力学模型跷跷板的教学研究

Research on Model of Teeterboard in Mechanics Teaching

跷跷板可以视为静力学中的杠杆模型,也可视为定轴转动的一维刚体模型。在刚体动力学研究的基础上,将力学模型跷跷板视为两端自由的均匀对称两跨Euler梁,计算了这类Euler梁的频率方程和位移函数,其频率方程和铰支-自由梁的频率方程完全相同,但是由于梁的跨数的增加,位移函数自变量的区间有所变化。位移函数是关于梁的中点反对称的,且在相差一个常数因子的条件下,位移函数是唯一的。

The teeterboard can be regarded as either a lever model in statics or a one-dimensional rigid body model rotating on a fixed axis.Based on the study of rigid body dynamics,in this paper,the mechanical model of teeterboard is considered as a homogeneous symmetrical two-span Euler beam with two free ends.The frequency equation and displacement function of this Euler beam are calculated.The frequency equation is identical to a pinned-free beam.However,the interval of the independent variables of displacement function changes with the increase of the span of the beam.The displacement function is anti-symmetric about the midpoint of the beam,and is unique under the condition of a constant factor difference.