Numerical Solution Of Simultaneous Equations Research Articles

Numerical solution of simultaneous equations based geometric error compensation for CNC machine tools with workpiece model reconstruction

This paper presents geometric error compensation of CNC machine tools based on workpiece model modification and the numerical solution of equation set. Firstly, the nominal cutter position is introduced to the geometric error model of machine tool as the compensation goal according to the kinematics of three-axis machine tools. With the polynomials of basic error components, the polynomials of the geometric error model are established, which are the functions about movements of all axes and nominal NC code. The accurate compensation is represented as solution of corresponding simultaneous equations. The high-efficiency numerical calculation can obtain the compensated NC code. Secondly, the polynomial equations about points of workpiece are established based on the simple relationship between workpiece model and NC codes. The compensated points are calculated using numerical solution. STL is chose as the format of reconstructed workpiece model. In order to obtain the precise workpiece model with compensated points, one conversion approach from CAD model to STL model is proposed. The number of points calculated with isoparametric method can control the precision of STL. The reconstructed STL model is inputted to CAM software to generate processing file for machining. The input of the proposed compensation is CAD model of workpiece rather than NC codes and CL data. It makes the compensation convenient and suitable for different three-axis machine tools. Finally, the experiments are carried out on Carver800T CNC machine tool to testify the effectiveness of proposed geometric error compensation.

Large deflection analysis of cantilever beam under end point and distributed loads

Although the deflection of beams has been studied for decades, the solutions were either linearized (i.e. small deflection) or based on elliptic integrals or functions (large deflection). The latter one includes the geometric nonlinearity but calculation of the deflection along the beam length requires numerical solution of simultaneous equations which is a significant drawback for optimization or reliability analysis. This paper is motivated to overcome these shortcomings by presenting an analytical solution for the large deflection analysis of a cantilever beam under free end point and uniform distributed loads. Direct nonlinear solution by use of homotopy analysis method was implemented to drive the semi-exact solution of trajectory position of any point along the beam length. For the purpose of comparison, the deflections were calculated and compared to those of finite element method which was taken as reference. It was found that the proposed solution is very accurate, efficient, and convenient for the discussed problem and can be applied to a large class of practical problems.

On the global convergence characteristics of numerically evaluated Jacobian matrices

Since locating all the fixed points of a nonlinear oscillator involves the numerical solution of simultaneous equations, it is useful to observe some of the global convergence characteristics of these techniques. Specifically, the popular Newton or quasi-Newton approaches require numerical evaluation of the Jacobian matrix of the Poincare map. This note focuses attention on the domains of attraction for a number of fixed point techniques applied to a single nonlinear oscillator with a single set of parameters. Clearly, there are many issues here, including proximity to bifurcations, order of the dynamical system, temporal convergence characteristics, i.e. CPU time, and so on, but it is instructive to observe a snapshot of the basins of attraction, the boundaries of which path-following routines seek to avoid when a parameter is changed.

Multiband least squares FIR filter design

This paper gives two methods for the least squares approximation design of FIR digital filters. They both allow multiple pass and stop bands and arbitrary transition bands, all with explicit control of band edges. The first method is as simple to use as a window method. This design method builds up an optimal multiband frequency response by sequentially adding and subtracting optimal lowpass filters with transition bands. However, it assumes the ideal response has spline transition functions and does not allow error weighting. The second method allows different weights in the multiple passbands, stopbands and transition bands but it requires numerical solution of simultaneous equations which may be ill-conditioned. These methods compliment the Parks-McClellan algorithm which minimizes the Chebyshev error.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimum Design of Dynamic Absorbers for a System Subjected to Random Excitation.

This paper presents equations and computational graphs that determine the optimally tuned and damped dynamic vibration absorber attached to a linear system subjected to stationary random excitation. The optimization criterion is to minimize a performance measure defined by the mean square acceleration response of the primary system to unit input acceleration (assumed to be ideal white noise). The optimum design conditions of dynamic absorber can be formulated if the primary system has no damping. If there is damping in the primary system, the optimum conditions are obtained from the numerical solution of simultaneous equations. The optimum conditions obtained here are different from the classical ones evaluated by the so-called Ormondroyd-Den Hartog theory. It is due to the difference in the performance measure in these theories. The optimum conditions are tested on a vibratory model. It can be shown that a properly designed dynamic absorber operates efficiently for reducing the vibration induced by random excitation.

An Analytical and Experimental Investigation of Turbulent Flow in Bearing Films Including Convective Fluid Inertia Forces

A means of handling both fluid film turbulence and the effects of fluid convective inertia in bearing films has been developed. The analysis employs a number of simplifying assumptions but has proved quite successful in the prediction of experimentally measured pressures. In this analysis turbulence is treated by means of turbulent viscosity correction factors. Convective inertia is accounted for by convective derivatives of mean fluid velocities averaged over the bearing film. Direct numerical solution of simultaneous equations representing conservation of momentum and fluid continuity is employed. The analysis developed is applied to pad bearings of arbitrary surface geometry, and has been incorporated in a computer program to calculate the film pressures developed in such bearings. Results from this program are compared with experimental pressure profile measurements made on a 12-in. dia, four pad, shrouded step journal bearing with silicone fluids as test lubricants. Good agreement is obtained for both concentric and eccentric operation.