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Synthesis of the Quadratic Near-optimal Closed of a Linear System with Disturbances

100%

Pełczewski J.

2002

tom Vol. 50, Nr 2

153--159

The paper deals with the synthesis procedure of a closed-loop quadratic, near-optimal control of a linear, time-invariant system, subject to known, deterministic, external disturbances. On the basis of the approximate, open-loop control obtained for that system by application of Legendre polynomials method the state-feedback matrix and the input signal vector were found.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

100%

Szybiński J. , Ruta P.

Studia Geotechnica et Mechanica

2019

tom Vol. 41, nr 1

1--12

This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

Quantitative Comparison of Cylindricity Profiles Measured with Different Methods Using Legendre-Fourier Coefficients

75%

Stępień K. , Makieła W. , Adamczak S. , Janecki D.

nr 3

Integrals of legendre polynomials and solution of some partial differential equations

63%

Belinsky R.

2000

tom Vol. 6, nr 2

259-282

We show a connection between the polynomials whose inflection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II).

Solution of fractional integro-differential equations using least squares and shifted legendre methods

63%

Mahdy A. M. S. , Nagdy A. S. , Mohamed D. S.

tom Vol. 23, nr 1

59-70

In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown.