Refining the Method of Averaging for a More Accurate Analytical Approximation to the Behaviors of a Damped Pendulum

KBM averaging is a widely used technique in the analysis of nonlinear dynamical systems. The KBM method allows complex systems to be approximated as perturbations of simple harmonic oscillator. In many cases, such as in otherwise linear systems with various forms nonlinear damping, the KBM method performs exceptionally well, with error proportional to the size of the perturbations. However, when the largest perturbation in the system arises from nonlinearities in the restoring force, the KBM method falls short, and the interesting effects of other nonlinear terms are drowned out by the approximation errors generated by the KBM method. By generalizing the notion of KBM averaging and approximating systems as perturbations the isoenergy contours of their corresponding Hamiltonian, a greater degree of accuracy can be obtained. We extend the work of several authors to show that not only is this method more accurate, but it is also simple to implement and generalizable to a wide range of nonlinear systems. As an illustrative example, the motion of a pendulum on a tilted platform is studied.