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梁在固有振动中的对偶关系

胡海岩

胡海岩. 梁在固有振动中的对偶关系[J]. 力学学报, 2020, 52(1): 139-149. DOI: 10.6052/0459-1879-20-019
引用本文: 胡海岩. 梁在固有振动中的对偶关系[J]. 力学学报, 2020, 52(1): 139-149. DOI: 10.6052/0459-1879-20-019
Hu Haiyan. DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149. DOI: 10.6052/0459-1879-20-019
Citation: Hu Haiyan. DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149. DOI: 10.6052/0459-1879-20-019
胡海岩. 梁在固有振动中的对偶关系[J]. 力学学报, 2020, 52(1): 139-149. CSTR: 32045.14.0459-1879-20-019
引用本文: 胡海岩. 梁在固有振动中的对偶关系[J]. 力学学报, 2020, 52(1): 139-149. CSTR: 32045.14.0459-1879-20-019
Hu Haiyan. DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149. CSTR: 32045.14.0459-1879-20-019
Citation: Hu Haiyan. DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149. CSTR: 32045.14.0459-1879-20-019

梁在固有振动中的对偶关系

基金项目: 1) 国家自然科学基金重点资助项目(11832005)
详细信息
    通讯作者:

    胡海岩

  • 中图分类号: O313

DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS

  • 摘要: 本文研究具有齐次边界的Euler-Bernoulli梁在固有振动中的对偶关系.将两种截面变化不同、但固有频率完全相同的梁定义为异截面对偶梁.通过位移描述和弯矩描述,指出具有齐次边界条件的变截面梁共有如下7类异截面对偶:一是自由-自由梁与固支-固支梁,二是滑支-自由梁与滑支-固支梁(及其镜像),三是铰支-自由梁与铰支-固支梁(及其镜像),四是铰支-滑支梁与铰支-滑支梁(及其镜像),五是滑支-滑支梁与滑支-滑支梁,六是铰支-铰支梁与铰支-铰支梁,七是固支-自由梁与自由-固支梁.在此基础上,将两种截面变化相同、固有频率也相同的梁定义为同截面对偶梁.研究表明,当且仅当梁的截面积函数和截面惯性矩函数具有特定指数函数形式时,前4类异截面对偶梁能成为同截面对偶梁.对于等截面梁,上述前3类同截面对偶仍可保持,而第4类同截面对偶退化为彼此镜像.此时,通过引入转角描述可发现等截面梁产生新对偶,即滑支-滑支梁与铰支-铰支梁对偶.上述等截面梁的对偶均具有如下特征,即对偶中的一种梁具有静定约束,另一种梁具有静不定约束.
    Abstract: The duality relations are studied in the paper for the Euler-Bernoulli beams with homogeneous boundaries in natural vibrations. A pair of beams is first defined as a dual of different cross-sections if they have the same natural frequencies, but different variations of cross-sections. The duals of different cross-sections are analyzed via a dual of displacement description and bending moment description, and the non-uniform beams with homogeneous boundaries can be classified as the following seven duals. They are (1) the dual of a free-free beam and a clamped-clamped beam, (2) the dual of a slipping-free beam and a slipping-clamped beam (and their mirrors), (3) the dual of a hinged-free beam and a hinged-clamped beam (and their mirrors), (4) the dual of two hinged-slipping beams, (5) the dual of two slipping-slipping beams, (6) the dual of two hinged-hinged beams, and (7) the dual of a clamped-free beam and a free-clamped beam. Then, a pair of beams is defined as a dual of identical cross-sections if they have the same natural frequencies and the same variations of cross-sections. It is proved that the first four duals of different cross-sections become the duals of identical cross-sections if and only if the area of cross-section and the inertial moment of cross-section of any beam in those duals take a specific form of exponential function. Afterwards, the first three duals of identical cross-sections are verified to keep the dual relations for uniform beams, whereas the fourth dual is degenerated to a pair of mirrors. Based on the dual of displacement description and slope description, a new dual of uniform beams is found for a slipping-slipping beam and a hinged-hinged beam. Finally presented is an important feature of all the duals of uniform beams. That is, one uniform beam in a dual has statically determinate constraints while the other uniform beam in the same dual has statically indeterminate constraints.
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  • 期刊类型引用(3)

    1. 常婷婷,沈峰,鲍四元. 基于两种梁理论对变幅锥形杆弯曲振动的特性分析及参数设计. 振动与冲击. 2024(02): 114-122 . 百度学术
    2. 张永康,鲍四元. 幂函数形式连续变截面梁振动的弯曲固有频率分析. 应用声学. 2024(02): 330-338 . 百度学术
    3. 薛纭,陈立群. Kirchhoff动力学比拟对弹性薄壳的推广. 力学学报. 2021(01): 234-247 . 本站查看

    其他类型引用(1)

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出版历程
  • 收稿日期:  2020-01-10
  • 刊出日期:  2020-02-09

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