• The Transactions of the Korean Institute of Electrical Engineers
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• v.39 no.7
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• pp.740-748
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• 1990

In this paper, we consider the problem of parameter estimation, i.e., definding the internal structure of a linear distribution parameter system from its input/output data. First, a linear partial differential equation describing the system is double-integrated with respect to two variables and then transformed into an integral equation. Next the Walsh Operation Matrix for Walsh function and their integration are introduced to transform the integral equation into algebraic simultaneous equations. Finally, we develop an algorithm to estimate the parameters of the linear distributed parameter system from the simple linear algebraic simultaneous equations. It is also shown that our algorithm could be effective in real time data processing since it uses the Fast Walsh Transform.

• Proceedings of the KIEE Conference
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• 1991.07a
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• pp.695-698
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• 1991

In this paper tried the estimation of parameter using of Cal-Sal functions. System equation given by the linear differential equation is converted into the integral equation, operation matrix for integral of Cal-Sal functions is used to find the estimation of parameter on the given system. Converting linear differential equation to linear algebraic equation, the method presented here computing time and required memory size can be reduced. Therefore real time data process can be possible.

• Coupled systems mechanics
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• v.12 no.6
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• pp.519-530
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• 2023

This paper presents a mathematical model suitable for the calculation of laminated glass, i.e. glass plates combined with an interlayer material. The model is based on a beam differential equation for each glass plate and a separate differential equation for the slip in the interlayer. In addition to slip, the model takes into account prestressing force in the interlayer. It is possible to combine the two contributions arbitrarily, which is important because the glass sheet fabrication process changes the stiffness of the interlayer in ways that are not easily predictable and could introduce prestressing of varying magnitude. The model is suitable for reformulation into an inverse procedure for calculation of the relevant parameters. Model consisting of a system of differential-algebraic equations, proved too stiff for cases with the thin interlayer. This novel approach covers the full range of possible stiffnesses of layered glass sheets, i.e., from zero to infinite stiffness of the interlayer. The comparison of numerical and experimental results contributes to the validation of the model.

• Structural Engineering and Mechanics
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• v.28 no.2
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• pp.221-238
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• 2008

Numerical solution to buckling analysis of beams and columns are obtained by the method of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for various support conditions considering the variation of flexural rigidity. The solution technique is applied to find the buckling load of fully or partially embedded columns such as piles. A simple semi- inverse method of DQ or HDQ is proposed for determining the flexural rigidities at various sections of non-prismatic column ( pile) partially and fully embedded given the buckling load, buckled shape and sub-grade reaction of the soil. The obtained results are compared with the existing solutions available from other numerical methods and analytical results. In addition, this paper also uses a recently developed technique, known as the differential transformation (DT) to determine the critical buckling load of fully or partially supported heavy prismatic piles as well as fully supported non-prismatic piles. In solving the problem, governing differential equation is converted to algebraic equations using differential transformation methods (DT) which must be solved together with applied boundary conditions. The symbolic programming package, Mathematica is ideally suitable to solve such recursive equations by considering fairly large number of terms.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.973-1008
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• 1997

We reconsider the problem of calssifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form $$ \ell_2(x)y"(x) + \ell_1(x)y'(x) = \lambda_n y(x). $$ We first obtain new (algebraic) necessary and sufficient conditions on the coefficients $\ell_1(x)$ and $\ell_2(x)$ for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classification of all classical orthogonal polynomials : up to a real linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi.cobi.

• Structural Engineering and Mechanics
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• v.17 no.1
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• pp.1-14
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• 2004

Numerical solution to linear bending analysis of circular plates is obtained by the method of harmonic differential quadrature (HDQ). In the method of differential quadrature (DQ), partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The method of HDQ that was used in the paper proposes a very simple algebraic formula to determine the weighting coefficients required by differential quadrature approximation without restricting the choice of mesh grids. Applying this concept to the governing differential equation of circular plate gives a set of linear simultaneous equations. Bending moments, stresses values in radial and tangential directions and vertical deflections are found for two different types of load. In the present study, the axisymmetric bending behavior is considered. Both the clamped and the simply supported edges are considered as boundary conditions. The obtained results are compared with existing solutions available from analytical and other numerical results such as finite elements and finite differences methods. A comparison between the HDQ results and the finite difference solutions for one example plate problem is also made. The method presented gives accurate results and is computationally efficient.

• Wind and Structures
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• v.32 no.1
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• pp.11-17
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• 2021

In recent years, there has been a tendency towards renewable energy sources considering the damages caused by non-renewable energy resources to nature and humans. One of the renewable energy sources is wind and energy is obtained with the help of wind turbines. To determine the behavior of wind turbines under earthquake loads, dynamic characteristics are required. In this study, the differential transformation method is proposed to determine the free vibration analysis of wind turbines with a variable cross-section. The wind turbine is modeled as an equivalent variable continuous flexural beam and blade weight is considered as a point mass at the top of the structures. The differential equation representing the free vibration of the wind turbine is transformed into an algebraic equation with the help of differential transformation method and the angular frequencies and the mode shapes of the wind turbine are obtained by the help of the differential transformation method. In the study, a sample taken from the literature was solved with the presented method and the suitability of the method was investigated. The same wind turbine example also modeled by finite element modelling software, ABAQUS. Results of the finite element model and differential transformation method are compared with each other and the results are in good agreement.

• Coupled systems mechanics
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• v.1 no.2
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• pp.155-163
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• 2012

This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.45 no.3
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• pp.399-406
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• 1996

This paper presents a method of hierarchical state estimation of the time-varying linear systems via Block-pulse function(BPF). When we estimate the state of the systems where noise is considered, it is very difficult to obtain the solutions because minimum error variance matrix having a form of matrix nonlinear differential equations is included in the filter gain calculation. Therefore, hierarchical approach is adapted to transpose matrix nonlinear differential equations to a sum of low order state space equation from and Block-pulse functions are used for solving each low order state space equation in the form of simple and recursive algebraic equation. We believe that presented methods are very attractive nd proper for state estimation of time-varying linear systems on account of its simplicity and computational convenience. (author). 13 refs., 10 figs.

• Journal of the Korean Society for Precision Engineering
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• v.12 no.5
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• pp.57-68
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• 1995

A general procedure for the numerical solution of coupled, nonlinear, differential two-point boundary-value problems, solutions of which are crucial to the controller design, has been developed and demonstrated. A fixed-end-points, free-terminal-time, optimal-control problem, which is derived from Pontryagin's Maximum Principle, is solved by an extension of Davidenko's method, a differential form of Newton's method, for algebraic root finding. By a discretization process like finite differences, the differential equations are converted to a nonlinear algebraic system. Davidenko's method reconverts this into a pseudo-time-dependent set of implicitly coupled ODEs suitable for solution by modern, high-performance solvers. Another important advantage of Davidenko's method related to the time-optimal problem is that the terminal time can be computed by treating this unkown as an additional variable and sup- plying the Hamiltonian at the terminal time as an additional equation. Davidenko's method uas used to produce optimal trajectories of a single-degree-of-freedom problem. This numerical method provides switching times for open-loop control, minimized terminal time and optimal input torque sequences. This numerical technique could easily be adapted to the multi-point boundary-value problems.