Influence of physical and geometrical system parameters uncertainties on the nonlinear oscillations of cylindrical shells

Abstract

This work investigates the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simplysupported cylindrical shells. These parameters are composed of both deterministic and random terms. Donnell's non-linear shallow shell theory is used to study the non-linear vibrations of the shell. To discretize the partial differential equations of motion, first, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all modes that couple with the linear modes through the quadratic and cubic nonlinearities. Then, a particular solution is selected which ensures the convergence of the response up to very large deflections. Finally, the in-plane displacements are obtained as a function of the transversal displacement by solving the inplane equations analytically and imposing the necessary boundary, continuity and symmetry conditions. Substituting the obtained modal expansions into the equation of motion and applying the Galerkin's method, a discrete system in time domain is obtained. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the physical and geometrical system parameters. Special attention is given to the influence of the uncertainties on the parametric instability and escape boundaries.

Influence of Physical and Geometrical Uncertainties in the Parametric Instability Load of an Axially Excited Cylindrical Shell

“…Solving analytically the partial differential equation, a consistent solution for the stress function̂is obtained. Finally, by substituting the adopted expansion for the transversal displacement together with the obtained stress function into (8) and by applying the stochastic Galerkin method…”

“…Theoretical and experimental results found in the literature show that cylindrical shells subjected to static loads are susceptible to buckling, and they may have a load capacity much lower than the theoretical critical load. This difference may be due to variations in physical and geometric properties [1][2][3], including geometric imperfections [4][5][6] or load noise [7,8]. It may occur in the manufacturing process or during the service life of such structures.…”

“…The nonlinear stochastic static analysis of cylindrical shells has been carried out by [19,20] and Silva et al [8] studied the influence of uncertainties on the nonlinear dynamic response, parametric instability boundary, bifurcation diagrams, and basins of attraction of cylindrical shells. However, none of these works uses the stochastic Galerkin method together with the expansion of the generalized polynomial chaos.…”

“…Usually, previous works were conducted studying numerically a very large set of samples so that the results considering uncertainties in a certain parameter can be statistically reliable. But this leads to a time consuming numerical procedure [1][2][3][4][5][6][7][8]. The Galerkin method seems to be a reliable alternative, especially when only the lower bound of buckling loads is desired.…”

Stochastic Natural Vibration Analyses of Free-Form Shells

“…However, the existing work mainly studies static characteristics and it is limited to the deterministic analysis without any consideration of uncertainty factors, which are unavoidable in practical engineering, such as geometrical imperfection and deviations of material properties. The previous stochastic studies on traditional shells have demonstrated the significant effect of uncertain factors on mechanical properties of structures [9][10][11][12][13]. The construction process of free-form shells is more complex when compared with traditional shells, which tends to lead more uncertainty factors.…”