Geodesic
Method for calculation of the stress-strain state for cable-membrane space reflector structures
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 62 (2019), pp. 5-18 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Cable-membrane space reflectors are widely used in the modern space industry. They are essential for communication, monitoring, and observation of the Earth and space objects. Experiments with actual reflector structures are quite expensive. Thus, effective calculation techniques should be applied to describe the reflector behavior under operative loads. A specific feature of such structures is its geometrical non-linear behavior, i.e., significant displacements of the elements under loads. Therefore, geometrical nonlinear governing equations of elasticity theory should be applied in describing the mathematical model of the reflector. The exact solution of these equations could be found only in the simplest cases. Thus, numerical methods for such equations should be used. This paper presents a two-stage calculation method of the stress-strain state for reflector structures based on force density and finite-element methods. The first stage embraces the calculation of the cable element shapes for reflector frontal (rear) nets by the nonlinear force density method. It has been proved that, in some cases, calculating the force density vector iteration step could be challenging due to the ill-conditioned matrix being a component part of this vector. To exclude this problem, the Moore-Penrose pseudoinverse matrix was applied. In the second stage, the calculated reflector frontal (rear) net shapes and corresponding values of cable tension were used as an initial estimate in determining the reflector node displacement field via the nonlinear finite-element method. The reflector stress-strain state is determined using a solution sequence in which every next solution involves the previous one as an initial estimation.
Keywords: force density method, finite-element method, geometrical nonlinearity, grid reflector, normal pseudosoloution.
Mots-clés : pseudoinverse matrix 
@article{VTGU_2019_62_a0,
     author = {A. V. Belkov and S. V. Belov and A. P. Zhukov and M. S. Pavlov and S. V. Ponomarev and S. A. Kuznetsov},
     title = {Method for calculation of the stress-strain state for cable-membrane space reflector structures},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {5--18},
     year = {2019},
     number = {62},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2019_62_a0/}
}
TY  - JOUR
AU  - A. V. Belkov
AU  - S. V. Belov
AU  - A. P. Zhukov
AU  - M. S. Pavlov
AU  - S. V. Ponomarev
AU  - S. A. Kuznetsov
TI  - Method for calculation of the stress-strain state for cable-membrane space reflector structures
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2019
SP  - 5
EP  - 18
IS  - 62
UR  - http://geodesic.mathdoc.fr/item/VTGU_2019_62_a0/
LA  - ru
ID  - VTGU_2019_62_a0
ER  - 
%0 Journal Article
%A A. V. Belkov
%A S. V. Belov
%A A. P. Zhukov
%A M. S. Pavlov
%A S. V. Ponomarev
%A S. A. Kuznetsov
%T Method for calculation of the stress-strain state for cable-membrane space reflector structures
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2019
%P 5-18
%N 62
%U http://geodesic.mathdoc.fr/item/VTGU_2019_62_a0/
%G ru
%F VTGU_2019_62_a0
A. V. Belkov; S. V. Belov; A. P. Zhukov; M. S. Pavlov; S. V. Ponomarev; S. A. Kuznetsov. Method for calculation of the stress-strain state for cable-membrane space reflector structures. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 62 (2019), pp. 5-18. http://geodesic.mathdoc.fr/item/VTGU_2019_62_a0/

[1] W. J. Lewis, Tension structures. Form, behavior, Thomas Telford, London, 2003, 201 pp.

[2] Y. N. Rabotnov, Mechanics of solids, Nauka, M., 1979

[3] O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, The finite element method for solid and structural mechanics, Butterworth-Heinemann, 2013, 714 pp. | DOI

[4] S. V. Ponomarev, A. P. Zhukov, A. V. Belkov, V. S. Ponomarev, S. V. Belov, M. S. Pavlov, “Stress-strain state simulation of large-sized cable-stayed shell structures”, IOP Conf. Series: Materials Science and Engineering, 71 (2015), 012070 | DOI

[5] S. V. Belov, A. V. Bel'kov, A. P. Zhukov, M. S. Pavlov, V. S. Ponomarev, S. V. Ponomarev, A. I. Velichko, V. I. Halimanovich, “Framework stress-strain state estimation for large-sized deployable space reflector”, Izvestiya vysshikh uchebnykh zavedenii. Fizika, 56:7–3 (2013), 131–133

[6] H. J. Schek, “The force density method for form finding and computation of general networks”, Computer Methods in Applied Mechanics and Engineering, 1974, no. 3, 115–134

[7] P. G. Malerba, M. Quagliaroli, “Flexible bridge decks suspended by cable nets. A constrained form finding approach”, Int. J. Solids and Structures, 50 (2013), 2340–2352

[8] S. Morterolle, B. Maurin, J. Quirant, C. Dupuy, “Numerical form-finding of geotensoid tension truss for mesh reflector”, Acta Astronautica, 76 (2012), 154–163

[9] G. Yang, D. Baoyan, Y. Zhang, D. Yang, “Uniform-tension form-finding design for asymmetric cable-mesh deployable reflector antennas”, Advances in Mechanical Engineering, 8:10 (2016), 1–7 | DOI

[10] A. G. Tibert, Deployable Tensegrity Structures for Space Applications, Doctoral Thesis http://www-civ.eng.cam.ac.uk/dsl/publications/TibertDocThesis.pdf

[11] A. P. Zhukov, The dynamics of the reflecting surface of a large-size umbrella-type reflector, Diss. in Physics and Mathematics, 2016

[12] A. V. Bel'kov, S. V. Belov, A. P. Zhukov, M. S. Pavlov, S. V. Ponomarev, “Initial estimation of geometrical nonlinear problem for mesh reflector”, Materials XXI International scientific. conference “Reshetnev reading”, 2017, 82–83

[13] V. M. Verzhbitskiy, Computational linear algebra, Vysshaya shkola, M., 2009

[14] M. S. Bukhtyak, A. V. Nikul?chikov, “An estimate for the mean-square deviation of the paraboloid reflector surface from the hexagonal frontal network”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 4(20), 2012, no. 4(20), 5–14