Asymptotic stability of a pendulum with quadratic damping

The equation considered in this paper is x '' + h(t) x' vertical bar x vertical bar| + omega(2)sin x = 0, where h(t) is continuous and nonnegative for and omega is a positive real number. This may be regarded as an equation of motion of an underwater pendulum. The damping force is proportional to the square of the velocity. The primary purpose is to establish necessary and sufficient conditions on the time-varying coefficient h(t) for the origin to be asymptotically stable. The phase plane analysis concerning the positive orbits of an equivalent planar system to the above-mentioned equation is used to obtain the main results. In addition, solutions of the system are compared with a particular solution of the first-order nonlinear differential equation u' + h(t) u vertical bar u vertical bar + 1 = 0. Some examples are also included to illustrate our results. Finally, the present results are extended to be applied to an equation with a nonnegative real-power damping force.