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.
-
Keywords:
- non-uniform beam /
- uniform beam /
- dual /
- natural vibration /
- description of bending moment /
- description of slope
-
-
[1] Krylov VV, Tilman FJBS . Acoustic 'black holes' for flexural waves as effective vibration dampers. Journal of Sound and Vibration, 2004,274:605-619 [2] Gao NS, Wei ZY, Zhang RH , et al. Low-frequency elastic wave attenuation in a composite acoustic black hole beam. Applied Acoustics, 2019,154:68-76 [3] Zargham S, Ward TA, Ramli R , et al. Topology optimization: a review for structural designs under vibration problems. Structural and Multidisciplinary Optimization, 2016,53:1157-1177 [4] Sun JL, Tian Q, Hu HY , et al. Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components. Journal of Sound and Vibration, 2019,448:83-108 [5] 胡海昌 . 多自由度结构固有振动理论. 北京: 科学出版社, 1987: 39-81 [5] ( Hu Haichang. Natural Vibration Theory of Structures of Multiple Degrees of Freedom. Beijing: Science Press, 1987: 39-81(in Chinese)) [6] 胡海岩 . 机械振动与冲击. 北京: 航空工业出版社, 1998: 159-160 [6] ( Hu Haiyan. Mechanical Vibrations and Shocks. Beijing: Press of Aeronautical Industry, 1998: 159-160(in Chinese)) [7] Rao SS . Mechanical Vibrations, Fourth Edition. New Jersey: Pearson Prentice Hall, 2004: 609-613 [8] Rao SS . Vibration of Continuous Systems. New York: John Wiley & Sons, 2007: 317-341 [9] Karnoppe BH . Duality relations in the analysis of beam oscillations. Zeitschrift für Angewandte Mathematik und Physik ZAMP, 1967,18:575-580 [10] Ram YM, Elhay S . Dualities in vibrating rods and beams: continuous and discrete models. Journal of Sound and Vibration, 1995,184:648-655 [11] Wang DJ, Wang QC, He BC . Qualitative Theory of Structural Mechanics. Singapore: Springer Nature, 2019: 129-132 [12] Chen Q, Zhu DM . Vibrational analysis theory and applications to elastic-viscoelastic composite structure. Computers and Structures, 1990,37:585-595 [13] Li XJ, Xu FY, Zhang Z . Symplectic method for natural modes of beams resting on elastic foundations. ASCE Journal of Engineering Mechanics, 2018,144:04018009 [14] Abrate S . Vibration of non-uniform rods and beams. Journal of Sound and Vibration, 1995,185:703-716 -
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)
计量
- 文章访问数: 1614
- HTML全文浏览量: 245
- PDF下载量: 193
- 被引次数: 4
下载:
