Bifurcation and chaos for a new model of trigonal centrifugal governor with nonsmooth control

TL;DR

This paper introduces a new trigonal centrifugal governor model with nonsmooth control, analyzing its complex nonlinear dynamics, bifurcations, and chaos, supported by theoretical, numerical, and experimental validation.

Contribution

The paper proposes a novel trigonal centrifugal governor model incorporating nonsmooth control, addressing modeling difficulties and analyzing its bifurcation and chaotic behaviors.

Findings

System exhibits pitchfork bifurcation and saddle-focus stability.

Chaotic thresholds are analytically derived using Melnikov method.

Experimental results confirm theoretical and numerical predictions.

Abstract

The flywheel ball and hexagonal structures in the design of the classical centrifugal governor systems lead to both modeling and analytical difficulties. In the present paper, a new trigonal centrifugal governor is proposed in an attempt to overcome both of these difficulties by introducing the radical nonlinearity and nonsmooth control strategy in the simple and clear formula. The nonlinear dynamical behaviors of this new model are investigated for both the autonomous and the non-autonomous cases. The three equations of motion of the TCG are presented based on Euler-Lagrange equation and the theorem of angular momentum. The velocity, nonlinear restoring force and nonsmooth torque surfaces are plotted to display the complex relationship of parameter change dependence. Secondely, the equilibrium bifurcation and stability analysis for autonomous system are investigated to show the pitchork bifurcation phenomena and the saddle-focus point respectively. It is found that the system bears significant similarities to the Duffing system with mono-stable and bi-stable characteristic. Finally, the three-dimensional Melnikov method is defined and employed to obtained the analytical chaotic thresholds for non-autonomous centrifugal governor system, and numerical results of chaotic behaviors verify the proposed theoretical criteria of the non-autonomous system. The experimental studies are carried out to validate the theoretical and numerical results.