Calculation of determinant of a polynomial matrix in complex basis

Varsamis, Dimitris

doi:10.12988/ces.2020.91606

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…One may recall that this technique could also have been applied directly to equation ( 1), but its extreme computational cost for large N values is not affordable nowadays with standard PCs. There exist some other methods that could perform this task such as the evaluation-interpolation technique (DCEI) and the evaluation-interpolation technique in complex basis (DCEIC) (Varsamis, 2020). These methods constitute lines of current research of the authors and will be considered in the future.…”

Section: Id Lumped Attachmentmentioning

confidence: 99%

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

Aguilar

Ruz

Blanco–Rodríguez

2023

Journal of Vibration and Control

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Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10−3 and 10−9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports.

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Section: Id Lumped Attachmentmentioning

confidence: 99%

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

Aguilar

Ruz

Blanco–Rodríguez

2023

Journal of Vibration and Control

View full text Add to dashboard Cite

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Calculation of Determinant of a two-variable Polynomial Matrix in Complex Basis

Varsamis

Kamilali

2022

Wseas Transactions on Systems and Control

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In this paper, an innovation in evaluation part of Evaluation Interpolation technique (E-I) is presented. The variables of this technique have been defined in Complex domain. The use of complex numbers instead of real ones in evaluation part, provides us with the opportunity of using conjugate properties. With these properties, the number of evaluations in evaluation part is being decreased. The interpolation part is common for both methods.

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Optimal Number of Required Points for Evaluation - Interpolation Technique in Complex Basis

Varsamis

Kamilali

2022

Wseas Transactions on Mathematics

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According to polynomial evaluation-interpolation technique, first of all, the set of required points, in which interpolation will be executed, must be defined in evaluation part. Taking into account that working in complex domain provides as with the advantage of conjugate properties, the number of evaluations at these points could be reduced. This paper presents the appropriate form of required points, aiming to an optimal reduction of their amount. The appropriate form of points for two variable polynomials depends on the degree of each variable. As a repercussion the number of evaluations could be reduced even up to the half.

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Calculation of determinant of a polynomial matrix in complex basis

Cited by 3 publications

References 7 publications

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

Calculation of Determinant of a two-variable Polynomial Matrix in Complex Basis

Optimal Number of Required Points for Evaluation - Interpolation Technique in Complex Basis

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