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Mehanika tverdogo tela. Lekcii.

V.A.Aleshkevich, L.G.Dedenko, V.A.Karavaev (Fizicheskii fakul'tet MGU)
Izdatel'stvo Fizicheskogo fakul'teta MGU, 1997 g. Soderzhanie

Proeciruya vektory L i $\omega$ na osi sistemy xyz, zhestko svyazannoi s tverdym telom, poluchim:

${\displaystyle \frac{\displaystyle {\displaystyle \partial L_{x} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{y} L_{z} - \omega _{z} L_{y} = 0;$

(3.49)

${\displaystyle \frac{\displaystyle {\displaystyle \partial L_{y} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{z} L_{x} - \omega _{x} L_{z} = 0;$

(3.50)

${\displaystyle \frac{\displaystyle {\displaystyle \partial L_{z} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{x} L_{y} - \omega _{y} L_{x} = 0;$

(3.51)

Poskol'ku osi Ox, Oy i Oz - glavnye osi inercii dlya tochki zakrepleniya, to $L_{x} = J_{x} \omega _{x} ,\; L_{y} = J_{y} \omega _{y} ,\; L_{z} = J_{z} \omega _{z}$ i iz (3.49-3.51) budem imet' sleduyushie uravneniya:

$J_{x} {\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{x} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{y} \omega _{z} \left( {\displaystyle J_{z} - J_{y} } \right) = 0;$

(3.52)

$J_{y} {\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{y} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{z} \omega _{x} \left( {\displaystyle J_{x} - J_{z} } \right) = 0;$

(3.53)

$J_{z} {\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{z} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{x} \omega _{y} \left( {\displaystyle J_{y} - J_{x} } \right) = 0,$

(3.54)

gde $J_{x}, J_{y}, J_{z}$ - glavnye momenty inercii tela. Obychno eti uravneniya nazyvayut uravneniyami Eilera pri otsutstvii momentov vneshnih sil.

V chastnom sluchae (ris. 3.14) $J_{x }= J_{y},$ i iz (3.52-3.54) poluchaem:

${\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{x} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{y} \omega _{0} = 0;$

(3.55)

${\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{y} }}{\displaystyle {\displaystyle \partial t}}} + \omega _{x} \omega _{0} = 0;$

(3.56)

${\displaystyle \frac{\displaystyle {\displaystyle \partial \omega _{z} }}{\displaystyle {\displaystyle \partial t}}} = 0,$

(3.57)

gde vvedeno oboznachenie

$\omega _{0} = \omega _{z} \cdot {\displaystyle \frac{\displaystyle {\displaystyle J_{z} - J_{y} }}{\displaystyle {\displaystyle J_{x} }}}.$

(3.58)

Iz (3.57) sleduet, chto $\omega _{z} = {\displaystyle \rm const},$ to est' proekciya vektora $\omega$ na os' simmetrii tela ostaetsya postoyannoi. Yasno, chto $\omega _{0}$ - takzhe postoyannaya velichina. Ee fizicheskii smysl stanovitsya ponyatnym, esli zapisat' reshenie uravnenii (3.55, 3.56):

$\omega _{x} = \omega _{ \bot } \cos \left( {\displaystyle \omega _{0} t + \varphi } \right); \omega _{y} = \omega _{ \bot } \sin \left( {\displaystyle \omega _{0} t + \varphi } \right);$

(3.59)

gde $\omega _{ \bot } = \sqrt {\displaystyle \omega _{x}^{2} + \omega _{y}^{2} }$ - proekciya vektora $\omega$ na ploskost' xy.

Takim obrazom, vektor $\omega$ sostavlyaet s os'yu simmetrii tela ugol $\theta = {\displaystyle \rm arctg}{\displaystyle \frac{\displaystyle {\displaystyle \omega _{ \bot } }}{\displaystyle {\displaystyle \omega _{z} }}}$ i vrashaetsya vokrug etoi osi, kak sleduet iz (3.59), s postoyannoi uglovoi skorost'yu $\omega _{0}$ Nachal'naya faza $\varphi$ etogo vrasheniya opredelyaetsya nachal'nymi usloviyami.

Posmotrim, kak budet vyglyadet' dvizhenie tverdogo tela v laboratornoi sisteme x₀y₀z₀. Poskol'ku nam izvestny znacheniya $\omega _{x} , \omega _{y}$ i $\omega _{z} ,$ to zakon dvizheniya tela (zavisimost' uglov Eilera ot vremeni) v principe mozhet byt' poluchen iz kinematicheskih uravnenii Eilera (1.30-1.32). Odnako eto svyazano s resheniem v obshem sluchae dovol'no slozhnyh differencial'nyh uravnenii, poetomu my ogranichimsya kachestvennym rassmotreniem dvizheniya tela. V silu togo, chto

$\bf L} = J_{x} \omega _{x} {\displaystyle \bf i} + J_{y} \omega _{y} {\displaystyle \bf j} + J_{z} \omega _{z} {\displaystyle \bf k$

(3.60)

(i, j, k - orty glavnyh osei inercii tela), a $J_{x} = J_{y},$ mozhno zapisat'

${\displaystyle \bf L} = J_{z} \omega _{z} {\displaystyle \bf k} + J_{x} \left( {\displaystyle \omega _{x} {\displaystyle \bf i} + \omega _{y} {\displaystyle \bf j}} \right) + J_{x} \omega _{z} {\displaystyle \bf k} - J_{x} \omega _{z} {\displaystyle \bf k}.$

(3.61)

Zdes' dobavlen i vychten chlen $J_{x} \omega _{z} {\displaystyle \bf k},$ chto pozvolyaet predstavit' (3.61) v vide

${\displaystyle \bf L} = \left( {\displaystyle J_{z} - J_{x} } \right)\omega _{z} {\displaystyle \bf k} + J_{x} \omega$

(3.62)

Otsyuda vidno, chto k (os' figury), L i $\omega$ lezhat v odnoi ploskosti. Iz (3.62) sleduet, chto

$\omega = \Omega - \omega _{0} k,$

(3.63)

gde

$\Omega = {\displaystyle \frac{\displaystyle {\displaystyle {\displaystyle \bf L}}}{\displaystyle {\displaystyle J_{x} }}}.$

(3.64)

est' sostavlyayushaya uglovoi skorosti po napravleniyu L. Ploskost', v kotoroi lezhat os' figury, $\omega$ i L, povorachivaetsya (precessiruet) vokrug napravleniya L s uglovoi skorost'yu $\Omega,$ nazyvaemoi skorost'yu precessii (ris. 3.16). Samo dvizhenie nazyvaetsya regulyarnoi precessiei svobodnogo simmetrichnogo volchka.

Ris. 3.16.

Otmetim, chto v sluchae veretenoobraznogo tela, izobrazhennogo na ris. 3.16, $J_{z} \lt J_{y} ,$ poetomu $\omega _{0} \lt 0$ (sm. (3.58)), i vektor - $\omega _{0} {\displaystyle \bf k}$ napravlen v tu zhe storonu, chto i k.

Zamechanie 1. Zakreplenie aksial'no simmetrichnogo tverdogo tela v centre mass mozhet byt' vypolneno ne tol'ko s pomosh'yu kardanova podvesa, no, naprimer, tak, kak pokazano na ris. 3.17. Massivnoe telo, sechenie kotorogo ploskost'yu risunka zashtrihovano, sharnirno zakrepleno v tochke O, sovpadayushei s centrom mass tela.

Ris. 3.17.

Zamechanie 2. Ispol'zuya postroenie Puanso (sm. lekciyu 2), regulyarnoi precessii svobodnogo simmetrichnogo volchka mozhno dat' naglyadnuyu geometricheskuyu interpretaciyu (ris. 3.18).

Ris. 3.18.

Moment impul'sa L, tela otnositel'no nepodvizhnogo centra mass O predstavlyaet soboi vektor, postoyannyi po velichine i napravleniyu. Ellipsoid inercii tela s centrom v tochke O, sechenie kotorogo izobrazheno na ris. 3.18, yavlyaetsya ellipsoidom vrasheniya. Kasatel'naya k ellipsoidu ploskost' BB' provedena cherez polyus R peresecheniya mgnovennoi uglovoi skorosti $\omega$ s ellipsoidom; eta ploskost' perpendikulyarna k vektoru L i v laboratornoi sisteme otscheta sohranyaet svoe polozhenie neizmennym. Pri regulyarnoi precessii volchka ellipsoid inercii tela katitsya po ploskosti BB' bez skol'zheniya, tak chto geometricheskim mestom polyusov R yavlyaetsya okruzhnost' radiusa $R ,$ prinadlezhashaya ploskosti BB'.

Zamechanie 3. Vo izbezhanie putanicy otmetim sleduyushee. Opisannoe vyshe dvizhenie svyazano s izmeneniem ugla precessii $\psi$ (sm. ris. 1.3), poetomu ono i bylo nazvano regulyarnoi precessiei (kinematicheskoe opredelenie). Odnako sushestvuyut opredeleniya precescii kak dvizheniya osi simmetrii tela pod deistviem momenta vneshnih sil (dinamicheskoe opredelenie, sm. lekciyu 4). Opisannoe zhe vyshe dvizhenie s tochki zreniya dinamicheskogo opredeleniya nazyvayut nutaciei.

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