加热弹性梁在热过屈曲构形附近的自由振动

FREE VIBRATION OF HEATED ELASTIC BEAMS IN THEVICINITY OF THE RMAL POST-BUCKLING CONFIGURATION

基于轴向可伸长梁的几何非线性理论建立了弹性直梁在热过屈曲静态大变形附近自由振动的几何非线性模型。在小振幅振动假设下,简化得到热过屈曲梁线性振动的控制方程。采用打靶法分别获得了两端不可移简支(pinned-pinned)和两端固定(fixed-fixed)梁的前四阶固有频率与升温之间的特征关系曲线。数值结果表明,梁在未屈曲时,各阶频率都随升温而单调下降。在过屈曲后,两端不可移简支梁的前两阶频率随升温单调上升,三、四阶频率随升温而单调下降。但是,两端固定梁在过屈曲后的各阶频率都随升温而单调增加。因此,可以通过温度调控来实现对结构固有频率的调整。

Based on the geometrically nonlinear theory for axially extensible beams, formulations of free vibrations in the vicinity of thermal post-buckling were derived. Then, by assuming that the vibration amplitude is small, linear version of the vibration problem was deduced. By using shooting method and in conjunction with numerical continuation, characteristic curves of the lower non-dimensional frequencies versus the temperature rise parameter for both pinned-pinned and fixed-fixed beams under uniformly heating were illustrated. The numerical results show that all the eigen frequencies of unbuckled beams decrease monotonously with the increment of the temperature rise. However, when the beam is in post-buckled state, apart from that the third-and the fourth-order frequencies of the pinned-pinned beam decrease all other frequencies increase along with the increment of temperature rise. In addition, it is found that all the curves are continuous but not smooth at the value of the critical temperature. This is because the beam goes into its secondary equilibrium path or post-buckling state at the critical temperature point. It also illustrates the features of bifurcation point at the frequency-inplane load curves. The results show that one can control the frequencies of constrained beams by adjusting the heating temperature.