One-to-one autoparametric resonances in infinitely long cylindrical shells

Abstract

The nonlinear response of infinitely long circular cylinders (rings) to a primary excitation of one of the flexural modes was analyzed, taking into account its interaction with its companion mode. Due to the complete circular symmetry of the cylindrical shell, each natural frequency corresponds to two orthogonal mode shapes. The mode with the same spatial variation as the external excitation is called the driven mode, while the other orthogonal mode is called the companion mode. A combination of symbolic manipulator (MACSYMA) and the method of multiple scales is used to derive four first-order ordinary differential equations for the modulation of the amplitudes and phases of the interacting modes. The fixed points of the modulation equations provide the frequency-response curves. There are two possible fixed-point solutions: a single-mode solution consisting of the driven mode only and a two-mode solution consisting of the driven and companion modes. The latter solution corresponds to traveling waves. As the excitation frequency varies, the fixed points of the single-mode solution suffer saddle-node collisions resulting in jumps. On the other hand, the fixed points of the two-mode solution can undergo Hopf bifurcations. Between the Hopf bifurcation frequencies, a numerical solution of the modulation equations shows that they possess limit-cycle or chaotic solutions. For a range of excitation frequencies, the periodic single-mode solution coexists with either a periodic or a periodically or chaotically modulated two-mode solution.