Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

Investigation of Model Configuration Parameters for Stick-Slip Modeling

Stick-slip vibrations modeling is an important topic within drillstring dynamics. Thus, a number of mathematical models has been suggested to describe behavior of drillstrings under torsional. Most of the models take similar approach with regard to, for instance, drillstring discretization, definition of external forces and velocity-weakening effect. Commonly, research papers focus on the models’ core — mathematical expressions that describe stick-slip oscillations and inherent to it negative damping. The results are usually presented in terms of downhole rotational velocity or displacement and reaction forces at various surface rotational velocities and applied external forces. However, little attention is paid to discussion and justification of selected model configuration, which includes definition of the following 1) total simulation time, time step, number of masses/elements, etc., 2) initial conditions and boundary conditions, and 3) numerical solver to obtain solution in time. This paper reviews commonly used configurations for stick-slip vibrations modeling and discusses selection criteria provided in the references. It also presents case studies to evaluate effect of the above-mentioned configuration properties on simulations outcome. A simple in-house 1DOF torsional dynamics model was used for that purpose, where one explicit and one implicit numerical solvers were applied to obtain solution in time. Three case studies are presented, which compare performance of two numerical solvers with respect to convergence and stability. The results from the case studies show, for example, that applied explicit numerical solver (Central Difference Method) introduces numerical damping, while implicit solver (Newton-Raphson Method) does not. Central Difference Method provides convergence when initial force is applied, while damping function has to be defined in case of Newmark-Raphson method to obtain convergence. Stability of the explicit numerical solver is determined by the time step, while selected implicit solver is unconditionally stable. A reasonably small time step has to be selected though to improve the accuracy of the results. Presented literature review and outcome from the case studies can be used by researchers within this area to select suitable configuration parameters for their models and critically evaluate the outputs. In addition, presented results have application in automated drilling where configuration parameters and calibration factors are updated in real time by control algorithms for continuous modeling of drillstring state with regard to stickslip. Understanding the effects of mentioned properties on system dynamics helps to select suitable combination of operational parameters to stabilize the drillstring.

A Two-Sub-Step Generalized Central Difference Method for General Dynamics

This paper proposes an implicit and unconditionally stable two-sub-step composite time integration method with controllable numerical dissipation for general dynamics called the two-sub-step generalized central difference (TGCD) method. The proposed method is established by performing the generalized central difference scheme in two sub-steps as the nondissipative and dissipative parts to ensure amplitude accuracy and controllable damping, respectively. It is accurate to the second order, with the amount of numerical dissipation controlled exactly by the spectral radius [Formula: see text]. In addition, the related parameters of the proposed method are determined by optimizing the amplitude and phase accuracy of the free vibration of a single degree-of-freedom system. Several representative linear and nonlinear numerical examples are analyzed to demonstrate the advantages of the proposed method in terms of accuracy, stability and efficiency, especially its stability in solving nonlinear problems.

SIMULATION OF MATHEMATICAL MODEL FOR MONORAIL SUSPENSION SYSTEM UNDER DIFFERENT TRACK CONDITIONS

Designing a new monorail suspension system for an existing monorail bogie to accommodate larger cars, locomotives and more passengers is a difficult and complicated problem to solve. This paper introduces a simulation of a mathematical model for a monorail suspension system that can be used as an analytical tool to investigate and predict the behavior of the model under different speeds and track conditions. In this paper, the simulation is performed to predict some dynamic characteristics monorail suspension system. This research work concentrates on the simulation of 15 degrees of freedom full-car Monorail suspension system. The model features the Monorail body, Front bogie, and rear bogie geometries, adopted equations of motion of the monorail suspension system and system matrices. Numerical Central Difference method was used to obtain the system responses subject to sinusoidal Track excitations. Three Track scenarios that have different loads and different driving speeds were conducted to investigate the monorail suspension system. The system results are analysed in terms of their dynamic responses. Fourier Fast transforms was used to calculate the frequency ranges of dynamic responses. As a result, some very important characteristics of the Monorail suspension system were revealed, with indicators that help to understand the effects of driving speeds and different loads, which can be used to better understand the system dynamic performance, to improve Monorail suspension system designs flaws detection.

An Efficient Nondissipative Higher-Order Single-Step Integration Method for Long-Term Dynamics Simulation

For long-term simulation of dynamic problems, it is desirable for the integration method to capture higher-order accuracy, while maintaining zero numerical damping. To this end, this paper proposes an efficient four-parameter single-step method (FPSM), which only adds two vector operations per step compared with the Newmark method. For linear analysis, these four parameters can be uniquely determined and the generated optimal scheme possesses several desirable attributes, such as nondissipation for undamped systems, fourth-order accuracy and an enough stability interval. Compared with existing nondissipative methods, such as the central difference method and the trapezoidal rule, the present method is recommended for general use due to its high accuracy and efficiency. Finally, several numerical examples are prepared to validate these superiorities of the proposed method.