Harmonic Vibration Formulas uye Mienzaniso
Kudengenyeka kweHarmonic inyaya inokosha mufizikisi, inowanzoonekwa muhurukuro dzekufamba kwenguva, mafungu, uye mashandisirwo einjiniya. Tinogona kuiwana mukufamba kwemasipiresi, mapendulum (emakona madiki), uye kunyange kudengenyeka kwemamorekuru ari muzvinhu. Inonzi "harmonic" nekuti kufamba kwayo kunotevera maitiro akajairwa uye kunogona kuenzaniswa nemabasa esine kana cosine. Chinyorwa chino chinokurukura tsananguro pfupi, mafomula akakosha, uye mienzaniso yematambudziko ane mhinduro kutibatsira kunzwisisa pfungwa yekufamba kweharmonic.
1. Kunzwisisa Harmonic Vibrations
Kudengenyeka kuri nyore kweharmonic (SHM) kufamba kwechinhu chinopfuura nepamwe nepamwe nepfungwa yekufananidza nechinhu chine simba rekudzorera rine hukuru hwakaenzana nekutama kwacho uye gwara rayo rinogara richienda papfungwa yekufananidza. Pamasvomhu, hunhu hwacho hunogona kunyorwa seizvi:
F = −kx
Chiratidzo chekusiya chinoratidza kuti simba rakananga rakapesana negwara rekutsauka. Kana chinhu chikadhonzwa kurudyi (positive x), simba rinodzoreredza riri kuruboshwe (negative), uye zvinopesana.
Masisitimu maviri anowanzo taurwa semienzaniso yeGHS ndeaya:
1. Sisitimu yechirimo–hukuru (hukuru pakupera kwechirimo).
2. Pendulum iri nyore (yemakona madiki, kazhinji < 15°).
2. Huwandu Hwakakosha muHarmonic Vibrations
Humwe huwandu hunogara huchionekwa muGHS:
– Kutsauka (x): daro rechinhu kubva panzvimbo yekuenzana (m).
– Amplitude (A): kutsauka kukuru (m).
– Nguva (T): nguva inodiwa kuti kudedera kukwane kamwe chete.
– Kuwanda (f): huwandu hwekudengenyeka kwesekondi (Hz).
– Kumhanya kweAngular / angular frequency (ω): parameter inokosha mu sine/cosine equation (rad/s).
– Chikamu (φ): ndicho chinoona mamiriro ekutanga ekufamba.
Hukama hwepakati pakati penguva nenguva:
f = 1/T
Uye hukama hwa ω na T na f:
ω = 2πf = 2π/T
3. Harmonic Vibration Equation
Chimiro chakajairika chekutsauka sebasa renguva chinogona kunyorwa:
x(t) = Chivi (ωt + φ)
kana
x(t) = A cos(ωt + φ)
Sarudzo ye sin/cos yakarurama zvakaenzana, zvichienderana nemamiriro ekutanga (semuenzaniso kana t = 0, nzvimbo yacho iri mu amplitude kana kuti pa equilibrium point).
Kumhanya uye Kukurumidza
Kumhanya ndiko kunobva pakutanga kwe displacement:
v(t) = dx/dt = Aω cos(ωt + φ) (kana x = A chivi)
kana kuti inogonawo kuva nechiratidzo chisina kunaka zvichienderana nechimiro chekutanga.
Kukurumidza ndiyo derivative yechipiri:
a(t) = d²x/dt² = −Aω² chivi(ωt + φ)
Sezvo x(t) = A sin(ωt + φ), saka:
a(t) = −ω² x(t)
Ichi chinhu chakakosha cheGHS: kukurumidza kunoenderana nekutama uye kwakapesana nekwakananga.
Kumhanya Kwakanyanya uye Kukurumidza Kwakanyanya
– v_max = Aω
– a_max = Aω²
Kumhanya kwakanyanya kunoitika kana chinhu chikapfuura nepakati pechiyero (x = 0). Kumhanya kwakanyanya kunoitika pahukuru (x = ±A).
4. Nguva yeKudengenyeka muSprings nePendulums
A. Chirimo–Misa
Kuti chitubu chive chakakodzera chine chitubu chisingachinji k uye huremu m:
T = 2π √(m/k)
ω = √(k/m)
Izvi zvinoreva kuti kana huremu hwakakura, nguva yacho inokura (inononoka). K yakakura (chitubu chakati sandara), nguva yacho inoderera (inokurumidza).
B. Pendulum iri nyore (Angle Diki)
Kune pendulum ine urefu hwetambo L uye kukurumidza kwegiravhiti g:
T = 2π √(L/g)
ω = √(g/L)
Zvinofadza kuti, mukuita angle diki, nguva yacho haitsamiri pahuremu hwependulum, asi inotsamira pahurefu negiravhiti.
5. Simba muHarmonic Vibrations
MuGHS, simba rose rinoramba riripo (kana pasina kukweshana):
– Simba rinogona kuwanikwa muchirimo:
Ep = ½ kx²
- Kinetic simba:
Ek = ½ mv²
– Simba rose remakanika:
E = Ek + Ep = ½ kA²
Kana x = ±A, v = 0 saka simba racho rinenge riripo zvachose. Kana x = 0, Ep = 0 uye simba racho rinenge riripo zvachose.
6. Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso 1: Kusarudza Nguva yeChirimo
Huremu hwe 0,5 kg hunoturikwa pachitubu chine k = 200 N/m. Sarudza nguva yekuzununguka!
Zvakapihwa: m = 0,5 kg, k = 200 N/m
Akabvunzwa: T
Mhinduro:
T = 2π √(m/k)
= 2π √(0,5 / 200)
= 2π √(0,0025)
= 2π (0,05)
= 0,1π s
≈ 0,314 s
Saka, nguva yekudedera kwechitubu inenge 0,314 s.
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Muenzaniso 2: Kuziva Frequency uye ω
Kubva pamubvunzo 1, tsvaga frequency (f) uye ω!
Mhinduro:
f = 1/T = 1/0,314 ≈ 3,18 Hz
ω = 2π/T = 2π/0,314 ≈ 20 rad/s
(kana zvakananga ω = √(k/m) = √(200/0,5) = √400 = 20 rad/s)
Saka, frequency yacho inenge 3,18 Hz uye ω = 20 rad/s.
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Muenzaniso 3: Kutsauswa kweEquation
Chinhu chinodedera zvinoenderana nehukuru hwe 0,10 m uye ω = 5 rad/s. Pa t = 0, chinhu chiri pa equilibrium uye chiri kufamba nenzira yakanaka. Sarudza displacement equation!
Zvapiwa: A = 0,10 m, ω = 5 rad/s
Mamiriro ekutanga: t = 0 → x = 0 uye v yekutanga iri positive.
Kana x(0) = 0, fomu yakakodzera iine sine:
x(t) = Chivi(ωt)
Sezvo v(t) = Aω cos(ωt), saka v(0) = Aω cos(0) = positive Aω, zvinoenderana.
Saka:
x(t) = 0,10 chivi(5t) (mamita)
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Muenzaniso 4: Kumhanya Kwakanyanya uye Kukurumidza Kwakanyanya
Kubva pamuenzaniso wechitatu, sarudza v_max uye a_max.
Mhinduro:
v_max = Aω = 0,10 × 5 = 0,50 m/s
a_max = Aω² = 0,10 × 25 = 2,5 m/s²
Saka, v_max = 0,50 m/s uye a_max = 2,5 m/s².
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Muenzaniso 5: Simba Muchirimo
Chitubu chine k = 100 N/m chinodedera nehukuru hweA = 0,20 m. Verenga simba rayo rose remakanika!
Mhinduro:
E = ½ kA²
= ½ (100)(0,20)²
= 50 × 0,04
= 2 J
Simba rose remagetsi rekudengenyeka i2 joules.
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Muenzaniso 6: Nguva yePendulum
Pendulum iri nyore yakareba 1 m. Kana g = 10 m/s², tsvaga nguva yayo.
Mhinduro:
T = 2π √(L/g)
= 2π √(1/10)
= 2π √0,1
≈ 2π (0,316)
≈ 1,99 s
Saka, nguva ye pendulum inenge 2,0 s.
7. Mhedziso
Kufamba kweharmonic kuri nyore inzira huru yekufamba kwenguva mufizikisi. Chinokosha pakuinzwisisa chiri muhukama huripo pakati pesimba rinodzoreredza simba nekutama (F = −kx), uye muenzaniso wayo wemasvomhu, unotevera basa resine/cosine. Nekunzwisisa mafomula enguva yechirimo nependulum, maequation ekutamisa-kumhanya-mhanya-kumhanya, uye pfungwa yesimba, uchaona zviri nyore kugadzirisa matambudziko ane chekuita nekudedera nemafungu.
Kana muchida, ndinogona kuwedzera mimwe mibvunzo yekudzidzira (pasina kukurukura pakutanga) kana kugadzira vhezheni pfupiso yemafomula akagadzirira bvunzo.
