Book Description

The nonlinear flexural vibrations of thin circular rings are analyzed by assuming two vibration modes and then applying Galerkin's procedure on the equations of motion. This procedure results in coupled nonlinear ordinary differential equations with time as the independent variable. The applied loading is taken to be harmonic in time, and approximate solutions to the equations are obtained by the method of averaging. One such solution involves the vibration of a single bending mode; a subsequent stability analysis shows that this single-mode response is valid only for certain combinations of amplitude, nonlinear coupling causes its companion mode to respond and participate in the motion. Approximate solutions are obtained for this coupled-mode case, and their stability is examined. The steady-state response curves contain an unusual "gap", where both the one- and two-mode solutions are unstable. These results were confirmed on an analog computer, and nonsteady vibrations were observed in the gap region. An experimental study of the problem was also conducted. Theory and experiment both indicate a nonlinearity of the softening type and the appearance of the companion mode. Measurements of the steady-state response are in good agreement with the calculated values, and the experimentally determined mode shapes agree with the form of the assumed deflection. The analytical and experimental results exhibit several features that are characteristic of nonlinear vibrations of axisymmetric systems in general and of circular cylindrical shells in particular.