Stabilization of a coupled inverted pendulums system via vertical forces

Abstract

To solve a number of practical problems (oscillations of the supporting circuits in construction, the problem of plasma stabilization, stabilization of synthesized biologic chains, etc.), model systems are used which are based on the laws of motion of basic coupled oscillators and their chains. This article analyses the mathematical model of a system consisting of two inverted pendulums with a flexible joint (a spring). The system is controlled by a controller in the form of a vertical oscillation of the attachment point of one of the pendulums. The article presents a detailed study of the dynamics of the described mechanical system and determines the conditions for its stabilization. Stability zones were identified in the initial parameter space. The paper describes the evolution of the stability zones depending on the spring’s stiffness. Spectra of solutions were obtained, showing that the motion of the system corresponds to almost periodic functions. It was established that unstable periodic regimes were present at the boundaries of the stability zones. The planes were determined that corresponded to the initial conditions which complied with the obtained periodic solutions. The key analytical results were obtained using the monodromy matrix. In our study we considered a situation when the system in the linear approximation is piecewise linear. For such a system the monodromy matrix can be given in explicit form. The article also presents the results of numerical experiments illustrating the dynamics of the system. It also demonstrates that when the parameters of the system are different from the initial values, the geometry of the stability zones also changes depending on the increase in the area of one of the zones. All figures illustrating the stability zones, evolution of the stability zones, solution spectra, pendulum motion graphs, and phase portraits were prepared using Wolfram Mathematica.