We achieve the periodic solutions \(p_{2} (\tau ,\mu^{ - 1} ),\,\,\dot{p}_{2} (\tau ,\mu^{ - 1} ),\,\,\gamma_{2} (\tau ,\mu^{ - 1} ),\,\,\dot{\gamma }_{2} (\tau ,\mu^{ - 1} )\) of system (14) when:
$$p_{2} (0,0) = \dot{p}_{2} (0,0) = \dot{\gamma }_{2} (0,\mu^{ - 1} ) = 0\,.$$
(16)
The generating system of (14) is obtained when \(\mu \to \infty\) in the form:
$$\ddot{p}_{2}^{(0)} + \omega^{2} p_{2}^{(0)} = 0,\quad \ddot{\gamma }_{2}^{(0)} + \gamma_{2}^{(0)} = 0.$$
(17)
So the periodic solutions of system (17), when the period \(T_{0} = 2\pi n\), become:
$$p_{2}^{(0)} = M_{1} \cos \,\omega \tau + M_{2} \sin \,\omega \tau ,\quad \gamma_{2}^{(0)} = M_{3} \cos \,\tau ,$$
(18)
where \(M_{i} ,\,\,\,i = 1,2,3\) are constants to be determined.
Assuming the following solutions of system (14) [9]:
$$\begin{aligned} & p_{2} (\tau ,\mu ) = \tilde{M}_{1} \cos \,\omega \tau + \tilde{M}_{2} \sin \,\omega \tau + \sum\limits_{k = 2}^{\infty } {\mu^{ - k} G_{k} (\tau )} , \\ & \gamma_{2} (\tau ,\mu ) = \tilde{M}_{3} \cos \,\tau + \sum\limits_{k = 2}^{\infty } {\mu^{ - k} H_{k} (\tau )} \,, \\ \end{aligned}$$
(19)
with a period \(T(\,\mu^{ - 1} ) = T_{0} + \alpha (\,\mu^{ - 1} )\) which reduces to (18) at \(\mu \to \infty\). Let us define the quantities \(\tilde{M}_{i} ,\,\,i = 1,2,3\) as follow:
$$\tilde{M}_{i} = M_{i} + \beta_{i} (\mu^{ - 1} ),\quad i = 1,2,3,$$
(20)
where \(\beta_{i}\) are functions of \(\mu^{ - 1}\) which represent the deviations of the initial values of \(p_{2} ,\,\,\dot{p}_{2} ,\,\,\gamma_{2}\) for system (14) from their initial values of generating system (17) such that \(\beta_{i} (\,0) = 0\).
Let us express the initial conditions (16) by the relations:
$$p_{2} (0,\mu^{ - 1} ) = \tilde{M}_{1} ,\quad \dot{p}_{2} (0,\mu^{ - 1} ) = \omega \tilde{M}_{2} ,\quad \gamma_{2} (0,\mu^{ - 1} ) = \tilde{M}_{3} ,\quad \dot{\gamma }_{2} (0,\mu^{ - 1} ) = 0.$$
(21)
We rewrite the periodic solutions (18) in the form:
$$p_{2}^{(0)} = E\cos \,(\omega \tau - \varepsilon ),\quad \gamma_{2}^{(0)} = M_{3} \cos \,\tau ,$$
(22)
where \(E = \sqrt {M_{1}^{2} + M_{2}^{2} }\) and \(\varepsilon = \tan^{ - 1} {{M_{2} } \mathord{\left/ {\vphantom {{M_{2} } {M_{1} }}} \right. \kern-0pt} {M_{1} }}\). Using (22) and (12), we get:
$$\begin{aligned} & s_{11}^{(0)} = E^{2} [(a\cos^{2} \varepsilon - 0.5\,) + b\omega^{2} A_{1}^{ - 2} (\sin^{2} \varepsilon - 0.5) + 0.5(b\omega^{2} A_{1}^{ - 2} - a)\cos 2(\omega \tau - \varepsilon )] \\ & \quad - \,2M_{3} [x^{\prime}_{0} (1 - \cos \tau ) + y^{\prime}_{0} \sin \tau ] - 0.5kM_{3}^{2} C_{1} (1 - \cos 2\tau ), \\ & s_{12}^{(0)} = \text{a}^{\text{2}} E\gamma^{\prime\prime}_{0} x^{\prime}_{0} (1 - a)^{ - 1} [\cos \varepsilon - \cos (\omega \tau - \varepsilon )] \\ & \quad + \,Ey^{\prime}_{0} b\gamma^{\prime\prime}_{0} A_{1}^{ - 1} \omega (1 - a^{ - 1} A_{1}^{ - 1} )[\sin \varepsilon + \sin (\omega \tau - \varepsilon )] \\ & \quad + \,aEM_{3} (\chi_{1} + a\gamma^{\prime\prime}_{0} k)\{ \cos \varepsilon - 0.5\cos [(\omega + 1)\tau - \varepsilon ] - 0.5\cos [(\omega - 1)\tau - \varepsilon ]\} \\ & \quad + \,\omega bA_{1}^{ - 1} EM_{3} (\chi_{2} A_{1}^{ - 1} - bk\gamma^{\prime\prime}_{0} )\{ 0.5\cos [(\omega + 1)\tau - \varepsilon ] - 0.5\cos [(\omega - 1)\tau - \varepsilon ]\} \\ & \quad + \,EM_{3} (z^{\prime}_{0} - \gamma^{\prime\prime}_{0} k)\{ a\cos \varepsilon + 0.5(b\omega A_{1}^{ - 1} - a)\cos [(\omega - 1)\tau - \varepsilon ] \\ & \quad - \,0.5(b\omega A_{1}^{ - 1} + a)\cos [(\omega + 1)\tau - \varepsilon ]\} , \\ & s_{21}^{(0)} = EM_{3} \{ a\cos \varepsilon + 0.5(b\omega A_{1}^{ - 1} - a)\cos [(\omega - 1)\tau - \varepsilon ] \\ & \quad - \,0.5(b\omega A_{1}^{ - 1} + a)\cos [(\omega + 1)\tau - \varepsilon ]\} , \\ & s_{22}^{(0)} = E^{2} \gamma^{\prime\prime}_{0} [a^{2} (\cos^{2} \varepsilon - 0.5) + b^{2} \omega^{2} A_{1}^{ - 2} (\sin^{2} \varepsilon - 0.5) \\ & \quad - \,0.5(a^{2} - b^{2} \omega^{2} A_{1}^{ - 2} )\cos 2(\omega \tau - \varepsilon )] + 0.5M_{3}^{2} (a\chi_{1} + bA_{1}^{ - 1} \chi_{2} )(1 - \cos 2\tau ) \\ & \quad + \,\gamma^{\prime\prime}_{0} M_{3} [a\,x^{\prime}_{0} (1 - a)^{ - 1} (1 - \cos \tau ) + by^{\prime}_{0} \,a^{ - 1} A_{1}^{ - 1} \sin \tau ]. \\ \end{aligned}$$
(23)
Substituting (22) and (23) into (16), we get:
$$\begin{aligned} & F_{2}^{(0)} = M_{1} L(\omega )\cos \omega \tau + M_{2} L(\omega )\sin \omega \tau + \ldots , \\ & \varPhi_{2}^{(0)} = M_{3} N(\omega )\cos \tau + \ldots , \\ & F_{3}^{(0)} = M_{1} K(\omega )\cos \omega \tau + M_{2} K(\omega )\sin \omega \tau + \ldots , \\ \end{aligned}$$
(24)
where:
$$\begin{aligned} & L(\omega ) = \omega^{2} [ - (aM_{1}^{2} + b\omega^{2} A_{1}^{ - 2} M_{2}^{2} ) + 0.25(M_{1}^{2} + M_{2}^{2} )(C_{1} A_{1}^{ - 1} + 3a + b\omega^{2} A_{1}^{ - 2} )] \\ & \quad + \,2\omega^{2} M_{3} x^{\prime}_{0} - \gamma^{\prime\prime}_{0} [z^{\prime}_{0} a^{ - 1} + a\chi_{1} (1 - \omega^{2} )] + k\{ A_{1} (\gamma_{0}^{\prime \prime 2} - 0.5M_{3}^{2} ) \\ & \quad + \,0.5M_{3}^{2} [a(A_{1} - 2\omega^{2} ) + \omega^{2} b]\} , \\ & N(\omega ) = - (aM_{1}^{2} + b\omega^{2} A_{1}^{ - 2} M_{2}^{2} ) - 0.5(M_{1}^{2} + M_{2}^{2} )[aB_{1} + \omega^{2} A_{1}^{ - 2} (1 - b)] \\ & \quad + \,2M_{3} x^{\prime}_{0} - \gamma^{\prime\prime}_{0} \,[z^{\prime}_{0} b^{ - 1} - a\chi_{1} (1 - \omega^{2} )] + k[M_{3}^{2} (b - a) - B_{1} \gamma_{0}^{\prime \prime 2} ], \\ & K(\omega ) = - 2\omega^{2} \gamma^{\prime\prime}_{0} \,[a^{2} x^{\prime}_{0} M_{1} (1 - a)^{ - 1} + \omega y^{\prime}_{0} bM_{2} (1 - a^{ - 1} A_{1}^{ - 1} )A_{1}^{ - 1} \\ & \quad + \,0.25a^{ - 1} A_{1}^{ - 1} C_{1} y^{\prime}_{0} (M_{1}^{2} + M_{2}^{2} )] - 2\omega^{2} aM_{1} M_{3} [\chi_{1} + a\gamma^{\prime\prime}_{0} k + z^{\prime}_{0} - \gamma^{\prime\prime}_{0} k] \\ & \quad + \,aM_{1} M_{3} [2A_{1} k\gamma^{\prime\prime}_{0} - z^{\prime}_{0} a^{ - 1} - aA_{1} k\gamma^{\prime\prime}_{0} (1 + B_{1} ) - \chi_{1} (1 + B_{1} )]. \\ \end{aligned}$$
(25)
Using (24) and (25), the following functions are obtained:
$$\begin{array}{*{20}l} {g_{2} (T_{0} ) = - \pi n\omega^{ - 1} M_{2} L(\omega ),} \hfill & {\dot{g}_{2} (T_{0} ) = \pi nM_{1} L(\omega ),} \hfill \\ {h_{2} (T_{0} ) = 0,} \hfill & {\dot{h}_{2} (T_{0} ) = \pi nM_{3} N(\omega ),} \hfill \\ {g_{3} (T_{0} ) = - \pi n\omega^{ - 1} M_{2} K(\omega ),} \hfill & {\dot{g}_{3} (T_{0} ) = \pi nM_{1} K(\omega ).} \hfill \\ \end{array}$$
(26)
Substituting by the initial conditions (21) into the first integration (17) when \(\tau = 0\), we get:
$$M_{3}^{2} + 2M_{3} \beta_{3} + \beta_{3}^{2} + 2\mu^{ - 1} a\gamma^{\prime\prime}_{0} M_{3} (M_{1} + \beta_{1} ) = 1 - \gamma_{0}^{\prime \prime 2} .$$
(27)
Let γ
″0
depends on μ−1, we get:
$$\gamma^{\prime\prime}_{0} = \mu^{ - 1} \varGamma ,\quad 0 < \varGamma < 1.$$
(28)
Taking into consideration, Eqs. (27) and (28), we get \(M_{3} ,\,\,\beta_{3}\) as follows:
$$M_{3} = 1,\quad \beta_{3} = - a\varGamma \mu^{ - 2} \tilde{M}_{1} - \frac{1}{2}\mu^{ - 2} \varGamma^{2} + \ldots .$$
(29)
The independent conditions for periodicity are:
$$\begin{aligned} & - (L_{1} (\omega ) - \omega^{2} N_{1} (\omega ))\pi n\omega^{ - 1} \tilde{M}_{2} + \mu^{ - 1} \,G_{3} (T_{0} ) + \ldots = 0, \\ & (L_{1} (\omega ) - \omega^{2} N_{1} (\omega ))\pi n\tilde{M}_{1} + \mu^{ - 1} \,\dot{G}_{3} (T_{0} ) + \ldots = 0, \\ & \mu^{ - 2} (\dot{H}_{2} (T_{0} ) + \mu^{ - 1} \dot{H}_{3} (T_{0} ))\tilde{M}_{3}^{ - 1} + \ldots = \alpha (\mu^{ - 1} ), \\ \end{aligned}$$
(30)
where \(L_{1} (\omega ),\,\,N_{1} (\omega )\) are obtained from \(L(\omega ),\,\,N(\omega )\) replacing \(M_{i}\) by \((M_{i} + \beta_{i} ),\,\,i = 1,2,3\) to get:
$$L_{1} (\omega ) - \omega^{2} N_{1} (\omega ) = W_{0} (\omega )(\tilde{M}_{1}^{2} + \tilde{M}_{2}^{2} ) - \gamma^{\prime\prime}_{0} [z^{\prime}_{0} \,W_{1} (\omega ) + k\gamma^{\prime\prime}_{0} \,W_{2} (\omega )] - kW_{3} (\omega )\tilde{M}_{3}^{2} ,$$
(31)
where:
$$\begin{array}{*{20}l} {W_{0} (\omega ) = (a - 1)(a + b - 2)/2b,} \hfill & {W_{1} (\omega ) = [3(a + b) - 2(2ab + 1)]/ab,} \hfill \\ {W_{2} (\omega ) = 2\omega^{2} [1 - (a + b)],} \hfill & {W_{3} (\omega ) = \omega^{2} b.} \hfill \\ \end{array}$$
(32)
For zeros approximation for power series of \(1/\mu\), Eq. (30) give:
$$M_{1} = M_{2} = 0\,.$$
(33)
Since the z-axis is directed along with the major or the minor axis of the ellipsoid of inertia of the body, we get: \(W_{0} (\omega ) > 0\) for all \(\omega\) under consideration.
Assume that:
$$\gamma^{\prime\prime}_{0} \,[z^{\prime}_{0} \,W_{1} + k\gamma^{\prime\prime}_{0} \,W_{2} ] + kW_{3} (\omega )M_{3}^{2} \ne 0\,.$$
(34)
Using (30), we get \(\beta_{1} ,\,\,\beta_{2}\) in power series expansions of powers less than \(\mu^{ - 2}\). Then for the rational values of the natural frequency \(\omega\) does not equal to \((1,2,1/2,3,1/3, \ldots )\), we get the required periodic solutions and the correction of the period \(\alpha (\mu^{ - 1} )\) as:
$$\begin{aligned} & p_{1} (\tau ,\mu^{ - 1} ) = \mu^{ - 1} [x^{\prime}_{0} (1 - a)^{ - 1} \gamma^{\prime\prime}_{0} + \chi_{1} M_{3} \cos \tau ] + \ldots , \\ & q_{1} (\tau ,\mu^{ - 1} ) = \mu^{ - 1} [y^{\prime}_{0} (1 - b)^{ - 1} \gamma^{\prime\prime}_{0} + A_{1}^{ - 1} M_{3} \chi_{2} \sin \tau ] + \ldots , \\ & r_{1} (\tau ,\mu^{ - 1} ) = 1 - 0.25\mu^{ - 2} M_{3} [kM_{3} C_{1} + 4x^{\prime}_{0} (1 - \cos \tau ) + y^{\prime}_{0} \sin \tau - kM_{3} C_{1} \cos 2\tau ] + \ldots , \\ & \gamma (\tau ,\mu^{ - 1} ) = M_{3} \cos \tau - 0.5\mu^{ - 2} \varGamma^{2} \cos \tau + \ldots , \\ & \gamma^{\prime}(\tau ,\mu^{ - 1} ) = - M_{3} \sin \tau + 0.5\mu^{ - 2} \varGamma^{2} \sin \tau + \ldots , \\ & \gamma^{\prime\prime}(\tau ,\mu^{ - 1} ) = \gamma^{\prime\prime}_{0} + \mu^{ - 2} M_{3} [x^{\prime}_{0} (1 - a)^{ - 1} \gamma^{\prime\prime}_{0} - 0.5M_{3} C_{1} \left( {\frac{{z^{\prime}_{0} }}{a + b - 1} + 0.5k\gamma^{\prime\prime}_{0} } \right) \\ & \quad - x^{\prime}_{0} (1 - a)^{ - 1} \gamma^{\prime\prime}_{0} \cos \tau + y^{\prime}_{0} (1 - b)^{ - 1} \gamma^{\prime\prime}_{0} \sin \tau + 0.25M_{3} C_{1} (\frac{{2z^{\prime}_{0} }}{a + b - 1} + k\gamma^{\prime\prime}_{0} )\cos 2\tau ] + \ldots , \\ \end{aligned}$$
(35)
$$\alpha (\mu^{ - 1} ) = 2\mu^{ - 2} \pi n\left\{ {M_{3} x^{\prime}_{0} - z^{\prime}_{0} \gamma^{\prime\prime}_{0} - 0.5k} \left[ {\gamma_{0}^{\prime \prime 2} (bB_{1} - aA_{1} ) + \,B_{1} \gamma^{\prime\prime}_{0} (1 - \gamma^{\prime\prime}_{0} )} - 0.125C_{1} M_{3}^{2} \right] \right\} + \ldots \,.$$
(36)
The obtained solutions (35) and (36) are considered as the generalization of the corresponding problem in gravity field which studied in previous works [10] (when k = 0), the deviations between them are given by:
$$\begin{aligned} & \Delta p_{1} = \mu^{ - 1} (1 - \omega^{2} )^{ - 1} kM_{3} \gamma^{\prime\prime}_{0} (A_{1} - \omega^{2} )\cos \tau + \ldots , \\ & \Delta q_{1} = \mu^{ - 1} A_{1}^{ - 1} M_{3} \left\{ {\left. {(1 - \omega^{2} )^{ - 1} [k(A_{1} - \omega^{2} )\gamma^{\prime\prime}_{0} ] - kA_{1} \gamma^{\prime\prime}_{0} } \right\}} \right.\sin \tau + \ldots , \\ & \Delta r_{1} = - 0.25\mu^{ - 2} M_{3} kC_{1} (1 - \cos 2\tau ) + \ldots , \\ & \Delta \gamma = \mu^{ - 2} [0] + \ldots ,\quad \Delta \gamma^{\prime} = \mu^{ - 2} [0] + \ldots , \\ & \Delta \gamma^{\prime\prime} = - 0.25\mu^{ - 2} M_{3}^{2} C_{1} k\gamma^{\prime\prime}_{0} (1 - \cos 2\tau ) + \ldots , \\ \end{aligned}$$
$$\Delta \alpha (\mu^{ - 1} ) = - \mu^{ - 2} \pi nk\left[ {\gamma_{0}^{\prime \prime 2} (bB_{1} - aA_{1} ) - 0.5M_{3}^{2} (b - a) + B_{1} \gamma^{\prime\prime}_{0} (1 - \gamma^{\prime\prime}_{0} )} \right] + \ldots \,\,.$$
(37)