Study of differential control method for solving chaotic solutions of nonlinear dynamic system

In this paper, we focus on the need to solve chaotic solutions of high-dimensional nonlinear dynamic systems of which the analytic solution is difficult to obtain. For this purpose, a Differential Control Method (DCM) is proposed based on the Mechanized Mathematics-Wu Elimination Method (WEM). By sampling, the computer time of the differential operator of the nonlinear differential equation can be substituted by the differential quotient of solving the variable time of the sample. The main emphasis of DCM is placed on substituting the differential quotient of a small neighborhood of the sample time of the computer system for the differential operator of the equations studied. The approximate analytical chaotic solutions of the nonlinear differential equations governing the high-dimensional dynamic system can be obtained by the method proposed. In order to increase the computational efficiency of the method proposed, a thermodynamics modeling method is used to decouple the variable and reduce the dimension of the system studied. The validity of the method proposed for obtaining approximate analytical chaotic solutions of the nonlinear differential equations is illustrated by the example of a turbo-generator system. This work is applied to solving a type of nonlinear system of which the dynamic behaviors can be described by nonlinear differential equations.     

数学规划加权残值法在非线性微分方程中的应用

本文研究数学规划加权残值法在非线性微分方程求解中的应用,利用数学规划加权残值法和ＬＰ模理论,把非线性微分方程边值问题转化为一个可微分的无约束非线性优化问题,从而运用成熟稳定的寻优方法求得问题的解。文中数字计算例子表明本文方法可以快速有效地求解非线性微分方程。     

Bifurcations at Combination Resonance and Quasiperiodic Vibrations of Flexible Beams

The nonlinear dynamic behavior of flexible beams is described by nonlinear partial differential equations. The beam model accounts for the tension of the neutral axis under vibrations. The Bubnov–Galerkin method is used to derive a system of ordinary differential equations. The system is solved by the multiple-scale method. A system of modulation equations is analyzed     

电磁场中载流薄板的弯曲行为分析

在建立载流薄板的非线性磁弹性运动方程、电动力学方程和Lorentz力表达式的基础上,通过变量代换,将描述薄板的磁弹性状态方程整理成含有10个基本函数的标准Cauchy型.采用差分法及准线性化方法,将标准Cauchy型非线性偏微分方程组,变换成为能够用离散正交法求解的准线性微分方程组.通过算例,得到了电磁场与机械载荷联合...     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

碳纳米管增强复合材料圆锥薄壳非线性自由振动

考虑碳纳米管复合材料作为功能梯度材料的不均匀性,基于连续介质理论以及哈密尔顿变分原理,建立了功能梯度碳纳米管增强复合材料开口圆锥薄壳结构的非线性运动偏微分控制方程,然后利用Galerkin法,将非线性偏微分控制方程转化为常微分控制方程,进而采用谐波平衡法求解了开口圆锥壳的非线性自由振动问题,并探讨了圆锥薄壳几何参数、碳纳米管参数对结构非线性自由振动的影响．数值研究表明结构的无量纲非线性自由振动频率与线性自由振动频率的比值随圆锥薄壳长厚比的增大而变小、并随圆锥角的增大而变大．     

The Cartan-Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders

We develop the Cartan-Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the first order is analyzed, and tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders. Published in Neliniini Kolyvannya, Vol. 10, No. 1, pp. 26–36, January–March, 2007.     

Large deflection analysis of rectangular plates by combined perturbation and finite strip method

The perturbation method and finite strip method are combined to solve the largedeflection bending problems of rectangular plates.Perturbation method is used to reducethe nonlinear differential equations into a series of linear differential equations.The finitestrip method is then employed to tackle these linear equations.Some calculation examplesare compared with those got by other methods.     

斜拉桥拉索-阻尼器系统非线性响应分析

考虑索的抗弯刚度、垂度及几何非线性的影响，得出了索一阻尼器系统的空间非线性振动偏微分方程，用中心差分法将微分方程在空间内离散，导出了系统的非线性振动常微分方程组。结合Newmark法及虚拟力法提出了一种用于求解非线性振动瞬态响应的杂交分析算法。并以典型的斜拉桥拉索为研究对象，给出了数值算例，并与Runge—Kutta直接积分法进行了比较，说明了杂交算法的准确性及有效性。     

Homotopy analysis and differential quadrature solution of the problem of free-convective magnetohydrodynamic flow over a stretching sheet with the Hall effect and mass transfer taken into account

This paper presents an analytical solution of the problem of free-convective magnetohydrodynamic flow over a stretched sheet with the Hall effect and mass transfer taken into account. A similarity transform reduces the Navier-Stokes, energy, Ohm law, and mass-transfer equations to a system of nonlinear ordinary differential equations. The governing equations are solved analytically using an analytical method for solving nonlinear problems, namely, the homotopy analysis method. The results are compared with the results of a promising numerical method of differential quadrature developed by the authors. It is shown that there is very good agreement between analytical results and those obtained by the differential quadrature method. The differential quadrature method was validated, and the effects of non-dimensional parameters on the velocity, temperature and concentration profiles were studied.     

Nonlinear transient response of stay cable with viscoelasticity damper in cable-stayed bridge

Taking the bending stiffness, static sag, and geometric non-linearity into consideration, the space nonlinear vibration partial differential equations were derived. The partical differential equations were discretized in space by finite center difference approximation, then the nonlinear ordinal differential equations were obtained. A hybrid method involving the combination of the Newmark method and the pseudo-force strategy was proposed to analyze the nonlinear transient response of the inclined cable-dampers system subjected to arbitrary dynamic loading. As an example, two typical stay cables were calculated by the present method. The results reveal both the validity and the deficiency of the viscoelasticity damper for vibration control of stay cables. The efficiency and accuracy of the proposed method is also verified by comparing the results with those obtained by using Runge-Kutta direct integration technique. A new time history analysis method is provided for the research on the stay cable vibration control.     

Stability of nonlinear time-varying perturbed differential equations

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

压电复合材料层合梁的分岔、混沌动力学与控制

研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分岔和混沌动力学响应. 基于vonKarman理论和Reddy高阶剪切变形理论,推导出了压电复合层合梁的动力学方程. 利用Galerkin法离散偏微分方程,得到两个自由度非线性控制方程,并且利用多尺度法得到了平均方程. 基于平均方程,研究了压电层合梁系统的动态分岔,分析了系统各种参数对倍周期分岔的影响及变化规律. 结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分岔解,从而阻止系统通过倍周期分岔进入混沌运动,并给出了控制分岔图.      

Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.     

On the Asymptotic Behavior of Solutions of a System of Nonlinear Differential Equations

We investigate the asymptotic behavior of a system of nonlinear differential equations of a special form at infinity. We also propose a method for the reduction of more general systems of nonlinear differential equations to this form, which enables one to study their asymptotic properties.     

Nonlinear stress and deformation analysis of thin current-carrying strip shells

The paper analyzes the nonlinear deformation of a current-carrying thin shell in coupled electromagnetic and mechanical fields. The nonlinear magnetoelastic kinetic equations, physical equations, geometric equations, electrodynamic equations, expressions for the Lorentz force of a current-carrying thin shell in a coupled field are given. The normal Cauchy form nonlinear differential equations that include ten basic unknown functions are obtained by the variable replacement method. The difference and quasi-linearization methods are used to reduce the nonlinear magnetoelastic equations to a sequence of quasilinear differential equations that can be solved by discrete orthogonalization. Numerical solutions for the stresses and strains in a current-carrying thin strip shell with two edges simply supported are obtained as an example. The dependence of the stresses and strains in the current-carrying thin strip shell on the electromagnetic parameters is discussed. In a special case, it is shown that the deformation of the shell can be controlled by changing the electromagnetic parameters     

Chaotic interaction dynamics of three structures: Two cylindrical shells nested into each other and their reinforcing local rib

This paper studies the chaotic dynamics of two cylindrical shells nested into each other with a gap and their reinforcing beam, also with a gap, which is subjected to a distributed alternating load. The problem is solved using methods of nonlinear dynamics and the qualitative theory of differential equations. The Novozhilov equations for geometrically nonlinear structures are used as the governing equations. Contact pressure is determined by Kantor’s method. Using finite elements in spatial variables, the partial differential equations for the beam and shells are reduced to the Cauchy problem, which is solved by explicit integration (Euler’s method). The chaotic synchronization of this system is studied.     

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.