Energy mechanism analysis for chaotic dynamics of gyrostat system and simulation of displacement orbit using COMSOL

doi:10.1016/j.apm.2020.11.015

Applied Mathematical Modelling

Volume 92, April 2021, Pages 333-348

https://doi.org/10.1016/j.apm.2020.11.015 Get rights and content

Highlights

•
The extremal ellipsoid surface reflecting the dynamic mechanism of the gyrostat system is derived.
•
The bifurcation of the Casimir power and energy leaps are found to be the indicators of different definitive dynamics.
•
The state changes of the displacement orbit of the gyrostat are simulated by the COMSOL finite element platform.

Abstract

Most of the existing research on gyrostat systems uses the Melnikov method to analyze chaotic dynamic characteristics. The mechanism of chaotic motion from the energy perspective of gyrostat systems has rarely been analyzed. The gyrostat system with positive damping disturbance is modeled in this paper. Casimir energy and power are given. The mechanism of the orbital expansion and contraction of the gyrostat is revealed through the sign change of Casimir power and the ellipsoid of Casimir energy. The bifurcations of the Casimir power corresponding to the state bifurcation and Lyapunov exponent spectrum reflect the evolution and transition of gyrostat's dynamics. The energy-level leap expressed by the variance of Casimir energy is found to determine the bifurcation of dynamics for the first time. The coexistence of the gyrostat is explained using Casimir power and local stability of equilibrium point. The COMSOL finite element simulation platform is built to study the state changes of the displacement orbit of the frame gyrostat model. A comparison is performed for the displacement results conducted by the COMSOL and the numerical simulation for the gyrostat system kinematic model. The different displacement states of the gyrostat system under free motion and in the presence of damping disturbance and external torques are obtained.

Introduction

The gyrostats have been widely used in aerospace and aviation, among which the complex dynamics have attracted much attention [1], [2], [3], [4]. The Melnikov method modified by Wiggins and Holmes [5,6] has been a prevalent approach in analyzing dynamics for a perturbed Hamiltonian system by calculating the distance between the stable and unstable manifolds. It is one of the main tools for analyzing the existence of Smale' horseshoe chaos [7]. The gyrostat is a type of rotational rigid body. The chaos existing in a rigid body moving on a circular orbit subject to a periodic disturbance in the sense of Smale' horseshoe has been proved [8]. The Melnikov function of the gyrostat chaotic attitude motion affected by nonlinear damping torque and external torque was derived and analyzed [9,10]. Similarly, the Melnikov function was found for the variable structure dual-spin gyrostat spacecraft with negative damping [11]. Other tools, like the Poincaré map method and the Lyapunov exponent method, have also been employed in analyzing the dynamics of gyrostats [12]. The following problems are identified:

(1)
Finding the Melnikov function is a challenging task.
(2)
This method is a rigorous mathematical approach instead of a physical analysis for finding the physical mechanism.
(3)
The method cannot analyze the boundary.
(4)
The Poincaré map, Lyapunov exponent and state bifurcation tools are numerical ways to display dynamics' evolution with the varying parameters. Still, these tools do not reflect the mechanism of the dynamics transition or bifurcation.

The gyrostat is a physically rigid body whose movement is operated by force and energy. Therefore, the energy analysis is necessary to reveal the movement mechanism. We find that energy bifurcation corresponds to the state bifurcation that reflects the evolution of different dynamics.

With the continuous development of chaos theory in different research fields, many scholars have begun to analyze the causes of chaotic dynamic behavior from the perspective of internal forces and energy of various systems. Both the frame gyrostat system or the dual-spin satellite system belong to a multi-rigid body dynamic system, of which the state has its physical meaning. In 1991, Arnold proposed a Kolmogorov model describing a dissipative dynamic system with a Hamiltonian function structure [13]. This model analyzed the dynamic operation mechanism of the system from the perspective of force. Based on the Kolmogorov model, Pasini and Pelino determined the energy cycle of the Lorenz system, and revealed the energy characteristics of the Lorenz system [14]. Qi et al. [15] have obtained the conversion relationship between the Qi four-wing chaotic system based on the Kolmogorov model and found the relationship between chaotic characteristics and the energy. Qi, et al. revealed the force and energy of actual physical models such as brushless DC motor, permanent magnet synchronous motor using the Kolmogorov model [16,17]. The Hamiltonian conservative chaotic systems of 4D rigid-bodies have been modeled using Casimir energy breaking [18,19]. Therefore, the energy analysis, especially the Casimir energy, can be an effective method to uncover the dynamical bifurcation mechanism and orbital expansion–contraction for the gyrostat system. The exchange between dissipated energy and the motor's supplied energy is one of the critical factors governing the different dynamic modes.

Dual-spin satellites and three-axis stable satellites are one of the actual mechanical models of gyrostats, which are used in navigation and space flight, and their dynamics have received attention in many studies [3,4,9,20]. Qi and Yang [21] have established a mathematical model of a three-axis rotor gyrostat system with a three-axis stable satellite as the background using Euler's kinetic equations, and the four types of torques were identified. The influence of different forms of torque in the gyroscope system on the dynamic behavior was studied in a negative damping disturbance. However, negative damping does not exist in the actual physical field. When the external torque acted by motor or other power on one of the axes, it can be regarded as the negative damping, so the negative damping in [21] is the external torque. The damping, also called positive damping, really exists in all axes; thereby, studying the impact of positive damping for the gyrostat is practical and significant. In [21], the gyrostat system with the negative damping produces a four-wing chaotic attractor, and the extremal energy surface is a hyperboloid. However, from the study, we find the gyrostat generates a double-wing chaotic attractor, and the extremal energy surface is ellipsoid.

Recently, the coexistence of dynamical systems has been studied heavily. The mechanism of coexistence relating to the Casimir power for the gyrostat system has not been studied.

Differing from the model in [21], this paper studies the gyrostat system in the presence of positive damping disturbance. Instead of the hyperboloid extremal surface, an ellipsoidal surface in the study is obtained. The relationship between the sign change of Casimir power and the orbital expansion-contraction is revealed. The Casimir variance bifurcation is used for energy-level leaping for the first time, which corresponds to the transitions of different dynamics of the gyrostat.

The COMSOL simulation platform is a specialized tool for solving partial differential equations of the physical systems using the finite element method. This software can powerfully simulate and display the physical settings, substance, configurations, and geometrical structure. The physical substance can be selected, like various biological and chemical reactions [22,23], the analysis of a rigid body [24], the temperature and pressure distributions in the thermal systems [25]. As far as our knowledge, we have not found the report studying COMSOL simulation for the gyrostat system.

This paper builds the gyrostat system model with a three-axis stable satellite background involving positive damping disturbance. Through the Kolmogorov model, the Casimir energy and Hamiltonian energy are given. The mechanism of orbital expansion-contraction of the gyrostat system is found by the sign change of Casimir power. The coexistence of the gyrostat system is studied using local stability and Casimir power. It is found that the Casimir energy-level leap is the bifurcation mechanism through Casimir energy variance. Therefore, this newly found method is energy-mechanism-based, which is different from the traditional approaches, like the state bifurcation, Lyapunov exponent spectrum that exhibits the bifurcation other than explaining the reason. The COMSOL finite element simulation platform is constructed. The consistency and difference between the Matlab and COMSOL are compared and explained, verifying that the modeled gyrostat system correctness.

The rest of this paper is as follows: Section 2 gives the dynamic model of the gyrostat system. Section 3 gives the Casimir power and energy analysis finding the mechanism of orbital expansion-contraction. Section 4 gives energy bifurcation analysis finding the mechanism of dynamics transitions for the gyrostat system. Section 5 constructs the COMSOL platforms to simulate the gyrostat system's displacement orbit, and a comparison with Matlab results is made. Finally, some conclusions are presented in Section 6.

Section snippets

Dynamic model of gyrostat system

A gyrostat rigid body is a rotating multi-rigid body structure. The new gyrostat system with a three-axis stable satellite as the background involves dissipation and external torque [Fig. 1].

A model of a gyrostat system with a negative damping disturbance and external torques has been given in [21]. The damping includes viscous drag, friction in mechanical systems, resistance in electronic circuits, and absorption and scattering of light in optical oscillators. Damping, also called positive

Casimir energy and power

There are four energy forms: kinetic energy, potential energy, dissipated energy, and externally supplied energy in the gyrostat system. The Hamiltonian energy of the system is given by combining Eqs. (8) and (9) $H = K + U = \frac{1}{2} (Π_{1} I_{x} ω_{x}^{2} + Π_{2} I_{y} ω_{y}^{2} + Π_{3} I_{z} ω_{z}^{2}) - \frac{h_{z} \sqrt{I_{z}}}{\sqrt{I_{x} I_{y}}} ω_{z} .$

The dissipation torque of the original gyrostat system is $Λ ω = {[\begin{matrix} μ_{x} ω_{x} & μ_{y} ω_{y} & μ_{z} ω_{z} \end{matrix}]}^{T}$ , and the external torque is $f = {[\begin{matrix} 0 & 0 & L_{z} \end{matrix}]}^{T}$ . The corresponding dissipated power and external power are $D = \frac{1}{2} 〈 \bar{ω}, \bar{Λ} \bar{ω} 〉 = \frac{1}{2} 〈 ω, Λ ω 〉 = \frac{1}{2} (μ_{x} ω_{x}^{2} + μ_{y} ω_{y}^{2} + μ_{z} ω_{z}^{2}),$ $G = 〈 \bar{ω}, \bar{f} 〉 = 〈 ω, f 〉 = L_{z} ω_{z} .$

The

Equilibrium properties

Let $\dot{ω} = {[\begin{matrix} {\dot{ω}}_{x} & {\dot{ω}}_{y} & {\dot{ω}}_{z} \end{matrix}]}^{T} = 0$ , we get the equilibrium points of system (1). $\begin{matrix} S_{1} = {[\begin{matrix} 0 & 0 & \frac{L_{z}}{u_{z}} \end{matrix}]}^{T}, \\ S_{2} = {[\begin{matrix} \frac{h_{z} a_{8}}{u_{x}} - a_{10} & - a_{8} & \frac{a_{4}}{2 a_{3}} \end{matrix}]}^{T}, S_{3} = {[\begin{matrix} a_{10} - \frac{h_{z} a_{8}}{u_{x}} & a_{8} & \frac{a_{4}}{2 a_{3}} \end{matrix}]}^{T}, \\ S_{4} = {[\begin{matrix} \frac{h_{z} a_{9}}{u_{x}} - a_{11} & - a_{9} & \frac{a_{5}}{2 a_{3}} \end{matrix}]}^{T}, S_{5} = {[\begin{matrix} a_{11} - \frac{h_{z} a_{9}}{u_{x}} & a_{9} & \frac{a_{5}}{2 a_{3}} \end{matrix}]}^{T}, \end{matrix}$ where $\begin{matrix} a_{1} = I_{x}^{2} h_{z}^{2} - 2 I_{x} I_{y} h_{z}^{2} - 4 u_{x} u_{y} I_{x} I_{y} + 4 u_{x} u_{y} I_{x} I_{z} + I_{y}^{2} h_{z}^{2} + 4 u_{x} u_{y} I_{y} I_{z} - 4 u_{x} u_{y} I_{z}^{2}, \\ a_{2} = u_{y} (I_{x} - I_{y}) (I_{y} - I_{z}), \\ a_{3} = I_{z}^{2} + I_{x} I_{y} - I_{x} I_{z} - I_{y} I_{z}, \\ a_{4} = I_{x} h_{z} + I_{y} h_{z} - 2 I_{z} h_{z} - \sqrt{a_{1}}, a_{5} = I_{x} h_{z} + I_{y} h_{z} - 2 I_{z} h_{z} + \sqrt{a_{1}}, \\ a_{6} = h_{z}^{2} u_{z} - I_{y} L_{z} h_{z} + I_{z} L_{z} h_{z} + u_{x} u_{y} u_{z}, \\ a_{7} = I_{z}^{2} L_{z} + I_{x} I_{y} L_{z} - I_{x} I_{z} L_{z} - I_{y} I_{z} L_{z} - I_{x} h_{z} u_{z} + I_{z} h_{z} u_{z}, \\ a_{8} = \sqrt{\frac{a_{6} + \frac{a_{7} a_{4}}{2 a_{3}}}{a_{2}}}, a_{9} = \sqrt{\frac{a_{6} + \frac{a_{7} a_{5}}{2 a_{3}}}{a_{2}}}, \\ a_{10} = \frac{(I_{y} - I_{z}) a_{8} a_{4}}{2 u_{x} a_{3}}, a_{11} = \frac{(I_{y} - I_{z}) a_{5} a_{9}}{2 u_{x} a_{3}} . \end{matrix}$

It can be seen

Theoretical basis of frame gyrostat

COMSOL is a software simulation platform using finite element analysis to solve partial differential equations. The numerical simulation of systems is solved by approximation using the model's temporal discretization and iterating calculation [28,29]. Compared with the numerical approximation algorithm, the COMSOL finite element simulation parameters come from the physical model itself, such as the model's geometrical structure, material properties and various fields. Before using the COMSOL

Conclusion

In this paper, the gyrostat system model with a three-axis stable satellite background involving positive damping disturbance and external torque was built. Through the Kolmogorov model transformation and decomposition of four torques, the Casimir energy and Hamiltonian energy were given. The mechanism of orbital expansion–contraction of the gyrostat system was revealed by the sign change of Casimir power that is the difference between the external power and dissipated power. The boundary

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61873186).

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View full text

Energy mechanism analysis for chaotic dynamics of gyrostat system and simulation of displacement orbit using COMSOL

Highlights

Abstract

Introduction

Section snippets

Dynamic model of gyrostat system

Casimir energy and power

Equilibrium properties

Theoretical basis of frame gyrostat

Conclusion

Declaration of Competing Interest

Acknowledgments

Appl. Math. Modell.

J. Sound Vibr.

Int. J. Nonlinear Mech.

Chaos, Solitons Fract.

Appl. Math. Model.

Appl. Math. Modell.

Commun. Nonlinear Sci. Numer. Simul.

J. Appl. Math. Mech.

Enzyme Microb. Technol.

Ann. Nucl. Energy