具有分数导数本构关系的粘弹性浅拱的非线性动力学行为

摘要

利用分数导数本构模型描述材料的粘弹性特性,建立了粘弹性浅拱在横向荷载作用下的动力学方程.利用Galerkin截断法并结合边界条件分别得到了一阶和二阶Galerkin系统的控制微分方程.通过数值计算,分析了简谐激励下一阶Galerkin系统的非线动力学行为.研究表明:随着外激励幅值的变化,粘弹性浅拱系统可以通过倍周期分岔或阵发性两条路径进入混沌;固定外激励幅值、频率以及阻尼系数等状态参数,不同初始条件下,系统可以出现多周期解共存、周期解与混沌解共存的现象.%The dynamics equation of a viscoelastic shallow arch under lateral loads, in which fractional derivative model is introduced to simulate the material characteristics, is established. The simplified differential equations of the first and second order Galerkin systems are developed combining Galerkin method with boundary conditions. Nonlinear dynamic behaviors of the first order Galerkin system under harmonic loads are discussed by numerical calculations. The results show that the system may lead to chaotic motion via period—doubling bifurcations or intermittent routes) In addition, the coexistences of multiple periodic solutions and the coexistences of periodic solutions with chaos solutions are found in this system when the excitation amplitude and frequency, damping and other state parameters are fixed but the initial conditions are changed.