Astronet > Kolebaniya i volny

po tekstam po klyuchevym slovam v glossarii po saitam perevod po katalogu

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

Svobodnye kolebaniya v dissipativnyh sistemah s vyazkim treniem.

V real'nyh sistemah vsegda proishodit dissipaciya energii. Esli poteri energii ne budut kompensirovat'sya za schet vneshnih ustroistv, to kolebaniya s techeniem vremeni budut zatuhat' i cherez kakoe-to vremya prekratyatsya voobshe.

Formal'no zatuhayushie kolebaniya opisyvayutsya uravneniem

$m\ddot {\displaystyle s} = F_{\tau } (s) + F_{tr} (\dot {\displaystyle s}),$

(1.46)

kotoroe, v otlichie ot (1.2), pomimo vozvrashayushei sily $F_{\tau },$ soderzhit i silu treniya $F_{tr}.$ Sila soprotivleniya dvizheniyu, voobshe govorya, zavisit kak ot napravleniya skorosti (naprimer, pri suhom trenii), tak i ot velichiny skorosti (pri dvizhenii v vyazkoi srede). Esli vozvrashayushaya sila proporcional'na smesheniyu: $F_{\tau } (s) = - ks,$ gde $k$ - koefficient proporcional'nosti (dlya pruzhinnogo mayatnika - zhestkost' pruzhiny), to uravnenie (1.46) mozhno perepisat' v vide

$\ddot {\displaystyle s} - {\displaystyle \frac{\displaystyle {\displaystyle F_{tr} }}{\displaystyle {\displaystyle m}}} + \omega _{0}^{2} s = 0,$

(1.47)

gde $\omega _{0} = \sqrt {\displaystyle {\displaystyle \frac{\displaystyle {\displaystyle k}}{\displaystyle {\displaystyle m}}}}$ - sobstvennaya chastota nezatuhayushih garmonicheskih kolebanii.

Vnachale my rassmotrim zatuhayushie kolebaniya v sluchae, kogda na koleblyusheesya telo deistvuet sila vyazkogo treniya, proporcional'naya skorosti: $F_{tr} = - \Gamma \dot {\displaystyle s}.$ Takaya situaciya mozhet imet' mesto, naprimer, pri kolebatel'nom dvizhenii tela v vozduhe ili zhidkosti, kogda chislo Reinol'dsa ${\rm Re} \sim 1$ ili ${\rm Re}\lt 1$ . Togda uravnenie (1.47) mozhno zapisat' v vide:

$\ddot {\displaystyle s} + 2\delta \dot {\displaystyle s} + \omega _{0}^{2} s = 0,$

(1.48)

gde $\delta = \Gamma / 2m$ - koefficient, ili pokazatel' zatuhaniya.

Obshaya ideya resheniya odnorodnyh lineinyh uravnenii tipa (1.48) zaklyuchaetsya v sleduyushem: v kachestve funkcional'noi zavisimosti $s(t)$ nado vybrat' takuyu, kotoraya pri differencirovanii po vremeni perehodit v samu sebya, to est' eksponentu: $s(t) = s_{0} e^{\lambda t}.$ Podstavim ee v uravnenie (1.48):

$s_{0} e^{\lambda t}(\lambda ^{2} + 2\delta \lambda + \omega _{0}^{2} ) = 0.$

(1.49)

Poskol'ku $e^{\lambda t} \ne 0,$ poluchaem tak nazyvaemoe "harakteristicheskoe" uravnenie:

$\lambda ^{2} + 2\delta \lambda + \omega _{0}^{2} = 0,$

(1.50)

kotoroe v dannom sluchae (dlya uravneniya vtorogo poryadka) imeet dva kornya

$\lambda _{1,2} = - \delta \pm \sqrt {\displaystyle \delta ^{2} - \omega _{0}^{2} },$

(1.51)

a samo uravnenie (1.48) - dva nezavisimyh resheniya: $s_{1} (t) = s_{01} e^{\lambda _{1} t}$ i $s_{2} (t) = s_{02} e^{\lambda _{2} t}.$ V silu lineinosti uravneniya (1.48) summa lyubyh ego reshenii takzhe yavlyaetsya resheniem, to est' spravedliv tak nazyvaemyi "princip superpozicii" reshenii, i obshim resheniem dannogo uravneniya yavlyaetsya

$s(t) = s_{01} e^{( - \delta + \sqrt {\displaystyle \delta ^{2} - \omega _{0}^{2} } )t} + s_{02} e^{( - \delta - \sqrt {\displaystyle \delta ^{2} - \omega _{0}^{2} } )t}.$

(1.52)

Reshenie soderzhit dve nezavisimye konstanty $s_{01}$ i $s_{02},$ kotorye opredelyayutsya iz nachal'nyh uslovii $s(0), v(0).$

V zavisimosti ot sootnosheniya $\delta$ i $\omega _{0}$ vozmozhny tri sluchaya.

Esli $\delta \lt \omega _{0},$ to $\sqrt {\displaystyle \delta ^{2} - \omega _{0}^{2} } = i\sqrt {\displaystyle \omega _{0}^{2} - \delta ^{2}},$ gde $i \equiv \sqrt {\displaystyle - 1}$ - "mnimaya" edinica. Reshenie yavlyaetsya kompleksnym¹, no, poskol'ku nachal'nye usloviya deistvitel'nye, to s pomosh'yu formuly Eilera:

$e^{i\varphi } = \cos \varphi + i\sin \varphi$

(1.53)

netrudno pokazat', chto obshee reshenie budet deistvitel'no i mozhet byt' zapisano v vide:

$s(t) = s_{0} e^{ - \delta t}\sin (\omega t + \varphi _{0} ),$

(1.54)

to est' predstavlyaet soboi zatuhayushie kolebaniya, chastota kotoryh $\omega$ men'she, chem u sobstvennyh nezatuhayushih kolebanii:

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

(1.55)

Kolebaniya, opisyvaemye (1.54), ne yavlyayutsya garmonicheskimi (ris. 1.14). Pod ih amplitudoi budem ponimat' velichinu

$A(t) = s_{0} e^{ - \delta t},$

(1.56)

kotoraya monotonno ubyvaet so vremenem. "Dlitel'nost'" kolebanii harakterizuetsya vremenem zatuhaniya

$\tau = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle \delta }}}.$

(1.57)

Ris. 1.14.

Esli podstavit' $\tau$ v (1.56), to legko videt', chto po istechenii vremeni zatuhaniya $\tau$ amplituda ubyvaet v e raz. Kolichestvo sovershennyh sistemoi kolebanii za vremya $\tau$ ravno otnosheniyu etogo vremeni k periodu zatuhayushih kolebanii $T = 2\pi / \omega .$ Esli zatuhanie v sisteme malo $(\delta \ll \omega _{0} ),$ to period kolebanii $T \approx 2\pi / \omega _{0},$ i chislo etih kolebanii veliko:

$N = {\displaystyle \frac{\displaystyle {\displaystyle \tau }}{\displaystyle {\displaystyle T}}} \approx {\displaystyle \frac{\displaystyle {\displaystyle \omega _{0} }}{\displaystyle {\displaystyle 2\pi \delta }}} \gg 1.$

(1.58)

Eksponencial'nyi zakon ubyvaniya amplitudy so vremenem pozvolyaet vvesti bezrazmernyi parametr - logarifmicheskii dekrement zatuhaniya $\theta,$ kotoryi raven logarifmu otnosheniya dvuh posledovatel'nyh otklonenii v odnu i tu zhe storonu:

$\theta = \ln {\displaystyle \frac{\displaystyle {\displaystyle A(t)}}{\displaystyle {\displaystyle A(t + T)}}} = \delta T.$

(1.59)

Iz (1.57), (1.58) i (1.59) nahodim:

$\theta = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle N}}}.$

(1.60)

Logarifmicheskii dekrement zatuhaniya mozhno ocenit', esli podschitat' chislo kolebanii, sovershennyh sistemoi za vremya zatuhaniya $\tau,$ to est' do umen'sheniya amplitudy kolebanii primerno v 3 raza. Chem bol'she chislo etih kolebanii, tem men'she poteri energii v sisteme.

Prosledim za ubyvaniem energii, zapasennoi oscillyatorom, s techeniem vremeni. Ispol'zuya (1.54), zapishem po analogii s (1.24) i (1.25) vyrazheniya dlya potencial'noi i kineticheskoi energii oscillyatora:

$E_{pot} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}ks_{0}^{2} e^{ - 2\delta t}\sin ^{2}\left( {\displaystyle \omega t + \varphi _{0} } \right),$

(1.61)

$E_{kin} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}m\omega ^{2}s_{0}^{2} e^{ - 2\delta t}\cos ^{2}(\omega t + \varphi _{0} ).$

(1.62)

Zametim, chto, strogo govorya, skorost' ravna

$v = \dot {\displaystyle s} = - s_{0} \delta e^{ - \delta t}\sin (\omega t + \varphi _{0} ) + s_{0} \omega e^{ - \delta t}\cos (\omega t + \varphi _{0} ).$

(1.63)

Ochevidno, chto esli $\delta \ll \omega,$ to pervym slagaemym v (1.63) mozhno prenebrech' i zapisat' vyrazhenie dlya kineticheskoi energii v vide (1.62). Summarnaya energiya oscillyatora ubyvaet so vremenem:

$E(t) = E_{pot} + E_{kin} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}s_{0}^{2} e^{ - 2\delta t}{\displaystyle \left[ {\displaystyle k\sin ^{2}(\omega t + \varphi _{0} ) + m\omega ^{2}\cos ^{2}(\omega t + \varphi _{0} )} \right]}.$

(1.64)

Primem vo vnimanie, chto pri $\delta \ll \omega _{0}$ chastota $\omega \approx \omega _{0} .$ Tak kak $k = m\omega _{0}^{2},$ to (1.64) okonchatel'no zapishetsya v vide

$E(t) = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}s_{0}^{2} m\omega _{0}^{2} e^{ - 2\delta t} = E_{0} e^{ - 2\delta t}.$

(1.65)

Polnaya energiya oscillyatora, ravnaya vnachale $E_{0} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2}}}s_{0} m\omega _{0}^{2},$ monotonno ubyvaet so vremenem po eksponencial'nomu zakonu i umen'shaetsya v e raz za vremya

$\tau _{E} = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle 2\delta }}} = {\displaystyle \frac{\displaystyle {\displaystyle \tau }}{\displaystyle {\displaystyle 2}}}.$

(1.66)

"Kachestvo" kolebatel'noi sistemy harakterizuyut bezrazmernym parametrom $Q,$ nazyvaemym dobrotnost'yu. Dobrotnost' proporcional'na otnosheniyu zapasennoi energii $E(t)$ k energii $\Delta E_{T},$ teryaemoi za period (ris. 1.15):

$Q = 2\pi {\displaystyle \frac{\displaystyle {\displaystyle E(t)}}{\displaystyle {\displaystyle \Delta E_{T} }}} = 2\pi {\displaystyle \frac{\displaystyle {\displaystyle E_{0} e^{ - 2\delta t}}}{\displaystyle {\displaystyle E_{0} e^{ - 2\delta t} - E_{0} e^{ - 2\delta (t + T)}}}} = {\displaystyle \frac{\displaystyle {\displaystyle 2\pi }}{\displaystyle {\displaystyle 1 - e^{ - 2\delta T}}}}.$

(1.67)

Esli chislo kolebanii veliko, to $\delta T = {\displaystyle \frac{\displaystyle {\displaystyle 1}}{\displaystyle {\displaystyle N}}} \ll 1.$ Togda

$Q = {\displaystyle \frac{\displaystyle {\displaystyle 2\pi }}{\displaystyle {\displaystyle 1 - e^{ - 2\delta T}}}} = {\displaystyle \frac{\displaystyle {\displaystyle 2\pi }}{\displaystyle {\displaystyle 1 - (1 - 2\delta T + \ldots)}}} \approx {\displaystyle \frac{\displaystyle {\displaystyle \pi }}{\displaystyle {\displaystyle \theta }}} = \pi N.$

(1.68)

Pri eksponencial'nom zakone ubyvaniya energii so vremenem dobrotnost' $Q$ okazyvaetsya postoyannoi velichinoi, kotoruyu, kak i logarifmicheskii dekrement zatuhaniya $\theta,$ mozhno legko ocenit' po chislu kolebanii $N_{Q} = \pi N \approx 3N,$ sovershennyh sistemoi do ih polnogo prekrasheniya (za vremya $3\tau$ amplituda kolebanii umen'shaetsya v $e^{3} \approx 20$ raz, to est' kolebaniya prakticheski polnost'yu zatuhayut).

Ris. 1.15.

Sleduet otmetit', chto dobrotnost' ne tol'ko harakterizuet zatuhanie kolebanii, no i yavlyaetsya vazhnoi velichinoi, opredelyayushei parametry vynuzhdennyh kolebanii, osushestvlyaemyh pod deistviem vneshnei periodicheskoi sily (sm. dalee).

¹Bolee podrobno metod kompleksnyh amplitud budet obsuzhdat'sya nizhe, pri rassmotrenii vynuzhdennyh kolebanii.

Nazad| Vpered

Publikacii s klyuchevymi slovami: kolebaniya - volny Publikacii so slovami: kolebaniya - volny
Sm. takzhe: Eho iz glubin krasnogo giganta

Obsudit' etu publikaciyu

Versiya dlya pechati

Astrometriya - Astronomicheskie instrumenty - Astronomicheskoe obrazovanie - Astrofizika - Istoriya astronomii - Kosmonavtika, issledovanie kosmosa - Lyubitel'skaya astronomiya - Planety i Solnechnaya sistema - Solnce