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Chen Xiang-Wei, Zhang Ye, Mei Feng-Xiang. An application of a combined gradient system to stabilize a mechanical system. Chinese Physics B, 2016, 25(10): 100201  
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An application of a combined gradient system to stabilize a mechanical system
Chen Xiang-Wei1, †, , Zhang Ye2, Mei Feng-Xiang3
Department of Physics and Information Engineering, Shangqiu Normal University, Shangqiu 476000, China
School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
School of Aerospace, Beijing Institute of Technology, Beijing 100081, China

 

† Corresponding author. E-mail: hnchenxw@163.com

Project supported by the National Natural Science Foundation of China (Grant Nos. 11372169 and 11272050).

Abstract
Abstract

A gradient system and a skew-gradient system can be merged into a combined gradient system. The differential equations of the combined gradient system are established and its property is studied. If a mechanical system can be represented as a combined gradient system, the stability of the mechanical system can be studied by using the property of the combined gradient system. Some examples are given to illustrate the applications of the results.

PACS: 02.30.Jr;45.20.Jj;11.30.–j
Keyword:gradient system;combined gradient system;mechanical system;stability
1. Introduction

Two important systems have been studied in chapter 9 “A wide range of nonlinear skills” of Ref. [1], one is the gradient system, and the other is the Hamilton system. The gradient system is an important problem of differential equations and dynamical systems, and is especially suitable for being studied by using the Lyapunov function. The skew-gradient system is given in Ref. [2], which is of great significance for studying the integral and stability of a mechanical system. In Refs. [3]–[16] the general gradient system was applied to the mechanical system, and in Refs. [17] and [18] the skew-gradient system was applied to the mechanical system. The gradient properties of the kinetic equation, not only play an important role in revealing the internal structure of a dynamical system, but also help to explore the dynamical behavior of the system, and are widely used in thermodynamics, optics, classical and quantum field theory and many other fields.[2,19] The semi-inverse method proposed by He[20] has been proved to be a most convenient and effective way to establish generalized variational principles, and has been applied successfully to many physical and mechanical fields.[21] In the present paper, the combination of the general gradient system and the skew-gradient system is called the combined gradient system. This kind of combined gradient system can be applied to the problem of the stability of the mechanical system which cannot be solved by the general gradient system nor by the skew-gradient system. Firstly, differential equations, properties and applications of the general gradient system and the skew-gradient system are introduced. Secondly, differential equations and some important properties of the combined gradient system are studied. Finally, some examples are given to illustrate the application to stabilize the mechanical system.

2. General gradient system
2.1. Differential equation

The differential equations of the general gradient system are written as[1,2]

where x = (x1, x2, …, xm and V = V (x) are potential functions.

2.2. Properties

The general gradient system (1) has the following important properties[1]

2.3. Applications

By the properties of the gradient system (1), we can study the stability of the solution of the system. The second order Lagrange system and the first order Lagrange system can become a gradient system under certain conditions, and the potential function V can become a Lyapunov function, then the stability of the system can be studied by using Lyapunov theorem, and the partial variable stability can be studied by using Rumyatsev theorem too.[10] At the same time, the stability can also be studied by using the third property of the gradient system. According to the third property and Lyapunov first approximation theory, the solution is asymptotically stable or unstable. In addition, we also know that this gradient system cannot be used to investigate the asymptotic stability of a damped oscillator which has plural characteristic roots.

3. Skew-gradient system
3.1. Differential equation

The differential equations of the skew-gradient system are[17]

where x = (x1, x2, …, xm), the same subscript represents summation, V = V(x) is an energy function[2] and matrix (bij (x)) is antisymmetrical, i.e.

3.2. Properties

The skew-gradient system has important properties as follows.

3.3. Applications

The skew-gradient system can be used to investigate the integral and the stability of the system, but cannot be used to study the asymptotic stability of the solution. The steady Hamilton system, the autonomous Birkhoff system and the generalized Hamilton system all are naturally skew-gradient systems, therefore, the Hamilton function and the Birkhoff function are the integrals of the system. If the Hamilton function and the Birkhoff function can become the Lyapunov functions, the solution is stable. The Lorenz equation is one of the earliest examples of chaos, and its Robbins model is a skew-gradient system.[17]

4. Combined gradient system
4.1. Differential equation

The combined gradient system is obtained by combining the gradient system (1) and the skew-gradient system (2). The differential equations of the system are

4.2. Properties
4.3. Application

If the differential equations of the mechanical system can be expressed as Eq. (4) of the combined gradient system, we can use the properties of the combined gradient system to investigate the stability of the solution of the mechanical system.

Example 1 We study a damped oscillator system with a single degree of freedom, in which the Lagrange function and the generalized force are expressed respectively as

where the quantities are dimensionless. Now, we merge them into a combined gradient system, and study the stability of the zero solution.

The differential equation of system (5) is

This is a damped oscillator system.

Let

Taking the derivative of x1 and x2, we have

It can be written as the form of a combined gradient system, i.e.,

where is positive-definite in the neighborhood of x1 = x2 = 0. Therefore, the solution x1 = x2 = 0 is asymptotically stable.

Example 2 The Lagrange function and the generalized force of the single degree of freedom system respectively are

The differential equation of system (6) is

Let

and take the derivative of x1 and x2, then we will have

They can be written as the form of a combined gradient system (4), i.e.,

where is positive-definite in the neighbourhood of x1 = x2 = 0. Therefore, the solution x1 = x2 = 0 is asymptotically stable.

Example 3 The single degree of freedom system is

where the quantities are dimensionless. Now we merge them into a combined gradient system (4), and study the stability of zero solution.

The Lagrange equation is

Let

Taking the derivative of x1 and x2, we have

They can be written as the form of combined gradient system (4), i.e.,

where is positive-definite in the neighbourhood of x1 = x2 = 0. Therefore, the solution x1 = x2 = 0 is asymptotic stable.

The three examples above are all for a Lagrange system, but these properties are also applicable to other mechanical systems. The three examples cannot be represented as the general gradient system nor as the skew-gradient system.

5. Conclusions

The study of the stability of a mechanical system usually needs one to construct the Lyapunov function. It is generally difficult to construct the Lyapunov function directly according to the differential equation. Gradient systems including the general gradient system, skew-gradient system, and combined gradient system, etc., are especially suitable for being studied by using the Lyapunov function. We can use the properties of gradient systems to investigate the stability of mechanical systems, if a mechanical system can be represented as a gradient system. The stabilities of solutions of these examples above cannot be studied by the general gradient system nor by the skew-gradient system, but can be investigated by the combined gradient system. In this paper, an indirect method of studying the stability of a mechanical system is presented.

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