• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1133-1143
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• 2009

In this work, we successfully extended dimensional differential transform method (DTM), by presenting and proving some new theorems, to solve the non-linear matrix differential Riccati equations(first and second kind of Riccati matrix differential equations). This technique provides a sequence of matrix functions which converges to the exact solution of the problem. Examples show that the method is effective.

• Journal of the Korean Institute of Telematics and Electronics
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• v.16 no.5
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• pp.18-26
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• 1979

This paper extends some well-known system theories on algebraic matrix Lyapunov and Riccati equations. These extended results contain two point boundary conditions in matrix differential equations and include conventional results as special cases. Necessary and sufficient conditions are derived under which linear systems are stabilizable with feedback gains derived from periodic two-point boundary matrix differential equations. An iterative computation method for two-point boundary differential Riccati equations is given with an initial guess method. The results in this paper are related to periodic feedback controls and also to the quadratic cost problem with a discrete state penalty.

• Journal of the Korean Chemical Society
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• v.52 no.3
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• pp.223-238
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• 2008

this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a Moore-Penrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices.

• East Asian mathematical journal
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• v.35 no.3
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• pp.341-349
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• 2019

In this paper, we are concerned with the minimal positive solution to system of the nonlinear matrix equations $A_1X^2+B_1Y +C_1=0$ and $A_2Y^2+B_2X+C_2=0$, where $A_i$ is a positive matrix or a nonnegative irreducible matrix, $C_i$ is a nonnegative matrix and $-B_i$ is a nonsingular M-matrix for i = 1, 2. We apply Newton's method to system and present a modified Newton's iteration which is validated to be efficient in the numerical experiments. We prove that the sequences generated by the modified Newton's iteration converge to the minimal positive solution to system of nonlinear matrix equations.

• 전기의세계
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• v.23 no.3
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• pp.43-52
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• 1974

The matrix inversion is very inefficient for computing direct solutions of the large spare systems of linear equations that arise in many network problems as a large electrical power system. Optimally ordered triangular factorization of sparse matrices is more efficient and offers the other important computational advantages in some applications with this method. The direct solutions are computed from sparse matrix factors instead of a full inverse matrix, thereby gaining a significant advantage is speed and computer memory requirements. In this paper, it is shown that the sparse matrix method is superior to the inverse matrix method to solve the linear equations of large sparse networks. In addition, it is shown that the sparse matrix method is superior to the inverse matrix method to solve the linear equations of large sparse networks. In addition, it is shown that the solutions may be applied directly to sove the load flow in an electrical power system. The result of this study should lead to many aplications including short circuit, transient stability, network reduction, reactive optimization and others.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.755-770
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• 2013

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix, and A, B and C are $n{\times}n$ given matrices with real elements. Another one is the matrix polynomial $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr$\acute{e}$chet derivative. Finally, we give some numerical examples.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.395-412
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• 2020

In this article, the values of internal force and deformation of a curved beam under any action with the firm or elastic supports are determined by using structural matrices. The article presents the general differential formulation of a curved beam in global coordinates, which is solved in an orderly manner using simple integrals, thus obtaining the transfer matrix expression. The matrix expression of rigidity is obtained through reordering operations on the transfer notation. The support conditions, firm or elastic, provide twelve equations. The objective of this article is the construction of the algebraic system of order twenty-four, twelve transfer equations and twelve support equations, which relates the values of internal force and deformation associated with the two ends of the directrix of the curved beam. This final algebraic system, expressed in matrix form, is divided into two subsystems: twelve algebraic equations of internal force and twelve algebraic equations of deformation. The internal force and deformation values for any point in the curved beam directrix are determined from these values in the initial position. The five examples presented show how to apply the matrix procedures developed in this article, whether they are curved beams with the firm or elastic support.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10a
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• pp.926-930
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• 1993

An efficient method of formulating the equations of motion of multibody systems is presented. The equations of motion for each body are formulated by using Newton-Eulerian approach in their generic form. And then a transformation matrix which relates the global coordinates and relative coordinates is introduced to rewrite the equations of motion in terms of relative coordinates. When appropriate set of kinematic constraints equations in terms of relative coordinates is provided, the resulting differential and algebraic equations are obtained in a suitable form for computer implementation. The system geometry or topology is effectively described by using the path matrix and reference body operator.

• Journal of Mechanical Science and Technology
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• v.14 no.2
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• pp.159-167
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• 2000

A computational method for the dynamic analysis of a constrained mechanical system is presented in this paper. The partial velocity matrix, which is the null space of the Jacobian of the constraint equations, is used as the key ingredient for the derivation of reduced equations of motion. The acceleration constraint equations are solved simultaneously with the equations of motion. Thus, the total number of equations to be integrated is equivalent to that of the pseudo generalized coordinates, which denote all the variables employed to describe the configuration of the system of concern. Two well-known conventional methods are briefly introduced and compared with the present method. Three numerical examples are solved to demonstrate the solution accuracy, the computational efficiency, and the numerical stability of the present method.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.6
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• pp.1351-1362
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• 1995

A method to estimate the torsional critical speed for practical rotors has been developed in this study. First, the rotor with a uniform shaft segment is modeled for undamped torsional motion analysis, while satisfying all the boundary conditions. This eventually generates governing equations for the torsional critical speeds of the system. The set of governing equations has the form of a sparse and banded matrix. The elements of banded matrix can be arranged in partitions, which correspond to the specific boundary of the rotor. This permits an automatic generation of the system matrix using a computer. In order to calculate the determinant generated by the simultaneous equations, which leads to the torsional critical speed, a recurring numerical algorithm for a (3*4) sub-matrix has been used. This numerical algorithm practically examines successive (3*4) sub-matrix, one at a time, instead of treating a huge matrix. The output of the program provides the mode shapes with continuous curves. The method has been implemented to three rotors given as examples : a simple rotor, Prohl's rotor, and Macmillan rotor.