Direct and inverse secondary resonance effects in the spherical motion of an asymmetric rigid body with moving masses
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O R I G I NA L PA P E R
Vladislav V. Lyubimov
Direct and inverse secondary resonance effects in the spherical motion of an asymmetric rigid body with moving masses
Received: 28 May 2019 / Revised: 13 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Secondary resonance effects in the spherical motion of a heavy asymmetrical rigid body with moving masses are reviewed in the case close to the Lagrange top. It is known that perturbations acting on the body, particularly a small displacement of the center of mass with respect to axis of symmetry and a small perturbing moment in a connected system of coordinates, lead to non-resonant evolutions of angular velocity. These evolutionary phenomena are called direct secondary resonance effects. They are visible in a dynamic system consisting of an asymmetric rigid body and several masses fixed relative to the body. However, the presence of moving masses connected to a rigid body by springs may lead to the stabilization of the angular velocity of the body. This dynamic phenomenon should be attributed to the inverse secondary resonance effect. The aim of this paper is to study the characteristic direct and inverse secondary resonance effects in a perturbed spherical motion of a rigid body with fixed and moving masses. The method of integral manifolds and the averaging method are used for the asymptotic analysis of the secondary resonance effects. The paper presents the numerical results of modeling the direct and inverse resonance effects.
1 Introduction Many recent studies focused on analyzing the evolution of the motion of an asymmetric rigid body relative to the center of mass [1–6], etc. It is known that if the frequency of an external force equals the frequency of the unperturbed oscillations of a rigid body, the rigid body may be captured into resonance. In this case, the long-term implementation of resonance leads to a significant increase in the nutation angle. In general, the probability of passage and capture into the resonance varies between one and zero. It depends on the initial condition near the separatrix [7]. The paper [5] gives valuable information on the resonance modes of the motion of a Lagrange top with small mass asymmetry at the passage through resonances. It should be noted that the phenomenon of the secondary resonance effects was discovered at the end of the twentieth century [8,9]. These phenomena can be considered as the evolution of slow variables of a dynamic system, induced by resonance in the non-resonant case. The use of a non-resonant averaging scheme [10] allows detecting resonant frequency ratios in the denominators of higher approximations of the averaging method. These denominators induce characteristic evolutionary phenomena called secondary resonance effects [9]. Secondary resonance effects during the motion of a rigid body around the center of mass were reviewed in cases close to the Lagrange top in the papers [11,12], with nonlinear and quasi-linear formulation, respectively. These resonant effects a
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