Bernoulli-Euler梁横向振动固有频率的轴力影响系数  

THE AXIAL LOAD INFLUENCE COEENFFICIT OF NATURAL FREQUENCIES FOR LATERAL VIBRATION OF A BERNOULLI-EULER BEAM

作　　者：张广芸[1] 张宏生[1] 陆念力[1] 

机构地区：[1]哈尔滨工业大学机电工程学院,哈尔滨150001

出　　处：《工程力学》2011年第10期65-71,共7页Engineering Mechanics

基　　金：国家科技支撑计划项目(2006BAJ12B03-2)

摘　　要：给出了考虑轴力对于Bernoulli-Euler梁横向振动固有频率影响系数的高精度表达式。与动力刚度法推导等截面梁自由振动分析的动态刚度阵不同,首先获得承受常轴力的Bernoulli-Euler梁横向自由振动微分方程的通解,并通过位移边界条件消去待定常数,得到精确形函数;使用有限元方法,建立了使用精确形函数表达等截面Bernoulli-Euler梁动态刚度阵的微分格式,该微分格式精确刚度阵与动力刚度法得到的刚度阵完全一致。仿照Timoshenko对压弯梁静态挠度表达中取用轴力影响因子的方法,提出了Bernoulli-Euler梁横向振动固有频率的轴力影响系数表达式,结合Wittrick-Williams算法和动态刚度阵证明了当轴力在±0.5倍第1阶欧拉临界力之间变化时,轴力影响系数表达式最大误差不超过2%,且随固有频率阶次的提高,误差越来越小。A high precision formula is proposed in order to consider the influence coefficient of natural frequencies for the lateral vibration of a Bernoulli-Euler beam subjected to axial loads. It is different from the dynamic stiffness matrix obtained by dynamic stiffness method for the free vibration of the uniform beam. According to the governing differential equation for the lateral vibration of a Bernoulli-Euler beam subjected to a constant axial load, the general solution is obtained. The exact shape function can be obtained by eliminating the undetermined constants of displacement boundary conditions. With the finite element method, the differential formulation for a dynamic stiffness matrix of a Bernoulli-Euler beam is proposed and expressed exactly in a dynamic shape function. The differential formulation＇s stiffness matrix is the same with the stiffness matrix obtained by a dynamic stiffness method. To follow the Timoshenko method used in the axial load influence factor of bending beam＇s static deflection formula, the formula to compute an axial load influence coefficient of natural frequencies for the lateral vibration of a Bernoulli-Euler beam is proposed. The Wittrick-Williams algorithm and dynamic stiffness matrix are used to prove that the maximum relative error of the proposed formula is less than 2%, when the axial load is between the positive and negative half of the first order Euler critical load. Furthermore, the higher the order of the natural frequencies is, the smaller the error is.

关 键 词：有限单元法 精确形函数 自由振动 Bernoulli-Euler梁 轴力影响系数 

分 类 号：TH113.1[机械工程—机械设计及理论]