An analytical solution to the equation of motion for the damped nonlinear pendulum

An analytical approximation of the solution to the differential equation describing the oscillations of the damped nonlinear pendulum at large angles is presented. The solution is expressed in terms of the Jacobi elliptic functions by including a parameter-dependent elliptic modulus. The analytical solution is compared with the numerical solution and the agreement is found to be very good. In particular, it is found that the points of intersection with the abscissa axis of the analytical and numerical solution curves generally differ by less than 0.1%. An expression for the period of oscillation of the damped nonlinear pendulum is presented, and it is shown that the period of oscillation is dependent on time. It is established that, in general, the period is longer than that of a linearized model, asymptotically approaching the period of oscillation of a damped linear pendulum.