粘弹性传动带横向非线性振动的稳定性与分岔现象

采用弹性力学法建立具有速度波动的横向非线性积分-偏微分控制方程,并对方程进行一阶Galerkin离散.首次理论性导出由平均速度和速度波动幅值共同决定的系统稳定区和超临界区的边界条件;然后,数值模拟分析粘弹性传动带运动系统的分岔现象和混沌运动.最后,利用分岔图和映射图重点分析平均速度、带速波动幅值对系统动力学的影响.结果表明:系统存在单周期、二倍周期、四倍周期和混沌运动,随着参数的增大,系统由单周期变为倍周期运动,最后进入混沌运动状态.通过数值模拟与理论公式计算出的分岔值进行对比,表明二者几乎一致,证明划分稳定性条件的正确性.

Stability and Bifurcation of Transverse Nonlinear Vibration in Viscoelastic Belt

The nonlinear integral-partial differential governing equations with speed fluctuation were built by elasticity. The 1^st order Galerkin was used to discredited the governing equations, the boundary conditions of the stability and super-critical regions determined by the average speed and speed fluctuation amplitude were firstly derived theoretically. Numerical simulations were conducted to analyze the phenomena of the bifurcation and chaos of viscoelastic moving belt, bifurcation diagrams and Poincare maps were used to analyze the impact of average speed and the amplitude of speed fluctuation on the dynamical system. It is indicated that the periodic, two-periodic,four-periodic and chaotic motions occur in the system; with increases of parameters, the system motion transforms from the single period into period-doubling, finally to chaos. The bifurcation value of both theoretical analysis and numerical simulation are almost the same, which validates the partition of stability condition.