Study On Periodic Flow Of Discontinuous Dynamical System Based On Collision And Friction

The impact and friction are universal in mechanical engineering, and are the com-mon and important contacts between two or more parts. Modeling of impact and friction in practical problems and research on their dynamical behaviors can help understand their mechanism, and further provide information for controlling or using them. There are many results on impact and friction, but these results mainly regard that impact or friction occur in the static domains and static boundaries, or use the theory on the continuous dynamical systems to study them. Thus the discussions on some realistic problems, for example the connection problems in mechanical engineering, are not ade-quate. Because of the clearance and relative motions between the moving parts, there are numerous impact or friction occurring in the connection problems. The motions of parts are strong nonlinear and discontinuous due to impact and friction, so their models are discontinuous dynamical systems. Recently a new theory, which is used to investi-gate the dynamical behaviors of discontinuous dynamical systems, is initially formed. It thinks that the domains or boundaries in which impact or friction onset are varying with time, which makes the dynamical behaviors of discontinuous dynamical systems more clearer. In this paper, using the new theory-the theory of the discontinuous dynamical systems, the connection models-the inclined impact pair and the horizontal impact pair with dry friction are investigated. The periodic flows of two oscillators are the foci of this paper. The main contents of this paper are following:1. The periodic motions without stick motions in inclined impact pair are investi-gated using the discrete mapping theory of the discontinuous dynamical systems. Based on the movements, two switching planes and four basic maps are defined, and the gov-erning equations of these maps are also given. And five moving modes of periodic flows with non-stick motions are determined by basic maps, where three periodic flows-the period-1 motion with the alternative two impacts on the two walls, the period-1 motion with only one impact on the lower wall, and the period-k motion with k impacts on the lower wall after double periodic bifurcations-are investigated. Using the governing equations of maps the parameter conditions of these periodic flows are obtained, and analytical predictions and numerical simulations on their occurring conditions are given. The results, that the alternatively symmetrical period-1 motion under N cycles of the base does not exist, are obtained and analytically proved. This result is more general than one in [78] because the previous result is obtained as N=1 and not proved. And it is the essential difference on the dynamical behaviors between the inclined impact pair and the horizontal impact pair. At last by Jacobian matrix and their eigenvalues of the maps the theoretical analysis on stability and bifurcations of periodic motions can be done. The numerical simulations can verify reasonability of the theoretical results.2. The general periodic flows in inclined impact pair are studied using the flows switchability theory and the mapping dynamics of the discontinuous dynamical systems. According to the motions the phase space is divided into several subdomains and their discontinuous boundaries. G functions on each boundaries are defined, and using them the sufficient and necessary conditions for the occurring or vanishing of the stick motions and the grazing flows are obtained and analytically proved. The numerical simulations are given to test the theoretical results. From the ranges of phase angle in which the stick motions onset it can be seen that the probabilities of two stick motions are different, but corresponding stick motions in horizontal impact pair have identical probabilities. It reveals another essential difference on the dynamical behaviors between the inclined impact pair and the horizontal impact pair. Further the basic maps with or without stick motions on discontinuous boundaries are defined, so the general mapping structures of periodic flows are obtained by basic maps. And then theoretical analysis for stability and bifurcations of periodic flows are conducted using Jacobian matrix and their eigenvalues of the mapping structures.3. The periodic flows in horizontal impact pair with dry friction are studied via the theory on the discontinuous dynamical systems. According to motion character of this pair, the phase space is divided into several subdomains and their boundaries, where the boundaries can fall into two categories:velocity boundaries and displacement boundaries due to their different properties. The continuous dynamical systems are determined in each subdomain, and they have different characters in two adjacent subdomains. Thus this pair is modeled as the discontinuous dynamical systems. Using the flows switchability theory of the discontinuous dynamical systems, the flows switching on the boundaries of two adjacent subdomains are studied, so the dynamical behaviors of this oscillator are analytically predicted. The main results are as following:the sufficient and necessary conditions for the occurring or vanishing of two stick motions and the existing of the grazing flows on velocity boundaries are obtained, and the preliminary results of the grazing flows on displacement boundaries are also obtained. From the theoretical analysis and numerical simulations, it can be seen that the friction has strong influence on the horizontal impact pair:in their second stage, the second stick motions on the displacement boundaries coincide with the first stick motions on the velocity boundaries, so two stick motions have identical vanishing conditions; whether the grazing flows on displacement boundaries occur depends on whether the conditions of passable flows on the velocity boundaries are satisfied, so the grazing flows on displacement boundaries might not exist, which are essentially different from corresponding ones of the horizontal impact pair without friction. Further the general mapping structures of periodic flows with or without stick motions are obtained by the mapping dynamics of the discontinuous dynamical systems, and theoretical analysis for stability and bifurcations of periodic flows are conducted.