Self-excited vibrations in time-variant systems

在机械动力学中，时变的，特别是周期的机械系统是很常见的。线性时间周期微分方程理论是由弗洛凯在大约一百年前提出的。在应用力学中，参数激励振动是用Floquet理论研究的，它与机械系统中运动方程的特殊结构一起导致特殊现象（如组合共振）。在过去，对特别保守和稳定的线性系统进行了详细的研究，这些系统可能由于额外的参数激励（参数共振）而变得不稳定。然而，在机械动力学中，通常这些效应只与极弱阻尼的系统有关，因此在现实中很少观察到。它们也往往只发生在参数激励的很窄的频率范围内。最近，时间周期机械系统在自激振动中也变得很重要。在许多工程系统中，自激以循环项（坐标比例力中的偏对称矩阵）的形式出现在运动方程中。在许多情况下，参数激励的频率远低于自激振动的频率，因此通常意义上的参数共振不起作用。即便如此，周期系数可能对稳定性至关重要。这种自激振动的一个例子是断裂尖叫。在数值分析中忽略周期系数通常会导致对结构变得不稳定的敏感性的高估，尽管在某些情况下它也可能被低估。在计划的项目中，将研究循环系统线性化运动方程的小周期扰动的影响。将研究非线性系统的亚临界和超临界Hopf分岔以及不同平稳解的吸引域。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，目的是允许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.