Robust stabilization and anti-resonance in parametric circulatory systems

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In this project we intend to study the stabilization of self-excited vibrations via parametric vibrations in circulatory systems. Dohnal coined the term parametric anti-resonance for stabilization of selfexcited vibrations through parametric excitation, although first examples were also found earlier by Tondl. Dohnal considered only systems with self-excitation due to nega¬tive damping and in phase parametric excitation in the stiffness matrix. In many cases of selfexcited systems, the self-excitation is however due to circulatory terms, rather than ‘negative damping’. Dohnal also explicitly excluded internal resonances and near resonances. These can however be quite important, since double eigenfrequencies are common in symmetric systems, and near internal resonances correspond to imperfectly balanced rotors, or other perturbations of symmetry, which are very common in engineering systems. Finally, the nonsynchronous parametric excitation, present in mechanical engineering systems with sliding contacts between parts, gives rise to very complex dynamical behavior (e.g. ‘total instability’ in the Cesari equations) and this has not been fully studied for circulatory systems. The cause of parametric resonance and anti-resonance was postulated by Dohnal to lie in an energy transfer between vibration modes, but never explored thoroughtly. This research should lead to a better understanding of so far not completely understood phenomena in self-excited systems (e.g. Why do brakes squeal at low speed only?) and can be of importance to many systems in engineering and possibly also in physics (see e.g. the particle trap, for which Wolfgang Paul received the Nobel Prize in 1989). Nonlinear parametrically excited systems will also be studied in this context, including the different bifurcations and limit cycles.