Intra-well and cross-well chaos in membranes and shells liable to buckling

Abstract

This work presents the mathematical modeling for the nonlinear vibration analysis of membrane and shell structures of arbitrary shape. These structures are usually the optimal form in many engineering applications. However, they may buckle under specific loading conditions and, in most cases, are sensitive to geometric imperfections. These structures when liable to unstable buckling present for load levels lower than the static critical load a multiwell potential functions, which has an underlying influence on the nonlinear dynamic behavior and stability of the structure in a dynamic environment. The energy barrier of each well is a key factor, and depending on the force control parameters and initial conditions, intra-well and cross-well motions may occur. Also coexisting attractors are the norm, influencing the structure's dynamic integrity. In these structures, escape from a potential well is preceded by global and local bifurcations, usually leading to chaos, which influence the number of coexisting attractor, their period and the topology of the basins of attraction. In the present work, three problems are addressed: an axially loaded cylindrical shell, a pressure-loaded spherical cap and a spherical membrane under internal pressure. These problems illustrate the possible multiwell functions observed in several membrane and shell problems, namely potential functions with one, two and three (symmetric or asymmetric) potential wells. A detailed parametric analysis is conducted through bifurcation diagrams of the Poincaré map, time responses and Floquet stability criterion to clarify the influence of the multiwell potential function on the bifurcation scenario, basins of attraction and system safety.

Appendix A

For the cylindrical shell model, equations of motion of the reduced 2DOF model after Galerkin discretization (20) are given by [60]:

$$ T_{11} \,\ddot{\zeta }_{11} \, + \,R_{11} \,\dot{\zeta }_{11} \, + \,\left( {U_{11} \, - \,\psi_{11} } \right)\,\zeta_{11} \, + \,U_{112} \,\zeta_{11} \,\zeta_{02} \, + \,\frac{1}{6}\,U_{1111} \,\zeta_{11}^{3} \, + \,\frac{1}{2}\,U_{1122} \,\zeta_{11} \,\zeta_{02}^{2} \, = \,0 $$

$$ T_{02} \,\ddot{\zeta }_{02} \, + \,R_{22} \,\dot{\zeta }_{02} \, + \,\left( {U_{22} \, - \,\psi_{22} } \right)\,\zeta_{02} \, + \,\,\frac{1}{2}\,U_{112} \,\zeta_{11}^{2} \, + \,\frac{1}{2}\,U_{1122} \,\zeta_{11}^{2} \,\zeta_{02} \,\, = \,0 $$

where

$$ U_{11} \, = \,8\frac{{\alpha^{2} \,m^{3} }}{{\pi^{2} n\,\left( {m^{2} \, + \,\gamma^{2} } \right)^{2} }}\, + \,\frac{2}{3}\,\frac{{\pi^{2} \beta^{4} \left( {m^{2} \, + \,\gamma^{2} } \right)^{2} }}{{m\,n\,\left( {1\, - \,\nu^{2} } \right)}} $$

$$ U_{22} \, = \,16\frac{{\alpha^{2} }}{{\pi^{2} \,m\,n}}\, + \,\frac{64}{3}\,\frac{{\pi^{2} \beta^{4} \,m^{3} }}{{n\,\left( {1\, - \,\nu^{2} } \right)}} $$

$$ U_{112} \, = \, - 32\frac{{\alpha^{3} \,n\,m^{3} }}{{\pi^{2} \left( {m^{2} \, + \,\gamma^{2} } \right)^{2} }}\, - 4\,\frac{{\alpha^{3} \,n}}{{\pi^{2} \,m}} $$

$$ U_{1111} \, = \,3\frac{{\beta^{4} \,\pi^{2} \left( {m^{4} \, + \,\gamma^{4} } \right)}}{m\,n} $$

$$ U_{1122} \, = \,64\frac{{\alpha^{4} \,n^{3} m^{3} }}{{\pi^{2} }}\left[ {\frac{1}{{\,\left( {9\,m^{2} \, + \,\gamma^{2} } \right)^{2} }}\, + \,\frac{1}{{\,\left( {m^{2} \, + \,\gamma^{2} } \right)^{2} }}} \right]\, $$

$$ \psi_{11} \, = \,8\,\frac{{\varGamma \,\alpha \,\beta^{2} \,m}}{{n\,\sqrt {3\,\left( {1\, - \,\nu^{2} } \right)} }} $$

$$ \psi_{22} \, = \,64\,\frac{{\varGamma \,\alpha \,\beta^{2} \,m}}{{n\,\sqrt {3\,\left( {1\, - \,\nu^{2} } \right)} }} $$

$$ T_{11} \, = \,16\,\frac{M\,h}{{\pi^{2} \,E\,n\,m}} $$

$$ T_{22} \, = \,32\,\frac{M\,h}{{\pi^{2} \,E\,n\,m}} $$

$$ R_{11} \, = \,8\frac{{h\,C_{1} }}{{\pi^{2} {\kern 1pt} E\,m\,n\,}}\, + \,8\,\frac{{h\,C_{2} \,\pi^{2} }}{{E\,L^{4} \,m\,n}}\left( {m^{2} \, + \,\gamma^{2} } \right)^{2} $$

$$ R_{22} \, = \,16\frac{{h\,C_{1} }}{{\pi^{2} {\kern 1pt} E\,m\,n\,}}\, + \,256\,\frac{{h\,C_{2} \,\pi^{2} \,m^{3} }}{{E\,L^{4} n}}. $$