Kev co kab ncaj: qhov elasticity ntawm cov khoom hauv lub cev yog raws li txoj cai ntawm hooke, thiab lub zog damping tsim thaum lub sijhawm txav mus los yog sib npaug rau thawj kab zauv ntawm qhov ceev dav dav (lub sijhawm derivative ntawm cov kev sib koom ua ke dav dav).
lub tswv yim
Lub kaw lus linear feem ntau yog ib qho qauv abstract ntawm kev co ntawm lub kaw lus tiag tiag. Lub kaw lus linear co siv lub hauv paus ntsiab lus superposition, uas yog, yog tias qhov teb ntawm lub kaw lus yog y1 nyob rau hauv qhov kev ua ntawm cov tswv yim x1, thiab y2 nyob rau hauv qhov kev ua ntawm cov tswv yim x2, ces qhov teb ntawm lub kaw lus nyob rau hauv qhov kev ua ntawm cov tswv yim x1 thiab x2 yog y1 + y2.
Raws li txoj cai superposition, ib qho kev nkag mus tsis raws cai tuaj yeem raug decomposed rau hauv qhov sib npaug ntawm cov impulses infinitesimal, thiab tom qab ntawd tag nrho cov lus teb ntawm lub kaw lus tuaj yeem tau txais. Qhov sib npaug ntawm cov khoom sib haum xeeb ntawm ib qho kev txhawb siab periodic tuaj yeem nthuav dav mus rau hauv ib qho ntawm cov khoom sib haum xeeb los ntawm Fourier transform, thiab cov nyhuv ntawm txhua qhov harmonic ntawm lub kaw lus tuaj yeem tshawb xyuas sib cais. Yog li ntawd, cov yam ntxwv teb ntawm cov kab ke linear nrog cov kev txwv tsis tu ncua tuaj yeem piav qhia los ntawm kev teb impulse lossis kev teb zaus.
Kev teb impulse yog hais txog kev teb ntawm lub cev rau lub unit impulse, uas piav qhia txog cov yam ntxwv teb ntawm lub cev hauv lub sijhawm. Kev teb zaus yog hais txog cov yam ntxwv teb ntawm lub cev rau lub unit harmonic input. Kev sib raug zoo ntawm ob qho yog txiav txim siab los ntawm Fourier transform.
kev faib tawm
Kev co linear tuaj yeem muab faib ua kev co linear ntawm ib-degree-of-freedom system thiab kev co linear ntawm ntau-degree-of-freedom system.
(1) kev co linear ntawm ib qho degree-of-free system yog kev co linear uas nws qhov chaw tuaj yeem txiav txim siab los ntawm kev sib koom ua ke dav dav. Nws yog qhov kev co yooj yim tshaj plaws uas ntau lub tswv yim yooj yim thiab cov yam ntxwv ntawm kev co tuaj yeem tau los ntawm. Nws suav nrog kev co harmonic yooj yim, kev co dawb, kev co attenuation thiab kev co yuam.
Kev co harmonic yooj yim: kev txav rov qab ntawm ib yam khoom nyob ze ntawm nws qhov chaw sib npaug raws li txoj cai sinusoidal nyob rau hauv qhov kev ua ntawm lub zog rov qab los sib piv rau nws qhov kev hloov chaw.
Kev co damped: kev co uas nws qhov amplitude pheej attenuated los ntawm kev muaj kev sib txhuam thiab dielectric tsis kam lossis lwm yam kev siv zog.
Kev co yuam: kev co ntawm lub kaw lus nyob rau hauv kev txhawb nqa tas li.
(2) qhov kev co linear ntawm lub multi-degree-of-freedom system yog qhov kev co ntawm lub linear system nrog n≥2 degrees ntawm kev ywj pheej. Ib lub system ntawm n degrees ntawm kev ywj pheej muaj n zaus ntuj thiab n hom tseem ceeb. Txhua qhov kev teeb tsa kev co ntawm lub system tuaj yeem sawv cev ua ke linear ntawm cov hom tseem ceeb. Yog li ntawd, txoj kev superposition hom tseem ceeb yog siv dav hauv kev tshuaj xyuas cov lus teb dynamic ntawm ntau-dof systems. Hauv txoj kev no, kev ntsuas thiab kev tshuaj xyuas ntawm cov yam ntxwv kev co ntuj ntawm lub system dhau los ua ib kauj ruam niaj hnub hauv kev tsim dynamic ntawm lub system. Cov yam ntxwv dynamic ntawm ntau-dof systems kuj tuaj yeem piav qhia los ntawm cov yam ntxwv zaus. Vim tias muaj kev ua haujlwm zaus ntawm txhua qhov nkag thiab tso zis, ib qho matrix yam ntxwv zaus yog tsim. Muaj kev sib raug zoo ntawm cov yam ntxwv zaus thiab hom tseem ceeb. Lub amplitude-zaus yam ntxwv nkhaus ntawm ntau-kev ywj pheej system yog txawv ntawm qhov ntawm ib-kev ywj pheej system.
Kev co linear ntawm ib qho degree ntawm kev ywj pheej system
Kev co linear uas qhov chaw ntawm ib lub cev 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 plaws kev co uas ntau lub tswv yim yooj yim thiab cov yam ntxwv ntawm kev co tuaj yeem tau los ntawm. Nws suav nrog kev co harmonic yooj yim, kev co damped thiab kev co yuam.
Kev co harmonic
Nyob rau hauv qhov kev ua ntawm kev rov qab kho lub zog proportional rau qhov kev hloov chaw, cov khoom rov qab ua ke hauv txoj kev sinusoidal ze ntawm nws qhov chaw sib npaug (DAIM DUAB 1). X sawv cev rau qhov kev hloov chaw thiab t sawv cev rau lub sijhawm. Cov lus qhia lej ntawm qhov kev co no yog:
(1)Qhov twg A yog tus nqi siab tshaj plaws ntawm kev hloov chaw x, uas hu ua qhov amplitude, thiab sawv cev rau qhov muaj zog ntawm kev co; Omega n yog qhov amplitude Lub kaum sab xis nce ntawm kev co ib ob, uas hu ua lub zaus kaum sab xis, lossis lub zaus voj voog; Qhov no hu ua theem pib. Hauv cov nqe lus ntawm f = n / 2, tus lej ntawm kev co ib ob hu ua zaus; Qhov rov qab ntawm qhov no, T = 1 / f, yog lub sijhawm nws siv los co ib lub voj voog, thiab qhov ntawd hu ua lub sijhawm. Amplitude A, zaus f (lossis zaus kaum sab xis n), theem pib, hu ua kev co harmonic yooj yim peb lub ntsiab lus.
DUAB 1 kab nkhaus kev co harmonic yooj yim
Raws li pom hauv daim duab 2, ib qho harmonic oscillator yooj yim yog tsim los ntawm qhov hnyav concentrated m txuas nrog los ntawm lub caij nplooj ntoos hlav linear. Thaum qhov kev hloov pauv ntawm kev co raug suav los ntawm qhov chaw sib npaug, qhov sib npaug ntawm kev co yog:
Qhov twg yog qhov tawv ntawm lub caij nplooj ntoos hlav. Cov lus teb dav dav rau qhov sib npaug saum toj no yog (1).A thiab tuaj yeem txiav txim siab los ntawm qhov chaw pib x0 thiab qhov ceev pib ntawm t = 0:
Tab sis omega n tsuas yog txiav txim siab los ntawm cov yam ntxwv ntawm lub kaw lus nws tus kheej m thiab k, tsis hais txog cov xwm txheej pib ntxiv, yog li omega n kuj tseem hu ua zaus ntuj.
DUAB 2 ib theem ntawm kev ywj pheej system
Rau ib qho oscillator harmonic yooj yim, qhov sib sau ua ke ntawm nws lub zog kinetic thiab lub zog muaj peev xwm yog qhov tsis hloov pauv, uas yog, tag nrho lub zog kho tshuab ntawm lub kaw lus raug khaws cia. Hauv cov txheej txheem ntawm kev co, lub zog kinetic thiab lub zog muaj peev xwm raug hloov pauv tas li rau ib leeg.
Kev co ntawm lub zog damping
Ib qho kev co uas nws qhov amplitude raug attenuated tas li los ntawm kev sib txhuam thiab dielectric tsis kam lossis lwm yam kev siv zog. Rau micro vibration, qhov ceev feem ntau tsis loj heev, thiab qhov nruab nrab tsis kam yog proportional rau qhov ceev rau thawj lub zog, uas tuaj yeem sau ua c yog tus coefficient damping. Yog li ntawd, qhov kev co ntawm ib degree ntawm kev ywj pheej nrog linear damping tuaj yeem sau ua:
(2)Qhov twg, m = c/2m hu ua qhov damping parameter, thiab. Cov lus teb dav dav ntawm cov mis (2) tuaj yeem sau ua:
(3)Kev sib raug zoo ntawm tus lej ntawm omega n thiab PI tuaj yeem muab faib ua peb qhov xwm txheej hauv qab no:
N > (yog tias muaj cov khoom me me uas ua rau muaj kev co me me) thiab cov khoom me me uas tsim tawm los ntawm kev co, cov qauv kev co yog:
Nws qhov amplitude txo qis raws li lub sijhawm raws li txoj cai exponential qhia hauv equation, raws li qhia hauv kab dotted hauv daim duab 3. Hais lus nruj me ntsis, qhov kev co no yog aperiodic, tab sis qhov zaus ntawm nws qhov siab tshaj plaws tuaj yeem txhais tau tias:
Hu ua tus nqi txo qhov loj, qhov twg yog lub sijhawm ntawm kev co. Lub logarithm ntuj ntawm tus nqi txo qhov loj hu ua logarithm rho tawm (amplitude) tus nqi. Pom tseeb, =, hauv qhov no, yog sib npaug rau 2/1. Ncaj qha los ntawm kev sim delta thiab, siv cov mis saum toj no tuaj yeem suav c.
Lub sijhawm no, qhov kev daws teeb meem ntawm kab zauv (2) tuaj yeem sau ua:
Nrog rau qhov kev taw qhia ntawm qhov ceev pib, nws tuaj yeem muab faib ua peb qhov xwm txheej tsis muaj kev co raws li qhia hauv daim duab 4.
N < (yog tias muaj kev cuam tshuam loj), cov lus teb rau kab zauv (2) tau qhia hauv kab zauv (3). Thaum lub sijhawm no, lub kaw lus tsis co lawm.
Kev co uas yuam
Kev co ntawm ib lub cev nyob rau hauv kev txhawb nqa tas li. Kev tshuaj xyuas kev co feem ntau tshawb nrhiav qhov teb ntawm lub cev rau kev txhawb nqa. Kev txhawb nqa tas li yog ib qho kev txhawb nqa tas li. Vim tias kev txhawb nqa tas li tuaj yeem raug rhuav tshem mus rau hauv qhov sib npaug ntawm ntau qhov kev txhawb nqa harmonic, raws li txoj cai superposition, tsuas yog qhov teb ntawm lub cev rau txhua qhov kev txhawb nqa harmonic xwb. Nyob rau hauv qhov kev ua ntawm kev txhawb nqa harmonic, qhov sib txawv ntawm kev txav mus los ntawm ib qib ntawm kev ywj pheej damped system tuaj yeem sau tau:
Cov lus teb yog qhov sib sau ua ke ntawm ob feem. Ib feem yog cov lus teb ntawm kev co damped, uas lwj sai sai nrog lub sijhawm. Cov lus teb ntawm lwm feem ntawm kev co yuam tuaj yeem sau ua:
DUAB 3 kab nkhaus kev co uas raug damped
DUAB 4 cov kab ntawm peb qhov xwm txheej pib nrog kev cuam tshuam tseem ceeb
Ntaus rau hauv
H /F0 = h (), yog qhov sib piv ntawm qhov teb ruaj khov rau qhov siab ntawm qhov excitation, piav qhia txog qhov siab ntawm qhov zaus, lossis qhov ua haujlwm nce; Cov khoom rau qhov teb ruaj khov thiab kev txhawb siab ntawm theem, piav qhia txog qhov zaus ntawm theem. Kev sib raug zoo ntawm lawv thiab qhov zaus ntawm excitation tau qhia hauv daim duab 5 thiab daim duab 6.
Raws li pom tau los ntawm qhov nkhaus amplitude-frequency (DAIM DUAB 5), yog tias muaj damping me me, qhov nkhaus amplitude-frequency muaj ib lub ncov. Qhov damping me dua, qhov ncov ntxhab dua; Qhov zaus sib xws rau lub ncov hu ua resonant frequency ntawm lub system. Yog tias muaj damping me me, qhov resonance frequency tsis txawv ntawm qhov natural frequency. Thaum qhov excitation frequency ze rau qhov natural frequency, qhov amplitude nce siab heev. Qhov xwm txheej no hu ua resonance. Thaum resonance, qhov nce ntawm lub system yog maximized, uas yog, qhov kev co yuam yog qhov muaj zog tshaj plaws. Yog li ntawd, feem ntau, ib txwm siv zog kom tsis txhob muaj resonance, tshwj tsis yog qee cov cuab yeej thiab khoom siv siv resonance kom ua tiav kev co loj.
DUAB 5 kab nkhaus zaus ntawm qhov siab
Pom tau los ntawm cov kab zaus theem (daim duab 6), tsis hais qhov loj ntawm kev damping, hauv omega xoom theem sib txawv bits = PI / 2, qhov cwj pwm no tuaj yeem siv tau zoo hauv kev ntsuas resonance.
Ntxiv rau qhov kev txhawb zog ruaj khov, qee zaum cov kab ke ntsib kev txhawb zog tsis ruaj khov. Nws tuaj yeem faib ua ob hom: ib qho yog qhov cuam tshuam tam sim ntawd. Qhov thib ob yog qhov cuam tshuam ntev ntawm kev tsis sib xws. Nyob rau hauv qhov kev txhawb zog tsis ruaj khov, cov lus teb ntawm lub kaw lus kuj tsis ruaj khov.
Ib qho cuab yeej muaj zog rau kev tshuaj xyuas qhov kev co tsis ruaj khov yog txoj kev teb impulse. Nws piav qhia txog cov yam ntxwv dynamic ntawm lub cev nrog rau qhov kev teb transient ntawm lub unit impulse input ntawm lub cev. Lub unit impulse tuaj yeem qhia ua delta function. Hauv kev tsim kho, delta function feem ntau yog txhais tias:
Qhov twg 0- sawv cev rau qhov taw tes ntawm t-axis uas mus txog xoom los ntawm sab laug; 0 ntxiv rau yog qhov taw tes uas mus rau 0 los ntawm sab xis.
DUAB 6. Kab nkhaus zaus theem
DUAB 7 txhua qhov kev nkag tuaj yeem suav tias yog qhov sib sau ua ke ntawm cov ntsiab lus ntawm lub zog
Lub kaw lus sib raug rau cov lus teb h (t) uas tsim los ntawm lub unit impulse ntawm t = 0, uas hu ua impulse response function. Xav tias lub kaw lus nyob ruaj khov ua ntej lub pulse, h (t) = 0 rau t < 0. Paub txog impulse response function ntawm lub kaw lus, peb tuaj yeem nrhiav tau cov lus teb ntawm lub kaw lus rau txhua qhov input x (t). Ntawm lub sijhawm no, koj tuaj yeem xav txog x (t) ua tus lej ntawm cov ntsiab lus impulse (DAIM DUAB 7). Cov lus teb ntawm lub kaw lus yog:
Raws li txoj cai superposition, tag nrho cov lus teb ntawm lub kaw lus sib raug rau x (t) yog:
Qhov kev sib xyaw ua ke no hu ua convolution integral lossis superposition integral.
Kev co linear ntawm ntau qib kev ywj pheej system
Kev co ntawm lub kaw lus linear nrog n≥2 degrees ntawm kev ywj pheej.
Daim Duab 8 qhia ob lub resonant subsystems yooj yim txuas nrog los ntawm lub caij nplooj ntoos hlav coupling. Vim tias nws yog ib qho system ob-degree-of-freedom, ob lub coordinates ywj pheej yog qhov xav tau los txiav txim siab nws qhov chaw. Muaj ob lub zaus ntuj hauv lub system no:
Txhua zaus sib raug rau ib hom kev co. Cov oscillators harmonic ua cov harmonic oscillations ntawm tib zaus, synchronously dhau los ntawm qhov chaw sib npaug thiab synchronously mus txog qhov chaw kawg. Hauv lub ntsiab vibration sib raug rau omega ib, x1 sib npaug rau x2; Hauv lub ntsiab vibration sib raug rau omega omega ob, omega omega ib. Hauv lub ntsiab vibration, qhov piv ntawm kev hloov chaw ntawm txhua pawg khaws ib qho kev sib raug zoo thiab tsim ib hom, uas hu ua hom tseem ceeb lossis hom ntuj. Lub orthogonality ntawm pawg thiab stiffness muaj nyob ntawm cov hom tseem ceeb, uas qhia txog kev ywj pheej ntawm txhua qhov kev co. Lub zaus ntuj thiab hom tseem ceeb sawv cev rau cov yam ntxwv ntawm kev co ntawm ntau qib ntawm kev ywj pheej system.
DUAB 8 lub kaw lus nrog ntau qib kev ywj pheej
Ib lub kaw lus uas muaj n degrees ntawm kev ywj pheej muaj n zaus ntuj thiab n hom tseem ceeb. Txhua qhov kev teeb tsa kev co ntawm lub kaw lus tuaj yeem sawv cev ua kev sib xyaw ua ke ntawm cov hom tseem ceeb. Yog li ntawd, txoj kev sib tshooj hom tseem ceeb yog siv dav hauv kev tshuaj xyuas cov lus teb dynamic ntawm ntau-dof systems. Hauv txoj kev no, kev ntsuas thiab kev tshuaj xyuas cov yam ntxwv ntawm kev co ntuj ntawm lub kaw lus dhau los ua ib kauj ruam niaj hnub hauv kev tsim qauv dynamic ntawm lub kaw lus.
Cov yam ntxwv dynamic ntawm ntau-dof systems kuj tuaj yeem piav qhia los ntawm cov yam ntxwv zaus. Vim tias muaj ib qho kev ua haujlwm ntawm cov yam ntxwv zaus ntawm txhua qhov input thiab output, ib qho matrix yam ntxwv zaus tau tsim. Lub amplitude-zaus yam ntxwv ntawm ntau-kev ywj pheej system txawv ntawm qhov ntawm ib-kev ywj pheej system.
Lub elastomer vibrates
Lub tshuab ntau theem ntawm kev ywj pheej saum toj no yog ib qho qauv siv tshuab kwv yees ntawm elastomer. Ib qho elastomer muaj ntau qib ntawm kev ywj pheej. Muaj qhov sib txawv ntawm qhov ntau thiab tsawg tab sis tsis muaj qhov sib txawv tseem ceeb ntawm ob qho. Txhua lub elastomer muaj ntau zaus ntuj thiab ntau hom sib xws, thiab muaj orthogonality ntawm hom ntawm pawg thiab kev tawv. Txhua qhov kev teeb tsa vibrational ntawm elastomer kuj tuaj yeem sawv cev ua ib qho linear superposition ntawm cov hom loj. Yog li ntawd, rau kev tshuaj xyuas dynamic teb ntawm elastomer, txoj kev superposition ntawm hom tseem ceeb tseem siv tau (saib linear vibration ntawm elastomer).
Xav txog qhov kev co ntawm ib txoj hlua. Cia peb hais tias ib txoj hlua nyias nyias uas muaj qhov hnyav m rau txhua qhov ntev, ntev l, raug rub ntawm ob qho kawg, thiab qhov kev rub yog T. Lub sijhawm no, qhov zaus ntuj ntawm txoj hlua yog txiav txim siab los ntawm cov qauv hauv qab no:
F = na/2l (n= 1,2,3…).
Qhov twg, yog qhov ceev ntawm nthwv dej transverse raws txoj hlua. Cov zaus ntuj ntawm cov hlua tshwm sim yog cov lej ntawm cov zaus tseem ceeb tshaj 2l. Qhov integer multiplicity no ua rau muaj cov qauv harmonic zoo. Feem ntau, tsis muaj qhov integer multiple relation ntawm cov zaus ntuj ntawm elastomer.
Peb hom thawj zaug ntawm txoj hlua nruj tau qhia nyob rau hauv daim duab 9. Muaj qee cov nodes ntawm qhov nkhaus hom tseem ceeb. Hauv qhov kev co tseem ceeb, cov nodes tsis co. Daim duab 10 qhia ntau hom ntawm lub phaj puag ncig uas txhawb nqa nrog qee cov kab nodal uas muaj cov voj voog thiab txoj kab uas hla.
Qhov kev tsim qauv tseeb ntawm qhov teeb meem kev co ntawm elastomer tuaj yeem xaus lus tias yog qhov teeb meem tus nqi ciam teb ntawm cov qauv sib txawv ib nrab. Txawm li cas los xij, qhov kev daws teeb meem tseeb tsuas yog pom tau hauv qee qhov xwm txheej yooj yim tshaj plaws, yog li peb yuav tsum siv qhov kev daws teeb meem kwv yees rau qhov teeb meem kev co ntawm elastomer nyuaj. Lub ntsiab lus ntawm ntau qhov kev daws teeb meem kwv yees yog hloov qhov tsis kawg mus rau qhov kawg, uas yog, kom faib cov kab ke ntau qib ntawm kev ywj pheej (txuas ntxiv mus) mus rau hauv cov kab ke ntau qib ntawm kev ywj pheej (txuas ntxiv). Muaj ob hom kev faib tawm uas siv dav hauv kev tshuaj xyuas engineering: txoj kev finite element thiab txoj kev modal synthesis.
DUAB 9 hom hlua
DUAB 10 hom ntawm lub phaj voj voog
Txoj kev finite element yog ib qho qauv sib xyaw ua ke uas muab ib qho qauv nyuaj ua ib qho lej ntawm cov ntsiab lus thiab txuas lawv ntawm ib qho lej ntawm cov nodes. Txhua chav tsev yog ib qho elastomer; Qhov kev faib tawm ntawm cov ntsiab lus yog qhia los ntawm kev ua haujlwm interpolation ntawm qhov kev hloov pauv ntawm node. Tom qab ntawd cov kev faib tawm ntawm txhua lub ntsiab lus raug tsom mus rau txhua lub node hauv ib hom ntawv, thiab tus qauv kho tshuab ntawm lub kaw lus discrete tau txais.
Kev tsim cov qauv (modal synthesis) yog kev rhuav tshem cov qauv nyuaj ua ntau yam qauv yooj yim dua. Raws li kev nkag siab txog cov yam ntxwv ntawm kev co ntawm txhua lub qauv, cov qauv no raug tsim ua cov qauv dav dav raws li cov xwm txheej sib koom tes ntawm qhov interface, thiab cov qauv kev co ntawm cov qauv dav dav tau txais los ntawm kev siv cov qauv kev co ntawm txhua lub qauv.
Ob txoj kev no sib txawv thiab muaj feem cuam tshuam, thiab siv tau ua piv txwv. Txoj kev modal synthesis kuj tseem siv tau ua ke nrog kev ntsuas kev sim los tsim ib txoj kev tshuaj xyuas theoretical thiab experimental rau kev co ntawm cov kab ke loj.
Lub sijhawm tshaj tawm: Plaub Hlis-03-2020
