A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams

Snježana Maksimović,Aleksandar Borković

doi:10.3390/math9151778

Snježana Maksimović, Aleksandar Borković

Open Access

https://doi.org/10.3390/math9151778

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Abstract

The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.

Highlights

Arc Length Parametrization and ItsArc-length parametrization can be considered to be the most natural of all possible parametrizations of a given curve [1]
There is a limited set of curves for which the arc-length parametrization can be expressed as an elementary function
If we take the derivative with respect to s in (4) and substitute it in (3), another necessary and sufficient condition for the arc-length parametrization is obtained x 02 (s) + y02 (s) = 1

Summary

Introduction

Arc-length parametrization can be considered to be the most natural of all possible parametrizations of a given curve [1]. It is proved in [2] that, on par with polynomials, it is possible to select a subclass of arc-length curves that has an arbitrary number of degrees of freedom These curves are useful in engineering applications, especially for the analysis of beam-like structures. It is of a particular interest to examine arc-length curves for which the governing equations of Bernoulli–Euler beam are significantly simplified, and analytical solutions are feasible. These solutions can provide valuable benchmark test results for the application of modern numerical methods to the analysis of free-form beams [5,6].

Preliminaries

Special Polynomials and Special Functions

Arc-Length Parametrization Using Special Functions

An Application to the Linear Static Analysis of Curved Beams

Conclusions

Methods