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

Document Type : Original Article

Author

1 Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, P. O. Box 5555, Makkah, Saudi Arabia.

2 Mathematics Department, Faculty of Science, Tanta University, P.O. Box 31527, Tanta, Egypt.

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 ro about the fixed z-axis of the
body. The periodic solutions of motion are obtained in neighborhood ro tends to ∞ . In
our work, we aim to find periodic solutions to the problem of motion in the neighborhood
of r0 tends to 0 . So, we give a new assumption that: ro 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 ro. 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 ( = 1, 2, 1/2, 3, 1/3, . . .) under the condition
 ′′
0
= cos 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.

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