Self-excited vibrations in time-variant systems

时变的，特别是周期性的机械系统是常见的机器动力学。线性时间周期微分方程的理论大约在一百年前由Floquet发展起来。在应用力学中，Floquet理论研究参数激励振动，它与机械系统中运动方程的特殊结构一起导致特殊现象（例如组合共振）。在过去，特别是保守和稳定的线性系统进行了详细的研究，这可能会变得不稳定，由于额外的参数激励（参数共振）。然而，在机器动力学中，这些效应通常只与阻尼极弱的系统相关，因此在现实中很少观察到。它们也倾向于仅在参数激励的非常窄的频率范围内发生。最近，时间周期性机械系统在自激振动的背景下也变得重要。在许多工程系统中，自激以循环项的形式出现在运动方程中（坐标比例力中的斜对称矩阵）。在这些情况中的许多情况下，参数激励的频率比自激振动的频率低得多，使得通常意义上的参数共振不起作用。即便如此，周期系数对于稳定性可能是至关重要的。这种类型的自激振动的一个例子是破碎尖叫声。在数值分析中忽略周期系数通常会导致高估结构变得不稳定的敏感性，尽管在某些情况下它也可能被低估。在计划的项目中，将研究循环系统运动的线性化方程的小周期扰动的影响。亚临界和超临界Hopf分岔以及不同的固定解决方案的吸引域将被检查的非线性系统。对于大型周期性系统（数千或数十万自由度），计划开发在FEM环境中处理问题的方法，目的是在这种环境中进行有效的稳定性分析。

Time-variant and in particular periodic mechanical systems are common in Machine Dynamics. The theory of linear time periodic differential equations was developed about a hundred years ago by Floquet. In applied mechanics, parametrically excited vibrations are studied with Floquet theory, which together with the particular structure of the equations of motion in mechanical systems leads to special phenomena (e.g. combination resonances). In the past, particularly conservative and stable linear systems were studied at length, which may become unstable due to additional parametric excitation (parametric resonance). In Machine Dynamics, however, usually these effects are only relevant for systems which are extremely weakly damped, and they therefore are only rarely observed in reality. They also tend to occur only in very narrow frequency ranges of the parametric excitation. More recently, time periodic mechanical systems have also become important in the context of self-excited vibrations. In a number of engineering systems, self-excitation appears in the equations of motion in the form of circulatory terms (skew symmetric matrices in the coordinate proportional forces). In many of these cases, the frequency of the parametrical excitation is much lower than that of the self-excited vibrations, so that parametric resonances in the usual sense do not play a role. Even so, the periodic coefficients may be crucial for stability. An example of this type of self-excited vibrations is break squeal. Ignoring the periodic coefficients in the numerical analysis usually leads to an overestimation of the susceptibility of a structure to become unstable, although in some cases it may also be underestimated. In the planned project, the influence of small periodic perturbations of the linearized equations of motion of circulatory systems will be studied. Subcritical and supercritical Hopf bifurcations as well as the domains of attraction of the different stationary solutions will be examined for nonlinear systems. For large periodic systems (many thousand or many hundred thousand degrees of freedom) it is planned to develop methods for dealing with the problem in a FEM environment, with the aim to allow an efficient stability analysis in this environment.