Influence shape imperfections on stochastic stability of elastic shell parametric vibrations

Mathematical modeling of stochastic parametric oscillations of an elastic cylindrical tank shell with real and modelled shape imperfections under the action of a random axial load was performed. Finite element models of the imperfect shell were generated in the NASTRAN software. A functional approach was applied to the formation of a reduced model of parametric shell oscillations in the form of a system of differential equations of the first Markov approximation with respect to moment functions of the second order, taking into account specific values ​​of the constant components of the parametric load. The stochastic component of the parametric load was given in the form of a delta-correlated random load. The members of the stiffness matrix of the reduced mathematical model are the squares of the frequencies of the natural oscillations of the perfect shell, obtained by the Lanczos method. The members of the reduced matrix of the geometric stiffness of the shell without and with shape imperfections were obtained using a two-stage calculation. At the first stage, the nonlinear statics problem under the action of the constant component of the parametric load was solved by the Newton-Raphson method. At the second stage, a modal analysis was performed using the Lanczos method taking into account the pre-stressed state of the shell. The influence of the stochastic component of the parametric load on the dynamic behavior of the shell was investigated using the fourth-order Runge-Kutta method. Response realizations and phase trajectories of the shell with real and simulated imperfections at a given frequency of the hidden periodicity of the stochastic load, damping coefficient, and correlation parameter were obtained. The stochastic stability of parametric oscillations of an imperfect shell was investigated using generalized Hill determinants. The stochastic stability problem was reduced to determining the characteristic indices of a linear autonomous system. The influence of real and simulated imperfections of the shell shape on the stability of parametric oscillations at different values ​​of the constant and stochastic components of the axial load was estimated.