In the process of transmission, the flexible membrane will be affected by the pretension of the guide roller and the external excitation force of the conductive silver slurry ejected by the nozzle. Because the large deflection deformation of the middle surface of the membrane caused by pretension and external force seriously affects the printing accuracy of the printed products, it is very important to study the geometric nonlinearity of the flexible membrane. In this paper, the transverse forced vibration model of the flexible membrane with pretension is established by Von Karman’s large deflection theory, and its vibration equation is discretized and solved by the Bubnov–Galerkin method and the fourth-order Runge Kutta method. Meanwhile, by analyzing the bifurcation diagram and TLE (Top Lyapunov Exponents) spectrum, the working state of the motion system with the change in parameters is obtained. In addition, the dynamic response of the system under the given parameters is obtained by comparing the phase diagram with the Poincaré mapping diagram.