Optimal Shape and First Integrals for Inverted Compressed Column

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column....

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Bibliographic Details
Published in:Mathematics (Basel) 2020-03, Vol.8 (3), p.334
Main Authors: Kacapor, Enes, Atanackovic, Teodor M., Dolicanin, Cemal
Format: Article
Language:English
Subjects:
Bifurcation theory
Boundary value problems
Control theory
Cross-sections
Eigenvalues
Equilibrium equations
first integrals
Gravitational fields
Integrals
Mathematical analysis
Nonlinear equations
Numerical integration
optimal shape
Optimization
pontryagin’s principle
Theorems
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Summary:We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8030334