Nonlinear Dynamics and Stability of Shallow Spherical Caps Under Pressure Loading

Abstract

The post-buckling and nonlinear dynamic response of shallow spherical caps subjected to external pressure are analyzed. The Novozhilov's nonlinear thin shell theory is used to express the strain-displacement relations. Following the Rayleigh-Ritz method, the displacement fields are expanded using a mixed series: Legendre polynomials in the meridional direction, harmonic functions in the circumferential direction. Once the linear analysis is completed, the displacement fields are re-expanded, and the nonlinear dynamic model is obtained by using the Lagrange equations. The response of clamped caps, made of isotropic and homogeneous material, is investigated. The bifurcation analyses of equilibrium points and periodic orbits are presented by using continuation techniques. Benchmark results are provided in terms of natural frequencies and critical buckling loads. The dynamic effects due to the interaction between static and dynamic pressure are investigated. Numerical results pointed out that, under particular load conditions, dynamic bifurcation results in non-negligible asymmetric states activation in the response of the structure.

Asymmetric vibrations and chaos in spherical caps under uniform time-varying pressure fields

“…In Fig. 2(a,b), the frequency-response curves obtained by directly integrating the ODEs are compared to continuation method results [32]. Starting from In order to provide further information for understanding the path-following analysis results, bifurcation diagrams of the Poincaré maps are here presented and discussed.…”

Asymmetric Vibrations and Chaos in Spherical Caps under Uniform Time-varying Pressure Fields

“…Consider a clamped shallow spherical cap, having a uniform thickness, made of steel. Using the 38 dofs nonlinear model developed in Ref [32], the nonlinear dynamic response of the cap under a time-varying harmonic pressure is investigated. For the sake of clarity, the linear mode shapes retained into the nonlinear ROM are listed in Table 1:it must be noted that both driven and companion vibration modes have been included for the asymmetric mode shapes (n=0).…”

“…where the occurrence of dynamic instabilities were proven through a pathfollowing analysis in Ref. [32].…”

“…The present work aims to address to some questions arisen recently in Ref. [32] on pressure loaded spherical caps, where the limits of axisymmetric models were shown using continuation techniques. Here the Novozhilov's geometrically nonlinear theory is considered.…”