Ngā Ariā Taketake o te Nekehanga Harmonic Māmā
Ko te nekehanga hāmoni māmā (SHM) he ariā taketake e ahu mai ana i ngā āhuatanga rerekē o te ahupūngao me te hangarau. Mai i ngā wiri o te pendulum ki ngā wiri o te aho kitā, ka whakaratohia e te SHM he turanga pakari mō te mārama ki te nekehanga o ngā mea i raro i ngā kaha whakaora. Ka rukuhia e tēnei tuhinga ngā mātāpono matua o te SHM, e whakamārama ana i ngā kupu matua, ngā whakatakotoranga pāngarau, me ngā pānga mahi.
He aha te Nekehanga Hāmoni Māmā?
Ko te nekehanga hāmoni māmā e pā ana ki tētahi momo nekehanga ā-wā kei reira te kaha whakaora e rite ana ki te nekehanga mai i te tūranga toharite, ā, e mahi ana i te ahunga e rerekē ana ki taua nekehanga. Ka puta tēnei momo nekehanga i roto i ngā pūnaha kei reira te kaha kupenga e mahi ana i runga i te mea ka taea te whakaahua e te Ture a Hooke, e kī ana he rite te kaha ki te kino o te nekehanga. Ko te tikanga, ko te SHM e tohuhia ana e te nekehanga sinusoidal e whakaatuhia ana i roto i ngā pūnaha pēnei i ngā puna, ngā pendulum, tae noa ki ngā wiri ngota.
Te Whakaora i te Kaha me te Nekehanga
I roto i te SHM, ka taea te whakaatu i te kaha whakaora (\(F\)) penei:
\[ F = -kx \]
ko \(k\) te pūmau kaha, ā, ko \(x\) te nekehanga mai i te tūranga taurite. Ko te tohu kino e tohu ana kei te anga tonu te kaha ki te ritenga kē atu i te nekehanga, me te whai kia whakahokia te mea ki tōna taurite.
Te Ture a Hooke i roto i te SHM
Ko tētahi o ngā pūnaha tino whakaahuatia i roto i te SHM ko te pūnaha puna-papatipu. E ai ki te Ture a Hooke:
\[ F = -kx \]
ko te \(k\) te pūmau puna, ā, e tohu ana i te pakari o te puna. Mēnā he papatipu \(m\) e piri ana ki tētahi puna, ka taurite te nekehanga e te kaha whakaora, ā, i roto i te wā, ka whakaatu te mea i te nekehanga wiriwiri huri noa i te tūranga taurite.
Te Hanga Pāngarau o te SHM
Ka taea te whakaahua i te whakaaturanga pāngarau o te SHM mā roto i ngā whārite rerekētanga. Ka taea te whakatauira i te nekehanga \(x(t)\) hei pānga o te wā \(t\) penei:
\[ x(t) = A \cos(\omegat + \phi) \]
kei hea:
– Ko te \(A\) te whānui, te nekehanga mōrahi mai i te tūranga taurite.
– Ko te auau koki ko \(\omega\).
– Ko te \(\phi\) te pūmau ā-wāhanga, e whakatau ana i te koki tīmatanga i \(t = 0\).
Auautanga Koki me te Wā
E pā ana te auau koki \(\omega\) ki ngā āhuatanga ā-tinana o te pūnaha wiriwiri:
\[ \omega = \sqrt{\frac{k}{m}} \]
ko \(m\) te papatipu o te mea e neke ana. Ko te wā \(T\), arā, te wā e pau ana mō te huringa katoa o te nekehanga, ka homai e:
\[ T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} \]
Ko te auau \(f\), arā, te maha o ngā wiriwiri mō ia wā, ko te taurite o te wā:
\[ f = \frac{1}{T} = \frac{\omega}{2\pi} \]
Āhuatanga me te Pūmau Āhuatanga
He mea nui te āhua \( \phi \) i roto i te whārite nekehanga \( x(t) = A \cos(\omega t + \phi) \) nā te mea ka whakatau i te tūranga tīmatanga o te matūriki i \( t = 0 \). I runga anō i te horopaki, ka taea te whakarerekē i \(\phi\) kia whakaata pai ai i ngā āhuatanga tīmatanga o te pūnaha.
Pūngao i roto i te Nekehanga Harmonic Māmā
Ko te pūngao miihini katoa (E) i roto i tētahi oscillator harmonic māmā ko te tapeke o ngā pūngao nekeneke me ngā pūngao pūmanawa, e noho pūmau ana mēnā kāore he kaha marara (pēnei i te waku).
Kaha Mahinga
Ko te pūngao pūmanawa \(U\) i roto i tētahi pūnaha puna ka hoatuhia e:
\[ U = \frac{1}{2} kx^2 \]
I te nekehanga mōrahi, kei te tihi te pūngao pūmanawa, i te tūnga taurite, he kore.
Mahinga Kinetic
Ko te pūngao nekeneke (K) o te papatipu e neke ana:
\[ K = \frac{1}{2} mv^2 \]
ko \( v \) te tere o te papatipu. Ko te pūngao nekeneke te mōrahi i te tūranga taurite, ā, ko te kore i ngā pito o te nekehanga.
Te Tiakitanga o te Energy
Ka taea te whakaatu i te mātāpono tiaki pūngao i roto i te SHM penei:
\[ E = \frac{1}{2} k A^2 = \frac{1}{2} kx^2 + \frac{1}{2} mv^2 \]
E whakaatu ana tēnei whārite, i te mea e wiri ana te papatipu, ka whakawhiti tonu te pūngao i waenga i ngā āhua nekeneke me ngā āhua pūmanawa engari ka noho pūmau tonu tō rāua tapeke.
Nekehanga Harmonic Kua Whakawhāititia, Kua Ārahihia
Ahakoa e whakaarohia ana e te nekehanga hawarite māmā ngā āhuatanga tino pai me te kore e ngaro te pūngao, he maha ngā wā ka pāngia ngā pūnaha o te ao tūturu e te whakaiti me ngā kaha akiaki o waho.
Nekehanga Harmonic Whakamarumaru
I roto i te oscillator harmonic damped, ka mahi ngā kaha ātete pēnei i te waku, te ātete hau rānei ki te nekehanga, ka heke te kaha o te oscillation i te roanga o te wā. He maha ngā wā ka whakatauirahia te kaha whakamātao penei:
\[ F_d = -bv \]
ko \(b\) te tauwehenga whakamākū, ā, ko \(v\) te tere. I runga i te nui o te whakamākū, ka taea te whakamākū iti iho i te pūnaha, te whakamākū nui rawa, te whakamākū nui rawa rānei.
Nekehanga Harmonic Pupuri
I roto i te nekehanga harmonic e peia ana, ka whakamahia he kaha ā-wā o waho \(F(t) = F_0 \cos(\omega_{d} t) \) hei pupuri i ngā wiri. Ko te urupare a te pūnaha e whakawhirinaki ana ki te whanaungatanga i waenga i te auau taraiwa \(\omega_d\) me te auau taiao \(\omega\). Ka puta te haruru ina \(\omega_d = \omega\), ka arahi pea ki ngā wiri nui.
Ngā Whakamahinga Whaihua o te SHM
He whānuitia ngā whakamahinga o te nekehanga harmonic māmā i roto i ngā mara maha:
– Karaka: Ka whakamahia e ngā karaka pendulum ngā mātāpono o te SHM hei pupuri i te pupuri tika i te wā.
– Hangarau: Ko te SHM te pūtake o te whakahaere o ngā pūnaha whakatārewatanga i roto i ngā waka, e whakarato ana i te whakamarie me te pumau.
– Ngā Pūnaha Whakawhitiwhiti Kōrero: Ka whakamahia e ngā orooro karaihe i roto i ngā hikohiko te SHM hei whakaputa auau pumau mō ngā taputapu whakawhitiwhiti kōrero.
– Ngā Taputapu Hauora: Ko ngā taputapu pēnei i ngā mīhini ultrasound e whakawhirinaki ana ki te nekehanga harmonic hei whakaputa ngaru oro mō te whakaahua.
Opaniraa
He mea nui te mārama ki ngā ariā taketake o te nekehanga harmonic māmā hei hopu i te nui o ngā āhuatanga ā-tinana. Ko te āhua ā-wā o te SHM, e tohuhia ana e te nekehanga sinusoidal me te whakahaeretia e ngā kaha whakaora, e whakarato ana i tētahi anga mō te tūhura i ngā pūnaha miihini uaua ake. Ahakoa i roto i te ahupūngao ariā, i te hangarau tono rānei, mā te mōhio ki ēnei mātāpono ka whakaratohia te tangata ki ngā taputapu hei tātari me te auaha i roto i ngā mara pūtaiao kanorau.
