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Impact oscillators : fundamentals and applications

Trzaska Z.

Journal of KONES

2015

Vol. 22, No. 1

307--314

The paper presents results of the analysis of nonlinear oscillators with nonsmooth elements and nonlinear systems with nonsmooth forcing components. Main attention has been focused on highlighting specific properties of nonsmooth systems compared to their smooth counterparts. Nonsmooth transformation of the time variable and the replacement of initial issues by boundary problems have been taken as the base for the analytical method. Results of numerical simulations and computing in the form of graphs of displacements and velocity waveforms and attractors are presented. To fully identify the system's behaviour and meet high performance specifications recourse to model all dynamics together with their interactions has been taken into account. Strong interactions among the parts of the system are considered and the phenomenon of the impact is exhibited. It has been found that non-smooth dynamical systems reveal significant wealth of nonlinear phenomena, including a chaotic, that are unique to this potentially important class of nonlinear systems. In non-smooth systems at small change of parameters, a sudden transition from a stable periodic oscillation to the full range of chaotic oscillations may often occur. The dynamics of nonsmooth oscillations with shock external forcing is analysed by using a relatively new mathematical tool, which appears to be hyperbolic algebra. The key idea of this tool is steeped in of non-smooth time transformations (NSTT) for strongly nonlinear, but still smooth models.

Dynamical processes in sequential-bipolar pulse sources supplying nonlinear loads

Trzaska Z.

Przegląd Elektrotechniczny

2014

R. 90, nr 3

147--152

A uniform basis for analysis of oscillations with essentially non-harmonic shapes, excited by sources of non-smooth or could be discontinuous in time signals is presented. The tool presented here employs non-smooth (impact) systems as a basis to describing not only impact, but also smooth or even linear dynamics. The focus is put on explicit links between impact dynamics and hyperbolic algebras analogously to the link between harmonic oscillations and conventional complex analyses. Illustrations and results of computer simulations are presented.

W artykule przedstawiono ujednolicone podstawy analizy drgań, głównie nieharmonicznych przebiegów, wzbudzanych źródłowymi sygnałami, niegładkimi lub nieciągłymi w czasie,. W analizie wykorzystano niegładkie (udarowe) systemy do wyznaczania nie tylko nieciągłej, lecz także ciągłej lub nawet liniowej dynamiki układu. Nacisk położony został na ustalenie powiązań między dynamiką uderzeniową i algebrą hiperboliczną, analogicznie do przypadku drgań harmonicznych i konwencjonalnych liczb zespolonych. Ilustracje i wyniki symulacji komputerowych są przedstawione.