旋转内接微梁的动力学建模及稳定性分析

Dynamic modelling and stability analysis of a rotating internal micro beam

在微尺度领域,材料力学性能存在尺度效应,使微梁动力学性态较传统宏观尺度柔性梁的动力学性态呈现明显不同.对固结于旋转刚环上内接微梁刚柔耦合动力学特性进行了研究,在精确描述微梁非线性变基础上,利用偶应力理论和Hamilton变分原理,在计入微梁由于横向变而引起的轴向变二耦合量条件下,推导出一次近似耦合模.首先忽略微梁纵向变影响,给出一次近似简化模,引入无量纲变量,对简化模做无量纲化处理,分析在非惯性系下内接微梁动力学响应,并与外接微梁进行比较;其次研究尺度效应对内接微梁动力特性影响.研究发现,与外接微梁只存在动力刚化效应不同,内接微梁还存在着动力柔化效应;本文给出了内接微梁无条件稳定临界径长比以及有条件稳定临界转速计算方法;尺度效应使微梁振动频率增大,振幅减小,提高了内接微梁失稳临界转速;随着模态断数增加,内接微梁失稳临界转速减小且有收敛值.

The dynamic properties of the micro beam are obviously different from those of the traditional macro beam due to the size effect of material on a micro scale. The rigid-flexible coupling dynamic properties analysis of an internal micro beam attached to a rotating hub is studied in this paper. Based on the accurate description of non-linear deformation of the flexible beam, the first-order approximation coupling model are derived from couple stress theory and Hamilton theory, taking the second-order coupling quantity of axial displacement caused by transverse displacement of the beam into account. The simplified first-order approximation coupling model which neglects the effect of axial deformation of a beam is presented. The simplified model is transformed into dimensionless form in which dimensionless parameters are identified. Firstly, the dynamic response of an internal micro beam is compared with that of an external micro beam, which are both in non-inertia system. Then, the influence of size effect on the dynamic properties of an internal micro beam is studied. Generally, it is pointed out that an internal micro beam has dynamic softening phenomena, which is different from the dynamic stiffening phenomena of an external micro beam. The critical ratio of the internal radius to the length of the beam for unconditional stability and the critical angular speed of conditional stability of an internal micro beam are derived. The size effect results in the descending of vibration amplitude and the increasing of vibration frequency and critical angular speed of conditional stability of the micro beam. As the number of modes increases, the critical rotating speed of an internal cantilever beam decreases, and it has a convergent value.