双凹摩擦摆支座自治系统及振动特性分析

Autonomous system of double concave friction pendulum and its vibration analysis

根据拉格朗日方程导出双凹摩擦摆支座的非线性自治振动微分方程;通过MATLAB对双凹摩擦摆支座的非线性自治振动微分方程进行数值求解,并绘制相平面图;利用非线性理论,对相平面图的拓扑结构做定性分析,判定运动类型、稳定性、奇点的位置、类型等,将双凹摩擦摆支座复杂的动力学行为直观地采用几何方法描述出来,刻划出双凹摩擦摆支座二阶自治系统的振动特性,并得出该系统停滞区、残余位移与系统参数、初始条件的关系式.研究结果表明双凹摩擦摆支座的相轨迹是朝原点趋近的螺线,系统的运动为衰减振动,运动类型属于全局渐进稳定,原点即为奇点,且属于稳定的焦点.

The nonlinearautonomous vibration differential equations of autonomous system of double concave friction pendulum was derived according to Lagrange equations,and the numerical solution of the equations were given by using of MATLAB software.The phase plane portrait was plotted.The topological structure of phase plane portrait was analyzed qualitatively by using of the theory of nonlinear vibrations,the movement type,stability,singularity position and type were judged.The complex dynamic behavior of the double concave friction pendulum were described intuitively by using of geometric method,and the vibration performance of the systems were depicted.The relationship of the stagnant area,residual displacement,system parameters and initial conditions were obtained.The results show that the phase path of double concave friction pendulum are the spiral with approaching the origin.The movement of the system as the damped vibration,and the movement type is the global asymptotic stability,the origin is the singular point and it belongs to a stable focus.