带有两个物体的Burridge-Knopoff模型的stick-slip运动

Stick-slip motion of two-block Burridge-Knopoff model

Burridge-Knopoff模型是研究地震机理的一种摩擦动力系统模型。考察了具有两个自由度的Burridge-Knopoff模型的摩擦激振问题。系统带有两个摩擦接触面,摩擦采用带有负斜率的速度依赖型摩擦曲线。因为摩擦引入的非光滑性,给系统的直接求解带来了很大的困难;又因为具有两个摩擦接触面,运动情况会更加复杂:物体一的2种形式stick或slip,物体二具有同样的2种形式,从而可组合成4种运动形式的交替运动。首先找到各个运动形式的显式解,再根据各运动形式的跃迁点即可得系统总体的运动。在小的驱动速度下把速度依赖型的摩擦对速度取平均值,近似为动摩擦系数;把位移和速度表达为时间的显式函数。再按位移和速度处于不同运动形式进行迭代求解待定系数。最后由确定的待定系数判断系统运动的性质(周期运动或是混沌运动)。

Burridge-Knopoff model is a friction dynamics model which is used to study the mechanism of earthquake. This paper investigates a two-block Burridge-Knopoff model that undergoes self-sustained oscillations induced by dry friction. There are two frictional contacts in the system, and the friction-velocity curve is chosen to have minus slope. Because of the non-smooth characteristics of friction, it＇s very difficult to solve it directly; and furthermore there are two friction contacts, and block one can have stick or slip motion, so can block two. There are four kinds of motion to alternately appear. This paper is intended to find the explicit solution of each kind of motion, and obtain the system motion according to the transition points between each motion. First the driving velocity is supposed to be small, and then the friction forces can be averaged by speed, which are constants. Then the displacements and velocities can be expressed as explicit functions of time, and the undetermined coefficients can be solved by iterating displacements and velocities according to their different kinds of motion. At the end, the characteristics of motion, i.e. whether it is periodic or chaotic can be determined by those coefficients.