﻿ 最简规范形的计算及其应用 Computation of the Simplest Normal Form and Its Application

Dynamical Systems and Control
Vol.05 No.03(2016), Article ID:18004,10 pages
10.12677/DSC.2016.53010

Computation of the Simplest Normal Form and Its Application

Shuping Chen

College of Applied Mathematics, Xiamen University of Technology, Xiamen Fujian

Received: Jun. 25th, 2016; accepted: Jul. 15th, 2016; published: Jul. 18th, 2016

ABSTRACT

This paper presents a new computation method for obtaining a significant refinement of the simplest normal form for four dimensional nonlinear systems. The formulae are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. By using the simplest normal form theory, the stability analysis of a simply-sup- ported functionally graded materials (FGMs) rectangular plate subject to the transversal and in- plane excitations is investigated. It is seen that the stability is exhibited under certain circumstances. The fourth-order Runge-Kutta algorithm is utilized to do numerical simulation of the stability behavior of the FGM rectangular plate.

Keywords:Functionally Graded Materials, Normal Form, Stability, Nonlinear Transformations

1. 引言

2. 四维非线性系统的最简规范形计算

， (1)

， (2)

，(3)

， (4)

。 (5)

。 (6)

。 (7)

。 (8)

， (9)

， (10)

，(11)

， ( 12 a )

， (12b)

， ( 12 c )

。 (12d)

。 (13)

。 (14)

，则有

。 (15)

。 (16)

。 (17)

。 (18)

3. 功能梯度材料矩形板模型

， ( 19a )

， (19b)

， ( 19c )

Figure 1. The model of a functionally graded materials rectangular plate

， (19d)

。(19e)

， ( 20a )

， (20b)

， ( 20c )

， (21)

， ( 22a )

， (22b)

， ( 23a )

， (23b)

， ( 23c )

。 (23d)

4. 功能梯度材料矩形板的稳定性分析

， (24)

(25)

。 (26)

。 (27)

， (28)

， (29)

(30)

(31)

(32)

5. 小结

Figure 2. Transition curves of the system (23) for the case of a non-semisimple double zero eigenvalues

(a) (b) (c) (d)(e)

Figure 3. Trajectory projection starting form initial point when

Computation of the Simplest Normal Form and Its Application[J]. 动力系统与控制, 2016, 05(03): 86-95. http://dx.doi.org/10.12677/DSC.2016.53010

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