Version 1
: Received: 25 May 2021 / Approved: 27 May 2021 / Online: 27 May 2021 (09:09:32 CEST)
Version 2
: Received: 23 September 2021 / Approved: 24 September 2021 / Online: 24 September 2021 (11:27:04 CEST)
How to cite:
Fogang, V. Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method. Preprints2021, 2021050660. https://doi.org/10.20944/preprints202105.0660.v2
Fogang, V. Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method. Preprints 2021, 2021050660. https://doi.org/10.20944/preprints202105.0660.v2
Fogang, V. Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method. Preprints2021, 2021050660. https://doi.org/10.20944/preprints202105.0660.v2
APA Style
Fogang, V. (2021). Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method. Preprints. https://doi.org/10.20944/preprints202105.0660.v2
Chicago/Turabian Style
Fogang, V. 2021 "Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method" Preprints. https://doi.org/10.20944/preprints202105.0660.v2
Abstract
This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.
Keywords
Axially functionally graded non-prismatic Euler-Bernoulli beam; finite difference method; additional points; vibration analysis; direct time integration method
Subject
Engineering, Civil Engineering
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Valentin Fogang
Commenter's Conflict of Interests: Author
Graphical abstract
Introduction