The dynamical response including the motion and the destruction of the electro-active polymer cylindrical shells subjected to the periodic pressure on the inner surface are studied within the framework of finite elasto-dynamics. It is proved that there exists a certain critical value of the internal pressure and the electric field through numerical computing and dynamic qualitative analysis based on the nonlinear differential equation for the motion of the inner surface of the shell. The motion of the shell is nonlinear quasi-periodic oscillation when the mean pressure of the periodic pressure and the voltage are less than their critical values, respectively. In contrast, the shell is destroyed when the pressure or the voltage exceeds the corresponding critical value. Moreover, the effect of the electric field and the pressure parameters on the critical values and the oscillation of the shell are then discussed.