Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

Abstract

In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.