基于绝对节点坐标法的变截面拱面外弯扭振动分析

Analysis of out of plane bending and torsional vibration of variable cross-section arch based on ANCF

基于绝对节点坐标法(absolute nodal coordinate formulation,ANCF)研究了均质变截面Euler-Bernoulli拱的面外弯扭振动的振动特性。建立了考虑面外弯曲和扭转的均质的振动模型,在总体惯性坐标系下,将它划分为若干个变截面拱单元,给出单元的动能、应变能以及外力势能的表达式,得到了单元质量矩阵和刚度矩阵,进而获得了拱的总体的质量矩阵和刚度矩阵。应用Lagrange方程建立了拱的弯扭振动微分方程。数值计算了两端固定的等截面圆弧拱和变截面圆弧拱的前三阶频率,画出了相应的振型图,分析了变截面圆弧拱的中心角、半径、高宽比以及均布径向载荷对其振动特性的影响规律;数值计算了两端固定的等截面和变截面抛物线型非圆弧拱的前三阶频率,画出了相应的振型图。

Based on the absolute node coordinate formulation(ANCF),vibration characteristics of a homogeneous Euler-Bernoulli arch with variable cross-section were studied.Firstly,a homogeneous vibration model considering out of plane bending and torsion was established.In the global inertial coordinate system,it was divided into several arch elements with variable cross-section.The expressions of element’s kinetic energy,strain energy and potential energy of external forces were derived to obtain element’s mass matrix and stiffness matrix,and furthermore the arch’s mass matrix and stiffness matrix were assembled.Secondly,the arch’s bending-torsional vibration differential equation was established by using Lagrange equation.Finally,the first three order natural frequencies of equal cross-section circular arc arch and variable cross-section circular arc arch fixed at two-end were calculated numerically and the corresponding modal shapes were plotted.Effect laws of variable cross-section circular arc arch’s center angle,radius,height-width ratio and uniformly distributed radial load on its vibration characteristics were analyzed.The first three order natural frequencies of equal cross-section and variable cross-section parabolic type non-circular arc arches fixed at two-end were calculated numerically and the corresponding modal shapes were plotted.