Accuracy and Reliability of Piecewise-Constant Method in Studying the Responses of Nonlinear Dynamic Systems

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Liming Dai ◽  
Xiaojie Wang ◽  
Changping Chen
Keyword(s):  
Numerical Methods ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamic Systems ◽  
Processing Unit ◽  
Kutta Method ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Accuracy and reliability of the numerical simulations for nonlinear dynamical systems are investigated with fourth-order Runge–Kutta method and a newly developed piecewise-constant (P-T) method. Nonlinear dynamic systems with external excitations are studied and compared with the two numerical approaches. Semianalytical solutions for the dynamic systems are developed by the P-T approach. With employment of a periodicity-ratio (PR) method, the regions of regular and irregular motions are determined and graphically presented corresponding to the system parameters, for the comparison of accuracy and reliability of the numerical methods considered. Central processing unit (CPU) time executed in the numerical calculations with the two numerical methods are quantitatively investigated and compared under the same computational conditions. Due to its inherent drawbacks, as found in the research, Runge–Kutta method may cause information missing and lead to incorrect conclusions in comparing with the P-T method.

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.

scholarly journals Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties

2009 ◽  
Vol 19 (3) ◽  
pp. 425-439 ◽  
Author(s):  
Andreas Rauh ◽  
Johanna Minisini ◽  
Eberhard Hofer
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Interval Arithmetic ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Operating Conditions ◽  
Nonlinear Dynamic Systems ◽  
Parameter Uncertainties ◽  
Analysis And Design ◽  
Verification Techniques

Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with UncertaintiesControl strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.

Periodicity and Nonlinearity of Nonlinear Dynamical Systems Under Periodic Excitation

2009 ◽  
Author(s):  
Lu Han ◽  
Liming Dai ◽  
Huayong Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Research Work ◽  
Nonlinear Dynamic Systems ◽  
Periodic Excitation ◽  
Nonlinear Dynamical

Periodicity and nonlinearity of nonlinear dynamic systems subjected to regular external excitations are studied in this research work. Diagnoses of regular and chaotic responses of nonlinear dynamic systems are performed with the implementation of a newly developed Periodicity Ratio in combining with the application of Lyapunov Exponent. The properties of the nonlinear dynamics systems are classified into four categories: periodic, irregular-nonchaotic, quasiperiodic and chaotic, each corresponding to their Periodicity Ratios. Detailed descriptions about diagnosing the responses of the four categories are presented with utilization of the Periodicity Ratio.

scholarly journals Comparison of Numerical Simulation of Epidemiological Model between Euler Method with 4th Order Runge Kutta Method

2021 ◽  
Vol 2 (1) ◽  
pp. 37-44
Author(s):  
Rizky Ashgi
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Sir Model ◽  
Euler Method ◽  
Kutta Method ◽  
System Of Differential Equations ◽  
Runge Kutta Method ◽  
Global Pandemic ◽  
The World ◽  
Runge Kutta

Coronavirus Disease 2019 has become global pandemic in the world. Since its appearance, many researchers in world try to understand the disease, including mathematics researchers. In mathematics, many approaches are developed to study the disease. One of them is to understand the spreading of the disease by constructing an epidemiology model. In this approach, a system of differential equations is formed to understand the spread of the disease from a population. This is achieved by using the SIR model to solve the system, two numerical methods are used, namely Euler Method and 4th order Runge-Kutta. In this paper, we study the performance and comparison of both methods in solving the model. The result in this paper that in the running process of solving it turns out that using the euler method is faster than using the 4th order Runge-Kutta method and the differences of solutions between the two methods are large.

scholarly journals Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Uğur Kadak ◽  
Muharrem Özlük
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Kutta Method ◽  
Science And Engineering ◽  
Runge Kutta Method ◽  
Wide Range ◽  
Runge Kutta

Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

scholarly journals Fast Recursive Filters for Simulating Nonlinear Dynamic Systems

2008 ◽  
Vol 20 (7) ◽  
pp. 1821-1846 ◽  
Author(s):  
J. H. van Hateren
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Feedforward Control ◽  
Horizontal Cell ◽  
Complex Model ◽  
Computational Scheme ◽  
Nonlinear Dynamic Systems ◽  
Processing Unit ◽  
Low Pass ◽  
System Variables

A fast and accurate computational scheme for simulating nonlinear dynamic systems is presented. The scheme assumes that the system can be represented by a combination of components of only two different types: first-order low-pass filters and static nonlinearities. The parameters of these filters and nonlinearities may depend on system variables, and the topology of the system may be complex, including feedback. Several examples taken from neuroscience are given: phototransduction, photopigment bleaching, and spike generation according to the Hodgkin-Huxley equations. The scheme uses two slightly different forms of autoregressive filters, with an implicit delay of zero for feedforward control and an implicit delay of half a sample distance for feedback control. On a fairly complex model of the macaque retinal horizontal cell, it computes, for a given level of accuracy, one to two orders of magnitude faster than the fourth-order Runge-Kutta. The computational scheme has minimal memory requirements and is also suited for computation on a stream processor, such as a graphical processing unit.

scholarly journals Comparative Study on Results of Euler, Improved Euler and Runge-Kutta Methods for Solving the Engineering Unknown Problems

2021 ◽  
Vol 3 (3) ◽  
Author(s):  
Khaing Khaing Lwin
Keyword(s):  
Numerical Methods ◽  
Comparative Study ◽  
Fourth Order ◽  
Teaching And Learning ◽  
Euler Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Engineering Problems ◽  
Runge Kutta ◽  
The Difference

The paper presents the comparative study on numerical methods of Euler method, Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications. The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems. This paper aims to enhance the teaching and learning quality of teachers and students for various levels. At each point of the interval, the value of y is calculated and compared with its exact value at that point. The next interesting point is the observation of error from those methods. Error in the value of y is the difference between calculated and exact value. A mathematical equation which relates various functions with its derivatives is known as a differential equation. It is a popular field of mathematics because of its application to real-world problems. To calculate the exact values, the approximate values and the errors, the numerical tool such as MATLAB is appropriate for observing the results. This paper mainly concentrates on identifying the method which provides more accurate results. Then the analytical results and calculates their corresponding error were compared in details. The minimum error directly reflected to realize the best method from different numerical methods. According to the analyses from those three approaches, we observed that only the error is nominal for the fourth-order Runge-Kutta method.

scholarly journals Resolução Numérica da equação diferencial de resfriamento de newton pelos métodos de EULER e RUNGE-KUTTA

2016 ◽  
Vol 2 (1) ◽  
pp. 10-25
Author(s):  
Andresa Pescador ◽  
Zilmara Raupp Quadros Oliveira
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Euler Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Analytical Resolution ◽  
Important Branch ◽  
First Order Differential Equations

This article presents the first-order differential equations, which are a very important branch of mathematics as they have a wide applicability, in mathematics, as in physics, biology and economy. The objective of this study was to analyze the resolution of the equation that defines the cooling Newton's law. Verify its behavior using some applications that can be used in the classroom as an auxiliary instrument to the teacher in addressing these contents bringing answers to the questions of the students and motivating them to build their knowledge. It attempted to its resolution through two numerical methods, Euler method and Runge -Kutta method. Finally, there was a comparison of the approach of the solution given by the numerical solution with the analytical resolution whose solution is accurate.

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