Nonlinear dynamical bifurcation and chaotic motion of shallow conical lattice shell

Abstract

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…Chaotic vibrations of shallow conical panels can be found in Refs. [81,178,179,205]. Krysko et al [81,178] found the Feigenbaum scenario for conical panels subjected to large harmonic excitations.…”

“…Awrejcewicz et al [179] discussed the routes to chaos of conical panels with non homogenous thickness. Wang et al [205] used the Melinkov method to discuss the chaotic response of a conical lattice shell modeled as a single degree of freedom system.…”