Studies On Multi Degree-of-Freedom Nonlinear System

Nonlinear science is one of the hot subjects which are studied by domesticand foreign experts and academicians. The piecewise-nonlinear self-exciteddynamic systems induced by dry friction are consisted in engineering fields, suchas the chatter of a cutting machine tool, the continuous creeping of a heavy-dutymachine tool, the "serpentuate" of train, the "shimmy" of vehicle, et. There aremany difficulties and complexities in analyzing this kind of system because of themulti-degree-of-freedom and large numbers of coupling factors. Thereby theinvestigations of this kind of system are mainly based on the numerical analysisand experiment research, furthermore, the analytical solution is only rested on thestudy of single-degree-of-freedom system. Therefore, in order to analyze thedynamical characteristics of this kind of multi-degree-of-freedom systemcompletely and effectually, it is essential to try to provide an effective method. Thevibration and water droplets spurting phenomenon of the dragon washbasin is atypical example of the shell-liquid coupled vibration induced by dry friction,therefore, the dragon washbasin is taken as the model in order to calculate andinvestigate the nonlinear self-excited vibration of piecewise-smooth withmulti-degree-of-freedom. The main work of this paper includes the followings:1,A set of approximate equations are discovered based on the characteristicsof the dragon washbasin's mathematic model equations. The numericalcalculations are done to the original equations and approximate ones, the resultsare compared, and the conclusion that the original equations can be replaced bythe approximate ones is obtained. Accordingly, the CPU space and calculate-timeare saved greatly.2,The stability of this multi-degree-of-freedom system's equilibrium pointsis analyzed by using the approximate equations. The stability region of thesystem's equilibrium points is computed by means of Hurwitz criterion, whichlays the foundation for the reasonable selection of parameters in analyticalcalculation.3,The method to confirm the subsection integral limits is put forward andsolved when the influences of the dragon washbasin's second modal and hands'vibration on the system are considered. The approximate analytical solutions onboth non-resonance and inner-resonance conditions are derived by means of KBMethod, the results of this paper agree well with which of thesingle-degree-of-freedom system and abundant nonlinear dynamicalcharacteristics of the dragon washbasin self-excited vibration system are obtained.Therefore, it provides an effective method to the analytical calculation of this kindof problems.4,The analytical solutions are analyzed by means of singularity theory, thetransition sets and many bifurcation models are obtained, which provide furtherexplanations to the dragon washbasin's dynamical characteristics.5,At the end of this paper, the main investigation results are summarized.