Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.

基于有限元和集中质量法的带阻尼环滚动轴承支撑的柔性转子动力学分析

将有限元与质量集中法相结合,建立了具有橡胶阻尼环滚动轴承支承的柔性转子的动力学模型。在该模型中,转子通过Timoshenko梁单元离散建模,支座和轴承外圈采用质量集中法建模,同时考虑了转子重力、不平衡力和非线性轴承力的影响。动力学方程采用精确积分和Runge-Kutta混合数值算法求解。为了验证建模方法的正确性,在转子-轴承试验台上进行了理论和实验分析,发现理论与实验研究的误差率小于10%;此外,还分析了橡胶阻尼环对转子-轴承耦合系统动力学特性的影响,所得结论与实际情况相吻合。在此基础上,分别研究了转速和橡胶阻尼环刚度对耦合系统的分岔和混沌行为的影响。结果表明,随着转速的增加,系统进入混沌状态的路径有激变、拟周期和间歇性分岔。然而,随着橡胶阻尼环刚度的变化,发现了激变、拟周期分岔和Hopf分岔到混沌的途径。此外,轴承间隙对转子系统的动态特性也有影响,过大的轴承间隙会使系统的周期运动变为混沌;橡胶阻尼环的刚度对系统的运动有很大的影响。