Variational methods of solving dynamic problems for fluid-containing bodies

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 37–77, October 2004.The study was partially sponsored by the German Research Fund (der Deutsche Forschungsgemeinschaft), Grant 436 UKR113/33/0-3.     

Non-iterative methods for dynamic analysis of nonlinear velocity-dependent problems

Nonlinear Dynamics - A class of nonlinear velocity-dependent problems must be solved iteratively for conventional integration methods since there exists no completely explicit integration method...     

Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series

The paper presents single-term Haar wavelet series (STHWS) approach to the solution of nonlinear stiff differential equations arising in nonlinear dynamics. The properties of STHWS are given. The method of implementation is discussed. Numerical solutions of some model equations are investigated for their stiffness and stability and solutions are obtained to demonstrate the suitability and applicability of the method. The results in the form of block-pulse and discrete solutions are given for typical nonlinear stiff systems. As compared with the TR BDF2 method of Shampine and Gill’s method, the STHWS turns out to be more effective in its ability to solve systems ranging from mildly to highly stiff equations and is free from stability constraints.     

非线性动力学方程的一种级数解

本文针对n维未知向量v的一阶微分议程v=Hv f(v,t)进行求解,其中Hv和f(v,t)分别是右端项的线性剂次部分和非线性部分,首先,将非线性部分f(v,t)在所论时刻tk处展成t-tk=τ的泰勒级数,并通过求原函数的方法直接给出每一项的积分,从而获得了待求微分方程在级数形式下的闭合争,它的具有不同精度的各次静似解可表示成τ的分段解析函数,便于研究非线性动力学行为与其物理参数的依赖关系。本文还用算例验证了各次近似解之间的数值关系,并和解析解等作了比较。     

STUDY ON PREDICTION METHODS FOR DYNAMIC SYSTEMS OF NONLINEAR CHAOTIC TIME SERIES＊

The prediction methods for nonlinear dynamic systems which are decided by chaotic time series are mainly studied as well as structures of nonlinear self-related chaotic models and their dimensions. By combining neural networks and wavelet theories, the structures of wavelet transform neural networks were studied and also a wavelet neural networks learning method was given. Based on wavelet networks, a new method for parameter identification was suggested, which can be used selectively to extract different scales of frequency and time in time series in order to realize prediction of tendencies or details of original time series. Through pre-treatment and comparison of results before and after the treatment, several useful conclusions are reached:High accurate identification can be guaranteed by applying wavelet networks to identify parameters of self-related chaotic models and more valid prediction of the chaotic time series including noise can be achieved accordingly.     

Interpolation perturbation method for solving nonlinear problems

I.IntroductionInthispaper,onthebasisofRefll],usingtheinterPotationPerturbationmethod,theauthorseekstosolveseveralnon-Iinearproblems.Itsmainpointsare:Introducinganinterpolationfunction,wedeterminethisfunctionbyusingtheperturbationmethodandthenproceedtoseek…     

THE GENERAL METHOD FOR SOLVING DYNAMIC PROBLEMS

ＴＨＥＧＥＮＥＲＡＬＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＤＹＮＡＭＩＣＰＲＯＢＬＥＭＳ￥(孙右烈)ＳｕｎＹｏｕｌｉｅ（ＳｈａｎｇｈａｉＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２０００７２，Ｐ．Ｒ．Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｔｈｅａｕ...     

A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics

In this paper a comparative study of five different stability computational methods based on the Floquet theory is presented. These methods are compared in terms of accuracy and CPU performance. Tests are performed on a set of nonlinear problems relevant to rotating machinery with rotor-to-stator contact and a variable number of degrees of freedom, whose periodic solutions are computed with the Harmonic Balance Method (HBM).     

Improved L-P method for solving strongly nonlinear problems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｏｎｐｒｏｂｌｅｍ¨ｕ ω20 ｕ =εｆ(ｕ , ｕ ,¨ｕ) , (1 )ｗｈｅｒｅｔｈｅｓｙｍｂｏｌ“·”ｄｅｎｏｔｅｓｔｈｅｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｈｅｉｎｄｅｐｅｎｄｅｎｔｖａｒｉａｂｌｅｔａｎｄω0ａｎｄεａｒｅａｌｌｔｈｅａｒｂｉｔｒａｒｙｃｏｎｓｔａｎｔｓ,ｈａｖｅｉｎｔｈｅｒｅｃｅｎｔｙｅａｒｓｓｅｖｅｒａｌｉｍｐｒｏｖｅｄＬ＿Ｐｍｅｔｈｏｄｓｗｈｅｎεｉｓｎｏｔｓｍａｌｌ (ｓｔｒｏ…     

Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.     

Block pivot methods for solving frictional contact problems

Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is “relative small”. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables. The project supported by the National Natural Science Foundation of China     

A comparative analysis of methods for solving equations in the nonlinear elasticity theory

An approach to solving a variational equation used in geometrically and physically nonlinear problems of deformable body mechanics is considered. This approach is based on the continuation of a solution with respect to the loading parameter. Large systems of nonlinear ordinary differential equations arise in such problems. Usually, these systems are solved by the Euler methods. It is proposed to use the Runge-Kutta and multistep methods and to estimate the total computational cost. A dependence of numerical errors on the number of integration steps is obtained. An optimal method for solving nonlinear problems is chosen on the basis of this dependence.     

The subharmonic bifurcation solution of nonlinear Mathieu's equation and Euler dynamic buckling problems

A new approach is presented in this paper on the basis of dynamic systems theory. This paper presents the form of a generic classification of stable response diagrams for the nonlinear Mathieu equation. In addition, a general method is presented for determining the topological type of the response diagram for a given equation. This method has been successfully applied to Euler dynamic buckling problems. Some new results are obtained.     

A simple and efficient fourth-order approximation solution for nonlinear dynamic system

In the present paper, based on the classical Magnus expansion, a simple and efficient fourth-order integrator is given for an arbitrary nonlinear dynamic system, which can preserve the qualitative properties of the exact solution. The proposed method can be considered an averaging technique, and only requires evaluations of exponentials of simple unidimensional integrals. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the method of this paper.     

求解非线性反问题的大范围收敛梯度正则化算法

基于同伦映射的思想,改进了求解非线性反问题的梯度正则化算法。通过路径跟踪有效地拓宽了梯度正则化算法求解的收敛范围。对于正则化参数的修正,通过引入拟Sigmoid函数,提出了一种下降速率可调的连续化参数修正方法,在保证迭代稳定的条件下,得到较好的计算效率,同时保证该算法具有很好的抵抗观测噪声能力。实际算例表明,该方法收敛范围宽,计算效率高,在存在较强观测噪声的条件下也能得到很好的反演结果。