He aha te wiri rārangi?

Wiritanga rārangi: kei raro i te ture a Hooke te ngāwari o ngā wāhanga o te pūnaha, ā, ko te kaha whakaiti i puta i te nekehanga he rite ki te whārite tuatahi o te tere whānui (te pānga wā o ngā taunga whānui).

ariā

Ko te pūnaha raina he tauira whakarāpopototanga o te wiri o te pūnaha tūturu. Ka whakamahia e te pūnaha wiri raina te mātāpono taupatupatu, arā, ki te mea ko te urupare a te pūnaha ko y1 i raro i te mahi a te tāuru x1, me y2 i raro i te mahi a te tāuru x2, ko te urupare a te pūnaha i raro i te mahi a te tāuru x1 me te x2 ko y1+y2.

I runga i te kaupapa o te taupatupatu, ka taea te wehewehe i tētahi tāurunga matapōkere ki te tapeke o tētahi raupapa o ngā akiaki iti rawa, kātahi ka taea te whiwhi i te urupare katoa o te pūnaha. Ka taea te whakawhānui i te tapeke o ngā wāhanga harmonika o tētahi whakaoho ā-wā ki tētahi raupapa o ngā wāhanga harmonika mā te whakawhiti Fourier, ā, ka taea te tirotiro motuhake i te pānga o ia wāhanga harmonika ki te pūnaha. Nō reira, ka taea te whakaahua i ngā āhuatanga urupare o ngā pūnaha raina me ngā tawhā pumau mā te urupare akiaki, te urupare auau rānei.

Ko te urupare āwhati e pā ana ki te urupare a te pūnaha ki te āwhati wae, e whakaahua ana i ngā āhuatanga urupare o te pūnaha i roto i te rohe wā. Ko te urupare auau e pā ana ki te āhuatanga urupare o te pūnaha ki te tāuru āwhati wae. Ko te ōritetanga i waenga i ngā mea e rua ka whakatauhia e te panoni Fourier.

whakarōpūtanga

Ka taea te wehewehe i te wiri rārangi ki te wiri rārangi o te pūnaha kotahi-tohu-o-te-herekore me te wiri rārangi o te pūnaha maha-tohu-o-te-herekore.

(1) Ko te wiri raina o tētahi pūnaha kotahi-tohu-o-te-herekore he wiri raina ka taea te whakatau i tōna tūranga mā te taunga whānui. Koinei te wiri māmā rawa atu e taea ai te ahu mai i ngā ariā me ngā āhuatanga taketake o te wiri. Kei roto ko te wiri harmonic ngāwari, te wiri kore utu, te wiri whakaiti me te wiri ā-kaha.

Te wiri hāmoni māmā: te nekehanga tauutuutu o tētahi mea i te taha o tōna tūranga taurite e ai ki te ture sinusoidal i raro i te mahi a te kaha whakaora e ōrite ana ki tōna nekehanga.

Wiri whakawhāiti: te wiri e whakaititia tonutia ana tōna kaha e te waku me te ātete hiko-kore, e ētahi atu whakapaunga pūngao rānei.

Wiri ā-kaha: te wiri o tētahi pūnaha i raro i te whakaoho tonu.

(2) ko te wiri raina o te pūnaha maha-tohu-o-te-herekore ko te wiri o te pūnaha raina me te n≥2 nekehanga o te herekore. He n ngā auau maori me te n ngā aratau matua o tētahi pūnaha o te n ngā nekehanga o te herekore. Ka taea te whakaatu i tētahi whirihoranga wiri o te pūnaha hei huinga raina o ngā aratau matua. Nō reira, ka whakamahia whānuitia te tikanga taupatupatu aratau matua i roto i te tātari urupare hihiri o ngā pūnaha maha-dof. Mā tēnei, ka noho te ine me te tātari i ngā āhuatanga wiri maori o te pūnaha hei taahiraa noa i roto i te hoahoa hihiri o te pūnaha. Ka taea hoki te whakaahua i ngā āhuatanga hihiri o ngā pūnaha maha-dof mā ngā āhuatanga auau. Nā te mea he mahi āhuatanga auau i waenga i ia tāuru me ia putanga, ka hangaia he matihiko āhuatanga auau. He whanaungatanga tino tika i waenga i te āhuatanga auau me te aratau matua. He rerekē te pihi āhuatanga whānui-auau o te pūnaha maha-herekore i tērā o te pūnaha kotahi-herekore.

Te wiri rārangi o te pūnaha kotahi te tohu o te herekoretanga

He wiri raina e taea ai te whakatau i te tūranga o tētahi pūnaha mā te whakamahi i tētahi taunga whānui. Koinei te wiri māmā rawa atu, tino taketake hoki, e taea ai te ahu mai i ngā ariā me ngā āhuatanga taketake maha o te wiri. Kei roto ko te wiri harmonic ngāwari, te wiri kua whakawhāitihia me te wiri ā-kaha.

Wiritanga hāmoni

I raro i te mahi a te kaha whakahoki e rite ana ki te nekehanga, ka taurite te mea i roto i te āhua sinusoidal tata ki tōna tūranga taurite (Pikitia 1). Ko X te nekehanga, ā, ko t te wā. Ko te whakaaturanga pāngarau o tēnei wiri ko:

(1)Ko A te uara mōrahi o te nekehanga x, e kiia nei ko te kaha, ā, e tohu ana i te kaha o te wiri; Ko Omega n te kaha Te pikinga koki o te wiri ia hekona, e kiia nei ko te auau koki, ko te auau porohita rānei; Ka kiia tēnei ko te wāhanga tīmatanga. Mō te f= n/2, ko te maha o ngā wiri ia hekona ka kiia ko te auau; Ko te whakahurihuri o tēnei, T=1/f, ko te wā e pau ana ki te wiri i te huringa kotahi, ā, ka kiia tēnā ko te wā. Te kaha A, te auau f (te auau koki n rānei), te wāhanga tīmatanga, e mōhiotia ana ko te wiri harmonic ngāwari e toru ngā huānga.

PIK. 1 te pihi wiri harmonic māmā

E ai ki te whakaaturanga i te Pikitia 2, ka hangaia he oscillator harmonic māmā e te papatipu kukū m e honoa ana e te puna rārangi. Ina tatauhia te nekehanga wiri mai i te tūranga taurite, ko te whārite wiri ko:

Kei hea te pakari o te puna. Ko te otinga whānui ki te whārite i runga ake nei ko (1).A, ā, ka taea te whakatau mā te tūranga tīmatanga x0 me te tere tīmatanga i t=0:

Engari ko te omega n ka whakatauhia anake e ngā āhuatanga o te pūnaha m me k, ahakoa ngā āhuatanga tīmatanga tāpiri, nō reira e mōhiotia ana hoki te omega n ko te auautanga taiao.

PIK. 2 pūnaha tohu kotahi o te herekoretanga

Mō tētahi oscillator harmonic māmā, he pumau te tapeke o tōna pūngao nekeneke me tōna pūngao pūmanawa, arā, ka tiakina te pūngao miihini katoa o te pūnaha. I roto i te tukanga wiri, ka hurihia tonutia te pūngao nekeneke me te pūngao pūmanawa ki a rātou anō.

Te wiri whakamākū

He wiri e whakaitihia tonutia ana tōna kaha e te waku me te ātete dielectric, e ētahi atu whakapaunga pūngao rānei. Mō te wiri moroiti, kāore te tere i te tino nui, ā, he rite te ātete waenga ki te tere ki te mana tuatahi, ka taea te tuhi ko c te tauwehenga whakamātao. Nō reira, ka taea te tuhi i te whārite wiri o te kotahi nekehanga o te herekoretanga me te whakamātao raina penei:

(2)Ko m =c/2m te tawhā whakamākū, ā. Ka taea te tuhi i te otinga whānui o te tātai (2):

(3)Ka taea te wehewehe i te whanaungatanga tau i waenga i te omega n me te PI ki ngā āhuatanga e toru e whai ake nei:

N > (mō te iti o te whakamākū) i puta ai te wiri whakaiti i te matūriki, ko te whārite wiri ko:

Ka heke tōna kaha i te wā e ai ki te ture taupū e whakaaturia ana i roto i te whārite, e whakaaturia ana i roto i te rārangi iraira i te Pikitia 3. He pono, he aperiodic tēnei wiri, engari ka taea te tautuhi i te auau o tōna tihi penei:

Ka kiia ko te tere whakaiti kaha, ko te wā o te wiri te wā. Ka kiia ko te logarithm maori o te tere whakaiti kaha ko te logarithm tango (kaha) tere. He mārama, ko te =, i tēnei take, he ōrite ki te 2/1. Mā te whakamahi i te tātai i runga ake nei ka taea te tatau i te c.

I tēnei wā, ka taea te tuhi i te otinga o te whārite (2):

I te taha o te ahunga o te tere tīmatanga, ka taea te wehewehe kia toru ngā āhuatanga kore-wiri e whakaaturia ana i te Pikitia 4.

N < (mēnā he nui te whakamākū), ka whakaaturia te otinga ki te whārite (2) i roto i te whārite (3). I tēnei wā, kāore te pūnaha e wiri ana.

Te wiri ā-kaha

Te wiri o tētahi pūnaha i raro i te whakaoho tonu. Ko te tātari wiri te mea nui e rangahau ana i te urupare a te pūnaha ki te whakaoho. Ko te whakaoho ā-wā he whakaoho auau noa. I te mea ka taea te wehewehe i te whakaoho ā-wā ki te tapeke o ngā whakaoho ā-hauora maha, e ai ki te mātāpono taupatupatu, ko te urupare anake a te pūnaha ki ia whakaoho ā-hauora e hiahiatia ana. I raro i te mahi a te whakaoho ā-hauora, ka taea te tuhi i te whārite rerekētanga o te nekehanga o tētahi pūnaha kua whakamātaohia te nekehanga kotahi:

Ko te urupare he tapeke o ngā wāhanga e rua. Ko tētahi wāhanga ko te urupare o te wiri kua whakamākūhia, e tere memeha haere ana i te wā. Ka taea te tuhi i te urupare o tētahi atu wāhanga o te wiri kua akiakihia:

PIK. 3 te pihi wiri kua whakamākūhia

PIK. 4 ngā kōpiko o ngā āhuatanga tīmatanga e toru me te whakamātao nui

Patohia te

Ko te H /F0= h (), ko te ōwehenga o te kaha urupare pumau ki te kaha whakaoho, e whakaatu ana i ngā āhuatanga o te kaha-auau, te mahi whakanui rānei; Ngā moka mō te urupare āhua pumau me te akiaki o te wāhanga, te whakaahuatanga o ngā āhuatanga o te auau wāhanga. Ko te whanaungatanga i waenga i a rātou me te auau whakaoho e whakaaturia ana i te Pikitia 5 me te Pikitia 6.

E kitea ana i te kōpiko aukume-auau (Pikitia 5), ​​i roto i te take o te whakaiti iti, kotahi te tihi o te kōpiko aukume-auau. Ko te iti o te whakaiti, ko te pari ake o te tihi; Ko te auau e rite ana ki te tihi ka kiia ko te auau oro o te pūnaha. I roto i te take o te whakaiti iti, kāore te auau oro i tino rerekē mai i te auau tūturu. Ina tata te auau whakaoho ki te auau tūturu, ka piki tere te aukume. Ka kiia tēnei āhuatanga ko te oro. I te oro, ka whakanuia te whiwhinga o te pūnaha, arā, ko te wiri āki te mea tino kaha. Nō reira, i te nuinga o te wā, me ngana tonu ki te karo i te oro, ki te kore ētahi taputapu me ngā taputapu e whakamahi i te oro hei whakatutuki i te wiri nui.

Pikitia 5 te pihi auau whānui

Ka kitea mai i te pihi auau ā-wāhanga (pikitia 6), ahakoa te rahi o te whakamāmā, i roto i ngā moka rerekētanga ā-wāhanga omega kore = PI / 2, ka taea te whakamahi whai hua i tēnei āhuatanga hei ine i te aukume.

Haunga te whakaoho pumau, ka tūtaki ētahi pūnaha ki te whakaoho kore pumau. Ka taea te wehewehe kia rua ngā momo: ko te tuatahi ko te pānga ohorere. Ko te tuarua ko te pānga roa o te koretake. I raro i te whakaoho kore pumau, ka kore pumau hoki te urupare a te pūnaha.

He taputapu kaha mō te tātari i te wiri kore pumau ko te tikanga urupare hihiri. E whakaahua ana i ngā āhuatanga hihiri o te pūnaha me te urupare tauwhiro o te whakaurunga hihiri wae o te pūnaha. Ka taea te whakaatu i te hihiri wae hei mahi delta. I roto i te hangarau, ka tautuhia te mahi delta penei:

Ko te 0- te pūwāhi i te tuaka-t e whakatata atu ana ki te kore mai i te taha maui; ko te 0 tāpiri te pūwāhi e haere ana ki te 0 mai i te taha matau.

Pikitia: Piko auau wāhanga 6

PIK. 7 ka taea te whakaaro ko tētahi tāuru he tapeke o tētahi raupapa o ngā huānga hihiri

E rite ana te pūnaha ki te urupare h(t) i hangaia e te ine ine i te t=0, e kiia nei ko te mahi urupare ine. Mēnā kei te tū tonu te pūnaha i mua i te ine, h(t)=0 mō te t<0. Mā te mōhio ki te mahi urupare ine o te pūnaha, ka kitea e tātou te urupare a te pūnaha ki tētahi tāuru x(t). I tēnei wā, ka taea e koe te whakaaro ki a x(t) hei tapeke o tētahi raupapa o ngā huānga ine (Pikitia 7). Ko te urupare a te pūnaha ko:

I runga i te mātāpono taupatupatu, ko te urupare katoa o te pūnaha e pā ana ki a x(t) ko:

Ka kiia tēnei taupū he taupū honohono, he taupū taupatupatu rānei.

Te wiri rārangi o tētahi pūnaha maha-tohu-o-te-herekore

Te wiri o tētahi pūnaha rārangi me te n≥2 nekehanga herekore.

E whakaatu ana te Pikitia 8 i ngā pūnaha iti e rua e honoa ana e te puna honohono. Nā te mea he pūnaha e rua ngā tohu-o-te-herekore, e rua ngā taunga motuhake e hiahiatia ana hei whakatau i tōna tūranga. E rua ngā auautanga taiao i roto i tēnei pūnaha:

Ia auau e rite ana ki tētahi momo wiri. Ka whakahaerehia e ngā oscillator harmonic ngā wiri harmonic o te auau kotahi, ka haere tahi i roto i te tūranga taurite, ka tae tahi ki te tūranga tino. I roto i te wiri matua e rite ana ki te omega kotahi, he ōrite te x1 ki te x2; I roto i te wiri matua e rite ana ki te omega rua, omega kotahi. I roto i te wiri matua, ka mau tonu te ōwehenga nekehanga o ia papatipu i tētahi whanaungatanga, ā, ka hanga i tētahi momo, e kiia nei ko te aratau matua, ko te aratau tūturu rānei. Kei roto i ngā momo matua te orthogonality o te papatipu me te pakari, e whakaata ana i te motuhaketanga o ia wiri. Ko te auau tūturu me te aratau matua e tohu ana i ngā āhuatanga wiri tūturu o te pūnaha maha-tohu herekore.

PIK. 8 pūnaha me ngā nekehanga herekoretanga maha

He pūnaha n ngā nekehanga herekore, ā, he n ngā auautanga maori me te n ngā aratau matua. Ka taea te whakaatu i tētahi whirihoranga wiri o te pūnaha hei huinga rārangi o ngā aratau matua. Nō reira, e whakamahia whānuitia ana te tikanga taupatupatu aratau matua i roto i te tātari urupare hihiri o ngā pūnaha maha-whakaaturanga. Mā tēnei, ka noho te ine me te tātari i ngā āhuatanga wiri maori o te pūnaha hei taahiraa noa i roto i te hoahoa hihiri o te pūnaha.

Ka taea hoki te whakaahua i ngā āhuatanga hihiri o ngā pūnaha maha-auau mā roto i ngā āhuatanga auau. Nā te mea he mahi āhuatanga auau kei waenganui i ia tāuru me ia putanga, ka hangaia he matihiko āhuatanga auau. He rerekē te pihi āhuatanga whānui-auau o te pūnaha maha-herekore i tērā o te pūnaha kotahi-herekore.

Ka wiri te elastomer

Ko te pūnaha maha-tohu herekore i runga ake nei he tauira ā-pūkaha tata o te elastomer. He mutunga kore te maha o ngā tohu herekore o te elastomer. He rerekētanga tau engari kāore he rerekētanga nui i waenga i ngā mea e rua. He mutunga kore te maha o ngā auau taiao me te mutunga kore o ngā aratau e rite ana o tētahi elastomer, ā, he orthogonality i waenga i ngā aratau o te papatipu me te pakari. Ka taea hoki te whakaatu i tētahi whirihoranga wiri o te elastomer hei taupatupatu raina o ngā aratau matua. Nō reira, mō te tātari urupare hihiri o te elastomer, ka taea tonu te whakamahi i te tikanga taupatupatu o te aratau matua (tirohia te wiri raina o te elastomer).

Tangohia te wiri o te aho. Me kī he aho angiangi, he taumaha m mō ia wae roa, ko te roa l, e kumea ana i ngā pito e rua, ā, ko te kume ko T. I tēnei wā, ka whakatauhia te auau maori o te aho e te whārite e whai ake nei:

F =na/2l (n= 1,2,3…).

Ko, te tere horapa o te ngaru whakawhiti i te ahunga o te aho. Ko ngā auau taiao o ngā aho he taurangi o te auau taketake i runga i te 2l. Ko tēnei taurangi maha ka hua ake he hanganga harmonic pai. I te nuinga o te wā, kāore he whanaungatanga taurangi maha i waenga i ngā auau taiao o te elastomer.

Ko ngā momo tuatahi e toru o te aho kua whakatōngia e whakaaturia ana i te PIK. 9. He pūnga kei te pihi aratau matua. I roto i te wiri matua, kāore ngā pūnga e wiri. E whakaatu ana te PIK. 10 i ētahi momo noa o te pereti porowhita e tautokona ana e te porowhita me ētahi rārangi pūnga i titoa mai i ngā porowhita me ngā diameter.

Ka taea te whakatau i te whakatakotoranga tika o te raruraru wiri elastomer hei raruraru uara rohe o ngā whārite rerekētanga ā-wāhanga. Heoi, ka kitea noa te otinga tika i roto i ētahi o ngā take māmā, nō reira me whakamahi tātou i te otinga tata mō te raruraru wiri elastomer uaua. Ko te ngako o ngā otinga tata maha ko te huri i te mutunga kore ki te mutunga kore, arā, ko te wehewehe i te pūnaha maha-tohu herekore kore-waenga (pūnaha tonu) ki roto i te pūnaha maha-tohu herekore mutunga kore (pūnaha motumotu). E rua ngā momo tikanga wehewehe e whakamahia whānuitia ana i roto i te tātari hangarau: te tikanga huānga mutunga kore me te tikanga whakahiato aratau.

PIK. 9 aratau aho

PIK. 10 aratau pereti porowhita

Ko te tikanga huānga mutunga kore he hanganga hiato e whakawehewehe ana i tētahi hanganga matatini ki roto i tētahi maha mutunga kore o ngā huānga, ā, ka hono i a rātou i tētahi maha mutunga kore o ngā pona. He elastomer ia waeine; Ko te nekehanga tohatoha o te huānga ka whakaatuhia e te mahi whakauru o te nekehanga pona. Kātahi ka arotahihia ngā tawhā tohatoha o ia huānga ki ia pona i roto i tētahi whakatakotoranga, ā, ka whiwhihia te tauira miihini o te pūnaha motumotu.

Ko te hanganga āhua ko te wehewehe i tētahi hanganga matatini ki ētahi hanganga iti māmā ake. I runga i te mārama ki ngā āhuatanga wiri o ia hanganga iti, ka hangaia te hanganga iti hei hanganga whānui e ai ki ngā tikanga taunga i runga i te atanga, ā, ka whiwhihia te āhua wiri o te hanganga whānui mā te whakamahi i te āhua wiri o ia hanganga iti.

He rerekē, he hono hoki ngā tikanga e rua, ā, ka taea te whakamahi hei tohutoro. Ka taea hoki te whakakotahi i te tikanga whakahiato aratau me te ine whakamātautau hei hanga i tētahi tikanga tātari ariā me te whakamātautau mō te wiri o ngā pūnaha nui.