轴向力作用下过屈曲Timoshenko梁与Euler-Bernoulli梁的自由振动特性对比  

Comparison of natural characters between buckling Timoshenko and Euler-Bernoulli beams under the axial force

作　　者：井洁 毛晓晔 丁虎[1] 陈立群[1] JING Jie;MAO Xiaoye;DING Hu;CHEN Liqun(Shanghai Institute of Applied Mathematics and Mechanics,School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444,China)

机构地区：[1]上海大学力学与工程科学学院上海市应用数学和力学研究所,上海200444

出　　处：《振动与冲击》2022年第24期33-40,71,共9页Journal of Vibration and Shock

基　　金：国家自然科学基金青年基金(12002195);国家自然科学基金杰出青年基金(12025204);上海市教育委员会科研创新计划(2019-01-07-00-09-E00018);上海市科学技术委员会浦江人才计划(A类)(20PJ1404000)。

摘　　要：首次研究了过屈曲轴向力下Timoshenko梁的自由特性,并且与Euler-Bernoulli梁对比。首先建立了受轴向压力作用的Timoshenko梁横向振动偏微分-积分控制方程,导出两端简支边界条件下临界屈曲轴向力和屈曲位形。与Euler-Bernoulli梁进行对比,判断了两种梁模型屈曲位形的稳定性,讨论了轴向力作用下参数对静态屈曲分岔的影响。运用Galerkin截断法对比了两种梁在过屈曲轴向力下的固有频率,着重讨论了参数对固有频率的影响及模型适用性。研究发现,与屈曲前相反,由于屈曲发生更早,Timoshenko梁屈曲后的基频在相同轴向力下总是大于Euler-Bernoulli梁;Timoshenko梁一部分振动能量存储在截面转动变形中,软化了弯曲刚度,这种软化作用在高阶频率上体现更明显,因此两种模型高阶频率差异要大于基频。对于两种梁模型,在屈曲发生后,除一阶频率随轴力增加而增加,高阶频率都不受轴力的影响,而屈曲前各阶频率都随轴向力增加而减小。对于长细比较大的梁,两种模型固有频率差异会越小,进一步说明了将Euler-Bernoulli梁用于细长梁研究、而将Timoshenko梁用于短粗梁研究的必要性。Natural characters of the buckling Timoshenko beam under the axial force was investigated comparing with the Euler-Bernoulli one for the first time.Firstly,the partial-differential-integral governing equation of the Timoshenko beam with simply supported boundaries was established.The critical buckling axial force and the buckling non-trivial configuration were derived analytically.The stability of the nontrivial configuration was discussed together with the Euler-Bernoulli beam.Meanwhile,the influence of the parameters on the buckling bifurcation was discussed.The Galerkin truncation method was applied to calculate the frequencies of the two models under their supercritical axial force.It was found that,after buckling,the frequency of the Timoshenko beam is always larger than that of the Euler-Bernoulli beam,for the critical axial force of the Timoshenko one is smaller.Part of the vibration energy of the Timoshenko beam was stored in the rotational deformation of the cross-section,which softens the bending stiffness.This softening effect is more obvious for higher order frequencies,which makes higher order frequencies more different than the fundamental frequency while comparing with the Euler-Bernoulli beam.Also,during the truncation,it can be found that higher order frequencies are no longer affected by the axial force in the supercritical region,except for the first order one.For beams with large slenderness ratio,the smaller the difference in the natural frequencies will be.This further illustrates the necessity of using the Euler-Bernoulli beam for slender structures and the Timoshenko beam for those short and thick.

关 键 词：TIMOSHENKO梁 屈曲 轴向力 固有频率 截断 

分 类 号：TH212[机械工程—机械制造及自动化] TH213.3