Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits

The paper demonstrates symmetric integral operator and interpolation based numerical approximations for linear and nonlinear ordinary differential equations (ODEs) with oscillatory factor x′=ψ(x)+χω(t), where the function χω(t) is an oscillatory forcing term. These equations appear in a variety of computational problems occurring in Fourier analysis, computational harmonic analysis, fluid dynamics, electromagnetics, and quantum mechanics. Classical methods such as Runge–Kutta methods etc. fail to approximate the oscillatory ODEs due the existence of oscillatory term χω(t). Two types of methods are presented to approximate highly oscillatory ODEs. The first method uses radial basis function interpolation, and then quadrature rules are used to evaluate the integral part of the solution equation. The second approach is more generic and can approximate highly oscillatory and nonoscillatory initial value problems. Accordingly, the first-order initial value problem with oscillatory forcing term is transformed into highly oscillatory integral equation. The transformed symmetric oscillatory integral equation is then evaluated numerically by the Levin collocation method. Finally, the nonlinear form of the initial value problems with an oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the Levin method. Results of the proposed methods are more reliable and accurate than some state-of-the-art methods such as asymptotic method, etc. The improved results are shown in the numerical section.

Numerical Damping Calibration Study of Particle Element Method-Based Dynamic Relaxation Approach for Modeling Longwall Top-Coal Caving

When using the explicit dynamic relaxation approach (DRA) to model the quasi-static rock breakage, fragmentation, and flow problems, especially the top-coal caving question, introducing numerical damping into the solution equation is inevitable for reducing the vibration frequency and impact speed of mesh nodes, which is significantly affect the fidelity of the computation results. Although the DRA has been widely adopted to simulate top-coal caving, the reasonable value and calibration method of numerical damping are still open issues. In this study, the calibration process of reasonable numerical damping for modeling top-coal caving is investigated by comparing with the experimental results, in which several geometry parameters of the drawing funnel are selected as the calibration indexes. The result shows that the most reasonable numerical damping value is 0.07 for the numerical modeling of interval top-coal caving in extra-thick coal seams. Finally, the correlation between the numerical damping and the physical top-coal drawing process is discussed. The numerical damping indirectly reflects the fragmentation in multi scale of coal mass and friction interaction between coal particles during the caving process, which reduces the vibration intensity of the top-coal caving system and dissipates the kinetic energy.

Study on Structural Error Compensation of 3-UPU Parallel Mechanism

Background: Compared with the traditional series mechanism, the parallel mechanism has a better kinematic performance. Structural size error is the main factor affecting the accuracy of parallel mechanisms. Objective: The paper mainly studies the compensation of the rod length error, the moving platform radius error and the fixed platform radius error of 3-UPU parallel mechanism. Methods: To establish a generalized forward and inverse solution equation with error compensation, the position change of the moving platform is measured by a laser interferometer, and the change amount of the three connecting rod lengths at the corresponding position is recorded. Optimized by least squares method, the optimized error compensation values are compensated to the kinematics algorithm of the numerical control system, and the positioning accuracy is improved. Results: The results show that the positioning accuracy is higher when the mechanism moves in the lower plane, and the positioning error in the z axis direction is smaller than x, y, y=x, y=-x axis. Conclusion: After the error compensation, the overall positioning accuracy of the mechanism is increased by 60%.

Efficient Planning and Solving Algorithm of S -Shape Acceleration and Deceleration

S -shape acceleration and deceleration are the most widely used flexible acceleration and deceleration method in the current CNC system, but its velocity solution equation contains irrational terms, which create a more complicated solution process. When analyzing the solution process of the S -shape acceleration and deceleration directly, using a traditional numerical solution method, the phenomenon of “solving the interval jump” arises, which is the main reason for low efficiency and poor stability of the solution. According to the S -curve profile and solution, the concept of separating the curve profile recognition from the velocity solution was proposed, and a method of quickly identifying the interval of the solution location was introduced. Through the method mentioned above, the complete acceleration and deceleration curve parameters can be obtained through a one-time plan and a one-time solution, and the solution efficiency and stability are guaranteed; solving the Newton problem depends too much on the initial value of Newton velocity, which not only retains the speed advantage of the Newton method but also uses the downhill factor to ensure its convergence. Through the simulation comparison and analysis, the efficiency, stability, and universality of the method are verified.

ADA Solve the Cubic Equation in a New Method with Engineering Application

Up to date the cubic equation or matrix tensor is consisting of nine values ​​such as stress tensor that turns into the cubic equation which has been used for solving classic method. This is to impose an initial root several times to get it when achieves the equation and any other party is zero. Then dividing the cubic equation on the equation of the root. After that dividing the cubic equation on the equation of the root and using the classical method to find the rest of the roots. This is a very difficult issue, especially if the roots are secret or large for those who are looking in a difficult field or even for those who are in the examination room. In this research, two equations were reached, one that calculates the angle and the other that calculates the three roots at high accuracy without any significant error rate. By taking advantage of the traditional method, not by imposing a value to get the root of that equation, but by imposing an equation to get the solution equation that gives the value of that root. After imposing that equation, the general equation was derived from which that calculated the three roots directly and without any attempts. The angle that was implicitly derived during the derive of the main equation is calculated by taking advantage of the constants that do not change (invariants) for the matrix tensor (T).

Doublel-robot coordination workspace Analysis based on Matlab

Aiming at the requirements of dual robot collaborative operation, a dual robot cooperation system model is established in SolidWorks2012 software to study the dual robot cooperation space. The D-H parameters are established, and the kinematics positive solution equation is obtained. The dual robot cooperative kinematics model is given. Based on the Monte Carlo method, the workspace of the dual robot is solved. The extreme value theory method is used to analyze and calculate, so as to extract the precise boundary contour of the common area of the dual robot workspace, and the collaborative space boundary surface and limit position of the dual robot are determined. The optimal coordinated working space of the dual robot end effector is obtained, which lays a theoretical foundation for the coordinated trajectory planning of the dual robot.

AN ANALYTICAL TEMPERATURE SOLUTION ANALYSIS FOR A MULTILAYER HEAT CONDUCTION PROBLEM

This paper presents a method of obtaining an analytic temperature solution for a two-layer heat conduction problem. Obtaining the temperature analytical solution for a multilayer heat conduction problem is not a direct method. The way to indentify the eigenvalues and to derive the Green function solution equation requires a different treatment since there are more than one domain to solve. This work presents a solution of a thermal twolayer problem based on Green’s functions.

Temperature Rise Model of Drum Brake

In order to analysis the temperature rising rules of vehicle, road conditions and vehicle running state were considered. Process of friction heat generation, radiation heat and convection current exchange during drum brake working process was researched, and temperature rising model of drum brake was established. Under the guidance of similarity theory and dimension analysis theory, convection current heat exchange test of drum brake was carried out, and the solution equation of convection current heat exchange was obtained. Real road test results shows that the temperature rising curves between calculations and test are basically identical and the maximum inaccuracy variables approximately 20%.

Analysis of Structural Parameter and Design of Elbow Joint Rehabilitation Parallel Robot

Elbow joint is one of the body's important joints, most of the activities of the human body are inseparable from the elbow joint, and including taking and holding movements.In order to increase the workspace of elbow joint, a novel elbow joint rehabilitation parallel robot based on 2-DOF orthogonal spherical parallel mechanism is proposed. First, the position inverse solution equation of elbow joint is established. Further, the workspace of elbow joint is analyzed. The optimal structural parameters are obtained by use of the objective function of optimization method. Finally, the virtual prototype of elbow joint rehabilitation parallel robot is designed using optimal structural dimensions parameters.