【作者】 陈玉明；

【导师】 李群宏；

【作者基本信息】 广西大学 ， 应用数学， 2011， 硕士

【摘要】 摩擦振子由于摩擦的非光滑性而蕴含着复杂的粘滑运动,这是摩擦振子的基本运动类型.本学位论文首先针对一单自由度干摩擦动力系统,讨论了其周期解的存在性及其应满足的条件,然后针对另一耦合的两自由度干摩擦动力系统,讨论了其在粘滑分界处应满足的必要条件.论文的主要工作如下：1.针对一单自由度带干摩擦的振动系统,在摩擦类型为库仑摩擦时,讨论了其粘滑周期运动,解析地得到了粘滑周期运动的存在性条件,并在此基础之上,半解析地得到了系统单个粘滑周期运动内,粘、滑运动分界处的时间及其运动状态,最后通过数值仿真更加直观地说明了所得到的理论结果,并且通过数值仿真,得到了在分岔曲线附近,周期运动的周期T与2π的比值T/2π与参数ζ的关系.由于系统关于速度具有对称性,当v0>0时可以完全类似的讨论,并且能得到和v,o<0时类似的结果.2.讨论了一类两自由度干摩擦系统的非光滑分岔和混沌现象,通过分析得到系统轨线在粘滑分界处应满足的必要条件,以及在切换流形上可能存在滑动现象的区域.通过数值仿真发现,当参数μm,较大时,系统均发生稳定的周期行为,然而当μm。减小到一定程度时,开始出现大片的混沌现象.在混沌参数区域中,系统夹杂着周期现象,除一般的简单周期运动外,其中还存在着极为复杂的周期运动.更多还原

【Abstract】 The stick-slip behavior in friction oscillators is very complicated due to the non-smooth of the dry friction, which is the basic form of motion of dynamical systems with friction. In this thesis, the stick-slip periodic solution in a single-degree-of-freedom oscillator with dry friction is investigated in detail, and for a two-degree-of-freedom system with dry friction, the necessary condition for the stick-slip boundary is obtained. The main results of the research involve:1. In a single-degree-of-freedom dynamical system with dry friction, under the assumption of kinetic friction being the Coulomb friction, the existence condition of the stick-slip periodic solution is analytically derived which gives out a new result in a class of friction systems. Moreover, the time and states of motion on the boundary of the stick and slip motions are semi-analytically obtained in a single stick-slip period. Finally, the theoretical results are validated by numerical simulation. At the same time, the relationship between the ratio T/2πwith period T as numerator and the parameterξnearby the bifurcation curve is obtained. Since the system is symmetrical with respect to the velocity, the case for v0>0 can be discussed in the same way as that of the case v0<0 which has been investigated in this paper, and it is clear that the similar conclusions hold.2. In a two-degree-of-freedom dynamical system with dry friction, the non-smooth bifurcation and chaos phenomena are investigated. By the theoretical analysis, the necessary condition which must be satisfied for the stick-slip orbits in the boundary of stick and slip motions is obtained, and in the switching manifold, we find out the possible area where the slip motion will happen. Using the numerical simulation, it is found that the system possesses steady periodic behaviors when the parameterμm takes rather large values. However, a large number of chaos phenomena appear when the parameterμm is descended to a certain extent. In the parameter region of chaos, there are also lots of periodic phenomena which include complicated periodic motions besides the simple ones.更多还原