利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。
Abstract
The elastic foundation beam model based on Bernoulli-Euler beam theory is widely used, but it has larger errors in calculations of higher order natural frequencies and deep beams.Here, based on the modified Timoshenko beam theory, a new vibration differential equation of an elastic foundation beam was established.Due to considering the moment of inertia caused by shear deformation based on Timoshenko beam theory, it has a better accuracy.The beam54 element of the finite element software ANSYS was used to do the finite element calculation for the beam’s vibration modes, and the results basically had no errors compared with the theoretical solution to verify the correctness of the proposed theoretical model.Based on the modified Timoshenko beam theory, the elastic foundation beam’s natural frequency transcendental equations and modal functions under various boundary conditions including free-free, pinned-pinned, pinned-free and clamped-clamped were deduced.The calculation results of an elastic foundation beam under different theories, different boundary conditions and different ratios of height to span were analyzed to verify the applicability of the proposed theoretical model.The relations among wave velocity, group velocity and wave number under different theories were analyzed.It was shown that constraint conditions and beam length have important effects on an elastic foundation beam’s vibration modes; foundation stiffness affects an elastic foundation beam’s natural vibration frequencies significantly.
关键词
弹性地基梁 /
剪切变形 /
转动惯量 /
边界约束 /
高频段 /
波速 /
群速度 /
振动模态
{{custom_keyword}} /
Key words
elastic foundation beam /
shear deformation /
moment of inertia /
boundary constraint /
high frequency section /
wave velocity /
group velocity /
vibration mode
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] 雷晓燕. 轨道过渡段刚度突变对轨道振动的影响[J].中国铁道科学, 2006, 27(5): 42-45.
LEI Xiaoyan. Influence of track stiffness transition on track vibration [J]. China Railway Science, 2006, 27(5): 42-45.
[2] 崔奕, 姜忻良, 鲍鹏. 变基床系数弹性地基梁解法及其应用[J].岩土力学, 2003, 24(4): 565-578.
CUI Yi, Jiang Xinliang, Bao Peng. Variable foundation bed coefficient elastic foundation beam method and its application [J]. Rock and Soil Mechanics, 2003, 24(2): 565-578.
[3] 彭丽, 丁虎, 陈立群. 粘弹性三参数地基梁横向自由振动[J]. 振动与冲击, 2014, 33(1): 101-105.
Peng Li, Ding Hu, Chen Liqun. Transverse free vibration of viscoelastic three parameter foundation beam [J]. Journal of Vibration and Shock, 2014, 33(1): 101-105.
[4] 夏桂云, 李传习, 曾庆元. Winkler地基上Timoshenko深梁的有限元分析[J]. 中南大学学报(自然科学版), 2010, 41(4): 1549-1555.
Xia Guiyun, Li Chuanxi, Zeng Qingyuan. Finite element analysis of Timoshenko deep beam on Winkler foundation [J]. Journal of Central South University (Science and Technology), 2010, 41(4): 1549-1555.
[5] Tang Y Q, Chen L Q, Yang X D. Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions [J]. International Journal of Mechanical Sciences, 2008, 50: 1448-1458.
[6] Magrab E B. Natural frequencies and mode shapes of Timoshenko beams with attachmens [J]. Journal of Vibration and Control, 2007, 13(7): 905-934.
[7] Morfidis K. Vibration of Timoshenko beams on three-parameter elastic foundation [J]. Computers and Structures, 2010, 88: 294-308.
[8] 彭丽, 丁虎, 陈立群.粘弹性三参数地基上Timoshenko梁横向自由振动[J]. 噪声与振动控制, 2013, 33(5): 107-188.
Peng, Li, Ding Hu, Chen Liqun. Transverse free vibration of Timoshenko beam on viscoelastic three parameter foundation [J]. Noise and Vibration Control, 2013, 33(5): 107-188.
[9] 夏桂云, 俞茂宏, 李传习, 等. 考虑水平摩阻的Winkler地基Timoshenko梁分析[J]. 土木工程学报, 2011, 44(6): 98-104.
Xia Guiyun, Yu Maohong, Li Chuanxi, et al. Timoshenko beam analysis of Winkler foundation considering horizontal friction [J]. China Civil Engineering Journal, 2011, 44(6): 98-104.
[10] 陈镕, 万春风, 薛松涛,等.Timoshenko梁运动方程的修正及其影响[J]. 同济大学学报, 2005, 33(6): 711―715.
Chen Rong, Wan Chunfeng, Xue Songtao, et al. Modification of motion equation of Timoshenko beam and its effect [J]. Journal of Tongji University, 2005, 33(6): 711―715.
[11] 夏桂云, 杨美良, 李传习, 等. 考虑底板与池壁相互作用和池壁剪切变形影响的圆形水池动力特性计算[J]. 工程力学, 2013, 30(10): 140―147.
Xia Guiyun, Yang Meiliang, Li Chuanxi, et al. Calculation of dynamic characteristics of circular cistern considering interaction between bottom plate and pool wall and shear deformation tank wall [J]. Engineering Mechanics, 2013, 30(10): 140―147.
[12] 王新敏. Ansys工程结构数值分析[M]. 北京: 人民交通出版社, 2007: 503-504.
Wang Xinmin. Numeric analysis of engineering structures by Ansys software [M]. Beijing: China Communications Press, 2007: 503-504.
[13] 盛宏玉. 结构动力学[M]. 合肥: 合肥工业大学出版社, 2005: 190―219.
Sheng Hongyu. Structural dynamics [M]. Hefei: HeFei University Technology Press, 2005: 190-219.
[14] 夏桂云, 李传习. 考虑剪切变形影响的杆系结构理论与应用[M]. 北京: 人民交通出版社, 2008: 242―244.
Xia Guiyun, Li Chuanxi. Theory and application of bar structure considering shear deformation . [M]. Beijing: China Communications Press, 2008: 242―244.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}