Self-excited vibrations in time-variant systems

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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.