He aha ka haʻalulu laina?

Haʻalulu laina: Ua pili ka elasticity o na mea i loko o ka nenoai i ke kanawai hooke, a ua like ka ikaika humuhumu i hanaia i ka oni ana me ka hoohalike mua o ka velocity laulaha.

manaʻo

ʻO ka laina laina maʻamau he kumu hoʻohālike abstract o ka vibration o ka ʻōnaehana maoli. ʻO ka linear vibration system e pili ana i ke kumu superposition, ʻo ia hoʻi, inā y1 ka pane o ka ʻōnaehana ma lalo o ka hana o ka hoʻokomo x1, a me y2 ma lalo o ka hana o ka hoʻokomo x2, a laila ka pane o ka ʻōnaehana ma lalo o ka hana o ka hoʻokomo x1 a me x2 ʻo y1+y2.

Ma ke kumu o ka manaʻo o ka superposition, hiki ke hoʻoheheʻe ʻia kahi hoʻokomo arbitrary i loko o ka huina o kahi moʻo o ka infinitesimal impulses, a laila hiki ke loaʻa ka pane a pau o ka ʻōnaehana. nā pūʻulu o nā ʻāpana harmonic e Fourier transform, a ʻo ka hopena o kēlā me kēia ʻāpana harmonic i ka ʻōnaehana hiki ke noiʻi ʻokoʻa. pane hoʻoulu a i ʻole pane pinepine.

ʻO ka pane impulse e pili ana i ka pane o ka ʻōnaehana i ka unit impulse, kahi e hōʻike ai i nā hiʻohiʻona pane o ka ʻōnaehana ma ka domain manawa. e ka hoololi Fourier.

hoʻokaʻawale

Hiki ke hoʻokaʻawale ʻia ka haʻalulu laina i loko o ka vibration linear o ka ʻōnaehana hoʻokahi-degree-of-freedom a me ka vibration linear o ka ʻōnaehana multi-degree-of-freedom.

(1) linear vibration o ka one-degree-of-freedom system is a linear vibration nona ke kūlana e hiki ke hoʻoholo ʻia e ka generalized coordinate. ʻO ka haʻalulu harmonic, ka haʻalulu manuahi, ka haʻalulu a me ka haʻalulu ikaika.

ʻO ka haʻalulu harmonic maʻalahi: ka neʻe pānaʻi ʻana o kahi mea ma kahi kokoke i kona kūlana kaulike e like me ke kānāwai sinusoidal ma lalo o ka hana a ka mana hoʻihoʻi e like me kona neʻe ʻana.

ʻO ka haʻalulu haʻahaʻa: ka haʻalulu i hoʻemi mau ʻia ka amplitude e ka hiki ʻana o ka friction a me ke kūpaʻa dielectric a i ʻole ka hoʻohana ʻana i ka ikehu.

ʻO ka haʻalulu ikaika: ka haʻalulu o kahi ʻōnaehana ma lalo o ka hauʻoli mau.

(2) ʻo ka haʻalulu laina o ka ʻōnaehana multi-degree-of-freedom ka haʻalulu o ka ʻōnaehana laina me nā n≥2 degere o ke kūʻokoʻa. ʻO ka ʻōnaehana hiki ke hōʻike ʻia ma ke ʻano he hui laina o nā ʻano nui. ʻO ka nānā ʻana i nā hiʻohiʻona haʻalulu kūlohelohe o ka ʻōnaehana e lilo i hana maʻamau i ka hoʻolālā hoʻolālā o ka ʻōnaehana. Ua kūkulu ʻia kahi matrix ʻano alapinepine. Aia kahi pilina ma waena o ka ʻano alapine a me ke ʻano kumu. ʻōnaehana.

Linear vibration o kahi degere hoʻokahi o ka ʻōnaehana kūʻokoʻa

ʻO ka vibration linear kahi e hiki ai ke hoʻoholo i ke kūlana o kahi ʻōnaehana e ka hui like ʻole. .

Haʻalulu harmonika

Ma lalo o ka ho'iho'i 'ana i ka ikaika e like me ka ho'one'e 'ana, ua pāna'i ka mea ma ke 'ano sinusoidal kokoke i kona kūlana kaulike (FIG. 1). Hō'ike ka X i ka ne'e 'ana a 'o t ka manawa. ʻO ka ʻōlelo makemakika o kēia haʻalulu:

(1)Ma kahi o A ka waiwai nui o ka neʻe ʻana x, i kapa ʻia ʻo ka amplitude, a e hōʻike ana i ka ikaika o ka haʻalulu; ʻO Omega n ka amplitude Angle hoʻonui o ka haʻalulu i kēlā me kēia kekona, i kapa ʻia ʻo ka angular frequency, a i ʻole ke alapine pōʻai; Ua kapa ʻia ka māhele mua. Ma nā huaʻōlelo o f= n/2, ua kapa ʻia ka helu o nā oscillations i kēlā me kēia kekona he alapine; pono ia e oscillate i hoʻokahi pōʻaiapuni, a ua kapa ʻia ʻo ia ka manawa. Amplitude A, alapine f (a i ʻole angular frequency n), ka pae mua, i ʻike ʻia he haʻalulu harmonic maʻalahi ʻekolu mau mea.

FIG. 1 ka pihi haʻalulu harmonic maʻalahi

E like me ka hoike ana ma ka FIG. 2, hoʻokumu ʻia kahi oscillator harmonic maʻalahi e ka m i hoʻopili ʻia e kahi puna laina.

ʻAuhea ka ʻoʻoleʻa o ka pūnāwai. ʻO ka hāʻina maʻamau i ka hoʻohālikelike ma luna nei (1).

Akā, hoʻoholo wale ʻia ka omega n e nā hiʻohiʻona o ka ʻōnaehana ponoʻī m a me k, kūʻokoʻa i nā kūlana mua hou, no laila ua ʻike ʻia ʻo omega n ʻo ke alapine maoli.

FIG. 2 hoʻokahi degere o ka ʻōnaehana kūʻokoʻa

No ka oscillator harmonic maʻalahi, ua mau ka huina o kona ikehu kinetic a me ka ikehu, ʻo ia hoʻi, mālama ʻia ka nui o ka ikehu mechanical o ka ʻōnaehana.

ʻO ka haʻalulu haʻalulu

He vibration nona ka amplitude e hoʻemi mau ʻia e ka friction a me ka dielectric resistance a i ʻole ka hoʻohana ʻana i ka ikehu ʻē aʻe. ka helu hoʻoheheʻe. No laila, hiki ke kākau ʻia ka haʻihaʻi haʻalulu o hoʻokahi degere o ke kūʻokoʻa me ka damping linear e like me:

(2)Ma hea, ua kapa ʻia ka m =c/2m ka ʻāpana hoʻoheheʻe, a. Hiki ke kākau ʻia ka hopena maʻamau o ke kumu (2):

(3)Hiki ke hoʻokaʻawale ʻia ka pilina helu ma waena o omega n a me PI i ʻekolu mau hihia:

N > (ma ke ʻano o ka hoʻoheheʻe liʻiliʻi) i hoʻopuka ʻia ka haʻalulu attenuation, ʻo ka hoohalike vibration:

E emi ana kona amplitude me ka manawa e like me ke kanawai exponential i hoikeia ma ka hoohalike, e like me ka hoikeia ma ka laina kiko ma FIG. 3. ʻO ka'ōlelo koʻikoʻi, he aperiodic kēia haʻalulu, akā hiki ke weheweheʻia ka pinepine o kona piko e like me:

Ua kapaia ka amplitude reduction rate, kahi o ka manawa o ka vibration.The natural logarithm of the amplitude reduction rate is called the logarithm minus (amplitude). ʻO ka delta hoʻokolohua hoʻokolohua a, me ka hoʻohana ʻana i ke ʻano o luna hiki ke helu ʻia c.

I kēia manawa, hiki ke kākau ʻia ka hopena o ka hoohalike (2):

Me ke kuhikuhi o ka velocity mua, hiki ke mahele ia i ekolu mau mea vibration e like me ka hoike ana ma FIG. 4.

N < (ma ka hoʻoheheʻe nui ʻana), hōʻike ʻia ka hopena o ka hoohalike (2) ma ka hoohalike (3).

ʻO ka haʻalulu ikaika

ʻO ka haʻalulu o kahi ʻōnaehana ma lalo o ka excitation mau. ʻO ka nānā ʻana o ka vibration e noiʻi nui i ka pane o ka ʻōnaehana i ka excitation. Periodic excitation he mea maʻamau maʻamau. pono ka pane o ka ʻōnaehana i kēlā me kēia hoʻohiwahiwa harmonic. degere o ke kuokoa damped system hiki ke kākau:

ʻO ka pane ka huina o nā ʻāpana ʻelua. ʻO kekahi hapa ka pane o ka haʻalulu damped, e pohō wikiwiki ana me ka manawa. Hiki ke kākau ʻia ka pane o kekahi ʻāpana o ka haʻalulu ikaika:

FIG. 3 puʻupuʻu haʻalulu haʻalulu

FIG. 4 nā pihi o ʻekolu kūlana mua me ka hoʻomoe koʻikoʻi

E kikokiko i ka

H /F0= h (), ʻo ia ka lakio o ka amplitude pane paʻa i ka amplitude excitation, e hōʻike ana i nā hiʻohiʻona amplitude-frequency, a i ʻole ka hana loaʻa; Hōʻike ʻia ke alapine hoʻouluulu ma FIG. 5 a me FIG. 6.

E like me ka mea i ʻike ʻia mai ka amplitude-frequency curve (FIG. 5), ma ke ʻano o ka damping liʻiliʻi, ʻo ka amplitude-frequency curve he kiʻekiʻe hoʻokahi. i kapa ʻia ke alapine resonant o ka ʻōnaehana. Ma ke ʻano o ka damping liʻiliʻi, ʻaʻole ʻokoʻa loa ka alapine resonance mai ke alapine maoli. pinepine, piʻi nui ka amplitude. Kapaʻia kēia hanana i ka resonance. Ma ka resonance, ua hoʻonuiʻia ka loaʻa o ka'ōnaehana,ʻo ia hoʻi,ʻo ka haʻalulu ikaika ka mea ikaika loa. haʻalulu.

FIG. 5 amplitude frequency curve

Hiki ke ʻike ʻia mai ka ʻāpana alapine (helu 6), me ka nānā ʻole i ka nui o ka hoʻoheheʻe ʻana, ma ka omega zero phase difference bits = PI / 2, hiki ke hoʻohana pono ʻia kēia ʻano i ke ana ʻana i ka resonance.

Ma waho aʻe o ka hauʻoli paʻa, ʻike ʻia nā ʻōnaehana i kekahi manawa me ka hauʻoli paʻa ʻole. Hiki ke hoʻokaʻawale ʻia i ʻelua mau ʻano: ʻo ka hopena wikiwiki. ʻO ka lua ka hopena mau loa o ka arbitrariness.

ʻO kahi mea hana ikaika no ka nānā ʻana i ka haʻalulu paʻa ʻole ʻo ia ke ʻano pane pane. Ua wehewehe pinepine ʻia ka hana penei:

Ma kahi o 0- e hoike ana i ke kiko ma ka axis t e hookokoke ana i ka zero mai ka hema; 0 plus ka kiko e hele ana i 0 mai ka akau.

FIG. 6 mahele alapine piʻi

FIG. 7 hiki ke noʻonoʻo ʻia kekahi mea hoʻokomo e like me ka huina o kahi pūʻulu o nā mea impulse

Hoʻopili ka ʻōnaehana i ka pane h (t) i hana ʻia e ka impulse unit ma t = 0, i kapa ʻia ʻo ka hana pane impulse. Ke manaʻo nei ua paʻa ka ʻōnaehana ma mua o ka pulse, h (t) = 0 no t<0. ʻIke ʻo ka hana pane pane o ka ʻōnaehana, hiki iā mākou ke ʻike i ka pane ʻana o ka ʻōnaehana i kekahi hoʻokomo x (t). I kēia manawa, hiki iā ʻoe ke noʻonoʻo i ka x (t) ma ke ʻano he pūʻulu o nā mea impulse (FIG. 7). ʻO ka pane o ka ʻōnaehana:

Ma muli o ke kumu hoʻohālikelike, ʻo ka huina pane o ka ʻōnaehana pili i x(t) ʻo ia:

Kapa ʻia kēia mea hoʻohui he hui hui a i ʻole he hui hoʻohui superposition.

ʻO ka haʻalulu laina o kahi ʻōnaehana multi-degree-of-freedom

ʻO ka haʻalulu o kahi ʻōnaehana laina me n≥2 degere o ke kūʻokoʻa.

Hōʻike ka Figure 8 i ʻelua mau subsystem resonant maʻalahi i hoʻopili ʻia e kahi puna hoʻohui.

ʻO kēlā me kēia alapine e like me ke ʻano o ka haʻalulu. Hoʻokō nā oscillators harmonic i nā oscillations harmonic o ka alapine like, synchronously hele ma ke kūlana kaulike a synchronously hiki i ke kūlana kiʻekiʻe. ʻO ka vibration nui e pili ana i ka omega omega ʻelua, omega omega hoʻokahi. kekahi pilina a hana i kekahi ʻano, i kapa ʻia ʻo ke ʻano nui a i ʻole ke ʻano kūlohelohe. Aia ka orthogonality o ka nuipa a me ka ʻoʻoleʻa i waena o nā ʻano kumu nui, e hōʻike ana i ke kūʻokoʻa o kēlā me kēia haʻalulu. o ka lehulehu-degere o ke kuokoa.

FIG. 8 pūnaewele me nā degere he nui o ke kūʻokoʻa

He ʻōnaehana n degere o ke kūʻokoʻa he n nā alapine kūlohelohe a me nā ʻano kumu nui. Hiki ke hōʻike ʻia kekahi hoʻonohonoho haʻalulu o ka ʻōnaehana ma ke ʻano he hui laina o nā ʻano nui. -dof systems. Ma kēiaʻano,ʻo ke ana a me ka nānāʻana i nā hiʻohiʻona vibration kūlohelohe o ka'ōnaehana e lilo i mea hana maʻamau i ka hoʻolālā hoʻoikaika o ka pūnaewele.

Hiki ke wehewehe ʻia nā hiʻohiʻona ikaika o nā ʻōnaehana multi-dof e nā hiʻohiʻona alapine. mai ka ʻōnaehana kūʻokoʻa hoʻokahi.

Haʻalulu ka elastomer

ʻO ka ʻōnaehana multi-degree o ke kūʻokoʻa ma luna nei he kumu hoʻohālike mechanical o elastomer. He helu pau ʻole ka elastomer o nā degere o ke kūʻokoʻa. Aia kahi ʻokoʻa nui akā ʻaʻohe ʻokoʻa koʻikoʻi ma waena o nā mea ʻelua. He helu pau ʻole o nā ʻano like ʻole, a aia ka orthogonality ma waena o nā ʻano o ka nuipa a me ka ʻoʻoleʻa. i hōʻike ʻia ma ke ʻano he linear superposition o nā ʻano nui.

E lawe i ka vibration o ke kaula. E olelo kakou ua hookuiia kekahi kaula lahilahi o ka m no ka wae hookahi, loihi l, ma na welau elua, a o ka hookuonoono T. I keia manawa, ua hooholoia ke alapine maoli o ke kaula e keia mau mea. hoohalike:

F =na/2l (n= 1,2,3…).

ʻO ka wikiwiki hoʻolaha o ka nalu transverse ma ke ala o ke kaula. ʻO nā alapine maoli o nā kaula he mau paʻa ia o ke alapine kumu ma luna o 2l. Ke alakaʻi nei kēia multiplicity integer i kahi ʻano harmonic leʻaleʻa. Ma ka laulā, ʻaʻohe ʻo ia ka pilina nui integer ma waena o nā alapine maoli o ka elastomer.

Hōʻike ʻia nā ʻano ʻekolu mua o ke kaula i hoʻopaʻa ʻia ma FIG. 9. Aia kekahi mau nodes ma ka pihi mode nui. I ka haʻalulu nui, ʻaʻole e haʻalulu nā nodes.FIG. Hōʻike ʻo 10 i kekahi mau ʻano maʻamau o ka pā pōʻai i kākoʻo ʻia me kekahi mau laina nodal i haku ʻia me nā pōʻai a me nā anawaena.

Hiki ke hoʻopau ʻia ka hoʻokumu pololei ʻana o ka pilikia vibration elastomer ma ke ʻano he pilikia waiwai palena o nā hoʻohālikelike ʻokoʻa hapa. pilikia haʻalulu. ʻO ke kumu o nā ʻano hoʻonā like ʻole e hoʻololi i ka palena ʻole i ka palena, ʻo ia hoʻi, e discretize ka lālā-emi. multi-degree of freedom system (continuous system) into a finite multi-degree of freedom system (discrete system). Aia ʻelua ʻano o nā ʻano discretization i hoʻohana nui ʻia i ka ʻenekinia kālailai: finite element method and modal synthesis method.

FIG. 9 ke ano o ke kaula

FIG. 10 ke ano o ka papa poepoe

ʻO ke ʻano o ka ʻeleʻele hope kahi ʻano hui e hoʻokaʻawale i kahi hale paʻakikī i loko o kahi helu palena o nā mea a hoʻopili iā lākou i kahi helu palena o nā nodes. ʻO kēlā me kēia ʻāpana he elastomer; Hoʻopili ʻia nā ʻāpana mahele o kēlā me kēia mea i kēlā me kēia node i kahi ʻano, a loaʻa ke kumu hoʻohālike mechanical o ka ʻōnaehana discrete.

ʻO ka synthesis Modal ka decomposition o kahi ʻano paʻakikī i loko o kekahi mau substructures maʻalahi. Loaʻa ka hale ma ka hoʻohana ʻana i ka morphology vibration o kēlā me kēia substructure.

Heʻokoʻa a pili nāʻanoʻelua, a hiki ke hoʻohanaʻia ma keʻano he kuhikuhi. Hiki ke hoʻohui maikaʻiʻia keʻano o ka synthesis modal me ke ana hoʻokolohua e hana i keʻano hoʻolālā a me ka hoʻokolohua hoʻokolohua no ka vibration o nā pūnaewele nui.