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端部带质量和弹簧约束悬臂梁振动响应的解析解

马斌捷 周书涛 贾亮 侯传涛 荣克林

马斌捷, 周书涛, 贾亮, 等 . 端部带质量和弹簧约束悬臂梁振动响应的解析解[J]. 北京航空航天大学学报, 2019, 45(5): 883-892. doi: 10.13700/j.bh.1001-5965.2018.0482
引用本文: 马斌捷, 周书涛, 贾亮, 等 . 端部带质量和弹簧约束悬臂梁振动响应的解析解[J]. 北京航空航天大学学报, 2019, 45(5): 883-892. doi: 10.13700/j.bh.1001-5965.2018.0482
MA Binjie, ZHOU Shutao, JIA Liang, et al. Vibration response analytical solutions of cantilever beam with tip mass and spring constraints[J]. Journal of Beijing University of Aeronautics and Astronautics, 2019, 45(5): 883-892. doi: 10.13700/j.bh.1001-5965.2018.0482(in Chinese)
Citation: MA Binjie, ZHOU Shutao, JIA Liang, et al. Vibration response analytical solutions of cantilever beam with tip mass and spring constraints[J]. Journal of Beijing University of Aeronautics and Astronautics, 2019, 45(5): 883-892. doi: 10.13700/j.bh.1001-5965.2018.0482(in Chinese)

端部带质量和弹簧约束悬臂梁振动响应的解析解

doi: 10.13700/j.bh.1001-5965.2018.0482
详细信息
    作者简介:

    马斌捷  男, 硕士, 研究员, 硕士生导师。主要研究方向:结构动力学

    周书涛  男, 博士, 高级工程师。主要研究方向:计算固体力学

    通讯作者:

    马斌捷, E-mail:mabj@163.com

  • 中图分类号: V214+.3

Vibration response analytical solutions of cantilever beam with tip mass and spring constraints

More Information
  • 摘要:

    根据端部带质量和弹簧约束悬臂梁的特征值条件,提出了一种特征变换方法,获得了带约束悬臂梁广义质量和振动响应的解析解。通过分析根部弯矩、端部位移、速度和加速度放大系数的变化特征可知,端部弹簧的刚度对静态和一阶载荷响应有明显的影响,减载设计时可以放宽对端部质量的限制,载荷响应分析阶次介于速度和加速度的分析阶次之间。提出的特征变换方法可应用于求解其他载荷分布、边界条件和端部约束悬臂梁的振动响应解析解。

     

  • 图 1  端部约束悬臂梁的振动示意图

    Figure 1.  Vibration schematic of cantilever beam with tip constraints

    图 2  悬臂梁的振型

    Figure 2.  Vibration types of cantilever beam

    图 3  不同约束悬臂梁的根部弯矩和端部位移放大系数

    Figure 3.  Amplification factors of root bending moments and tip displacements for cantilever beam with different constraints

    图 4  不同约束时火箭模型和等直梁的根部弯矩放大系数

    Figure 4.  Root bending moment amplification factors of rocket model and constant section beam with different constraints

    图 5  悬臂梁的4种振动响应放大系数

    Figure 5.  Four amplification factors of vibration response for cantilever beam

    表  1  端部带不同质量和弹簧约束悬臂梁的特征值λn

    Table  1.   Eigenvalues λn of cantilever beam with different tip mass and spring constraints

    特征值阶次无约束硬弹簧中等弹簧中等质量软弹簧
    小质量大质量小质量大质量
    一阶1.875 222.623 231.622 631.552 261.710 281.048 66
    二阶4.694 174.149 423.950 553.983 544.114 543.949 92
    三阶7.854 617.192 997.082 587.102 837.190 487.082 54
    四阶10.995 610.298 810.219 910.234 110.298 510.219 9
    下载: 导出CSV

    表  2  端部带不同质量和弹簧约束悬臂梁无量纲广义质量的解析解和数值解

    Table  2.   Analytical and numerical solutions of dimensionless generalized mass for cantilever beam with different tip mass and spring constraints

    特征值阶次无约束硬弹簧中等弹簧中等质量软弹簧
    小质量大质量小质量大质量
    一阶0.250 01.314 8/1.314 63.375 8/3.375 31.244 3/1.244 20.568 0/0.568 10.681 9/0.681 8
    二阶0.250 00.271 60.251 60.253 60.259 90.251 4
    三阶0.250 00.254 10.250 50.251 20.253 70.250 5
    四阶0.250 00.252 00.250 20.250 60.252 00.250 2
    注:表格中“/”前数据为解析解,“/”后数据为数值解,无“/”栏中的解析解与数值解相同。
    下载: 导出CSV
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出版历程
  • 收稿日期:  2018-08-16
  • 录用日期:  2018-11-16
  • 刊出日期:  2019-05-20

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