变速粘弹性传送带非线性动力稳定性与分岔

Non-linear Dynamic Stability of Viscoelastic Transmission Belt

基于粘弹性材料本构方程及带运动方程建立变速粘弹性传送带非线性动力学模型;利用多重尺度理论讨论变速粘弹性传送带在不同情况下的非线性动力稳定性。研究表明:当传送带速度的波动频率远离0或系统固有频率2倍时,系统的振幅有界、相位为时间尺度的对数函数,并且当带的稳态速度接近系统临界速度时,非线性对系统的频率影响强烈;当传送带速度波动频率接近0时,系统振幅有界;当传送带速度波动频率接近系统固有频率2倍时,系统出现失稳与分岔现象。

In this paper, the non-linear dynamic stability of viscoelastic transmission belt with time-dependent velocities is investigated. Based on the constitutive description of Kelvin viscoelastic material and the morion equation of the axially moving belt, the nonlinear dynamic model that dominates the transverse vibration of the viscoelastic transmission belt is established.Then the non-linear dynamic stability is studied by using multiple scale method. It is found that： 1） for fluctuation frequency away from zero or two times the natural frequency, the amplitudes of vibration are bounded and the phases are the logarithm functions in time. The non-linear effects become important for velocities close to critical velocity. 2） for fluctuation frequencies close to zero, the amplitudes of vibration are bounded. 3） for fluctuation frequencies close to two times the natural frequency, two non-trivial solutions bifurcate from the trivial steady-state solution, There are two supercritical pitchfork bifurcation points. Passing through the first bifurcation point, the trivial solution looses stability and a non-trivial solution is obtained. If frequency is increased and pass through the second bifurcation point, the trivial solution is again stale and non-trivial solutions disappear.