree-of-freedom system.

(1) linear vibration ntawm ib-degree-of-freedom system yog ib txoj kab kev co uas nws txoj haujlwm tuaj yeem txiav txim siab los ntawm kev sib koom ua ke. harmonic vibration, dawb vibration, attenuation vibration thiab yuam kev co.

Yooj yim harmonic vibration: lub reciprocating motion ntawm ib yam khoom nyob rau hauv ib ncig ntawm nws equilibrium txoj hauj lwm raws li txoj cai sinusoidal nyob rau hauv qhov kev txiav txim ntawm ib tug restoration quab yuam proportional rau nws tsiv.

Damped kev co: kev vibration uas nws lub amplitude yog txuas mus ntxiv los ntawm qhov muaj kev sib txhuam thiab dielectric tsis kam lossis lwm yam kev siv zog.

Forced vibration: kev co ntawm lub system nyob rau hauv tas li excitation.

(2) cov kab kev vibration ntawm ntau-degree-of-freedom system yog kev co ntawm linear system nrog n≥2 degrees ntawm kev ywj pheej.A system ntawm n degrees ntawm txoj kev ywj pheej muaj n ntuj frequencies thiab n lub ntsiab hom.Txhua yam kev co configuration ntawm qhov system tuaj yeem sawv cev raws li kev sib xyaw ua ke ntawm cov qauv loj.Yog li ntawd, txoj kev tseem ceeb ntawm superposition txoj kev yog dav siv nyob rau hauv dynamic teb tsom xam ntawm multi-dof systems.Qhov no, kev ntsuas thiab tsom xam ntawm lub ntuj vibration yam ntxwv ntawm lub Lub kaw lus yuav dhau los ua cov kauj ruam niaj hnub hauv kev tsim cov khoom siv dynamic.Cov yam ntxwv dynamic ntawm ntau lub tshuab dof kuj tuaj yeem piav qhia los ntawm cov yam ntxwv zaus.Vim tias muaj cov yam ntxwv zaus ntawm txhua qhov kev tawm tswv yim thiab cov zis, qhov zaus tus yam ntxwv matrix yog tsim. yog ib qho kev sib raug zoo ntawm tus yam ntxwv zaus thiab lub hom phiaj tseem ceeb.Qhov amplitude-frequency yam ntxwv nkhaus ntawm ntau txoj kev ywj pheej yog txawv ntawm qhov kev ywj pheej ib leeg.

Linear vibration ntawm ib theem ntawm kev ywj pheej system

Ib txoj kab kev vibration uas txoj haujlwm ntawm lub kaw lus tuaj yeem txiav txim siab los ntawm kev sib koom ua ke dav dav.Nws yog qhov yooj yim tshaj plaws thiab tseem ceeb tshaj kev co los ntawm ntau lub tswv yim thiab cov yam ntxwv ntawm kev co tuaj yeem muab tau. .

Harmonic kev co

Nyob rau hauv qhov kev txiav txim ntawm kev rov qab quab yuam proportional rau qhov kev hloov, cov khoom reciprocates nyob rau hauv ib tug sinusoidal yam nyob ze nws txoj hauj lwm equilibrium (Daim duab 1).X sawv cev rau qhov kev hloov thiab t sawv cev rau lub sij hawm.Qhov kev qhia lej ntawm qhov kev vibration no yog:

(1)Qhov twg A yog tus nqi siab tshaj plaws ntawm kev xaav x, uas yog hu ua amplitude, thiab sawv cev rau qhov kev siv ntawm kev co; Omega n yog lub amplitude Angle increment ntawm kev vibration ib ob, uas yog hu ua angular zaus, los yog lub voj voog zaus; Qhov no yog hu ua theem pib.Hais ntawm f = n / 2, tus naj npawb ntawm oscillations ib ob yog hu ua zaus; Qhov inverse ntawm no, T = 1 / f, yog lub sij hawm nws yuav siv sij hawm rau oscillate ib lub voj voog, thiab qhov ntawd yog hu ua lub sij hawm.Amplitude A, zaus f (los yog angular zaus n), thawj theem, hu ua yooj yim harmonic vibration peb lub ntsiab.

FIG.1 yooj yim harmonic vibration nkhaus

Raws li qhia hauv FIG.2, ib qho yooj yim harmonic oscillator yog tsim los ntawm cov concentrated mass m txuas los ntawm ib tug linear caij nplooj ntoos hlav.Thaum lub vibration displacement yog xam los ntawm lub equilibrium txoj hauj lwm, qhov sib npaug vibration yog:

Qhov twg yog qhov nruj ntawm lub caij nplooj ntoos hlav.Qhov kev daws teeb meem rau qhov sib npaug saum toj no yog (1).A thiab tuaj yeem txiav txim siab los ntawm qhov chaw pib x0 thiab pib tshaj tawm ntawm t = 0:

Tab sis omega n tsuas yog txiav txim siab los ntawm cov yam ntxwv ntawm lub cev nws tus kheej m thiab k, ywj siab ntawm cov txheej txheem pib ntxiv, yog li omega n tseem hu ua lub ntuj zaus.

FIG.2 ib leeg degree ntawm kev ywj pheej system

Rau ib qho yooj yim harmonic oscillator, qhov sib npaug ntawm nws lub zog kinetic thiab lub zog muaj peev xwm yog qhov tsis tu ncua, uas yog, tag nrho cov tshuab hluav taws xob ntawm lub cev tau txais kev cawmdim.Nyob rau hauv cov txheej txheem ntawm kev co, lub zog kinetic thiab lub zog muaj peev xwm hloov mus tas li rau ib leeg.

Lub damping vibration

Ib qho kev vibration uas nws lub amplitude yog txuas ntxiv attenuated los ntawm kev sib txhuam thiab dielectric tsis kam los yog lwm yam kev siv zog.Rau micro vibration, qhov tshaj tawm feem ntau tsis loj, thiab qhov nruab nrab kuj yog proportional rau qhov tshaj tawm mus rau thawj lub zog, uas tuaj yeem sau tau raws li c yog lub damping coefficient.Yog li ntawd, qhov sib npaug ntawm kev vibration ntawm ib qib ntawm kev ywj pheej nrog linear damping tuaj yeem sau tau raws li:

(2)Qhov twg, m = c / 2m yog hu ua damping parameter, thiab.Qhov kev daws teeb meem ntawm cov mis (2) tuaj yeem sau tau:

(3)Kev sib raug zoo ntawm tus lej ntawm omega n thiab PI tuaj yeem muab faib ua peb yam hauv qab no:

N > (nyob rau hauv cov ntaub ntawv ntawm me me damping) particle tsim attenuation kev co, qhov sib npaug vibration yog:

Nws amplitude txo nrog lub sij hawm raws li txoj cai exponential qhia nyob rau hauv kab zauv, raws li qhia nyob rau hauv cov kab dotted hauv FIG.3. Hais lus nruj me ntsis, qhov kev vibration no yog aperiodic, tab sis qhov zaus ntawm nws lub ncov tuaj yeem txhais tau tias:

Yog hu ua lub amplitude txo tus nqi, qhov twg yog lub sij hawm ntawm kev vibration.Lub ntuj logarithm ntawm lub amplitude txo tus nqi yog hu ua lub logarithm rho tawm (amplitude) rate.Obviously, =, nyob rau hauv cov ntaub ntawv no, yog sib npaug zos rau 2/1.Drectly los ntawm lub kev sim ntsuas delta thiab, siv cov qauv saum toj no tuaj yeem xam c.

Lub sijhawm no, kev daws teeb meem ntawm kev sib npaug (2) tuaj yeem sau tau:

Nrog rau kev taw qhia ntawm qhov pib tshaj tawm, nws tuaj yeem muab faib ua peb qhov tsis muaj kev vibration raws li qhia hauv FIG.4.

N < (nyob rau hauv cov ntaub ntawv ntawm damping loj), qhov kev daws teeb meem rau kab zauv (2) yog qhia nyob rau hauv kab zauv (3).Thaum lub sij hawm no, lub kaw lus tsis vibrating.

quab yuam kev vibration

Kev vibration ntawm lub kaw lus nyob rau hauv tas li excitation.Vibration tsom xam feem ntau tshawb xyuas cov lus teb ntawm lub system rau excitation.Periodic excitation yog ib txwm excitation.Vim lub sij hawm excitation yeej ib txwm decomposed rau hauv cov sum ntawm ntau harmonic excitation, raws li lub superposition txoj cai, tsuas Cov lus teb ntawm lub kaw lus rau txhua qhov kev sib haum xeeb yog xav tau.Raws li qhov kev txiav txim ntawm harmonic excitation, qhov sib txawv sib npaug ntawm kev txav ntawm ib qho kev ywj pheej damped system tuaj yeem sau tau:

Cov lus teb yog cov lej ntawm ob ntu.Ib feem yog cov lus teb ntawm damped vibration, uas decays sai nrog rau lub sij hawm.Cov lus teb ntawm lwm qhov ntawm yuam kev vibration tuaj yeem sau tau:

FIG.3 damped vibration nkhaus

FIG.4 curves ntawm peb qhov chaw pib nrog qhov tseem ceeb damping

Ntaus hauv

H / F0 = h (), yog qhov sib piv ntawm cov lus teb tsis tu ncua amplitude rau excitation amplitude, characterizing amplitude-frequency cov yam ntxwv, los yog nce muaj nuj nqi; Cov khoom rau lub xeev khov kho thiab kev txhawb siab ntawm theem, characterization ntawm theem zaus yam ntxwv. excitation zaus yog qhia nyob rau hauv FIG.5 thiab FIG.6.

Raws li tuaj yeem pom los ntawm amplitude-frequency nkhaus (Fig. 5), nyob rau hauv cov ntaub ntawv ntawm me me damping, lub amplitude-frequency nkhaus muaj ib tug ncov.Qhov me me lub damping, lub steeper lub ncov; Qhov zaus sib xws rau lub ncov yog hu ua lub resonant zaus ntawm lub system.Thaum me me damping, lub resonance zaus tsis ntau txawv ntawm lub ntuj zaus.Thaum lub excitation zaus nyob ze rau lub ntuj zaus, lub amplitude nce sharply.Qhov tshwm sim no hu ua resonance.Thaum resonance, qhov nce ntawm qhov system yog maximized, uas yog, qhov yuam kev vibration yog qhov khaus tshaj plaws.Yog li ntawd, feem ntau, ib txwm siv zog kom tsis txhob resonance, tshwj tsis yog qee cov twj paj nruag thiab cov cuab yeej siv resonance kom ua tiav qhov loj. kev vibration.

FIG.5 amplitude zaus nkhaus

Pom tau los ntawm theem zaus nkhaus (daim duab 6), tsis hais qhov loj ntawm damping, hauv omega xoom theem sib txawv me ntsis = PI / 2, tus yam ntxwv no tuaj yeem siv tau zoo hauv kev ntsuas resonance.

Ntxiv nrog rau qhov ua kom tsis tu ncua, cov tshuab qee zaum ntsib qhov tsis khov kho excitation.Nws tuaj yeem muab faib ua ob hom: ib qho yog qhov cuam tshuam tam sim ntawd.Qhov thib ob yog qhov cuam tshuam ntev ntawm arbitrariness.Unsteady excitation, cov lus teb ntawm lub kaw lus kuj tsis ruaj khov.

Cov cuab yeej muaj zog rau kev txheeb xyuas qhov kev vibration tsis ruaj khov yog txoj kev ua haujlwm impulse.It piav qhia cov yam ntxwv ntawm lub cev nrog cov lus teb hloov pauv ntawm lub zog impulse input ntawm lub system.The unit impulse tuaj yeem qhia tau tias yog delta muaj nuj nqi.In engineering, lub delta. kev ua haujlwm feem ntau txhais tau tias:

Qhov twg 0- sawv cev rau lub ntsiab lus ntawm t-axis uas zero ntawm sab laug; 0 ntxiv yog qhov taw tes uas mus rau 0 ntawm sab xis.

FIG.6 theem zaus nkhaus

FIG.7 txhua qhov kev tawm tswv yim tuaj yeem suav tau tias yog cov lej ntawm cov ntsiab lus ntawm impulse

Lub kaw lus sib raug rau cov lus teb h(t) generated los ntawm chav tsev impulse ntawm t = 0, uas yog hu ua lub impulse teb muaj nuj nqi.Puas hais tias lub system nyob ruaj ruaj ua ntej lub mem tes, h(t) = 0 rau t<0.Knowing lub impulse teb muaj nuj nqi ntawm lub system, peb yuav nrhiav tau cov lus teb ntawm lub system rau tej input x(t).Thaum lub sij hawm no, koj yuav xav txog x(t) raws li cov sum ntawm ib tug series ntawm impulse ntsiab (Daim duab. 7) .Cov lus teb ntawm qhov system yog:

Raws li txoj cai superposition, tag nrho cov lus teb ntawm lub kaw lus sib raug rau x (t) yog:

Qhov no yog hu ua convolution integral los yog superposition integral.

Linear vibration ntawm multi-degree-of-freedom system

Kev vibration ntawm ib txoj kab nruab nrog n≥2 degrees ntawm kev ywj pheej.

Daim duab 8 qhia txog ob qhov yooj yim resonant subsystems txuas nrog lub caij nplooj ntoos hlav sib txuas.Vim nws yog ob-degree-of-freedom system, yuav tsum muaj ob txoj haujlwm ywj pheej los txiav txim siab nws txoj haujlwm.Nws muaj ob qhov ntau zaus hauv qhov system no:

Txhua zaus sib raug mus rau hom kev vibration.Lub harmonic oscillators nqa tawm harmonic oscillations ntawm tib zaus, synchronously dhau los ntawm qhov sib npaug ntawm txoj hauj lwm thiab synchronously ncav cuag qhov chaw huab.In lub ntsiab kev co sib xws rau omega ib, x1 yog sib npaug rau x2; Lub ntsiab kev vibration sib xws rau omega omega ob, omega omega one.In lub ntsiab kev co, qhov sib txawv txav ntawm txhua pawg khaws ib qho kev sib raug zoo thiab tsim ib hom hom, uas yog hu ua lub ntsiab hom lossis hom ntuj.Lub orthogonality ntawm huab hwm coj thiab Kev nruj muaj nyob ntawm cov hom tseem ceeb, uas qhia txog kev ywj pheej ntawm txhua qhov kev vibration.Lub ntuj zaus thiab lub ntsiab hom sawv cev rau cov yam ntxwv ntawm kev vibration ntawm ntau qib kev ywj pheej.

FIG.8 system nrog ntau qib ntawm kev ywj pheej

Lub kaw lus ntawm n degrees ntawm kev ywj pheej muaj n ntuj frequencies thiab n lub ntsiab hom.Txhua yam kev co configuration ntawm lub system yuav sawv cev raws li ib tug linear ua ke ntawm cov loj hom.Yog li ntawd, lub ntsiab hom superposition txoj kev yog dav siv nyob rau hauv dynamic teb tsom xam ntawm ntau -dof systems.Nyob rau hauv txoj kev no, kev ntsuas thiab kev tsom xam ntawm cov yam ntxwv ntawm lub cev kev co ntawm lub cev ua ib kauj ruam niaj hnub nyob rau hauv lub dynamic tsim ntawm lub system.

Cov yam ntxwv dynamic ntawm multi-dof systems kuj tuaj yeem piav qhia los ntawm cov yam ntxwv zaus.Vim tias muaj cov yam ntxwv ua haujlwm zaus ntawm txhua qhov kev tawm tswv yim thiab cov khoom tawm, ib qho kev sib txawv ntawm cov qauv matrix yog tsim.Qhov amplitude-frequency yam ntxwv nkhaus ntawm ntau txoj kev ywj pheej yog txawv. los ntawm txoj kev ywj pheej ib leeg.

Lub elastomer vibrates

Qhov saum toj no multi-degree ntawm txoj kev ywj pheej yog ib qho piv txwv ntawm cov qauv ntawm elastomer.Ib elastomer muaj ib tug infinite tus naj npawb ntawm degrees ntawm kev ywj pheej.Nws muaj ib tug ntau yam sib txawv tab sis tsis muaj qhov tseem ceeb sib txawv ntawm ob.Ib yam elastomer muaj ib tug infinite tooj ntawm ntuj frequencies thiab tus naj npawb ntawm cov qauv sib thooj, thiab muaj qhov sib txawv ntawm cov qauv ntawm qhov loj thiab qhov tawv nqaij.Txhua yam kev vibrational ntawm lub elastomer kuj tuaj yeem sawv cev raws li ib txoj kab superposition ntawm cov hom loj.Yog li ntawd, rau dynamic teb tsom xam ntawm elastomer, txoj kev superposition ntawm lub ntsiab hom tseem siv tau (saib linear vibration ntawm elastomer).

Siv lub vibration ntawm ib txoj hlua. Cia peb hais tias ib txoj hlua nyias ntawm qhov loj m ib chav tsev ntev, ntev l, yog tensioned ntawm ob hnub, thiab qhov nro yog T.Tam sim no, lub ntuj zaus ntawm txoj hlua yog txiav txim los ntawm cov hauv qab no. kev sib npaug:

F = na/2l (n= 1,2,3…).

Qhov twg, yog qhov kev tshaj tawm tshaj tawm ntawm cov nthwv dej ntawm kev sib hloov raws li cov kev taw qhia ntawm txoj hlua.Lub ntuj frequencies ntawm cov hlua tshwm sim los ua ntau yam ntawm lub hauv paus zaus tshaj 2l.Qhov no integer multiplicity ua rau ib tug qab ntxiag harmonic qauv. Feem ntau, tsis muaj xws li integer ntau kev sib raug zoo ntawm lub ntuj frequencies ntawm elastomer.

Thawj peb hom ntawm txoj hlua tensioned yog qhia hauv FIG.9. Muaj qee qhov ntawm lub ntsiab hom nkhaus.Nyob rau hauv lub ntsiab kev co, cov nodes tsis vibrate.FIG.10 qhia tau hais tias muaj ntau hom qauv ntawm lub voj voog ncig lub voj voog nrog qee cov kab nodal uas muaj cov voj voog thiab txoj kab uas hla.

Qhov tseeb formulation ntawm qhov teeb meem elastomer vibration yuav xaus tau raws li tus ciam teb tus nqi teeb meem ntawm ib feem ntawm qhov sib npaug sib npaug.Txawm li cas los xij, cov kev daws teeb meem tsuas tuaj yeem pom nyob rau hauv qee qhov ntawm qhov yooj yim tshaj plaws, yog li peb yuav tsum tau mus rau qhov kwv yees kev daws teeb meem rau lub complex elastomer. Cov teeb meem kev vibration.Lub ntsiab lus ntawm ntau yam kev daws teeb meem yog los hloov qhov infinite mus rau qhov finite, uas yog, discrete lub limb-tsawg multi-degree ntawm kev ywj pheej system (txuas ntxiv system) mus rau hauv ib tug finite multi-degree ntawm kev ywj pheej system (discrete system) .Nws muaj ob hom kev discretization siv dav hauv engineering tsom xam: finite element method thiab modal synthesis method.

FIG.9 mode txoj hlua

FIG.10 hom ntawm lub voj voog phaj

Finite element method yog cov qauv sib xyaw uas paub daws teeb meem cov qauv rau hauv ib qho kev txiav txim siab ntawm cov ntsiab lus thiab txuas rau lawv ntawm tus lej ntawm cov nodes.Txhua chav tsev yog ib qho elastomer; Qhov kev faib tawm ntawm cov khoom yog qhia los ntawm kev cuam tshuam ntawm cov node hloov chaw. kev faib tsis tau ntawm txhua lub ntsiab lus yog concentrated rau txhua qhov ntawm ib hom ntawv, thiab tau txais cov qauv kev sib cais ntawm qhov system.

Modal synthesis yog decomposition ntawm ib tug complex qauv rau hauv ob peb yooj yim substructures.Rau lub hauv paus ntawm kev nkag siab txog kev co yam ntxwv ntawm txhua lub substructure, lub substructure yog synthesized rau hauv ib tug dav qauv raws li cov kev sib koom tes nyob rau hauv lub interface, thiab kev co morphology ntawm lub general. qauv yog tau los ntawm kev siv lub vibration morphology ntawm txhua substructure.

Ob txoj kev sib txawv thiab muaj feem cuam tshuam, thiab tuaj yeem siv tau raws li kev siv.Cov qauv kev sib xyaw ua ke kuj tseem tuaj yeem ua ke nrog cov kev sim ntsuas los tsim ib txoj kev tshawb fawb thiab kev sim rau kev vibration ntawm cov tshuab loj.