On the global geometric structure of the dynamics of the elastic pendulum. (English) Zbl 0963.70013

Georgiou, Ioannis T.

Nonlinear Dyn. 18, No. 1, 51-68 (1999).

The authors studies the dynamics of planar elastic pendulum by considering it as a singular perturbation of uncoupled pendulum. The equations of motion are \(\ddot\theta+ {2\dot\theta \dot R\over 1+R}+ {\sin\theta \over 1+R}=0\) and \(\ddot R+({\omega_s \over\omega_p})^2 R-(1+R_)\dot \theta^2+1-\cos\theta=0\), where \(\omega_p\) and \(\omega_s\) denote respectively natural frequencies of the pendulum and radial oscillator. The author determines the global geometric structure of the dynamics in terms of two-dimensional invariant manifolds of motion. A general analytic study is carried out and confirmed by numerical experiments.

Reviewer: S.Nocilla (Torino)

MSC:

70K05	Phase plane analysis, limit cycles for nonlinear problems in mechanics
70K20	Stability for nonlinear problems in mechanics
70K50	Bifurcations and instability for nonlinear problems in mechanics
70K60	General perturbation schemes for nonlinear problems in mechanics

Keywords:

planar elastic pendulum; singular perturbation; global geometric structure; two-dimensional invariant manifolds of motion