Stability of the Parametric Vibrations of a Shell in the Form of a Hyperbolic Paraboloid

A numerical technique for constructing a reduced model of the stability of the parametric vibrations of a hyperbolic paraboloidal shallow shell with negative Gaussian curvature is presented. To form the reduced matrices of mass, damping, stiffness, and geometrical stiffness, finite-element software routines are employed. The nonlinear analysis of static and dynamic behavior of a hyperbolic paraboloid made it possible to reveal the differences in its behavior from that of shallow shells with positive Gaussian curvature. By analyzing the influence of the constant component of the parametric loading on the natural frequencies, it is established that the shell losses stability in a certain loading range, followed by stabilization. To study this feature, it is proposed to use an additional reduced model of the stability of the parametric vibrations of a hyperbolic paraboloid.