A Numerical Study for a Flexible Euler-Bernoulli Beam with a Force Control in Velocity and a Moment Control in Rotating Velocity

Jump To References Section

Authors

  • Abro Goh Andre-Pascal
     Institut National Polytechnique Houphouet-Boigny de Yamoussoukro ,CI
  • Bomisso Gossrin Jean-Marc
     Universite Nangui Abrogoua d'Abobo-Adjame ,CI
  • Toure Kidjegbo Augustin
     Institut National Polytechnique Houphouet-Boigny de Yamoussoukro ,CI
  • Coulibaly Adama
     Universite Felix Houphouet Boigny de Cocody ,CI

DOI:

https://doi.org/10.18311/jims/2023/33044

Keywords:

Beam Equation, Existence and Uniqueness, Higher Regularity, Finite Element Method, Galerkin Method, Priori Estimates.

Abstract

In this paper, we numerically study a exible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity. First, we show the existence and uniqueness of the weak solution using Faedo-Galerkin's method with the intermediate spaces. Then, we use the finite elements method with the cubic Hermite polynomials for the approximation of (1.1){(1.5) in space such that the semi-discrete scheme obtained is stable and convergent. In addition, an a-priori error estimate is obtained. Finally, we perform numerical simulations in order to validate this method.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Downloads

Published

2023-03-24

How to Cite

Andre-Pascal, A. G., Jean-Marc, B. G., Augustin, T. K., & Adama, C. (2023). A Numerical Study for a Flexible Euler-Bernoulli Beam with a Force Control in Velocity and a Moment Control in Rotating Velocity. The Journal of the Indian Mathematical Society, 90(1-2), 125–148. https://doi.org/10.18311/jims/2023/33044

Issue

Volume 90, Issue 1-2, January-June 2023

Section

Articles

License

Copyright (c) 2023 Abro Goh Andre-Pascal, Bomisso Gossrin Jean-Marc, Toure Kidjegbo Augustin, Coulibaly Adama

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

 

References

Paul J. Allen, A fundamental theorem of homomorphism for semirings, Proc. Amer. Math. Soc., 21 (1969), 412–416. DOI: https://doi.org/10.1090/S0002-9939-1969-0237575-4

Shahabaddin Ebrahimi Atani, The ideal theory in quotients of commutative semirings, Glasnik Matematicki, 42 (62)(2007), 301 - 308. DOI: https://doi.org/10.3336/gm.42.2.05

A. P. Goh Abro, J. M. Gossrin Bomisso, A. Kidjgbo Tour, and Adama Coulibaly, A Numerical Method by Finite Element Method (FEM) of an Euler-Bernoulli beam to Variable Coefficients, Advances in Mathematics: Scientific Journal, 9 (2020), 8485 - 8510. DOI: https://doi.org/10.37418/amsj.9.10.77

H. T. Banks and I. G. Rosen, Computational methods for the identification of spatially varying stiffness and damping in beams, Theory and advanced technology, 3 (1987), 1 - 32.

M. Basson and N. F. J. Van Rensburg, Galerkin finite element approximation of general linear second order hyperbolic equations, Numer. Func. Anal. Opt. 34(9) (2013), 976 - 1000. DOI: https://doi.org/10.1080/01630563.2013.807286

M. Basson, B. Stapelberg and N. F. J. Van Rensburg, Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation, Numerical Functional Analysis and Optimization, 38(4) (2017), 466 - 485. DOI: https://doi.org/10.1080/01630563.2016.1254655

Bomisso G. Jean Marc, Tour´e K. Augustin and Yoro Gozo, Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation, Journal of Mathematics Research, 9 (2017), 30 - 48. DOI: https://doi.org/10.5539/jmr.v9n4p30

H. Brezis, Fonctional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. DOI: https://doi.org/10.1007/978-0-387-70914-7

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

J. L. Lions and E. Magenes, Probl`emes aux limites non homog`enes et Applications, Dunod, 1 1968.

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems with applications, Communications and control Engineering series, Springer-Verlag London Ltd, London, 1999. DOI: https://doi.org/10.1007/978-1-4471-0419-3

Mensah E. Patrice, On the numerical approximation of the spectrum of a flexible Euler-Bernoulli beam with a force control in velocity and a moment control in rotating velocity, Far East Journal of Mathematical Sciences (FJMS), 126(1)(2020), 13 - 31. DOI: https://doi.org/10.17654/MS126010013

M. Miletic and A. Arnold, A piezoelectric Euler-Bernoulli beam with dynamic boundary control: Stability and dissipative FEM, Acta Applicandae Mathematicae, (2014), 1 - 37. DOI: https://doi.org/10.1007/s10440-014-9965-1

B. Rao, Uniform stabilization of a hybrid system of elasticity, Siam J. Control and Optimization 33(2) (1995),440 - 454. DOI: https://doi.org/10.1137/S0363012992239879

A. Shkalikov, Boundary problem for ordinary differential operators with parameter in boundary conditions, Journal of Soviet Mathematics, 33 (1986), 1311 - 1342. DOI: https://doi.org/10.1007/BF01084754

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, Springer-Verlag, New York, 68 1988. DOI: https://doi.org/10.1007/978-1-4684-0313-8

Tzin Wang, A Hermite cubic immersed finite element space for beam design problems, Thesis, Blacksburg Virginia, 2005.