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.

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.

scholarly journals Numerical calculation for coupling vibration system by Piecewise-Laplace method

2020 ◽  
Vol 103 (3) ◽  
pp. 003685042093855
Author(s):  
Pan Fang ◽  
Kexin Wang ◽  
Liming Dai ◽  
Chixiang Zhang
Keyword(s):  
Numerical Computation ◽  
Laplace Transformation ◽  
Original System ◽  
Continuity Condition ◽  
Kutta Method ◽  
Laplace Method ◽  
Time Step ◽  
Coupling Vibration ◽  
Runge Kutta Method ◽  
Runge Kutta

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method.

Dynamic Response of SFD-Sliding Rigid Bearing Rotor -System

2013 ◽  
Vol 706-708 ◽  
pp. 1310-1313
Author(s):  
Ji Yan Wang
Keyword(s):  
Dynamic Response ◽  
Dynamic Model ◽  
Periodic Motion ◽  
Rotor System ◽  
Rigid Rotor ◽  
Kutta Method ◽  
Sliding Bearing ◽  
Runge Kutta Method ◽  
Stable Periodic ◽  
Runge Kutta

This paper establishes a linking dynamic model of SFD-sliding bearing rigid rotor system by employing Runge-Kutta method to solve dynamic question of the above systems. The study has shown: rigid rotor system can keep stable periodic motion in a certain scope with the following quasi-periodicity bifurcation.

Dynamic Characteristics and Load Coefficient Analysis of Involute Spline Couplings

2014 ◽  
Vol 889-890 ◽  
pp. 450-454
Author(s):  
Xiang Zhen Xue ◽  
San Min Wang
Keyword(s):  
Dynamic Model ◽  
Fourth Order ◽  
Dynamic Characteristics ◽  
Dynamic Load ◽  
Dynamic Equations ◽  
Kutta Method ◽  
Meshing Stiffness ◽  
Runge Kutta Method ◽  
Spline Coupling ◽  
Involute Spline

As one of the important components of aviation and space transmission systems, dynamic characteristics of involute spline couplings influence its lifetime and reliability seriously. Here, taking the backlash of spline joint into account, considering the meshing stiffness varying with the teeth engaged, established the dynamic model with varying stiffness and dynamic equations, and calculated the number of actual meshing teeth and comprehensive meshing stiffness while bearing the varying torque, then, solved dynamic equations using the fourth order Runge - Kutta method, finally, get the teeth meshing number is 23,and the maximum dynamic load coefficient gets smaller from 1.19 to1.15 with the decrease of . This provides a numerical basis for wear`s studying and lifetime`s forecasting of involute spline coupling.

scholarly journals Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

2018 ◽  
Vol 11 (2) ◽  
pp. 575-592 ◽  
Author(s):  
Thomas Rößler ◽  
Olaf Stein ◽  
Yi Heng ◽  
Paul Baumeister ◽  
Lars Hoffmann
Keyword(s):  
Integration Scheme ◽  
Kutta Method ◽  
Time Step ◽  
Third Order ◽  
Truncation Errors ◽  
The Third ◽  
Trajectory Calculations ◽  
Integration Schemes ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.

Symplectic analysis on orbit-attitude coupling dynamic problem of spatial rigid rod

2020 ◽  
Vol 26 (17-18) ◽  
pp. 1614-1624 ◽  
Author(s):  
Weipeng Hu ◽  
Tingting Yin ◽  
Wei Zheng ◽  
Zichen Deng
Keyword(s):  
Dynamic Model ◽  
Canonical Form ◽  
Dynamic Problem ◽  
Kutta Method ◽  
Dynamic Behaviors ◽  
Structure Preserving ◽  
Runge Kutta Method ◽  
Coupling Dynamic ◽  
Rigid Rod ◽  
Runge Kutta

An orbit-attitude coupling dynamic model for the spatial rigid rod that is abstracted from the large-stiffness slender components widely used in spatial structures is established, and the symplectic method is used to estimate the validity of the dynamic model by analyzing the coupling dynamic behaviors of the rod in this work. Based on the Hamiltonian variational principle, the orbit-attitude dynamic model of the spatial rigid rod is proposed, and the canonical form of the model is presented first. Then, the symplectic Runge–Kutta method is developed, and the structure-preserving properties of the canonical form, including the conservation law of energy and conservative property in the phase space, are investigated to illustrate the validity of the numerical results obtained by the symplectic Runge–Kutta method subsequently. Finally, the effects of the nonspherical perturbations of the Earth on the coupling dynamic behaviors are investigated numerically. From the simulation results, it can be concluded that the main orbit-attitude coupling dynamic behaviors of the spatial large-stiffness slender component excited by the nonspherical perturbation can be described by the proposed dynamic model ignoring the deformation as well as the transverse vibration of the slender component, which provides an approach for simplifying rapid dynamic analysis on the spatial large-stiffness slender component. In addition, the validity and the structure-preserving properties of the symplectic Runge–Kutta method for the orbit-attitude coupling dynamic problem of the spatial rigid rod are also illustrated.

scholarly journals Application of Differential Transformation Method for Solving HIV Model with Anti-Viral Treatment

2020 ◽  
Vol 14 (3) ◽  
pp. 378-388
Author(s):  
Esther Y. Bunga ◽  
Meksianis Z. Ndii
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Kutta Method ◽  
System Of Differential Equations ◽  
Time Step ◽  
Differential Transformation ◽  
Runge Kutta Method ◽  
Runge Kutta

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily found and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM)  to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters.. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time step.

scholarly journals Structure-Preserving Analysis of Skeleton Structure of Solar Sail in the Deploying Process

2018 ◽  
Vol 36 (2) ◽  
pp. 302-307
Author(s):  
Tingting Yin ◽  
Zichen Deng ◽  
Weipeng Hu ◽  
Xindong Wang
Keyword(s):  
Dynamic Model ◽  
Phased Array ◽  
Lagrange Equation ◽  
Solar Sail ◽  
Symplectic Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
System Energy ◽  
Runge Kutta ◽  
Skeleton Structure

For the simplified dynamic model of the skeleton structure of solar sail in the solar power satellite via arbitrarily large phased array system (SPS-ALPHA) in the deploying process, the symplectic method is employed to simulate the dynamic behaviors of the skeleton structure of solar sail and the characteristic of vibration, the constraints default as well as the energy-preserving of the system are all discussed in this paper.Firstly, the simplified dynamic model of the skeleton structure is established based on the variational principle, which is rewritten in the form of the associated canonical equation in Hamilton framework from the Lagrange equation that describes the deploying process of the skeleton structure of solar sail. And then, the equation is numerically simulated by the symplectic Runge-Kutta method and the classical Runge-Kutta method respectively. Comparing with the classical Runge-Kutta method, the symplectic Runge-Kutta method employed in this paper can preserve the displacement constraint and the system energy well with excellent numerical stability.

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.

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