外激励作用下弹性支撑浅拱的非线性动力学分析

NONLINEAR DYNAMICAL BEHAVIOURS OF ELASTICALLY SUPPORTED SHALLOW ARCH UNDER EXTERNAL EXCITATIONS

研究了外激励下两端采用转动弹簧约束的铰支浅拱在发生1∶1内共振时的非线性动力学行为.通过引入基本假定和无量纲化变量得到浅拱的动力学控制方程,将阻尼项、外荷载项和非线性项去掉后,所得线性方程及对应边界条件即可确定考虑转动弹簧影响的频率和模态,发现转动约束取不同刚度值时系统存在模态交叉与模态转向两种内共振形式.对动力方程进行Galerkin全离散,并采用多尺度法对内共振进行了摄动分析,得到了极坐标和直角坐标两种形式的平均方程,其中平均方程系数与转动弹簧刚度一一对应.最低两阶模态之间1∶1内共振的数值研究结果表明:外激励能激发内共振模态的非线性相互作用,参数处于某一范围时系统存在周期解、准周期解和混沌解窗口,且通过(逆)倍周期分岔方式进入混沌.

The nonlinear resonance dynamics of the hinged shallow arches,both ends of which are con- strained by torsional springs, under external excitation are investigated. The dimensionless dynamic equa- tions are obtained by employing the basic assumptions of shallow arch. By removing the damper, external load and non-linear terms, the obtained linear equation under given boundary conditions is solved to determine the frequencies and modes. Two internal resonance types of crossing and veering are found when the torsional constraints adopt different stiffness values. Further, the dynamic equation is solved by a full-basis Galerkin discretization,and the multiple scale method is used to study the internal resonances by perturba- tion analysis, which leads to both the polar- and Cartesian-form averaging equations whose coefficients have a one-to-one correspondence with the stiffness of torsional spring. The numerical results for 1 ： 1 internal resonance between the two lowest modes show that the nonlinear interactions can be excited by external excitation. Moreover, when the parameters are controlled in a certain range, there are periodic, quasi-period- ic and chaotic windows in the system,which can enter into chaos by the （inverse） period-doubling bifurcation.