受单点横向非定常约束梁的响应分析  

On the response of a beam with a one-point transverse rheonomic restraint

作　　者：丁维高 谢进[1] DING Weigao;XIE Jin(School of Mechanical Engineering,Southwest Jiaotong University,Chengdu 610031,China)

机构地区：[1]西南交通大学机械工程学院,成都610031

出　　处：《振动与冲击》2021年第2期176-184,共9页Journal of Vibration and Shock

基　　金：国家自然科学基金(51575457)。

摘　　要：为研究梁上任意一点受横向非定常约束的稳态响应问题。提出直接使用第一类拉格朗日方程与欧拉-伯努利梁理论建立梁的动力学方程,从而可以使用简单边界下梁的模态函数表示约束作用于任一位置时的梁的响应;以约束为谐波函数为例推导了梁响应的解析表达式,并通过算例验证了该方法的正确性。研究结果表明:受单点横向非定常约束梁的共振频率与非定常约束作用点的位置相关;共振时各阶模态频响会同时达到峰值,模态频响曲线呈现多峰值特征;而在各阶主频附近,对应阶模态响应没有明显的峰值,而其余阶模态响应会达到极小值;利用求解响应峰值频率比计算公式,得到非定常约束作用位置与梁动态响应峰值和极小值之间关系的解曲线;对解曲线的分析表明:当非定常位移约束作用在模态函数零点位置时,模态频响曲线会发生峰值与极小值合并的现象。The response of a beam under a one-point transverse rheonomic restraint was focused on.The first kind Lagrange equation was employed to establish the dynamics equation of the system,so that the displacement response was able to be represented by modal functions under simple boundary conditions.An analytic solution was obtained under the condition that the rheonomic restraint was represented by a harmonic function.An example was provided to prove the correctness of the present analytic solution.The results show that the resonant frequencies of the beam with one-point transverse rheonomic restraint are higher than those of the beam without the restraint(the main frequencies),and are dependent upon the location of the rheonomic restraint.On the frequency response curve of each mode,there are naturally multi peaks and each modal response arrives at its peaks at same frequencies.At the main frequencies of each mode,there is no obvious peak on the curve corresponding to the modal response of this order,however the modal response of the remaining order all reaches its minimum value.With the method of estimating the resonant frequency for the beam under rheonomic restrained condition,the graphs of solution were plotted to represent the relationship of the location of the constraint with the peaks and minimum values of the response.Based on the analysis of the graphs,it is found that the minimums of the modal frequency response will merge with the maximums of the modal frequency response when the rheonomic restrained modal function is zero.

关 键 词：非定常约束 欧拉-伯努利梁 模态响应 解析解 共振频率 

分 类 号：TB123[理学—工程力学]