Computational Analysis of Curved Beams - In and Out-of-Plane Vibrations

Raposo F.O.^1,

¹Structural Mechanics Laboratory, FCT - Nova University of Lisbon, 2829-516 Caparica, Portugal

Cite this paper:
Raposo F.O.. Computational Analysis of Curved Beams - In and Out-of-Plane Vibrations. Journal of Mechanical Design and Vibration. 2021; 9(1):1-12. doi: 10.12691/jmdv-9-1-1

Abstract

In the present work are shown results obtained from a computational model for the dynamic analysis of planed curved beams with constant cross-section and curve radius, based on the exact solution of the differential equilibrium equations of a beam element. This model allows the estimation of the forced harmonic motion, both in-plane and out-of-plane of curvature on a free-free support condition. The input data of the model is the geometry section of the beam, material properties, type of motion, consideration of the shear stiffness, and rotatory inertia effects. The output result is possible either in a frequency response of a fixed parameter curved beam or a plot representation of the natural frequencies curves for an opening angle range, from an approximated straight beam to a closed opened ring. Each of those curves is defined as family of modes and performed on present work. The objective of this work is to clarify the general behaviour of curved beams with the mentioned conditions by using an accurate model showing results as a reference for the scientific community. The model was validated by testing and comparing results with three beam specimens previously studied in other publications. The biggest aim is also to present new results and hypothesis, contesting other authors regarding the natural frequencies of curved beams.

Keywords:
planar curved beams natural frequency dynamic analysis frequency response function exact solution family of modes

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	V. H. R. Hoppe, Vibrationen eines Ringes in seiner Ebene, 1871 (73) (1871) 158-170.
[2]	A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publications, 1944.
[3]	Seidel, B. S. and E. A. Erdélyi, On the vibrations of a thick ring in its own plane, J. Eng. Ind., 86 (3) (1964) 240-244.
[4]	Rao S. S. and V. Sundararajan, In plane flexural vibrations of circular rings, ASME J. App Mech, 36 (3) (1969) 620-625.
[5]	Timoshenko S. and D. H. Young and W. Weaver, Vibrations Problems in Engineering, John Wiley & Sons, 1974.
[6]	Wang T. M. and A. Lasky and M. Ahmad, Natural Frequencies for out-of-plane Vibrations of Continuous Curved Beams, Int J. Sol Struct, 20 (3) (1984) 257-263.
[7]	A. P. V. Urgueira, Análise Dinâmica de Vigas Curvas – Desenvolvimento de um Modelo Analítico, Instituto Superior Técnico, Lisboa, Portugal, 1985.
[8]	Montalvão e Silva, J. M. and A. P. V. Urgueira, Out of Plane Dynamic Response of Curved Beams – An Analytical Model, Int. J. Sol Struct., 24 (3) (1988) 271-284.
[9]	Tufekçi, E. and A. Arpaci, Exact Solution of In-Plane Vibrations of Circular Arches with Account Taken of Axial Extension, Transverse Shear and Rotatory Inertia Effects, J. Sound Vib. 209 (5), (1998) 845-856.
[10]	Tufekçi, E. and O. Y. Dogruer, Exact Solution of Out Of Plane Free Vibrations of a Circular Beam with Uniform Cross Section, Faculty of Mechanical Engineering Istanbul Technical University, Istanbul, Turkey, 2002.
[11]	Tufekçi, E., T. Bostanci, O. Ozdemirci, and O. Oldac, Experimental Verification of Numerical and Analytical Solutions of Free Vibrations of Curved Beams, Faculty of Mechanical Engineering Istanbul Technical University, Istanbul, Turkey, 2006.
[12]	Tufekçi, E. and Dogruer O. Y., Out-of-plane free vibration of a circular arch with uniform cross-section: Exact solution, J. Sound Vib. 291 (3-5) (2006) 525-538.
[13]	Raposo, F. O. Análise Dinâmica de Vigas Curvas – Desenvolvimento de um Programa Computacional, Faculdade de Ciências e Tecnologia – Universidade Nova de Lisboa, Caparica, Portugal, 2018 (in Portuguese). (https://run.unl.pt/handle/10362/56415).
[14]	Lee, B.K. Oh, S.J., Mo, J.M., Lee, T.E., Out-of-plane free vibrations of curved beams with variable curvature, J. Sound Vib. 318 (1-2) (2008), 227-246.