IMPACT OSCILLATORS: FUNDAMENTALS AND APPLICATIONS

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.

[1]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos with Student Solutions Manual , 2016 .

[2]  Helmut Podhaisky,et al.  Bifurcations and synchronization of singularly perturbed oscillators: an application case study , 2012 .

[3]  Wieslaw Marszalek,et al.  Mixed numerical and analytical analysis of nonlinear circuits with nonsmooth inputs: A hyperbolic algebra approach , 2014, 2014 IEEE 57th International Midwest Symposium on Circuits and Systems (MWSCAS).

[4]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[5]  Z. Trzaska.  Efficient Approach to Determine the Steady-State Energy in Networks with Periodic Discontinuous Excitations , 2007 .

[6]  Benoit B. Mandelbrot,et al.  Fractals and Chaos , 2004 .

[7]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[8]  Jan Awrejcewicz,et al.  Changes in the gait characteristics caused by external load, ground slope and velocity variation , 2011 .

[9]  D. Lathrop.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 2015 .

[10]  Z. Trzaska.  Dynamical processes in sequential-bipolar pulse sources supplying nonlinear loads , 2014 .

[11]  J. Awrejcewicz,et al.  On continuous approximation of discontinuous systems , 2005 .

[12]  Z. Trzaska.  Nonsmooth analysis of the pulse pressured infusion fluid flow , 2014 .

[13]  Jan Awrejcewicz,et al.  Asymptotic Approaches to Strongly Non-linear Dynamical Systems , 2003 .

[14]  Nicholas J. Higham,et al.  Matlab guide, Second Edition , 2005 .

[15]  David J. Wagg,et al.  Periodic sticking motion in a two-degree-of-freedom impact oscillator , 2005 .

[16]  P. Olejnik.  Metody numeryczne rozwiązywania, analizy i kontroli nieciągłych układów dynamicznych , 2013 .