Robust stabilization and anti-resonance in parametric circulatory systems

在这个项目中，我们打算通过参数振动来研究循环系统中自激振动的稳定化。Dohnal创造了参数反共振这一术语，通过参数激励来稳定自激振动，尽管第一个例子也由Tondl早些时候发现。Dohnal只考虑由于负阻尼和刚度矩阵中的同相参数激励而具有自激的系统。然而，在许多自激系统的情况下，自激是由于循环项，而不是“负阻尼”。Dohnal还明确地排除了内部共振和近共振。然而，这些可能是相当重要的，因为双本征频率在对称系统中很常见，而近内部共振对应于不完全平衡的转子，或其他对称扰动，这在工程系统中很常见。最后，非同步参数激励存在于部件之间有滑动接触的机械工程系统中，会产生非常复杂的动力学行为（例如Cesari方程中的“完全不稳定性”），这一点尚未对循环系统进行充分研究。Dohnal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。这项研究将有助于更好地理解迄今为止尚未完全理解的自激系统中的现象（例如，为什么刹车只在低速时发出尖叫声？），并可能对许多工程系统和物理学系统具有重要意义（例如，沃尔夫冈·保罗因粒子陷阱而获得1989年诺贝尔奖）。非线性参数激励系统也将在此背景下进行研究，包括不同的分岔和极限环。

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.