Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes

Abstract

The authors derived a mathematical model of geometrically nonlinear vibrations of three-layer shells, which describes the vibrations of the structure with amplitudes comparable to its thickness. The high-order shear theory is used in the derivation of this model. Rotational inertia is also taken into account. At the same time, the middle layer is a honeycomb structure made thanks to additive FDM technologies. In addition, each shell layer is described by five variables (three displacement projections and two rotation angles of the normal to the middle surface). The total number of unknown variables is fifteen. To obtain a model of nonlinear vibrations of the structure, the method of given forms is used. The potential energy, which takes into account the quadratic, cubic, and fourth powers of the generalized displacements of the structure, is derived. All generalized displacements are decomposed by generalized coordinates and eigenforms, which are recognized as basic functions. It is proved that the mathematical model of shell vibrations is a system of nonlinear non-autonomous ordinary differential equations. A numerical procedure is used to study nonlinear periodic vibrations and their bifurcations, which is a combination of the continuation method and the shooting method. The shooting method takes into account periodicity conditions expressed by a system of nonlinear algebraic equations with respect to the initial conditions of periodic vibrations. These equations are solved using Newton's method. The properties of nonlinear periodic vibrations and their bifurcations in the area of subharmonic resonances are numerically studied. Stable subharmonic vibrations of the second order, which undergo a saddle-node bifurcation, are revealed. An infinite sequence of bifurcations leading to chaotic vibrations is not detected.