The periodic rotary motions of a rigid body in a new domain of angular velocity

Abstract

In the previous works, the limiting case for the motion of a rigid body about a fixed point in a Newtonian force field, which comes from a gravity center lies on Z -axis, is solved. The authors apply the small parameter technique which is achieved giving the body a sufficiently large angular velocity component r o about the fixed z -axis of the body. The periodic solutions of motion are obtained in neighborhood r o tends to $$\infty$$ ∞ . In our work, we aim to find periodic solutions to the problem of motion in the neighborhood of r 0 tends to $$0$$ 0 . So, we give a new assumption that: r o is sufficiently small. Under this assumption, we must achieve a large parameter and search for another technique for solving this problem. This technique is named; a large parameter technique instead of the small one well known previously. We see the advantage of the new technique which appears in saving high energy used to begin the motion and give the solution of the problem in another domain. The obtained solutions by the new technique depend on r o . We consider that the center of mass of this body does not necessarily coincide with the fixed point O. We reduce the six nonlinear differential equations of the body and their three first integrals to a quasilinear autonomous system of two degrees of freedom and one first integral. We solve the rational case when the frequencies of the generating system are rational except $$(\,\omega = \,1,\,2,1/2,3,1/3, \ldots )$$ ( ω = 1 , 2 , 1 / 2 , 3 , 1 / 3 , … ) under the condition $$\gamma^{\prime\prime}_{0} = \cos \theta_{o} \approx 0$$ γ 0 ″ = cos θ o ≈ 0 . We use the fourth-order Runge–Kutta method to find the periodic solutions in the closed interval of the time t and to compare the analytical method with the numerical one.