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The Convolution Of Series And Its Application On Bar Structure

Biri Salah
Alhusain O.
Holnapy D.
Revision date


The cascade model is widely applied in water-currents flow calculations. The mathematical background of the cascade models is convolution. The convolution, especially in the case of continuous functions, is usually solved by Laplace transformation, which is handled with considerable difficulty. This study principally deals with convolution of series, rather than continuous functions. The convolution is traced back to the multiplication of the series’ characteristic polynomials. To find a suitable solution for this task, instead of the traditional definition of the linear space a more general definition was adopted. According to this adopted definition, the linear combination created by elements of module from vectors under addition in a way that the external relationship -instead of multiplying the vectors with real numbers- is solved by square matrixes. To cast the external relationship into a matrix multiplication form is possible because the algebraic structure found in the external relationship sufficient to be ‘ring’ according to the general definition of the linear space, and it is not required to be ‘field’ (body) as what used to be the common concept in traditional engineering practice.

The method can be applied on any model that can be described by the linear differential equations with constant coefficients. To make it easy to follow each step of implementing the procedure it was demonstrated by application on one of the most simple bar structures, that is a Kelvin-Voigt type material model of cantilever.

*Engineering > Mechanical and Structural Engineering
*Mathematics > Algebra > Field and Ring Theory

cascade model, series, convolution, bar structure

CONVOLUTION OF SERIES.pdf (162.5 KB) - PDF Document