Asymptotic stability of a pendulum with quadratic damping
Abstract
The equation considered in this paper is where h( t) is continuous and nonnegative for and ω 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 timevarying 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 abovementioned equation is used to obtain the main results. In addition, solutions of the system are compared with a particular solution of the firstorder nonlinear differential equation Some examples are also included to illustrate our results. Finally, the present results are extended to be applied to an equation with a nonnegative realpower damping force.
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 October 2014
 DOI:
 10.1007/s000330130361x
 Bibcode:
 2014ZaMP...65..865S