Robust stabilization and anti-resonance in parametric circulatory systems

在这个项目中，我们打算通过循环系统中的参数振动来研究自激振动的稳定性。多纳尔创造了参数反共振一词，用于通过参数激发来稳定自激振动，尽管第一个例子也是由Tondl在更早的时候发现的。Dohnal只考虑了刚度矩阵中由于负阻尼力和同相参数激励而产生的自激系统。然而，在许多自激系统中，自激是由循环项引起的，而不是“负阻尼项”。多纳尔还明确排除了内部共振和接近共振。然而，这可能是非常重要的，因为双特征频率在对称系统中很常见，而近内共振对应于不完全平衡的转子，或其他对称扰动，这在工程系统中非常常见。最后，机械工程系统中部件间存在滑动接触的非同步参数激励会导致非常复杂的动力学行为(例如，Cesari方程中的“完全不稳定”)，这在循环系统中还没有得到充分的研究。多纳尔假设参数共振和反共振的原因在于振动模式之间的能量传递，但从未深入探讨过。这项研究应该会更好地理解自激系统中迄今尚未完全理解的现象(例如，为什么刹车只在低速时发出尖叫声？)并且对工程中的许多系统以及可能在物理学中也是重要的(例如，见粒子陷阱，沃尔夫冈·保罗因此而在1989年获得诺贝尔奖)。在此背景下，还将研究非线性参数激励系统，包括不同的分叉和极限环。

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.