Abstract
The flexible elastic rods undergoing large deflections are widely used in the engineering practice. Analytic solutions for flexible rods are known for some special cases but there are cases when the exact solution does not exist. In such cases, it is expedient to use numerical methods, and the most known is a finite element method (FEM). This paper presents a new numerical method based on second-degree splines of the defect 1 allowing to solve the linearized nonlinear equations with high accuracy for large deflections of a thin elastic rod. The method efficiency is evaluated on the test problem of a pure bending (not shear) of a thin elastic rod. This paper shows that the method ensures the accuracy with a relative error does not exceed 1 × 10−6 for a sufficiently dense number of nodes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Popov EP (1948) Nonlinear problems of the thin rods statics. Gostekhizdats, Leningrad, Moscow
Popov EP (1986) Theory and calculation of flexible rods. Nauka, Moscow
Feodos’ev VI (1979) Strength of materials. Nauka, Moscow
Timoshenko SP, James GM (2002) Mechanics of materials. Izd-vo Lan’, St. Petersburg
Timoshenko SP, James GM (1961) Theory of elastic stability. McGraw-Hill, New York
Kudoyarova VM, Kishalov AE (2016) Solution of applied heat transfer and gas dynamics problems in ANSYS. UGATU, Ufa
Basov KA (2009) CATIA and ANSYS. Solid state simulation. DMK Press, Moscow
Kaplun AB, Morozov EM, Olfer’eva MA (2003) ANSYS in the engineer’s hands. URSS, Moscow
Basov KA (2017) Graphic interface of ANSYS. Profobrazovanie, Saratov
Pavlov VP (2016) Analysis of the spectrum of frequencies of own fluctuations of a rod by the method of splines. Vestn UGATU 20–4(74):16–22
Pavlov VP (2017) Transverse vibrations of a rod with variable cross sections and calculation of its eigenfrequencies by the method of splines. Vestnik UGATU 21–2(76):3–16
Zhernakov VS, Pavlov VP, Kudoyarova VM (2017) The enhanced spline-method for numerical results of natural frequencies of beams. Proc Eng 176C:438–450. https://doi.org/10.1016/j.proeng.2017.02.343
Pavlov VP (2016) The method of splines in the calculation of the studs for stability. Vestn UGATU 20–4(74):45–53
Kudoyarova VM, Pavlov VP (2016) The spline method for the solution of the transient heat conduction problem with nonlinear initial and boundary conditions for a plate. Proc Eng 150:1419–1426. https://doi.org/10.1016/j.proeng.2017.10.541
Pavlov VP (2003) Method of spline and other numerical me-ODS for solving one-dimensional problems of mechanics deformable solids. UGATU, Ufa
Abdrakhmanova AA, Pavlov VP (2007) A fiberglass core’s intense condition’s mathematical modelling at various rigidities of supports. Vestn UGATU 9–5(23):87–92
Pavlov VP, Abdrakhmanov AA, Abdrakhmanova RP (2013) The problem of rods calculation by the spline of the fifth degree of defect 2. Math Notes YSU 20–1:50–59
Zhernakov VS, Pavlov VP, Kudoyarova VM (2017) The spline-method for numerical calculation of the natural-vibration frequency of a beam with variable cross-section. Proc Eng 206C:710–715. https://doi.org/10.1016/j.proeng.2017.10.542
Pavlov VP, Kudoyarova VM (2017) A numerical method based on spline for solution heat conduction nonlinear problems. Proc Eng 206C:704–710. https://doi.org/10.1016/j.proeng.2017.10.541
Smirnov VI (1956) Course of higher mathematics. Gos. iz-vo tekhn. teor. lit-ry, Moscow
Acknowledgements
The authors acknowledge receiving support base part of funded research program of Russian Foundation for Basic Research (RFBR) and Government of the Republic of Bashkortostan in the framework of a scientific project No. 17-48-020824_p_a.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Zhernakov, V.S., Pavlov, V.P., Kudoyarova, V.M. (2019). Deformation of Thin Elastic Rod Under Large Deflections. In: Radionov, A., Kravchenko, O., Guzeev, V., Rozhdestvenskiy, Y. (eds) Proceedings of the 4th International Conference on Industrial Engineering. ICIE 2018. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-95630-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-95630-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95629-9
Online ISBN: 978-3-319-95630-5
eBook Packages: EngineeringEngineering (R0)