复弯矩表示的非圆截面杆平衡的Schrdinger方程及其半解析解

Schrdinger Equation of Equilibrium for a Thin Elastic Rod with Non-Circular Cross Section in Terms of Complex Bending Moment and Its Half Analytical Solution

弹性细杆的平衡和稳定性问题的研究在工程和分子生物学中有重要的应用背景。利用文中提出的复柔度概念,建立了用复弯矩表示的非圆截面杆平衡的Schrdinger方程。借助复曲率概念,导出以杆的曲率、挠率和截面相对Frenet坐标系的扭角为未知变量的2阶常微分方程,此方程与传统使用的Kirchhoff方程等价。文献中仅适用于圆截面杆平衡问题的Schrdinger方程为本文导出方程的特例。对于准对称截面杆,用小参数法分别建立了零次和一次近似方程,其中零次近似方程存在解析解。对于截面的主轴坐标轴与中心线的Frenet坐标轴重合的无扭转杆特殊情形,Schrdinger方程转化为Duffing方程,应用数值方法作出了Duffing杆变形后的三维几何图形。

The study on equilibrium and stability of a thin elastic rod has an important background in engineering and molecular biology. The equilibrium equations of a rod with non circular cross section in the form of Schrdinger equation was established using the complex flexinility and expressed in terms of complex bending moment of the cross section. The Schrdinger equation with curvature, twisting and torsion angle of the cross section as unknown variables can be used instead of the traditional Kirchhoff equation. The published Schrdinger equation suitable only for the rod with circular cross section is a special case of the derived equation. For the case of a quasi symmetrical cross section, zero and one order approximation for the equation were derived by means of the method of small parameter. The zero order approximation is of the same form as in circular cross section and has analytical solution. In a special case of a twistless rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centerline, the Schrdinger equation is transformed to the Duffing equation. Three dimensional geometric shapes of the elastic rod after deformation were given on the basis of the numerical computation.