变速粘弹性传送带混沌运动

Chaotic Motion of Viscoelastic Transmission Belt with Time-dependent Velocity

基于Kelvin粘弹性材料本构方程及带运动方程建立了同时具有速度和横向力扰动的粘弹性传送带非线性动力学模型。利用Galerkin’s方法将系统简化为参数激励的单模态Duffering振子,并得到系统的音叉分岔点、同宿轨道;利用Melnikov函数法讨论了不同参数(如带稳态速度、扰动速度、扰动力、材料性能等)对系统混沌域的影响。结果表明:1)速度和力扰动同时存在时,混沌域位于第一象限一直线上方,且混沌域随速度扰动频率增加而变小,随力扰动频率增加而变大,当力或速度扰动频率不变时直线都过定点,与之对应不能通过改变力或速度扰动幅值改变混沌域;2)对速度低频扰动可通过增加带速度并保持较大扰动振幅避免混沌,对速度高频扰动通过减小带速度避免混沌;3)材料粘性增加混沌域变小,材料刚度增加混沌区域变增大。

The non-linear dynamic stability and chaotic motion of viscoelastic transmission belt with time-dependent velocities and subjected to a transverse distributed varying extemal excitation is investigated. Based on the constitutive description of Kelvin viscoelastic material and the motion equation of the axially moving belt, the nonlinear dynamic model that dominates the transverse vibration of the viscoelastic transmission belt is established. And then one-mode approximation of the goveming equation is obtained by the Galerkin＇s method. This approximation leads to a parametrically excited Duffing＇s oscillator which exhibits a symmetric pitchfork bifurcation as the axial velocity of the belt is varied beyond a critical value. Finally, Melnikov＇s criterion is employed to fmd out the parameter regime （such as steady velocity, fluctuation velocity, extcmal force and material property etc） in which chaos occurs. It is found that： 1） Chaos region is above a line in the first quadrant if both velocity and external fluctuation coexist. And chaos region become large with the increasing of frequency of external excitation, but chaos region become small with the increasing of frequency of velocity fluctuation. Meanwhile the line always passes a fixed point if the frequency of extemal excitation or velocity fluctuation is not change. At this time chaos can not be controlled by the adjustment of external force or speed magnitude. 2） For lower speed fluctuation frequencies, one can employ higher steady speeds and maintain the amplitude of speed fluctuations to avoid chaos, however, for higher speed fluctuation frequencies, one can only avoid chaos by decreasing the steady velocity of the belt. 3） Chaos region become small with the increasing of material viscosity, but chaos regions become large with the increasing of material stiffness.