徐银,朱泉冰,谭凯中,杜秀彬,尼浩,周伟,王龙

• 1:中国石油大学(华东)理学院

摘要(Abstract)：

在动态法测量固体材料的杨氏模量实验中，学生难以理解模型并确定基频共振频率。针对此问题，本文利用有限元软件COMSOL:Multiphysics(简称COMSOL)进行数字化课程资源建设的探索，完成了等截面、变截面细长杆弯曲振动的App开发。通过输入几何参数、材料参数等，可自动完成几何建模、网格划分，并基于有限元计算及后处理可视化，输出细长杆特征频率(包括基频和高阶)、节点位置、相应振型及谐响应等。仿真后特征频率以及节点位置与理论公式计算值进行对比。结果表明，仿真结果具有准确性、可靠性。App经过编译可不依赖于COMSOL直接运行，不仅对教学具有良好的辅助作用，同时在实际工程结构设计、振动控制中具有重要的参考应用价值。

关键词(KeyWords)： COMSOL;弯曲振动;基频共振频率;App开发

Abstract:

Keywords:

基金项目(Foundation): 中国石油大学(华东)探究性实验项目建设(TJ-202113,TJ-202314);中国石油大学(华东)研究生教育教学改革项目(YJG2023051);; 山东省大学生创新创业训练计划项目(S202410425036);山东省大学生创新创业训练计划项目(S202410425002);; 校级大学生创新创业训练计划项目(202511027CX)

作者(Author): 徐银,朱泉冰,谭凯中,杜秀彬,尼浩,周伟,王龙

参考文献(References)：

• [1] 李书光，张亚平，朱海丰.大学物理实验[M].北京：科学出版社，2018:229-235.LI S G,ZHANG Y P,ZHU H F.College physics experiment[M].Beijing:Science Press,2018:229-235.(in Chinese)
• [2] 樊聪聪，刘华扬，程敏熙.动态法测杨氏模量实验的探索与改进[J].大学物理实验，2016,(1):51-59.FAN C C,LIU H Y,CHENG M X.Exploration and improvement of dynamic method for measuring Young's modulus[J].College Physics Experiment,2016,29(1):51-59.(in Chinese)
• [3] 沈春根，盛雪德，陈寒松，等.机械动力学结构模态分析实验的教学改革[J].实验科学与技术，2010,(4):106-108.SHEN C G,SHENG X D,CHEN H S,et al.Teaching reform of modal analysis experiment in mechanical dynamics structure[J].Experiment Science and Technology,2010,8(4):106-108.(in Chinese)
• [4] 李东波，赵冬，华军，等.基于ANN的碳纤维楠竹锚杆锚固力预测研究[J].力学与实践，2013,(2):40-45.LI D B,ZHAO D,HUA J,et al.Prediction of the anchoring force of carbon fiber bamboo bolts based on ANN[J].Mechanics and Practice,2013,35(2):40-45.(in Chinese)
• [5] 王战，吴炜，楼旭阳，等.输入约束下的欧拉-伯努利梁的鲁棒自适应边界振动控制[J].南京理工大学学报，2023,(6):774-781.WANG Z,WU W,LOU X Y,et al.Robust adaptive boundary vibration control of an Euler-Bernoulli beam under input constraints[J].Journal of Nanjing University of Science and Technology,2023,47(6):774-781.(in Chinese)
• [6] 王龙，李东霞，马红章，等.动态法测量杨氏模量的理论分析与Matlab辅助研究[J].大学物理，2015,(10):39-42.WANG L,LI D X,MA H Z,et al.Theoretical analysis and Matlab-assisted research on dynamic method for measuring Young's modulus[J].College Physics,2015,34(10):39-42.(in Chinese)
• [7] CLOUGH R W,PENZIEN J.Dynamics of structures[M].New York:McGraw-Hill,1975.
• [8] MALASHIN A A.Problems of boundary control over the transverse-longitudinal vibrations of strings[J].DOKL PHYS,2011,56(9):502-505.
• [9] 侯祥林，范炜，贾连光.变截面压杆临界载荷的迭代算法[J].哈尔滨工业大学学报，2011,(S1):237-240.HOU X L,FAN W,JIA L G.Iterative algorithm for critical load of variable cross-section compression rods[J].Journal of Harbin Institute of Technology,2011,43(S1):237-240.(in Chinese)
• [10] FEDOTOV I A,POLYANIN A D,SHATALOV Y M,et al.Longitudinal vibrations of a Rayleigh-Bishop rod[J].Doklady Physics,2010,55(12):609-614.
• [11] KULESHOV A A.Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod with a free or fixed right end consisting of two segments with different densities and elasticities[J].Doklady Mathematics,2012,85(1):80-82.
• [12] IL'IN V A.Longitudinal vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical travel times on each segment[J].Doklady Mathematics,2009,80(3):910-913.
• [13] 王江选.运行时变结构动力学系统模态振型辨识方法研究[D].北京：北京理工大学，2017.WANG J X.Research on modal shape identification method for time-varying structural dynamic systems during operation[D].Beijing:Beijing Institute of Technology,2017.(in Chinese)
• [14] 齐阳，赵于博.既有建筑结构抗震鉴定与加固设计分析[J].建设科技，2023,(2):27-30.QI Y,ZHAO Y B.Seismic identification and reinforcement design analysis of existing building structures[J].Construction Science and Technology,2023,(2):27-30.(in Chinese)
• [15] 韩强，胡珉磊，刘英培，等.大跨度桥梁抗震分析中的整体有限元法及其应用[J/OL].北京工业大学学报，2024,(12):1448-1457.HAN Q,HU M L,LIU Y P,et al.Application of global finite element method in seismic analysis of long-span bridges[J/OL].Journal of Beijing University of Technology,2024,(12):1448-1457.(in Chinese)