本文对Bernoulli-Euler梁受轴向载荷时的横向振动特性进行分析。首先获得满足受轴向载荷梁自由振动微分方程的挠度函数的解析表达式，并给出两端为任意弹性约束梁固有频率方程的解析形式，从而可求解出各阶频率和对应的各阶模态。其次，得到5种经典边界条件下受轴向力梁频率方程的简洁形式，选取其中4种边界条件梁进行数值计算。所得结果与利用动刚度法与Wittrick-Williams算法的结果一致。然后，对于5种含有弹性约束边界且受轴向载荷梁，给出其固有频率方程的解析形式及特定轴向力下频率的数值解。所得解析频率方程是精确的，可校核其他数值方法的精度，且在振动优化和模态分析上具有较好的参考价值。结果表明:在轴拉作用下梁的无量纲频率明显比在轴压作用下梁的无量纲频率大；且轴拉时梁无量纲频率随着轴向拉力的变大而逐渐变大；轴压时梁的无量纲频率随轴压力的变大而变小,其中轴压力的值应不超过欧拉临界力。本文还给出若干种边界条件时轴力对梁基频影响系数的实用公式，误差不超过1%。
Abstract
The transverse vibration characteristics of Bernoulli-Euler beam subjected to axial loads is given. Firstly, the analytical expression of deflection is obtained to satisfy the differential equation for free vibration of beam subjected to axial force, and the natural frequency equations of beam’s transverse vibration under general elastic boundary conditions are presented. The natural frequencies can be solved and the corresponding mode for each order can be obtained. Secondly, The brief forms of the frequency equation for beams subjected to axial forces under five kinds of classical boundary conditions are obtained, and the numerical computation is carried out for four kinds of boundary conditions. The obtained results are found to be coincided with those by using the dynamic stiffness method and Wittrick-Williams algorithm. Then for five kinds of boundary conditions including elastic constraint at the beam end, the analytical frequency equation are obtained and some numerical results of natural frequency for beams under axial force are presented. The analytical form of the frequency equation is accurate in theory, which can be used to check the accuracy of other numerical methods, and has good reference in vibration optimization and modal analysis. The influence of axial forces on structural vibration response is analyzed, and the results show that the dimensionless frequency of beams under axial tension is obviously larger than that under axial compression. The dimensionless frequency of beams under axial tension increases with the increase of axial tension. The beam’s dimensionless frequency decreases with the increase of axial compression, in which the value of the axial compression should be less than the critical Euler force. For beams subjected to axial force with several kinds of boundary conditions, practical approximate equations is proposed with the errors less than 1%.
关键词
Bernoulli-Euler梁 /
轴向载荷 /
自由振动 /
固有频率
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Key words
Bernoulli-Euler beam /
axial load /
free vibration /
natural frequency
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