scholarly journals Spectral Fixed Point Method for Nonlinear Oscillation Equation with Periodic Solution

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ding Xu ◽  
Xian Wang ◽  
Gongnan Xie
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Oscillation ◽  
Fixed Point Method ◽  
Point Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Linear Oscillation ◽  
Oscillation Equation

Based on the fixed point concept in functional analysis, an improvement on the traditional spectral method is proposed for nonlinear oscillation equations with periodic solution. The key idea of this new approach (namely, the spectral fixed point method, SFPM) is to construct a contractive map to replace the nonlinear oscillation equation into a series of linear oscillation equations. Usually the series of linear oscillation equations can be solved relatively easily. Different from other existing numerical methods, such as the well-known Runge-Kutta method, SFPM can directly obtain the Fourier series solution of the nonlinear oscillation without resorting to the Fast Fourier Transform (FFT) algorithm. In the meanwhile, the steepest descent seeking algorithm is proposed in the framework of SFPM to improve the computational efficiency. Finally, some typical cases are investigated by SFPM and the comparison with the Runge-Kutta method shows that the present method is of high accuracy and efficiency.

The Analysis of the Dynamic Behavior of the Nonlinear Oscillations of a Fluttering Plate Based on the Periodicity Ratio Method

2012 ◽  
Author(s):  
Liming Dai ◽  
Xiaojie Wang
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Oscillation ◽  
Nonlinear Oscillations ◽  
Ratio Method ◽  
Kutta Method ◽  
Governing Equations ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper, an investigation of the dynamic behavior of a fluttering plate is performed. The governing equations of the nonlinear oscillation derived using Galerkin’s method are presented and solved numerically by the Runge-Kutta method. Four modes are used for accurate results. The periodicity ratio method is briefly introduced and applied to generate the region diagram for the motion of the plate. The corresponding motions under different parameters following the region diagram are illustrated. The results demonstrate the efficiency and accuracy of periodicity ratio method.

scholarly journals Modeling of Deadtime Events in Power Converters with Half-Bridge Modules for a Highly Accurate Hardware-in-the-Loop Fixed Point Implementation in FPGA

2021 ◽  
Vol 11 (14) ◽  
pp. 6490
Author(s):  
Roberto Saralegui ◽  
Alberto Sanchez ◽  
Angel de Castro
Keyword(s):  
Fixed Point ◽  
Real Time ◽  
Power Converters ◽  
Power Converter ◽  
Hardware In The Loop ◽  
Kutta Method ◽  
Time Step ◽  
Worst Case ◽  
Runge Kutta Method ◽  
Runge Kutta

Hardware-in-the-loop (HIL) simulations of power converters must achieve a truthful representation in real time with simulation steps on the order of microseconds or tens of nanoseconds. The numerical solution for the differential equations that model the state of the converter can be calculated using the fourth-order Runge–Kutta method, which is notably more accurate than Euler methods. However, when the mathematical error due to the solver is drastically reduced, other sources of error arise. In the case of converters that use deadtimes to control the switches, such as any power converter including half-bridge modules, the inductor current reaching zero during deadtimes generates a model error large enough to offset the advantages of the Runge–Kutta method. A specific model is needed for such events. In this paper, an approximation is proposed, where the time step is divided into two semi-steps. This serves to recover the accuracy of the calculations at the expense of needing a division operation. A fixed-point implementation in VHDL is proposed, reusing a block along several calculation cycles to compute the needed parameters for the Runge–Kutta method. The implementation in a low-cost field-programmable gate arrays (FPGA) (Xilinx Artix-7) achieves an integration time of 1μs. The calculation errors are six orders of magnitude smaller for both capacitor voltage and inductor current for the worst case, the one where the current reaches zero during the deadtimes in 78% of the simulated cycles. The accuracy achieved with the proposed fixed point implementation is very close to that of 64-bit floating point and can operate in real time with a resolution of 1μs. Therefore, the results show that this approach is suitable for modeling converters based on half-bridge modules by using FPGAs. This solution is intended for easy integration into any HIL system, including commercial HIL systems, showing that its application even with relatively high integration steps (1μs) surpasses the results of techniques with even faster integration steps that do not take these events into account.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

Application of the piecewise constant method in nonlinear dynamics of drill string

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jialin Tian ◽  
Jie Wang ◽  
Yi Zhou ◽  
Lin Yang ◽  
Changyue Fan ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Drill String ◽  
Vibration Velocity ◽  
Kutta Method ◽  
Time Step ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

Sign in / Sign up

Export Citation Format

Share Document