Abstract
In this work, we present advanced investigations and treatments for the problem of a restricted vibrating motion of a connected gyrostat with a spring. It is supposed that the gyrostat spins slowly about the minor or major principal axis of the inertia ellipsoid. The gyrostat is acted upon by a gyrostatic couple vector besides the action of Newtonian and electromagnetic fields. The approach of the large parameter is applied to obtain the periodic solutions for the governing system of equations of motion of the gyrostat. A geometric illustration using the angles of Euler is given for such motion to evaluate and analyze the gyrostatic motion at any instant. The analysis of the obtained solutions is considered in terms of numerical data throughout computer programs. Characterized parametric data are assumed through one of the numerical methods for obtaining numerical solutions that prove the validity of the analytical obtained periodic solutions. The obtained solutions, besides the phase diagrams, have been drawn to describe these solutions' periodicity and stability procedures. The novelty of this work comes from the imposition of a new initial condition that does not restrict movement around the dynamic symmetry axis. This assumption allows the use of a new technique for the solution called the large parameter. This technique gives solutions in a completely new domain that are different from the ones studied in previous works.