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

An analytical approximation of the solution to the differential equationdescribing the oscillations of the damped nonlinear pendulum at large anglesis presented. The solution is expressed in terms of the Jacobi elliptic functionsby including a parameter-dependent elliptic modulus. The analytical solutionis compared with the numerical solution and the agreement is found to bevery good. In particular, it is found that the points of intersection with theabscissa axis of the analytical and numerical solution curves generally differby less than 0.1%. An expression for the period of oscillation of the dampednonlinear pendulum is presented, and it is shown that the period of oscillationis dependent on time. It is established that, in general, the period is longer thanthat of a linearized model, asymptotically approaching the period of oscillationof a damped linear pendulum.