On rotational solutions for elliptically excited pendulum

Abstract

The author considers the planar rotational motion of the mathematical pendulum with its pivot oscillating both vertically and horizontally, so the trajectory of the pivot is an ellipse close to a circle. The analysis is based on the exact rotational solutions in the case of circular pivot trajectory and zero gravity. The conditions for existence and stability of such solutions are derived. Assuming that the amplitudes of excitations are not small while the pivot trajectory has small ellipticity the approximate solutions are found both for high and small linear damping. Comparison between approximate and numerical solutions is made for different values of the damping parameter.

Dynamics of a Pendulum of Variable Length and Similar Problems

“…( 5 ) Using this substitution in equation (4) and multiplying it by 1 + εϕ(τ) we obtain the equation for η asη…”

“…Its stability with respect to the variable η is equivalent to that of the equation (4) with respect to θ due to relation (5). According to Lyapunov's theorem on stability based on a linear approximation for a system with periodic coefficients the stability (instability) of the solution η = 0 of equation (6) is determined by the stability (instability) of the linearized equation…”

“…The dynamics of these mechanical systems is described by similar equations and is studied with the use of common methods. The material of the chapter is based on publications of the authors [1][2][3][4][5][6][7] with the renewed analytical and numerical results. The methodological peculiarity of this work is in the assumption of quasi-linearity of the systems which allows us to derive higher order approximations by the averaging method.…”

The motion characteristics of the double‐pendulum system with skew walls

“…The authors documented the parametric resonances and the effect of the high‐frequency motion on the system stabilization. Belyakov reported the planar rotational motion of a mathematical pendulum with an elliptical excitation analysis, based on the exact solutions. The derivation of existence and stability of these solutions allows approximate solutions for high and low linear damping to be found.…”

Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

“…It documents the parametric resonance causing the stable downward vertical equilibrium and the stabilization of the unstable upward equilibrium state by subjecting the whole system to the high-frequency motion. The issue of excited pendulum was also elaborated by Belyakov in [12]. He presented planar rotational motion of the mathematical pendulum with an elliptical excitation analysis, based on the exact solutions for the circular trajectory and no gravity.…”