非对称强非线性振动特征分析

提出了构造一类非线性振子解析逼近周期解的的初值变换法.用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以振幅、角频率和偏心距为独立变量的不完备非线性代数方程组;关键是考虑初值变换,增加补充方程,构成了以角频率、振幅和偏心距为变量的完备非线性代数方程组.作为例子利用初值变换法求解了相对论修正轨道方程的六种分岔周期解.给出了非对称振动的幅频曲线和偏频(偏心距与角频率的关系)曲线.发现了固有角频率漂移现象.     

Nonlinear stabilization in infinite dimension

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infinite-dimensional setting. First we present designs for nonlinear ODEs with constant, time-varying or state-dependent input delays, which arise in numerous applications of control over networks. Second, we present a design for nonlinear ODEs with a wave (string) PDE at its input, which is motivated by the drilling dynamics in petroleum engineering. Third, we present a design for systems of (two) coupled nonlinear first-order hyperbolic PDEs, which is motivated by slugging flow dynamics in petroleum production in off-shore facilities. Our design and analysis methodologies are based on the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping. We present several simulation examples that illustrate the design methodology.     

A modification of Davidenko''s method for nonlinear systems

In 1953 Davidenko showed how to reduce the solving of a system of nonlinear equations to the integration of a system or ordinary differential equations. This method also required the solution of linear algebraic equations at each integration step, which is computationally undesirable. We show how to reduce the solving of a system of nonlinear equations solely to the integration of a system of ordinary differential equations, avoiding entirely the solution of linear algebraic equations. A numerical example shows interesting differences between the two approaches in the neighborhood of a bifurcation point.     

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

Numerical solution of nonlinear differential equations using computer algebra

This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.     

On the modelling of actuator dynamics and the computation of prescribed trajectories

The dynamics of actuator mechanisms is presented using a multibody modelling approach to concisely express the structure of the system equations. The Lagrange equations are used to obtain the Newton–Euler equations to which constraint equations are augmented to form a system of differential algebraic equations. The differential algebraic equations are cast as ordinary differential equations and computed using the numerical integrator LSODAR of Petzold and Hindmarsh. Constraint compliance is investigated to ensure the accuracy of the results. Animation of an excavator and wheel loader system is presented and graphs of constraint forces show the nature of the actuator dynamics involved in maintaining specified bucket trajectories. The model is general in nature and caters for arbitrary mechanism connectivity and physical properties.     

A Robust Numerical Scheme For Pricing American Options Under Regime Switching Based On Penalty Method

This paper is devoted to develop a robust numerical method to solve a system of complementarity problems arising from pricing American options under regime switching. Based on a penalty method, the system of complementarity problems are approximated by a set of coupled nonlinear partial differential equations (PDEs). We then introduce a fitted finite volume method for the spatial discretization along with a fully implicit time stepping scheme for the PDEs, which results in a system of nonlinear algebraic equations. We show that this scheme is consistent, stable and monotone, hence convergent. To solve the system of nonlinear equations effectively, an iterative solution method is established. The convergence of the solution method is shown. Numerical tests are performed to examine the convergence rate and verify the effectiveness and robustness of the new numerical scheme.     

Triangularizing kinematic constraint equations using Gröbner bases for real-time dynamic simulation

Real-time simulation is an essential component of hardware- and operator-in-the-loop applications, such as driving simulators, and can greatly facilitate the design, implementation, and testing of dynamic controllers. Such applications may involve multibody systems containing closed kinematic chains, which are most readily modeled using a set of redundant generalized coordinates. The governing dynamic equations for such systems are differential-algebraic in nature—that is, they consist of a set of ordinary differential equations coupled with a set of nonlinear algebraic constraint equations—and can be difficult to solve in real time. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing a recursively solvable system for calculating the dependent generalized coordinates given values of the independent coordinates. The proposed approach can be used to generate computationally efficient simulation code that avoids the use of iteration, which makes it particularly suitable for real-time applications.     

Preconditioning for a Class of Spectral Differentiation Matrices

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton’s method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator     

An integral-equation solution for the finite deflection of clamped skew sandwich plates

The large-deflection behaviour of skew sandwich plates is governed by a system of five coupled nonlinear partial differential equations which are highly complex in nature. In the reported study, this problem is analysed using an integral-equation approach. The integral equations of beams along the skew directions is used with appropriate boundary conditions to transform the governing nonlinear partial differential equations into a set of nonlinear algebraic equations. These equations are then solved using an iterative scheme suggested by Brown. The results obtained by this method are compared with available results of other investigators and the agreement is found to be good. Load-deflection characteristics have been presented for clamped skew sandwich plates.     

EASY-FIT: a software system for data fitting in dynamical systems

EASY-FIT is an interactive software system to identify parameters in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, that depends on the solution of the dynamical system. The software system is implemented in form of a Microsoft Access database running under MS-Windows 95/98/NT/2000. The underlying numerical algorithms are coded in Fortran and are executable independently from the interface. Model functions are either interpreted and evaluated symbolically by a Fortran-similar modeling language, that allows in addition automatic differentiation of nonlinear functions, or by user-provided Fortran subroutines. Received December 30, 2000     

From Chronological Calculus to Exponential Representations of Continuous and Discrete-Time Dynamics: A Lie-Algebraic Approach

In this paper, formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived. It is shown that the chronological exponential admits an ordinary exponential representation, the exponent being given by an explicitly computable Lie series expansion. The results are then used to describe controlled dynamics, dynamics under sampling and forced discrete-time dynamics. The study emphasizes the role of Lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations, dynamics under sampling and forced discrete-time dynamics up to hybrid ones.     

Analytical and numerical study of photocurrent transients in organic polymer solar cells

This article is an attempt to provide a self consistent picture, including existence analysis and numerical solution algorithms, of the mathematical problems arising from modeling photocurrent transients in organic polymer solar cells (OSCs). The mathematical model for OSCs consists of a system of nonlinear diffusion–reaction partial differential equations (PDEs) with electrostatic convection, coupled to a kinetic ordinary differential equation (ODE). We propose a suitable reformulation of the model that allows us to prove the existence of a solution in both stationary and transient conditions and to better highlight the role of exciton dynamics in determining the device turn-on time. For the numerical treatment of the problem, we carry out a temporal semi-discretization using an implicit adaptive method, and the resulting sequence of differential subproblems is linearized using the Newton–Raphson method with inexact evaluation of the Jacobian. Then, we use exponentially fitted finite elements for the spatial discretization, and we carry out a thorough validation of the computational model by extensively investigating the impact of the model parameters on photocurrent transient times.     

轮-带驱动系统稳态周期响应谐波平衡分析

研究含有单向离合器、两滑轮及附件的轮-带驱动系统稳定稳态周期响应.通过单向离合器连接从动轮与附属系统,并计入传送带的横向振动的影响,导出了由偏微分-积分方程与分段常微分方程组成的连续-离散型非线性耦合方程组.利用Galerkin方法将连续非线性方程组截断为一组非线性常微分方程组,再运用谐波平衡法得到轮-带驱动系统耦合非线性振动的稳态响应.通过比较有无单向离合器装置的系统稳定稳态幅频响应曲线,研究了单向离合器对驱动系统以及轮-带系统非线性动态特性的影响.并首次研究了高频激励下轮-带系统的稳态响应.最后,运用Runge-Kutta方法对比验证了基于谐波平衡法得到的稳态响应.     

压电复合材料梁的全局分叉、混沌动力学分析

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、分叉和混沌动力学响应.基于yon Karman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,得到二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...     

Design of maximum thrust plug nozzles with variable inlet geometry

An optimization analysis is presented for axisymmetric plug nozzles with varible inlet geometry. The analysis is based on the governing gas dynamic relations for a rotational flow of a frozen or equilibrium gas mixture. The problem is formulated to maximize the axial thrust produced by the plug nozzle for a general isoperimetric constraint, such as constant nozzle length or constant nozzle surface area. The effects of base pressure and ambient pressure are included in the thrust expression to be maximized. The governing gas dynamic equations and the differential and integral constraints that the solution must satisfy are incorporated into the formulation by means of Lagrange multiples. The formalism of the calculus of variations is applied to the resulting functional to be maximized. The results of the optimization analysis are a set of partial differential equations for determining the Lagrange multipliers in the region of interest and a set of equations for determining the necessary boundary conditions for the solution. The complete set of equations for the gas dynamic properties and the Lagrange multipliers are system of first order, quasi-linear, non-homogeneous partial differential equations of the hyperbolic type, which can be treated by the method of charac- teristics. The characteristic and compatibility equations for the system are presented. A numerical solution procedure is presented to determine wether or not a given plug nozzle geometry is an optimal solution. An iteration technique is developed which systematically adjusts the plug nozzle geometry until the optimal solution is obtained. Selected parametric studies are presented. These studies illustrate the effect of the specific heat ratio, the design pressure ratio and the base pressure model on the thrust peformance and nozzle geometry of optimal, fixed length, plug nozzles.     

The BEM for numerical solution of partial fractional differential equations

A numerical method is presented for the solution of partial fractional differential equations (FDEs) arising in engineering applications and in general in mathematical physics. The solution procedure applies to both linear and nonlinear problems described by evolution type equations involving fractional time derivatives in bounded domains of arbitrary shape. The method is based on the concept of the analog equation, which in conjunction with the boundary element method (BEM) enables the spatial discretization and converts a partial FDE into a system of coupled ordinary multi-term FDEs. Then this system is solved using the numerical method for the solution of such equations developed recently by Katsikadelis. The method is illustrated by solving second order partial FDEs and its efficiency and accuracy is validated.     

Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey–predator model with stability analysis

The Journal of Supercomputing - A coupled nonlinear prey–predator system is presented. The system formulation is based on nonlinear ordinary differential equations with imprecise parameter...     

ATOMFT: solving ODEs and DAEs using Taylor series

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODEs into a FORTRAN 77 object program that is compiled, linked with the ATOMFT runtime library, and run to solve the problem. We review the use of the ATOMFT system for nonstiff and stiff ODEs, the propagation of global errors, and applications to differential algebraic equations arising from certain control problems, to boundary value problems, to numerical quadrature, and to delay problems.