BIOGEOGRAPHY AND ORIGIN OF LILIVM LONGIFLORUM AND L. FORMOSANUM (LILIACEAE) ENDEMIC TO THE RYUKYU ARCHIPELAGO AND TAIWAN AS DETERMINED BY ALLOZYME DIVERSITY

In this paper, a dynamic model is proposed for analysis of nonlinear vibrations of rolling mills with fixed and time-varying clearances. Self-excited vibrations of the system that is basically involved with piece-wise nonlinearity and discontinuities are investigated via asymplotic method. It is shown by numerical results obtained tor the nonlinear system with a time-varying clearance that different forms of nonlinear vibrations appear lo be periodic, quasi-periodic and chaotic. Influence of the system parameters on the nonlinear vibration hehaviors is examined by applying the Poincare sections, the bifurcation are observed in nonlinear motions of the rolling phenomena are observed in nonlinear motions of the rolling mill mechanism and are of significant importance for design of this type of mechanical systems.