曲梁几何方程推导

GEOMETRIC EQUATION DERIVATION OF CURVED BEAM

曲梁几何方程的精确描述是曲梁静力分析和动力分析时建立微分方程的关键。现有研究中关于曲梁几何方程的论述与推导,存在着坐标系、内力矢量和位移矢量正负规定不明确或混乱的现象。该文基于“剪心”与“形心”重合假定,对曲梁的几何方程进行了严格的推导和阐述:首先明确规定了坐标系、内力矢量以及位移矢量的正方向,而后分别推导了曲梁微段的面内变形和面外变形,其中,面内变形包括轴向拉压应变和径向弯曲,面外变形则包括竖向弯曲和扭转。同时验证了所得曲梁几何方程的正确性,为曲梁运动力学微分方程的建立和求解奠定了基础。研究结果可为相关理论研究及工程应用提供有益参考。

The key to deduce differential equations is accurate geometric equations in the static and dynamic analyses of a curved beam.In the current study,there are still many problems to be solved,such as confused coordinate system and ambiguous positive-negative regularity of internal forces and deformation.Assume that the shear centre overlaps the centroid of the cross section,the strict deductions of the geometric equations are obtained.The coordinate system and positive-negative regularity of internal forces and deformation are firstly defined,then the in-plane and out-plane deformations of a curved beam are deduced.The in-plane deformation includes axial strain and radial bending,while the out-plane deformation includes vertical bending and torsion.At the same time,the correctness of the geometric equations of a curved beam is verified,and the correctness of the geometric equations of a curved beam is verified,which lays a foundation for the establishment and solution of curved beam's dynamics differential equations.The results could provide helpful suggestions for the relevant theoretical research and engineering application.