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Kolebaniya i volny. Lekcii.

V.A.Aleshkevich, L.G.Dedenko, V.A.Karavaev (Fizicheskii fakul'tet MGU)
Izdatel'stvo Fizicheskogo fakul'teta MGU, 2001 g. Soderzhanie

Vynuzhdennye kolebaniya s proizvol'noi chastotoi.

Budem iskat' reshenie uravneniya (2.10) v kompleksnom vide:

$\hat {\displaystyle s}(t) = \hat {\displaystyle s}_{0} e^{i\omega t}$

(2.26)

Vynuzhdayushuyu silu v pravoi chasti (2.10) takzhe zapishem v kompleksnoi forme:

$\hat {\displaystyle F}(t) = \hat {\displaystyle F}_{0} e^{i\omega t},$

(2.27)

gde $\hat {\displaystyle F}_{0} = F_{0}$ - deistvitel'noe chislo, poskol'ku dlya prostoty my polozhili, chto nachal'naya faza v vyrazhenii dlya sily (2.5) ravna nulyu.

Togda uravnenie (2.10) mozhno zapisat' v vide:

$\ddot {\displaystyle \hat {\displaystyle s}} + 2\delta \dot {\displaystyle \hat {\displaystyle s}} + \omega _{0}^{2} \hat {\displaystyle s} = {\displaystyle \frac{\displaystyle {\displaystyle \hat {\displaystyle F}_{0} }}{\displaystyle {\displaystyle m}}}e^{i\omega t}.$

(2.28)

Kompleksnuyu amplitudu $\hat {\displaystyle s}_{0} = s_{0} e^{i\varphi _{0} }$ legko nahodim podstanovkoi (2.26) v (2.28):

$( - \omega ^{2} + 2i\delta \omega + \omega _{0}^{2} )\hat {\displaystyle s}_{0} e^{i\omega t} = {\displaystyle \frac{\displaystyle {\displaystyle \hat {\displaystyle F}_{0} }}{\displaystyle {\displaystyle m}}}e^{i\omega t}.$

(2.29)

Otsyuda poluchaem:

$\hat {\displaystyle s}_{0} = {\displaystyle \frac{\displaystyle {\displaystyle \hat {\displaystyle F}_{0} }}{\displaystyle {\displaystyle m(\omega _{0}^{2} - \omega ^{2} + 2i\delta \omega )}}}.$

(2.30)

Iz (2.30) netrudno naiti amplitudu kolebanii $s_{0} = {\displaystyle \left| {\displaystyle \hat {\displaystyle s}_{0} } \right|} :$

$s_{0} = {\displaystyle \frac{\displaystyle {\displaystyle F_{0} }}{\displaystyle {\displaystyle m\sqrt {\displaystyle (\omega _{0}^{2} - \omega ^{2})^{2} + 4\delta ^{2}\omega ^{2}} }}}$

(2.31)

i fazu $\varphi _{0} = \arg \hat {\displaystyle s}_{0} :$

$tg\varphi _{0} = {\displaystyle \frac{\displaystyle {\displaystyle Im\hat {\displaystyle s}_{0} }}{\displaystyle {\displaystyle Re\hat {\displaystyle s}_{0} }}} = {\displaystyle \frac{\displaystyle {\displaystyle 2\delta \omega }}{\displaystyle {\displaystyle \omega ^{2} - \omega _{0}^{2} }}},$

(2.32)

polnost'yu opredelyayushie vynuzhdennye kolebaniya (2.25).

Zavisimost' amplitudy $s_{0}$ ot chastoty $\omega ,$ zadavaemaya formuloi (2.31), nazyvaetsya amplitudno-chastotnoi harakteristikoi (AChH), a zavisimost' $\varphi _{0} (\omega ),$ opisyvaemaya formuloi (2.32), nazyvaetsya fazo-chastotnoi harakteristikoi (FChH). Na ris. 2.3 izobrazhena AChH, kotoraya otobrazhaet narastanie amplitudy $s_{0}$ pri priblizhenii $\omega$ k $\omega _{0} .$ Eto yavlenie poluchilo nazvanie rezonansa smeshenii. Interesno, chto maksimal'noe znachenie amplitudy, v $Q$ raz prevoshodyashee staticheskoe smeshenie ${\displaystyle F_{0} } / {\displaystyle k,}$ dostigaetsya na chastote

$\omega _{s} = \sqrt {\displaystyle \omega _{0}^{2} - 2\delta ^{2},}$

(2.33)

kotoraya neskol'ko men'she kak sobstvennoi chastoty $\omega _{0} ,$ tak i chastoty zatuhayushih kolebanii $\sqrt {\displaystyle \omega _{0}^{2} - \delta ^{2}} .$ Dlya prakticheskih celei dlya chastot $\omega ,$ lezhashih vblizi chastoty $\omega _{0} ,$ formula (2.31) mozhet byt' znachitel'no uproshena. Tak, mozhno polozhit'

$(\omega _{0}^{2} - \omega ^{2})^{2} = (\omega _{0} - \omega )^{2}(\omega _{0} + \omega )^{2} \approx (\omega _{0} - \omega )^{2} \cdot 4\omega _{0}^{2} ;$	(2.34)
$4\delta ^{2}\omega ^{2} \approx 4\delta ^{2}\omega _{0}^{2} .$

Ris. 2.3.

S uchetom priblizhenii (2.34) formula (2.31) primet vid:

$s_{0} (\omega ) = {\displaystyle \frac{\displaystyle {\displaystyle F_{0} }}{\displaystyle {\displaystyle k}}}Q{\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle \sqrt {\displaystyle \left( {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0} - \omega }}{\displaystyle {\displaystyle \delta }}}} \right)^{2} + 1} }}}.$

(2.35)

V fizike bezrazmernuyu funkciyu

$L(\omega ) = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle \sqrt {\displaystyle \left( {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0} - \omega }}{\displaystyle {\displaystyle \delta }}}} \right)^{2} + 1} }}}$

(2.36)

nazyvayut Lorencevoi, a grafik etoi funkcii nazyvayut Lorencevym konturom. Shirinu $\Delta \omega$ etogo kontura, opredelyayushuyu ostrotu rezonansa, nahodyat iz usloviya ubyvaniya vdvoe energii kolebatel'noi sistemy, proporcional'noi kvadratu amplitudy $s_{0} (\omega )$ v (2.35), chto ekvivalentno priblizhennomu sootnosheniyu

${\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle \sqrt {\displaystyle \left( {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle {\displaystyle {\displaystyle \Delta \omega } / {\displaystyle 2}}}}{\displaystyle {\displaystyle \delta }}}} \right)^{2} + 1} }}} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle \sqrt {\displaystyle 2} }}} \approx 0,7,$

(2.37)

kotoroe poyasnyaetsya risunkom 2.4. Pri etom uslovii ${\displaystyle \frac{\displaystyle {\displaystyle \Delta \omega }}{\displaystyle {\displaystyle 2}}} = \delta ,$ t.e. $\Delta \omega = 2\delta .$ Shirina Lorenceva kontura harakterizuet polosu propuskaniya kolebatel'noi sistemy, t.e. takuyu oblast' chastot vneshnei sily, dlya kotoryh sistema effektivno otklikaetsya na garmonicheskoe vneshnee vozdeistvie. Legko videt', chto dobrotnost' sistemy ravna

$Q = {\displaystyle \frac{\displaystyle {\displaystyle \pi }}{\displaystyle {\displaystyle \delta T}}} = {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0} }}{\displaystyle {\displaystyle \Delta \omega }}},$

(2.38)

t.e. obratno proporcional'na polose propuskaniya.

Ris. 2.4.

S umen'sheniem koefficienta $\delta$ AChH menyaet svoyu formu, kak eto izobrazheno punktirom na ris. 2.3 dlya ${\displaystyle \delta }' \lt \delta .$ Polosa propuskaniya $\Delta \omega$ umen'shaetsya, dobrotnost' ${\displaystyle Q}'$ vozrastaet, i rezonans stanovitsya bolee ostrym.

Fazo-chastotnaya harakteristika dlya dvuh razlichnyh koefficientov zatuhaniya izobrazhena na ris. 2.5. Fizicheskoe soderzhanie zavisimosti $\varphi _{0} (\omega )$ my podrobno obsudili dlya treh razlichnyh rezhimov vynuzhdennyh kolebanii. Otmetim lish', chto s umen'sheniem zatuhaniya $\delta$ krivaya $\varphi _{0} (\omega )$ stanovitsya bolee "chuvstvitel'noi" k izmeneniyu chastoty vblizi rezonansa.

Ris. 2.5.

Naryadu s rezonansom smeshenii, mozhno govorit' o rezonanse skorostei $\dot {\displaystyle s}$ i rezonanse uskorenii $\ddot {\displaystyle s}.$

Skorost' koleblyusheisya massy ravna:

$\dot {\displaystyle s} = s_{0} \omega \sin (\omega t + \varphi _{0} + {\displaystyle {\displaystyle \pi } / {\displaystyle 2}}),$

(2.39)

a ee uskorenie:

$\ddot {\displaystyle s} = s_{0} \omega ^{2}\sin (\omega t + \varphi _{0} + \pi ),$

(2.40)

t.e. amplitudno-chastotnaya harakteristika dlya skorosti poluchaetsya umnozheniem AChH (2.31) na $\omega ,$ a dlya uskoreniya - na $\omega ^{2}$ :

$v_{0} = s_{0} \omega = {\displaystyle \frac{\displaystyle {\displaystyle F_{0} }}{\displaystyle {\displaystyle m\sqrt {\displaystyle \left( {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0}^{2} - \omega ^{2}}}{\displaystyle {\displaystyle \omega }}}} \right)^{2} + 4\delta ^{2}} }}},$

$w_{0} = s_{0} \omega ^{2} = {\displaystyle \frac{\displaystyle {\displaystyle F_{0} }}{\displaystyle {\displaystyle m\sqrt {\displaystyle \left( {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0}^{2} }}{\displaystyle {\displaystyle \omega ^{2}}}} - 1} \right)^{2} + 4{\displaystyle \frac{\displaystyle {\displaystyle \delta ^{2}}}{\displaystyle {\displaystyle \omega ^{2}}}}} }}}.$

Na ris. 2.6 izobrazheny chastotnye zavisimosti amplitud skorosti $v_{0} = s_{0} \omega$ i uskoreniya $w_{0} = s_{0} \omega ^{2}.$

Ris. 2.6.

Harakterno, chto rezonans skorosti proishodit na chastote $\omega _{{\displaystyle s}'} = \omega _{0} ,$ a rezonans uskoreniya - pri $\omega _{{\displaystyle s}''} \gt \omega _{0} .$ Otmetim, chto vse rezonansnye chastoty svyazany mezhdu soboi:

$\omega _{s} \cdot \omega _{{\displaystyle s}''} = \omega _{{\displaystyle s}'}^{2} = \omega _{0}^{2} .$

(2.41)

Otmetim takzhe, chto po prichinam, rassmotrennym ranee, v oblasti nizkih chastot maly kak uskorenie, tak i skorost'. V oblasti vysokih chastot uskorenie konechno $(s_{0} \omega ^{2} \to {\displaystyle {\displaystyle F_{0} } / {\displaystyle m}})$ i obespechivaetsya lish' vneshnei siloi. Odnako skorost' po-prezhnemu neznachitel'na, poskol'ku telo ne uspevaet razognat'sya.

Ne predstavlyaet truda narisovat' samostoyatel'no fazo-chastotnye harakteristiki dlya skorosti i dlya uskoreniya, pol'zuyas' formulami (2.39) i (2.40), poskol'ku oni poluchayutsya prostym sdvigom FChH dlya smesheniya (2.32), izobrazhennoi na ris. 2.5, vverh sootvetstvenno na $\pi / 2$ ili na $\pi .$

V zaklyuchenie rassmotrim vopros o podvode energii k oscillyatoru pri proizvol'noi chastote vynuzhdayushei sily. Srednyaya za period moshnost' etoi sily ravna

$N = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle T}}}{\displaystyle \int\limits_{0}^{T} {\displaystyle F(t)\dot {\displaystyle s}(t)dt} } = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle T}}}{\displaystyle \int\limits_{0}^{T} {\displaystyle F_{0} \sin \omega t \cdot v_{0} \sin \left( {\displaystyle \omega t + \varphi _{0} + {\displaystyle \frac{\displaystyle {\displaystyle \pi }}{\displaystyle {\displaystyle 2}}}} \right)dt} } = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle T}}}F_{0} v_{0} {\displaystyle \int\limits_{0}^{T} {\displaystyle \sin \omega t\sin (\omega t + \psi _{0} )dt} } = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle T}}}F_{0} v_{0} {\displaystyle \int\limits_{0}^{T} {\displaystyle \sin ^{2}\omega t\cos \psi _{0} dt} } + {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle T}}}F_{0} v_{0} {\displaystyle \int\limits_{0}^{T} {\displaystyle \sin \omega t\cos \omega t\sin \psi _{0} dt} } = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}F_{0} v_{0} \cos \psi _{0} ,$

gde $\psi _{0} = \varphi _{0} + {\displaystyle \frac{\displaystyle {\displaystyle \pi }}{\displaystyle {\displaystyle 2}}}$ - sdvig faz mezhdu skorost'yu i siloi. My vidim, chto maksimum podvodimoi k oscillyatoru moshnosti dostigaetsya na chastote $\omega _{0} ,$ poskol'ku pri etom maksimal'ny i amplituda skorosti $v_{0},$ i $\cos \psi _{0} (\psi _{0} = 0).$ Pri drugih chastotah vynuzhdayushei sily eta moshnost' bystro umen'shaetsya i stremitsya k nulyu, kak pri $\omega \to 0,$ tak i pri $\omega \to \infty .$

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