The Stability of Certain Motion of a Charged Gyrostat in Newtonian Force Field

This work aims to study the stability of certain motions of a rigid body rotating about its fixed point and carrying a rotor that rotates with constant angular velocity about an axis parallel to one of the principal axes. This motion is presumed to take place due to the combined influence of the magnetic field and the Newtonian force field. The equations of motion are deduced, and moreover, they are expressed as a Lie–Poisson Hamilton system. The permanent rotations are calculated and interpreted mechanically. The sufficient conditions for instability are presented employing the linear approximation method. The energy-Casimir method is applied to gain sufficient conditions for stability. The regions of linear stability and Lyapunov stability are illustrated graphically for certain values of the parameters.

Geometric Models for Lie–Hamilton Systems on ℝ2

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

Dynamic Analysis of Flexible Rotor Based on Transfer Symplectic Matrix

In view of the numerical instability and low accuracy of the traditional transfer matrix method in solving the high-order critical speed of the rotor system, a new idea of incorporating the finite element method into the transfer matrix is proposed. Based on the variational principle, the transfer symplectic matrix of gyro rotors suitable for all kinds of boundary conditions and supporting conditions under the Hamilton system is derived by introducing dual variables. To verify the proposed method in rotor critical speed, a numerical analysis is adopted. The simulation experiment results show that, in the calculation of high-order critical speed, especially when exceeding the sixth critical speed, the numerical accuracy of the transfer symplectic matrix method is obviously better than that of the reference method. The relative errors between the numerical solution and the exact solution are 0.0347% and 0.2228%, respectively, at the sixth critical speed. The numerical example indicates the feasibility and superiority of the method, which provides the basis for the optimal design of the rotor system.

Singularity Study of Sectorial Domain by Weak Formulation and Fractal Finite Element in Hamilton System

The weak formulation of mixed state equations including boundary conditions are presented in polar coordinate system, mixed variational formulation is established in sectorial domain. The fractal finite element method is used to analyse the sector domain problem. The present result is exactly analogous to the Hamiltonian mechanics for a dynamic system by simulating time variable t with coordinate variable r. The stress singularity at singular point is investigated by means of the fractal finite element method. The present study satisfies the continuity conditions of stresses and displacements at the interfaces. The principle and method suggested here have clear physical concepts. So this method would be easily popularized in dynamics analysis of elasticity.

Study on the Multi-Body Dynamic Characteristics of FPSO Soft Yoke Mooring System Based on Symplectic Algorithm

The soft yoke single-point mooring (SYMS) system is the main mooring approach for the floating production storage and offloading (FPSO) unit. As a typical multi-rigid-body system, a SYMS consists of the single-point turret, yoke, mooring legs, and mooring support. It releases the rotational degrees of freedom of an FPSO through the combined effects of multiple joint structures, so as to deliver the weather-vane effect of the FPSO. In this paper, a multi-body dynamics model of the soft yoke mooring system was established. To deal with the difficult integration in the process of solving differential-algebraic equations, a symplectic numerical integration method was proposed on the basis of the Zu Chongzhi method. The proposed solution format had simple symplectic property automatically satisfying the Hamilton system, as well as a high accuracy in solving nonlinear systems. The measured data of the FPSO’s six degrees of freedom (6DoF) under two different sea conditions were selected, and the mooring restoring force of the SYMS was calculated. The calculated results showed that the symplectic solution method could the actual stress state of the structures with more obvious dynamic characteristics. Furthermore, the displacement and stress state of the single-body structures, such as the mooring legs and yoke, and the analysis result could comprehensively evaluate the overall working state of the SYMS.

Bounding the Number of Zeros of Abelian Integral for a Class of Integrable non-Hamilton System

This paper investigates the planar differential systems [Formula: see text], [Formula: see text] under the perturbations of polynomials of [Formula: see text] with degree [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. The upper bound for the number of zeros of Abelian integral from the period annulus around the center is obtained.