Упругие балки минимального веса, при наличии нескольких видов изгибающих нагрузок

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А.А. Гурченков, Н.Т. Вилисова, И.М. Герман, А.М. Романенков
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Elastic beams of minimum weight in the presence of several
types of bending loads
© A.A.Gurchenkov1,2, N.T. Vilisova1, I.M. German2, A.M. Romanenkov2
1Bauman Moscow State Technical University, Moscow, 105005, Russia
2MATI – Russian State Technological University named after K.E. Tsiolkovsky,
Moscow, 109387, Russia
The article considers the problem of optimizing the loaded beam thickness, i.e. minimiz-
ing weight of the structure, for given boundary conditions and restrictions of strain ca-
pacity. It was found that the mathematical model in this case is the boundary value prob-
lem for ordinary differential equation of 4th order. Solving the optimization problem is
based on two different approaches. The first one is the classical variational method based
on studying the variation of the minimized functional and analyzing the stationary point
of the functional. In the second method, the Pontryagin maximum principle is applied to
the problem with fixed left and right ends.
Numerical experiments carried out for different types of bending loads, are illustrated by
graphs. Comparison of the results shows the equivalence of the two approaches. This
significantly extends the range of optimization problems, for solution of which software
with models of complex systems is developed.
Keywords: beam thickness optimization, variational method, principle of maximum.
REFERENCES
[1] Banichuk N.V. Izvestiya AN SSSR. Mekhanika tverdogo tela – Proceedings of
the USSR AS. Mechanics of Rigid Body, 1974, no. 4, pp. 44–51.
[2] Moiseev N.N., Ivanilov Yu.P., Stolyarova E.M. Metody optimizatsii
[Optimization Techniques]. Moscow, Nauka Publ., 1978.
[3] Samarskiy A.A. Vvedenie v teoriyu raznostnykh skhem [Introduction to the
Theory of Difference Schemes]. Moscow, Nauka Publ., 1971.
[4] Tsvey . Balki i plity na uprugom osnovanii [Beams and Plates on Elastic
Foundation]. Moscow, MADI Publ., 2014, 96 p.
[5] Vasserman N.N., et al. Soprotivlenie materialov [Strength of Materials]. Perm,
Perm National Research Polytechnic University Publ., 2011, 365 p.
[6] Makarov E.G. Kursovaya rabota po metodu konechnykh elementov [Term paper
on the finite element method]. St. Petersburg, Baltic State Technical University
"Voenmeh" Publ., 2011, 49 p.
[7] Sankin Yu.N., Yuganova N.A. Nestatsionarnye kolebaniya sterzhnevykh system
pri soudarenii s prepyatstviem [Unsteady Oscillations of Rod Systems in a
Collision with an Obstacle]. Ulyanovsk, Ulyanovsk State Technical University
Publ., 2010, 174 p.
[8] Isaev V.I. Matematicheskie modeli sterzhney, balok i plit v zadachakh
sosredotochennogo udara [Mathematical Models of Rods, Beams and Plates
in Problems of the Centric Impact]. Ph.D. Thesis (Phys.-Math.). Moscow,
2007, 155 p.
[9] Atamuratov Molodoy uchenyy – Young Scientist, 2014, no. 1, pp. 13–18.
[10] Andreev V.I., Barmenkova E.V., Matveeva A.V. Vestnik MGSU – Herald of
the National Research University Moscow State University of Civil Engineering,
2014, no. 1, pp. 25–32.