考虑翘曲效应的空间曲梁耦合振动研究

The coupled vibrational behavior of curve beams with different geometry forms and material properties is analytically investigated in this project, in which theoretical analysis and numerical simulation in combination with typical specimen test will be adopted. The differential equations of motion for curve beams with noncircular cross-sections, which is a coupled system consisting of 14 second order partial differential equation with variable coefficients, are derived. In the formulations, the warping effects upon natural frequencies and vibration mode shapes are first studied in addition to considering the rotary inertia, the shear and axial deformation effects. Additionally, some results that are different from those of traditional beam theories will be found. The purpose of this subject is to reveal variation rules of warping effect with respect to parameters of curve beams, including geometric and material parameters. This project is established in improving and consummating the traditional beam theories to build a more general curve beam theory, which provide new scientific research reference and design principles to the curve beam designers.

本课题拟采用理论分析、数值模拟和典型试件试验相结合的手段，对具有不同几何形态及不同材料特性的曲梁的耦合振动特性进行系统的研究。拟推导由14个二阶变系数偏微分方程耦合而成的多种曲梁的运动微分方程，在方程中,不仅考虑传统的转动惯量、轴向和剪切变形的影响，而且拟首次考虑了横截面的翘曲变形对曲梁自由振动特性的影响，并从中发现了与传统的梁理论不同的结论。本课题目的是揭示翘曲效应随多种几何参数和材料参数的变化规律，立足于对传统梁理论进行有益的改进与完善，以形成一套更普遍的曲梁理论，为工程中曲梁设计人员提供新的科研依据和设计准则。

本项目以多种具有不同几何形态、不同材料属性的曲梁为研究对象，推导出多组新的描述曲梁耦合振动的运动微分方程，方程中不仅考虑了横截面的初始扭曲、转动惯量、轴向变形和横向剪切变形的效应，而且首次加入了与扭转有关的翘曲变形对曲梁固有频率的影响。这些方程组由14 个一阶变系数偏微分方程强烈耦合而成，且加入了广义翘曲坐标和广义翘曲力矩两个自由度后，方程呈现出很强的刚性，采用改进的Riccati传递矩阵法求解。. 传统观念认为“翘曲变形是可以忽略的非重要因素”。本项目的研究表明：对于某些材料参数、边界参数及几何参数，横截面的翘曲变形对曲梁的固有频率将产生很大影响，是不能忽略的重要因素。这是一个崭新的结论。且翘曲变形对曲梁耦合振动的影响是否显著，会随着曲梁设计参数（横截面宽高比、横截面的几何形状、横截面的放置方法、梁的跨高比、轴线的曲率和扭率、材料属性等）的不同而变化。通过对各个设计参数进行敏感性分析和对比研究，本项目回答了“何种工况下翘曲效应是必须考虑的；何种工况下翘曲效应是可以忽略的？”这一关键科学问题。同时发现了一些与传统的圆形截面曲梁所不同的特性。