Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum

Abstract

In this paper, the authors have studied dynamic responses of a parametric pendulum by means of analytical methods. The fundamental resonance structure was determined by looking at the undamped case. The two typical responses, oscillations and rotations, were investigated by applying perturbation methods. The primary resonance boundaries for oscillations and pure rotations were computed, and the approximate analytical solutions for small oscillations and period-one rotations were obtained. The solution for the rotations has been derived for the first time. Comparisons between the analytical and numerical results show good agreements.

Parametrically excited helicopter ground resonance dynamics with high blade asymmetries

“…In the present work, great attention is paid to parametrically excited systems [7,9]. The presence of parametric and periodic terms can subject the system to parametric instability conditions.…”

“…Taking into account the above solvability conditions and replacing them in Eqs. (9) to (11), the responses u n 1 (T 0 , T 1 , T 2 ) are obtained by solving the inhomogeneous second order differential equations (see Appendix A).…”

“…pendulum dynamics in parametric excitations [7,8,9] are all examples of problems in which these methods have been used. In the latter applications, the authors observed that parametric excitations lead to the occurrence of parametric resonances that in turn lead to unstable oscillation conditions.…”

Multiple Bifurcations of a Cylindrical Dynamical System

“…2(b). The double connected homoclinic orbits of first type denoted by hom1, for h = 0, can be written in the parametric form as: (8) x hom1…”

“…For example, the chaotic dynamics and subharmonic bifurcations in a pendulum-like system were studied in [7]. The approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum were discussed in [8]. The bifurcation in an inverted pendulum with the high frequency excitation was proved by using analytical and experimental investigations in [9].…”