(Okỉkịk — ////ito*fiCOS9?*sin9?i—/u@ksm<pk,
ỳ k = - f i H 2T}k c o s 2 d k - / * C O S 0 * + ~ ^ t c o s ỡ * ,
■í2**?rA = fẲH2íỉkilkCOs6ksindk+fLH2lk sỉndk— /i^sinớ*,
9^* = ^**+9?*» 0* =
These equations have the standard form, for which the method of perturbation theory can conveniently be used. Following this method, in the first approximation we can re place (7.6) by equations: ềk+ — M, (0*cosỹ*Ị, z a)* ^ Uk ỉ k ệ k = - / í W , {<p* s i n ỹ * |, ỳ k = — ^ r/* — M ,{/*cosẽ*Ị+ M .Ị^ co sÕ * }, ' £*»7*0* = iu //2A/,|/*sin0*)-iuA/t |!//Jksin0*).
These are obtained by averaging the right-hand sides of (7.6) in time.
Limiting our investigation to the one-frequency regimes of oscillations of a beam, we assume that any natural frequency, for example 0)j, is in the resonance. Then, the solution (7.3) may be written in the form:
y = Xj(x)Tj(t),
(7-8) V
2 = 2 j X k(x)Ik[t), k = l
and Eqs. (7.7) become:
(7.9) 1 , - - ^ + ^ , « , ^ ) .
“>JĨJ<PJ = - / * Pì(ỉ j,<Pj)>
where
P i = A /, (Ắc#;), ^ 2 = M ị { $ / S Ì n ỹ '/ } { ik- 9ik- i| - 0 ( * # » .
By way of example, we consider the transverse oscillations of a beam with hinged ends,
for which xk-= sin — X, assuming th a t Q ( x , r ) — P s ỉ n ^ Ỵ - x s i n r / and that V2—CO* — /4.1.
Then we have:
£/ = - y ( - H i + - 7 P]sin2<p\ ỉj,
wjtj<Pj = 2 [ ~ ^ + ^ ( 2 P /+ P /c o s2 9^ +3^)]^ , where
Nonlinear connected oscillations o f rigid bodies 329
The stationary values of Ệj and (Pj are determined by the equations Ệj = ộ)j = 0. Hence, we obtain:
ỉ ) = -3^ ( A - 2 P Ỉ L ± ] / p f ĩ J ^ H Ĩ ? ) i
sin2<PJ = Hiv/ LPj, cos2( f j = ~ 7 P2 ]/ P j L 2 — H i v 2 .
L r j
It has been proved that only the upper sign corresponds to the stability of oscillations of the beam. Thus, in the first approximation, we have:
(7.10)when when y = Ệjsin(ơ)jt+<pj)sin X, CO z = ^ / k( i ) s i n - ^ - ^ , A « 1 1 tj = ] / 3^ - 2P ] , itilfpj = -J.—>-*-» = --- --- ■=“ 17 / 2 ^ '- J