A Linearised θ Numerical Scheme for the Vibrations of Inextensible Beams

Authors

  • Theodosios K. Papathanasiou Department of Civil and Environmental Engineering, Brunel University London, Uxbridge UB8 3PH, UK https://orcid.org/0000-0003-2130-5172

DOI:

https://doi.org/10.13052/ejcm1779-7179.3015

Keywords:

flexural vibrations; inextensibility; finite element method; large deflections; time-integration; structural elements

Abstract

A linearised finite element numerical scheme for the vibration of inextensible beams is developed. The proposed scheme is based on the methodology introduced by S. Bartels [15] and satisfies a linearised form of the inextensibility constraint. The time m arching procedure is based on repeated use of the theta-parameter integration quadrature. Three parameters are introduced in total and appropriately selected such that the energy conservation features are improved compared to the Bartels algorithm while the inextensibility constraint is satisfied as accurately as possible. Cubic Hermite polynomials are employed for the spatial discretisation. The Bartels algorithm is retrieved as a special case. Several numerical experiments are presented demonstrating the theoretically predicted enhanced inextensibility mimicking and optimum values of the method parameters are identified.

Downloads

Download data is not yet available.

Author Biography

Theodosios K. Papathanasiou, Department of Civil and Environmental Engineering, Brunel University London, Uxbridge UB8 3PH, UK

Theodosios K. Papathanasiou is Lecturer (Structures Engineering) at the department of Civil and Environmental Engineering, Brunel University London. He specialises in Computational Mathematics/Mechanics and in particular the development of Finite Element schemes for Coupled-Field problems (e.g. Hydroelasticty, Thermoelasticity). Dr Papathanasiou is a Mechanical Engineer (National Technical University of Athens) and holds two MSc degrees (Computational Mechanics/Applied Mechanics) and a PhD from the NTUA school of Applied Mathematical and Physical Sciences. He has co-authored more than 30 publications in peer reviewed journals, more than 30 publications in international scientific conferences, and has participated in numerous funded research projects. Dr Papathanasiou was a Marie Curie Research Fellow (IAPP) at the University of Trento (Italy), where he conducted research on the thermomechanical response of refractory ceramics.

References

Maddocks JH. (1984) Stability of nonlinearly elastic rods. Archive for Rational Mechanics and Analysis, 85:311–54.

Bosi F, Misseroni D, Dal Corso F, Bigoni D. (2014) An elastica arm scale. Proc. R. Soc. A 470,20140232, http://dx.doi.org/10.1098/rspa.2014.0232

Bigoni D, Bosi F, Dal Corso F, Misseroni D. (2014) Instability of a penetrating blade. J. Mech. Phys.Solids 64, 411–425, http://dx.doi.org/10.1016/j.jmps.2013.12.008

Xiao J, Chen X. (2013) Buckling morphology of an elastic beam between two parallel lateral constraints: implication for a snake crawling between walls. J. R. Soc. Interface 10, 20130399, https://doi.org/10.1098/rsif.2013.0399

Dal Corso, F., Misseroni, D., Pugno, N.M., Movchan, A.B., Movchan, N.V., Bigoni, D. (2017) Serpentine locomotion through elastic energy release. J. R. Soc. Interface, 14, 20170055, doi: 10.1098/rsif.2017.0055

Rus D, Tolley, MT. (2015) Design, fabrication and control of soft robots. Nature 521, 467–475, https://doi.org/10.1038/nature14543

Laschi C, Cianchetti M, Mazzolai B, Margheri L, Follador M, Dario P. (2012) Soft robot arm inspired by the octopus. Adv. Robotics, 26, 709–727, https://doi.org/10.1163/156855312X626343

Armanini, C., Dal Corso, F., Misseroni, D., Bigoni, D. (2017). From the elastica compass to the elastica catapult: an essay on the mechanics of soft robot arm. Proc. R. Soc. A, 473, 20160870, https://doi.org/10.1098/rspa.2016.0870

Grothaus, M. & Marheineke, N. (2015) On a nonlinear partial differential algebraic system arising in technical textile industry: analysis and numerics. IMA J. Numer. Anal. 36, 4, 1783–1803, https://doi.org/10.1093/imanum/drv056

Vlahopoulos N. and Bernitsas, M. M., (1988) Three dimensional nonlinear dynamics of nonintegral riser bundle, Journal of Ship Research, 35, 1, 40–57, ISSN 0022-4502.

Vlahopoulos, N. and Bernitsas, M. M., (1988) Static Dynamic and Eigen Analysis of Non-Integral Production Risers, Applied Ocean Research, 10, 3, 144–154, https://doi.org/10.1016/S0141-1187(88)80014-2

Liwen He, Jia Lou, Jianke Du, Huaping Wu (2018) Voltage-driven nonuniform axisymmetric torsion of a tubular dielectric elastomer actuator reinforced with one family of inextensible fibers. Eur J Mech A Solids, 71, 386–393, https://doi.org/10.1016/j.euromechsol.2018.06.004

Caflisch, R. E. & Maddocks, J. H. (1984) Nonlinear dynamical theory of the elastica. Proc. Roy. Soc. Edinb. Sect. A, 99, 1–23, https://doi.org/10.1017/S0308210500025920

Manning, R. S. (2014) A catalogue of stable equilibria of planar extensible or inextensible elasticords for all possible Dirichlet boundary conditions. J. Elasticity, 115, 105–130, https://doi.org/10.1007/s10659-013-9449-y

Bartels, S. (2016) A simple scheme for the approximation of elastic vibrations of inextensible curves, IMA J. Numer. Anal, 36, 3, 1051–1071, https://doi.org/10.1093/imanum/drv054

Bartels, S. (2013) A simple scheme for the approximation of the elastic flow of inextensible curves. IMA J. Numer. Anal., 33, 4, 1115–1125, https://doi.org/10.1093/imanum/drs041

Lee P., Kim S., (2020) A variable-θ

method for parabolic probems of nonsmooth data, Computers & Mathematics with applications, 79(4), 962–981, https://doi.org/10.1016/j.camwa.2019.08.006

Bartels S., Reiter Ph., and Riege J. (2018). A simple scheme for the approximation of selfavoiding inextensible curves. IMA J. Numer. Anal., 38, 2, 543–565, https://doi.org/10.1093/imanum/drx021

Published

2021-08-18

How to Cite

Papathanasiou, T. K. (2021). A Linearised θ Numerical Scheme for the Vibrations of Inextensible Beams. European Journal of Computational Mechanics, 30(1), 121–144. https://doi.org/10.13052/ejcm1779-7179.3015

Issue

Section

Original Article