两自由度的Burridge-Knopoff模型的一种近似分析方法
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摘要
Burridge-Knopoff模型是速度驱动的多质量的质量弹簧系统。它是用来进行地震机理研究的一种摩擦动力系统的模型。本文考查了具有两个自由度的Burridge-Knopoff模型的摩擦激振问题。摩擦的不光滑性给系统的求解带来了很大的困难。又系统带有两个摩擦接触面,运动情况更加复杂。本文通过对摩擦力取平均的方法,得到了系统在各个运动形式下的通解,采用半解析的方法对此两自由度的系统进行了研究。通过以上方法,给出了一种周期运动的具体运动形式以及位移和速度的时程关系;并且发现在不同驱动速度的情况下,随着驱动速度的增大,系统的运动形式存在简单和复杂交替出现的现象,并给出了不同系统参数下的分岔图。
Burridge-Knopoff model is a velocity driven multi-block mass spring system.It is a friction dynamical model which is used to study the mechanism of earthquake.The two block model was investigated that undergoes self-sustained oscillations induced by dry friction.The non-smooth characteristic of friction makes the solving of the system very difficult.And there are two fricional contacts in the system,so the motion of system is more complex.The average method for dynamic friction force was carried out,and the general solutions were obtained for different kinds of motion,and then a semi-analytic method was used to study the system.Through the method,a specific kind of motion and state of motion with time history were given.It is found that,the simple motion and complex motion occur alternately as the driving speed varies,and the bifurcation diagram was also given for different systematic parameters.
Burridge-Knopoff model is a velocity driven multi-block mass spring system.It is a friction dynamical model which is used to study the mechanism of earthquake.The two block model was investigated that undergoes self-sustained oscillations induced by dry friction.The non-smooth characteristic of friction makes the solving of the system very difficult.And there are two fricional contacts in the system,so the motion of system is more complex.The average method for dynamic friction force was carried out,and the general solutions were obtained for different kinds of motion,and then a semi-analytic method was used to study the system.Through the method,a specific kind of motion and state of motion with time history were given.It is found that,the simple motion and complex motion occur alternately as the driving speed varies,and the bifurcation diagram was also given for different systematic parameters.
引文
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[10]Galvanetto U.Bifurcations and chaos in a four-dimensional mechanical system with dry friction[J].Journal of Sound and Vibration,1997,204(4):690-695.
[2]冯奇,张相庭.单自由度摩擦系统离散模型[J],应用数学和力学,2001,22(8):853-861.
[3]Feng Q,A discrete model of a stochastic friction system[J].Comput Methods Appl Mech Engrg,2003,192:2339-2354.
[4]Leine R I,van Campen D H,Kraker de.Stick-slip vibrations induced by alternate friction models[J].Nonlinear Dynamics,1998,16:41-54.
[5]Awrejcewicz J,Dzyubak L,Grebogi C.Estimation of chaotic and regular(stick-slip and slip-slip)oscillations exhibited by coupled oscil-lators with dry friction[J].Nonlinear Dynamics,2005,42:383-394.
[6]van de Vrande B L,van Campen D H,de Kraker A.An approximate analysis of dry-friction-induced stick-slip vibrations by a smoothingprocedure[J].Nonlinear Dynamics.1999,19:157-169.
[7]Galvanetto U,Knudsen.Event maps in a stick-slip system[J].Nonlinear Dynamics,1997,13:99-115.
[8]Nussbaum J,Ruina A.A two degree-of-freedom earthquake model with static/dynamic friction[J].Pure and Applied Geophysics,1987,125(4):629-656.
[9]Galvanetto U.Some remarks on the two-block symmetric Burridge-Knopoff model[J].Physics Letters A.2002,293:251-259.
[10]Galvanetto U.Bifurcations and chaos in a four-dimensional mechanical system with dry friction[J].Journal of Sound and Vibration,1997,204(4):690-695.
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