Chaos in vibrating systems with a limited power-supply

Abstract

New models and properties of forced oscillations of the various physical systems (pendulum and piezoceramic transducer) due to the interaction with the excitation device of limited power-supply are investigated in details. Using an analysis of the largest Lyapunov exponent for a complicated system-vibrating subsystem and exciter-the regions for three steady state regimes are determined, namely: stationary, periodic and chaotic.

Deterministic chaos of a spherical pendulum under limited excitation

“…2e) has a spiral structure and resembles, to a certain extent, the chaotic attractors discovered in the case of a plane pendulum [13,15]. It should be noted that the phase portrait of these chaotic attractors resembles (sometimes substantially) the phase portraits of cycles with a large number of strokes whose bifurcations lead to the birth of chaotic attractors.…”

“…This approach, together with the wide application of various reduction methods, enables one to reduce the order of dynamical systems used for the investigation of oscillations of pendulums. However, in many cases, this idealization leads to substantial errors in the qualitative and quantitative description of dynamical modes [9,[13][14][15]. The radical change in the world outlook caused by the discovery of deterministic chaos in dynamical systems considerably extended our notion of possible stable oscillation modes in pendulum systems.…”

Synchronization and sommerfeld effect as typical resonant patterns

“…This implies a very reasonable physical argument. With an increase in base vibrations, for example, the geometric nonlinearity of the elastic base should brightly manifest itself, so that this assuredly may lead to the so-called phenomenon of "pulling", or even, chaotic oscillations [7][8][9]. However, a more detailed mathematical study can demonstrate that the dynamic phenomena associated with the Sommerfeld effect are of more subtle nature.…”

Deterministic Chaos in Generator-Piezoceramic Transduser System

“…мощности, сравнимой с мощностью, излучаемой или потребляемой пьезокерамическими телами при их деформациях. Настоящая статья посвящена анализу эффектов взаимодействия, в совокупности названных эффектом Зоммерфельда-Кононенко [4,5,6,18,19,20,21,23,24], колебаний пьезокерамического излучателя и механизма возбуждения этих колебанийгенератора электрического тока, имеющего ограниченную мощность. С этой целью строится новая математическая модель, описывающая процесс взаимодействия пьезокерамического излучателя, нагруженного на гидросреду с сопротивлением, и генератора.…”