气动力及热载荷作用下功能梯度材料矩形板的非线性动力学
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摘要
本文分析了四边简支功能梯度材料矩形板在气动热载荷作用下的非线性动力学。本文基于Reddy的三阶剪切变形薄板理论和von-Karmann型大挠度非线性几何关系,得到了功能梯度矩形板弹性力学问题的基本方程,以此为基础通过Hamilton原理建立了热载荷和气动力作用下板的弹性动力学问题五个自由度的数学模型。选择横向位移的模态函数使其满足板四边简支的边界条件,进行二阶模态截断,用比较消元法和Galerkin方法得到板横向位移的非线性常微分控制方程组。本文假设材料参数沿厚度呈幂律变化,材料物性与温度相关,温度分布符合一维热传导定律,用活塞理论导出了高超音速非线性气动力的具体形式。模型考虑系统受线性气动力和非线性气动力作用两种不同的情况。
文章用多尺度法得到系统在不同内共振条件下的平均方程,以Potter的观点为基础,将1:2内共振条件下系统的解分为平凡解、单模态解、双模态解、行波解、驻波解、调幅行波解六类,讨论了各类解存在和稳定的条件,以Aluminum- Alumina功能梯度材料板为例,用半解析法和数值方法研究了温度变化和组分指数对响应类型及稳定性的影响。数值实验最初只激发一个模态,系统因所受热载荷不同或有不同的材料组分指数,而可能出现双模态振动、单模态振动、倍周期振动等不同类型的响应,数值结果与半解析法的结果定性相同,但存在定量差异,文章讨论了可能引起这些差异的原因。
本文分别分析了线性气动力和非线性气动力作用下板的气动热弹性响应,用多尺度法得到了系统在1:2内共振条件下的平均方程,受线性气动力作用时板没有非零定常解,受非线性气动力作用时当组分指数为0.2或5时平均系统中存在稳定的非零平衡点和非零周期解,但在原系统中没有找到对应的行波解和调幅行波解,数值结果得到的响应不是收敛到零就是发散到无穷。由于将四边简支FGM矩形板的双三角级数形式模态函数进行二阶截断的形式比较特殊,经Galerkin离散后各模态的线性项不耦合,因此模态发生屈曲前,不论受到哪种气动力作用,系统的零平衡点始终稳定。文中细致地分析了模态函数的截断形式对零平衡点稳定性的影响。
本文分析了板的几何参数和组分指数对模态频率的影响,两个共振项系数异号是1:2内共振系统存在非零周期解的必要条件,文中分析了几何参数变化对共振项系数符号的影响。数值结果中有些情况下模态的Poincare截面有自相似结构,且有宽频连续频谱,有混沌运动的特点,本文用Melnikov法得到,系统的Hamilton量很小,不足以产生同宿轨道,没有Smale马蹄意义下的混沌。
Nonlinear dynamics of functionally graded plate subjected to aero-thermal load is analyzed in this paper. Based on Reddy’s third-order shear deformation plate theory and von-Karman- type large deflection nonlinear geometric relations, basic equations of functionally graded rectangular plate’s elasticity are obtained. Using Hamilton’s principle, elastodynamic model of simply-supported functionally graded plate subjected to aero-thermal load is established, which consists of five degrees of freedom. Transversal motion’s modal function satisfying simply-supported boundary is truncated to two modes. Nonlinear ordinary differential equations of transversal oscillation of the plate are derived by using elmination methods and Galerkin’s method. Material properties vary continously in the thickness direction according to simple power-law. Temperature distribution is in line with one-dimensional heat conduction law. Hypersonic aerodynamics pressure is approximated by higher-order poiston theory. Supersonic as well as hypersonic aerodynamic pressure is considered in this paper.
Average equations in different internal resonance are obtained by using multi-scale method. Based on Potter’s idea, basic stationary solutions are divided into six types, including trivial state, pure modes, mixed modes, traveling waves, standing waves, and modulated traveling waves. Condition for the existence and stability of each type of solutions are discussed. Paticular results of the Aluminum-Alumina functionally graded rectangular plate are given. Effects of temperature and volume fraction index on nonlinear behaviors and stabilities of the plate are analyzed. In numerical experiments only one mode is excited at the beginning, however, many kinds of response, e.g. pure-mode oscillation, mixed-mode oscillation, n-periodic oscillation, may emerge with different temperature or different volume fraction index. Numerical results are qualitatively as same as theorical results, but are quantitatively different, possible reasons are discussed in the paper.
Nonlinear aero-thermal elastodynamics of the plate subjected to linear and nonlinear aero-dynamic pressure are respectively analyzed. Average equations in 1:2 internal resonance of the plate subjected to aero-thermal load are obtained by using multi-scale method. There is no stationary solution if the plate is subjected to linear aero-dynamic pressure. With nonlinear aero-dynamic pressure, if fraction index is equal to 0.2 or 5, average system has stable equilibriums and stable periodic solutions, but relevant traveling waves and modulated traveling waves in original system have not been found. Numerical results show that the response either converges to zero or diverges to infinity. Because of the special form of the truncated two-mode function of the transversal motion, discreted by Galerkin method, linear terms of the two modes are uncoupled. So before buckling, zero equilibrium is always stable, no matter which kind of aerodynamic pressure the plate subjected to. Effects of the form of truncated two-mode function on zero-equilibrium’s stability are discussed detailedly in this paper.
Effects of geometrical parameters and fraction index on natural frequencies of the two modes are analyzed. Coefficients of the two resonant terms have opposite sign is a necessary condition for the existence of non-zero stationary solutions in 1:2 internal resonance. Effects of geometrical parameters on coefficients of the resonant terms are discussed. Numerical results indicate that first mode’s Poincare map has self-similar structure sometimes; its spectrum has a continuous, broad-band nature. In order to recognize chaos, Melnikov’s method is used. Hamilton quantity is obtained and is too small to generate homoclinic orbits, so it is concluded that chaos in the sense of Smale horseshoe is absent.
文章用多尺度法得到系统在不同内共振条件下的平均方程,以Potter的观点为基础,将1:2内共振条件下系统的解分为平凡解、单模态解、双模态解、行波解、驻波解、调幅行波解六类,讨论了各类解存在和稳定的条件,以Aluminum- Alumina功能梯度材料板为例,用半解析法和数值方法研究了温度变化和组分指数对响应类型及稳定性的影响。数值实验最初只激发一个模态,系统因所受热载荷不同或有不同的材料组分指数,而可能出现双模态振动、单模态振动、倍周期振动等不同类型的响应,数值结果与半解析法的结果定性相同,但存在定量差异,文章讨论了可能引起这些差异的原因。
本文分别分析了线性气动力和非线性气动力作用下板的气动热弹性响应,用多尺度法得到了系统在1:2内共振条件下的平均方程,受线性气动力作用时板没有非零定常解,受非线性气动力作用时当组分指数为0.2或5时平均系统中存在稳定的非零平衡点和非零周期解,但在原系统中没有找到对应的行波解和调幅行波解,数值结果得到的响应不是收敛到零就是发散到无穷。由于将四边简支FGM矩形板的双三角级数形式模态函数进行二阶截断的形式比较特殊,经Galerkin离散后各模态的线性项不耦合,因此模态发生屈曲前,不论受到哪种气动力作用,系统的零平衡点始终稳定。文中细致地分析了模态函数的截断形式对零平衡点稳定性的影响。
本文分析了板的几何参数和组分指数对模态频率的影响,两个共振项系数异号是1:2内共振系统存在非零周期解的必要条件,文中分析了几何参数变化对共振项系数符号的影响。数值结果中有些情况下模态的Poincare截面有自相似结构,且有宽频连续频谱,有混沌运动的特点,本文用Melnikov法得到,系统的Hamilton量很小,不足以产生同宿轨道,没有Smale马蹄意义下的混沌。
Nonlinear dynamics of functionally graded plate subjected to aero-thermal load is analyzed in this paper. Based on Reddy’s third-order shear deformation plate theory and von-Karman- type large deflection nonlinear geometric relations, basic equations of functionally graded rectangular plate’s elasticity are obtained. Using Hamilton’s principle, elastodynamic model of simply-supported functionally graded plate subjected to aero-thermal load is established, which consists of five degrees of freedom. Transversal motion’s modal function satisfying simply-supported boundary is truncated to two modes. Nonlinear ordinary differential equations of transversal oscillation of the plate are derived by using elmination methods and Galerkin’s method. Material properties vary continously in the thickness direction according to simple power-law. Temperature distribution is in line with one-dimensional heat conduction law. Hypersonic aerodynamics pressure is approximated by higher-order poiston theory. Supersonic as well as hypersonic aerodynamic pressure is considered in this paper.
Average equations in different internal resonance are obtained by using multi-scale method. Based on Potter’s idea, basic stationary solutions are divided into six types, including trivial state, pure modes, mixed modes, traveling waves, standing waves, and modulated traveling waves. Condition for the existence and stability of each type of solutions are discussed. Paticular results of the Aluminum-Alumina functionally graded rectangular plate are given. Effects of temperature and volume fraction index on nonlinear behaviors and stabilities of the plate are analyzed. In numerical experiments only one mode is excited at the beginning, however, many kinds of response, e.g. pure-mode oscillation, mixed-mode oscillation, n-periodic oscillation, may emerge with different temperature or different volume fraction index. Numerical results are qualitatively as same as theorical results, but are quantitatively different, possible reasons are discussed in the paper.
Nonlinear aero-thermal elastodynamics of the plate subjected to linear and nonlinear aero-dynamic pressure are respectively analyzed. Average equations in 1:2 internal resonance of the plate subjected to aero-thermal load are obtained by using multi-scale method. There is no stationary solution if the plate is subjected to linear aero-dynamic pressure. With nonlinear aero-dynamic pressure, if fraction index is equal to 0.2 or 5, average system has stable equilibriums and stable periodic solutions, but relevant traveling waves and modulated traveling waves in original system have not been found. Numerical results show that the response either converges to zero or diverges to infinity. Because of the special form of the truncated two-mode function of the transversal motion, discreted by Galerkin method, linear terms of the two modes are uncoupled. So before buckling, zero equilibrium is always stable, no matter which kind of aerodynamic pressure the plate subjected to. Effects of the form of truncated two-mode function on zero-equilibrium’s stability are discussed detailedly in this paper.
Effects of geometrical parameters and fraction index on natural frequencies of the two modes are analyzed. Coefficients of the two resonant terms have opposite sign is a necessary condition for the existence of non-zero stationary solutions in 1:2 internal resonance. Effects of geometrical parameters on coefficients of the resonant terms are discussed. Numerical results indicate that first mode’s Poincare map has self-similar structure sometimes; its spectrum has a continuous, broad-band nature. In order to recognize chaos, Melnikov’s method is used. Hamilton quantity is obtained and is too small to generate homoclinic orbits, so it is concluded that chaos in the sense of Smale horseshoe is absent.
引文
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