Solution Of Simultaneous Algebraic Equations Research Articles

二重動吸振器の最大振幅倍率最小化設計

The optimal design method of double-mass dynamic vibration absorbers (DVAs) is discussed with respect to the minimization of the maximum amplitude magnification factor. The performance of a double-mass DVA is superior to a single-mass DVA with the same mass ratio, although the design methods are still the subject of studies. The design optimizations of double-mass DVAs that are arranged in parallel or series are formulated precisely using an optimality criteria approach in which the optimal parameters are obtained as the numerical solutions of simultaneous algebraic equations. The primal equations are derived using Vieta's formula with the assumption that the optimal design is realized with three resonant points of equal height. The additional equations are derived as the determinants of Jacobian matrices that are defined using the primal equations. After rearrangement, these formulations realize the direct numerical solution of the design optimization via the solution of simultaneous algebraic equations. Examples are provided that prove the effects of double-mass DVAs. The formulations used in this study are variants of the algebraic approach developed by O. Nishihara that realized the closed-form algebraic exact solutions to a popular design optimization of a single-mass DVA. The well-known design formulae that use the fixed points by J. P. Den Hartog and J. E. Brock correspond to an approximate solution to this problem.

A Matrix Formulation for the Moment Distribution Method Applied to Continuous Beams

The Moment Distribution Method is a quite powerful hand method of structural analysis, in which the solution is obtained iteratively without even formulating the equations for the unknowns. It was formulated by Professor Cross in an era where computer facilities were not available to solve frame problems that normally require the solution of simultaneous algebraic equations. Its relevance today, in the era of personal computers, is in its insight on how a structure reacts to applied loads by rotating its nodes and thus distributing the loads in the form of member-end moments. Such an insight is the foundation of the modern displacement method. This work has a main objective to present an exact solution for the Moment Distribution Method through a matrix formulation using only one equation. The initial moments at the ends of the members and the distribution and carry-over factors are calculated from the elementary procedures of structural analysis. Four continuous beams are investigated to illustrate the applicability and accuracy of the proposed formulation. The use of a matrix formulation yields excellent results when compared with those in the literature or with a commercial structural program.

Ｎｅｗｔｏｎ法によるＬａｎｄａｕ−Ｌｉｆｓｈｉｔｚ−Ｇｉｌｂｅｒｔ方程式の解法

The use of Newton’s method to solve the Landau-Lifshitz-Gilbert (LLG) equation discretized according to the Crank-Nicolson method is examined from the viewpoint of accuracy and computation time. Magnetization reversal and Bloch wall motion in a thin Permalloy film were calculated as an example. Although the convergence speed of Newton’s method is high, the solution of simultaneous algebraic equations requires a computation time of O(N3) in principle, when we denote the numbers of computing cells as N. If we take only the nearest-neighbor magnetostatic interaction into consideration in the implicit treatment, the coefficient matrix for the algebraic equation and the computation time are reduced to tridiagonal and O(N), respectively, in the case of one-dimensional calculation. Newton’s iteration can be repeated several times without increasing the order of computing time since the time required to calculate the demagnetizing field is O(N log N) if we use FFT. On detailed comparison with the Runge-Kutta method, a typical method for ordinary differential equations, the proposed method is found useful from the viewpoint of time-step dependence of numerical error and reliability.

Iterative techniques for the solution of complex DC-rail-traction systems including regenerative braking

The solution of simultaneous algebraic equations describing a DC-traction power system is based on an iterative technique because of their nonlinearity. Two iterative methods specified for DC-load-flow problems are described and their mathematical and geometrical explanations are presented. The methods have been tested for DC-fed traction power systems and comparisons made of the computational time and iteration count. The DC-traction software which has been developed enables the situation with trains operating in regenerative-braking mode to be simulated. A condition in which convergence problems are to be expected occurs when there is a surplus of power produced during regenerative braking. In this case two different modelling approaches to represent regenerative trains in the simulation are investigated. A representative DC-fed-traction power system with train regenerative braking was simulated and the preferred method is selected by analysing the simulation results obtained and the computation times expended.

A globally convergent algorithm for the solution of the steady-state semiconductor device equations

An iterative method for solving the discretized steady-state semiconductor device equations is presented. This method uses Gummel’s block iteration technique to decouple the nonlinear Poisson and electron-hole current continuity equations. However, the main feature of this method is that it takes advantage of the diagonal nonlinearity of the discretized equations, and solves each equation iteratively by using the nonlinear Jacobi method. Using the fact that the diagonal nonlinearities are monotonically increasing functions, it is shown that this method has two important advantages. First, it has global convergence, i.e., convergence is guaranteed for any initial guess. Second, the solution of simultaneous algebraic equations is avoided by updating the value of the electrostatic and quasi-Fermi potentials at each mesh point by means of explicit formulae. This allows the implementation of this method on computers with small random access memories, such as personal computers, and also makes it very attractive to use on parallel processor machines. Furthermore, for serial computations, this method is generalized to the faster nonlinear successive overrelaxation method which has global convergence as well. The iterative solution of the nonlinear Poisson equation is formulated with energy- and position-dependent interface traps. It is shown that the iterative method is globally convergent for arbitrary distributions of interface traps. This is an important step in analyzing hot-electron effects in metal-oxide-silicon field-effect transistors (MOSFETs). Various numerical results on two- and three-dimensional MOSFET geometries are presented as well.

The reduction of one-dimensional eigenvalue problems to the solution of simultaneous algebraic equations with one nonlinearity

A method is developed to obtain numerically eigenvalues associated with ordinary differential equations and variational formulations. The numerical operation required is the solution of a set of linear algebraic equations, with one nonlinearity, i.e., the transcendental dependence on the eigenvalue. This operation is the same for all problems resulting from equations of the same order, and therefore can easily be programmed; the particulars of a specific problem are then just input for the program.

The indefinite Z-transform technique and application to analysis of difference equations

The indefinite Z-transform technique is proposed. The method for solving linear difference equations using indefinite Z-transforms is compared with the methods employing the infinite one-sided Z-transforms and the finite Z-transforms. The distinct advantage of the method presented in this paper is that the desired solutions are obtained without employing standard inverse Z-transform techniques, such as the convolution theorem, or extensive Z-transform tables. All that is needed here for the derivations of the desired solutions is the application of Cramer's rule for the solution of simultaneous algebraic equations using the characteristic values. Therefore, this technique could be also readily used by those who have not studied the familiar Z-transform technique.

Numerical Integration of the Spherical-Harmonics Equations

A method is formulated for numerical integration of the spherical-harmonics equations in the case of cylindrical geometry. This method avoids many of the difficulties of the usual analytical techniques and allows space-varying sources as well as regions of low neutron cross section and large physical size. The usual spherical-harmonic equations (truncated) are presented in cylindrical geometry. To obtain a set of equations which (because they are more intuitive in form) lead to readily manageable numerical solution, the equations are converted to the discrete ordinate form in cylindrical geometry. From the discrete-ordinate equations, one may readily discuss inward- and outward-going neutrons. Based on this, reflection matrices are introduced at each radius r, one describing the reflection of inwardly directed neutrons by the medium inward of r and the other describing the reflection of outwardly directed neutrons by the medium outward of r. The complete source-independent properties of the medium are described by these reflection matrices. Furthermore, the matrices can be obtained by numerical integration in a single pass, one by integrating from the center out and the other by integrating from the outside in. The source can be treated by considering at each radius r the flux that escapes outward due to sources inward of r and by considering separately the flux that goes inward due to sources outward of r. The first of these escape fluxes is obtained by integration outward from the origin, using the corresponding reflection matrix, the second by integration inwards. Once the above quantities have been found, the fluxes are obtained by solution of simultaneous algebraic equations (no further integrations). Numerical results necessary for the use of this method in the P3 approximation are also given.

Some Applications of a New Approach to Loss Minimization in Electrical Utility Systems

1. In Appendix I, a computational procedure for the solution of economic load dispatching problems is developed. The equations are in generalized form in that the variables at any entering node to the network may be either the vector potential above ground or else the vector current entering the electric network. Generating station losses as well as electric network losses are included. 2. An example is presented in the body of the paper which involves three generators and two loads. The equations presented in Appendix I are adapted to fit this specific problem. In so doing, it was elected to replace the generalized variables by voltages above ground at all entering nodes on the electric network. 3. Numerical results for the example are presented in Appendix II. 4. The method presented in this paper requires the solution of simultaneous algebraicequations, some of which involve nonlinearities. Experience with the example suggests: (a) A Taylor's series expansion technique is applicable; (b) The selection of a starting point, in the multidimensional domain of the variables, that will lie near the point representing the actual solution, is of vital importance in assuring the rapid convergence of the calculations; (c) The selection of such a point may be achieved by computing a common incremental rate Pa? at the generating stations while neglecting the electric network losses, and then computing the electric potentials plus the Lagrange multipliers while assuming equal but unspecified incremental power rates for the generators.