Asymptotic stability of a pendulum with quadratic damping

Abstract

The equation considered in this paper iswhere h(t) is continuous and nonnegative for t ≥ 0 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 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 equationSome 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.Mathematics Subject Classification (2010). Primary 34D23 · 34D45; Secondary 34C10 · 37C75 · 70K05.

The Krylov–Bogoliubov–Mitropolsky method for modeling a forced damped quadratic pendulum oscillator

“…For α = 1 and f = 0, the quadratically unforced damped pendulum equation is recovered. 34 Moreover, for ε = f = 0, the conserved unforced undamped pendulum equation is recovered,…”

An analysis of pendulum motion in the presence of quadratic and linear drag

“…An analytical approximation of the solution to the differential equation describing a linearly damped pendulum undergoing large-angle swings is presented in [5]. Asymptotic stability of pendulums under quadratic damping with a time-varying damping coefficient has also been investigated [6]. The free and forced oscillations of a torsion spring pendulum damped by viscous and dry friction were investigated in [7].…”