Nonlinear oscillation analysis of a pendulum wrapping on a cylinder

In this paper, the nonlinear oscillation of a pendulum wrapping and unwrapping on two cylindrical bases is studied, and an analytical solution is obtained using the multiple scales method. The equation of motion is derived based on an energy conservation technique. By applying the perturbation method to the differential equation, the nonlinear natural frequency of the system is calculated, along with its time response. Analytical results are compared with numerical findings and good agreement is found. The effect of large amplitude and radius of cylinders on system frequency is evaluated. The results indicate that as the radius of the cylinder increases, the system frequency is increased. Also, it is illustrated that initial amplitude plays a dual role in the frequency. As the initial amplitude increases up to a certain point, the frequency is increased, while by increasing it to higher values, the system frequency decreases.