A Note on Bifurcation of Equilibrium Forms of a Gas Balloon Parachute

Zgorzelska, Anita; Guze, Hanna

doi:10.2478/amsil-2025-0009

Abstract

We will be concerned with deformations of a free elastic top rim of a parachute of a gas balloon. The top rim is connected with the circular deflation port of the balloon envelope by heavy duty flexible load tapes. The inside part of the balloon is filled with compressed gas. Equilibrium forms of the parachute may be found as solutions of a certain nonlinear functional-differential equation with two physical parameters: an elasticity coefficient of tapes and a physical parameter describing compressed gas. This equation possesses radially symmetric solutions corresponding to circular shapes of the top rim. Our goal is to study the existence of symmetry breaking bifurcation of the top rim of parachute.

References

S.S. Antman, Nonlinear Problems of Elasticity, Appl. Math. Sci., 107, Springer-Verlag, New York, 1995.
Search in Google Scholar Back to article
A.Yu. Borisovich, J. Dymkowska, and Cz. Szymczak, Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation, J. Math. Anal. Appl. 307 (2005), no. 2, 480–495.
Search in Google Scholar Back to article
A.Yu. Borisovich and J. Janczewska, Stable and unstable bifurcation in the von Kármán problem for a circular plate, Abstr. Appl. Anal. 2005, no. 8, 889–899.
Search in Google Scholar Back to article
I. Chueshow and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics, Springer Monogr. Math., Springer, New York, 2010.
Search in Google Scholar Back to article
P.G. Ciarlet, Mathematical Elasticity. Volume III: Theory of Shells, Stud. Math. Appl., 29, North-Holland Publishing Co., Amsterdam, 2000.
Search in Google Scholar Back to article
M.G. Crandall and P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340.
Search in Google Scholar Back to article
H. Guze and J. Janczewska, Symmetry-breaking bifurcation for free elastic shell of biological cluster, part 2, Milan J. Math. 82 (2014), no. 2, 331–342.
Search in Google Scholar Back to article
H. Guze and J. Janczewska, Subcritical bifurcation of free elastic shell of biological cluster, Nonlinear Anal. Real World Appl. 24 (2015), 61–72.
Search in Google Scholar Back to article
J. Janczewska, Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point, Cent. Eur. J. Math. 2 (2004), no. 4, 561–572.
Search in Google Scholar Back to article
J. Janczewska, Multiple bifurcation in the solution set of the von Kármán equations with S¹-symmetries, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 109–126.
Search in Google Scholar Back to article
N.F. Morozov, Selected Two-Dimensional Problems of Elasticity Theory (in Russian), Univ. Press, Leningrad, 1978.
Search in Google Scholar Back to article
J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, Inc., Hoboken, New Jersey, 2002.
Search in Google Scholar Back to article
Yu. I. Sapronov, Branching of solutions of smooth Fredholm equations, in: Yu. G. Borisovich (Ed.), Equations on Manifolds (in Russian), Novoe Global. Anal., Voronezh. Gos. Univ., Voronezh, 1982, pp. 60–82.
Search in Google Scholar Back to article
I.I. Vorovich, Mathematical Problems in the Nonlinear Theory of Shallow Shells (in Russian), Nauka, Moscow, 1989.
Search in Google Scholar Back to article

A Note on Bifurcation of Equilibrium Forms of a Gas Balloon Parachute

Abstract

Paradigm

My account