scholarly journals Solving Fractional Dynamical System with Freeplay by Combining Memory-Free Approach and Precise Integration Method

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Q. X. Liu ◽  
X. S. He ◽  
J. K. Liu ◽  
Y. M. Chen ◽  
L. C. Huang
Keyword(s):  
Integration Method ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Precise Integration ◽  
Linear Ordinary Differential Equations ◽  
First Order ◽  
Precise Integration Method ◽  
Runge Kutta Method ◽  
Predictor Corrector ◽  
Fractional Dynamical System

The Yuan-Agrawal (YA) memory-free approach is employed to study fractional dynamical systems with freeplay nonlinearities subjected to a harmonic excitation, by combining it with the precise integration method (PIM). By the YA method, the original equations are transformed into a set of first-order piecewise-linear ordinary differential equations (ODEs). These ODEs are further separated as three linear inhomogeneous subsystems, which are solved by PIM together with a predictor-corrector process. Numerical examples show that the results by the presented method agree well with the solutions obtained by the Runge-Kutta method and a modified fractional predictor-corrector algorithm. More importantly, the presented method has higher computational efficiency.

Assessment of numerical integration algorithms for nonlinear vibration of railway vehicles

2016 ◽  
Vol 231 (6) ◽  
pp. 729-739 ◽  
Author(s):  
Chao Yang ◽  
Duo Li ◽  
Tao Zhu ◽  
Shoune Xiao
Keyword(s):  
Integration Method ◽  
Railway Vehicle ◽  
Kutta Method ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Adams Method ◽  
Runge Kutta Method ◽  
Vehicle Systems ◽  
Runge Kutta ◽  
Predictor Corrector

The primary purpose of this paper was to choose appropriate time integration methods for the simulation of nonlinear railway vehicle systems. The nonlinear elements existing in railway vehicle systems were summarized, and the relevant mathematical expressions were provided. A newly developed integration method, which is the corrected explicit method of double time steps, was implemented in five typical nonlinear examples of nonlinear railway vehicle systems. The Newmark method, the Wilson-θ method, the Runge–Kutta method, the predictor-corrector Adams method, the Zhai method, and the precise integration method were also employed for comparison purpose. Finally, the scope of application of these methods was pointed out . The results show that the Newmark method and the Wilson-θ method should not be applied to nonlinear railway vehicle systems as these methods result in errors. In contrast to the predictor-corrector Adams method and the precise integration method, the Runge–Kutta method with error control (RK45) is not applicable to the non-smooth problems although the RK45 possesses high accuracy. In addition, the application of the RK45 and the predictor-corrector Adams method with error control may result in spurious tiny oscillation in the vehicle system related to nonlinear vertical wheel–rail forces. The corrected explicit method of double time steps, the Zhai method, the standard Runge–Kutta method, the precise integration method, the RK45, and the predictor-corrector Adams method which possess tight error tolerances are recommended for nonlinear railway vehicle systems according to the requirements of accuracy and computational efficiency.

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

Extension of the Wittrick-Williams Algorithm to Mixed Variable Systems

1997 ◽  
Vol 119 (3) ◽  
pp. 334-340 ◽  
Author(s):  
Zhong Wanxie ◽  
F. W. Williams ◽  
P. N. Bennett
Keyword(s):  
Timoshenko Beam ◽  
Stiffness Matrix ◽  
Integration Method ◽  
Dynamic Stiffness ◽  
Integration Algorithm ◽  
Precise Integration ◽  
Matrix Computations ◽  
Precise Integration Method ◽  
Count Method ◽  
Mixed Variable

A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

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