The Hopf-Hopf Bifurcation Of A Two-Degree-Of-Freedom Self-Excited System With Dry Friction

The vibration induced by the dry friction universally exists in engineering. The self-excited vibration caused by the dry frictions has significant impacts to the mechanical structures. In order to reveal these phenomena, many models have been established. In most of these models, there were just few of qualitative analysis or some calculations, but systematic investigation on the dry friction models exists scantly in literature. Whereas the vibration phenomena caused by the dry frictions is abundant and multiform, it is necessary to do thorough studies. In addition, the dry friction vibration is classified as a sort of non-smooth dynamics systems, so the research on this aspect reflects the people's attention and widespread interesting in the non-smooth dynamics systems.In the theory of normal form, the ordinary differential equations are simplified through the method of coordinate transformation. In general, the normal form equations derived are simple, and they can keep the important qualities information of original equation moreover. Because the computation of normal form which is canonical and uniform, is convenient for computer calculation, the normal form theory has been made a great progress and widely applied in physic with the development of electronic technology, and has became an elementary tool for the dynamic research of ordinary differential equations.In this paper, the Hopf-Hopf bifurcations of a two-degreed-freedom self-excited system with dry friction were studied, and the three order truncated normal form in nonresonance was obtained using the PB normal form theory. First, considering the extensive use in practice, the dry friction function was chosen as a discontinuous vector field slip-friction function. Because of it's discontinuity, the friction function can't be transformed to the normal form equation as usual; hence, the series expanding method was proposed so that the discontinuous friction function is transformed to continuous function near the balance point. In this way, the normal form of the two-degreed-freedom self-excited system in nonresonance case is educed, and the explicit expression of coefficient of the system's three order truncated normal form is obtained. Based on the explicit expression of coefficient, the stability of the invariant circle or torus bifurcation can be judged through theory analysis and logical deduction without any numerical calculation, also the complexity of the phase diagram analysis is reduced. From the system's three order truncated normal form analysis, it was shown that, the systems with small damp and small critical velocity only possess of instability Hopf invariant circle and instability Hopf-Hopf torus, and the 1:1 strong resonant case is impossible to appear. Finally, by virtue of normal form equations, the Hopf bifurcation and Hopf-Hopf torus bifurcation phenomena were analyzed in detail, the stability region and corresponding parameter range of each bifurcation are calculated, the coefficient and related bifurcation regions of the system's three order truncated normal form are obtained. The results reveal the complicate dynamic actions of the dry friction system. The method of series expanding and PB normal form theory, which may be generalized, can be applied to analysis the Hopf and Hopf-Hopf bifurcation of the system in all cases. The numerical simulations verify the correctness of all theory results of this paper.