修正Timoshenko梁系统的归一化与对偶关系

蔡汶秀; 郑罡; 唐宇; 孙测世; 叶念雨; 薛文琪

doi:10.21656/1000-0887.450292

修正Timoshenko梁系统的归一化与对偶关系

doi: 10.21656/1000-0887.450292

1.
重庆交通大学省部共建山区桥梁及隧道工程国家重点实验室，重庆 400074
2.
重庆交通大学土木工程学院，重庆 400074
3.
四川路桥华东建设有限责任公司，成都 610299

基金项目:

国家自然科学基金 52378284

详细信息

作者简介:
蔡汶秀(1997—)，女，博士生

郑罡(1972—)，男，研究员，博士，博士生导师(E-mail: zhenggang@cqjtu.edu.cn)

中图分类号: Q342
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- 被引次数: 0
出版历程
- 收稿日期: 2024-10-28
- 修回日期: 2025-11-17
- 刊出日期: 2025-12-01

Normalization and Duality Relations of Modified Timoshenko Beams

1.
State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R. China
2.
School of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R. China
3.
Sichuan Road & Bridge East China Construction Co., Ltd., Chengdu 610299, P.R. China

摘要

摘要: 为研究修正Timoshenko梁系统的对偶条件和分类，讨论其理论意义. 首先通过引入时间和空间缩放变换，实现了对修正Timoshenko梁动力学方程的标准化；其次，基于该归一化方程，论证了在任意相同边界条件下参数型对偶关系的存在性；然后，探讨了不同截面类型参数型对偶关系的特点；最后，在固支-铰支、固支-固支和固支-自由这三种边界条件下，求解标准化方程，提出了构造对偶梁的方法，并通过文献算例展示了修正Timoshenko梁参数型对偶性的特征. 结果表明：归一化算法求解修正Timoshenko对偶梁的频率完全相同，且基于该算法可论证其修正Timoshenko梁的参数型对偶条件. 动力学特性的参数型对偶关系为修正Timoshenko梁的一种本质属性，时空缩放变换为揭示该特性的有效方法.
- 修正Timoshenko梁 /
- 对偶关系 /
- 归一化算法 /
- 动力学方程
Abstract: The duality conditions and classification of the modified Timoshenko beam systems were investigated, with its theoretical significance highlighted. First, the modified dynamic equations for the Timoshenko beam were normalized through time and space scaling transformations. Based on the normalized equations, the existence of parametric duality relations was established under arbitrary identical boundary conditions. Next, the parametric dual characteristics corresponding to different cross-sectional geometries were analyzed. Finally, under clamped-hinged, clamped-clamped, and clamped-free boundary conditions, the normalized equations were solved, to develop a novel method for building dual beams. Through examples from the literatures, the parametric dual characteristics of modified Timoshenko beams were further elucidated. The results demonstrate that, the frequencies of the modified Timoshenko dual beams derived via the normalization algorithm are identical, which confirms the parametric dual conditions of the modified beams. This study reveals that the parametric duality of dynamic properties is an intrinsic feature of modified Timoshenko beams, and the time and space scaling transformations provide a robust framework for uncovering these properties.
- modified Timoshenko beam /
- duality relation /
- normalization algorithm /
- dynamic equation

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图 1 Timoshenko梁典型单元

Figure 1. A typical element of the Timoshenko beams

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图 2 在不同的边界条件下无量纲频率-梁长(ω-l)曲线

Figure 2. Dimensionless frequency-length (ω-l) curves in different boundary conditions

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图 3 修正Timoshenko梁的归一化算法流程图

Figure 3. The flow chart of the normalization algorithm for the modified Timoshenko beam

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图 4 修正Timoshenko对偶梁的构造流程图

Figure 4. Construction flowchart of the modified Timoshenko dual beam

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表 1 不同边界条件下修正Timoshenko梁的频率方程

Table 1. Frequency equations of the modified Timoshenko beam under different boundary conditions

boundary condition	end condition	frequency equation
clamped-hinged	φ(0)=0	$\left(\lambda_2-\frac{N \omega^2}{\lambda_2}\right) \tanh \left(\lambda_1 l\right)-\left(\lambda_1+\frac{N \omega^2}{\lambda_1}\right) \tan \left(\lambda_2 l\right)=0$
	θ(0)=0
	φ(l)=0
	θ′(l)=0
clamped-clamped	φ(0)=0	$2-2 \cosh \left(\lambda_1 l\right) \cos \left(\lambda_2 l\right)+\left(\frac{\lambda_1+\frac{N \omega^2}{\lambda_1}}{\lambda_2-\frac{N \omega^2}{\lambda_2}}-\frac{\lambda_2-\frac{N \omega^2}{\lambda_2}}{\lambda_1+\frac{N \omega^2}{\lambda_1}}\right) \sinh \left(\lambda_1 l\right) \sin \left(\lambda_2 l\right)=0$
	θ(0)=0
	φ(l)=0
	θ(l)=0
clamped-free	φ(0)=0	$2+\left(\frac{\lambda_1}{\lambda_2}-\frac{\lambda_2}{\lambda_1}\right) \sin \left(\lambda_2 l\right) \sinh \left(\lambda_1 l\right)+\left(\frac{\lambda_1}{\lambda_2} \frac{\lambda_1+\frac{N \omega^2}{\lambda_1}}{\lambda_2-\frac{N \omega^2}{\lambda_2}}+\frac{\lambda_2}{\lambda_1} \frac{\lambda_2-\frac{N \omega^2}{\lambda_2}}{\lambda_1+\frac{N \omega^2}{\lambda_1}}\right) \cos \left(\lambda_2 l\right) \cosh \left(\lambda_1 l\right)=0$
	θ(0)=0
	φ′(l)－θ(l)=0
	θ′(l)=0

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表 2 修正Timoshenko梁的基本物理参数

Table 2. The basic parameters of the modified Timoshenko beams

numerical example	beam dimension (length，width，height)/m	density/(kg/m³)	elastic modulus/GPa	shear modulus/GPa	Poisson’s ratio	shear coefficient
ref. [11]	2×0.1×0.1	8 000	260	100	0.3	5/6
ref. [11]	2×0.1×0.1	8 000	260	100	0.3	5/6
ref. [12]	0.8×0.1×0.1	8 000	260	100	0.3	5/6

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表 3 不同边界条件下的修正Timoshenko梁频率

Table 3. Frequencies of modified Timoshenko beams under different boundary conditions

boundary condition	i	$\widetilde{\omega}_{\mathrm{n}}$/(rad/s)	$\widetilde{\omega}_{\mathrm{r}}$^[11]/(rad/s)	e/%
clamped-hinged	1	631.232	630.412	0.13
	2	2 018.847	2 014.308	0.23
	3	4 125.726	4 112.848	0.31
	4	6 864.657	6 838.452	0.38
	5	10 135.553	10 091.089	0.44
	6	13 837.133	13 769.815	0.49
	7	17 875.455	17 782.256	0.52
	8	22 168.834	22 047.127	0.55
	9	26 649.650	26 498.326	0.57
	10	31 264.002	31 082.446	0.58
clamped-clamped	1	915.671	912.938	0.30
	2	2 488.636	2 476.375	0.50
	3	4 775.607	4 744.422	0.66
	4	7 676.644	7 615.902	0.80
	5	11 090.608	10 990.008	0.92
	6	14 916.089	14 767.862	1.00
	7	19 060.313	18 857.965	1.07
	8	23 443.475	23 183.168	1.12
	9	28 000.130	27 679.656	1.16
	10	32 678.554	32 297.026	1.18
clamped-free	1	893.027 6	892.977	0.01
	2	5 227.950 2	5 228.583	-0.01
	3	13 349.451 4	13 361.775	-0.09
	4	23 453.124 4	23 521.191	-0.29
	5	34 658.126 3	34 843.732	-0.53
	6	46 382.805 5	46 733.804	-0.75
	7	58 312.566 9	58 848.124	-0.91
	8	70 281.369 7	71 003.211	-1.02
	9	82 211.315 8	83 112.545	-1.08
	10	94 070.323 5	95 140.946	-1.13

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表 4 对偶梁基本物理参数

Table 4. Basic physical parameters of the dual beams

beam	beam dimension/m	elastic modulus/GPa	density/(kg/m³)	shear coefficient	Poisson’s ratio	shear modulus/GPa
ref. [11]	$\begin{gathered}\tilde{l} \times \tilde{b} \times \tilde{h}= 2 \times 0.1 \times 0.1\end{gathered}$	260	8 000	5/6	0.3	100
beam #1	$\begin{gathered}\tilde{l} \times \tilde{b} \times \tilde{h}= 2 \times 0.3 \times 0.1\end{gathered}$	208	6 400	5/6	0.3	80
beam #2	$\begin{aligned} & \tilde{l}=1.998 ; \\ & \widetilde{D}=0.119\end{aligned}$	245.029	8 000	9/10	0.3	94.242

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表 5 不同对偶梁的频率

Table 5. The dual beam frequency comparison

beam	boundary condition	i	ω/(rad/s)	α_t	α_x	$\widetilde{\omega}$/(rad/s)
ref. [11]	clamped-hinged	1	1.317E-2	2.086E-5	5.859E-2	631.232
		2	4.212E-2			2 018.847
		3	8.607E-2			4 125.726
		4	1.432E-1			6 864.657
beam #1	clamped-hinged	1	1.317E-2	2.086E-5	5.859E-2	631.232
		2	4.212E-2			2 018.847
		3	8.607E-2			4 125.726
		4	1.432E-1			6 864.657
beam #2	clamped-hinged	1	1.317E-2	2.086E-5	5.854E-5	631.232
		2	4.212E-2			2 018.847
		3	8.607E-2			4 125.726
		4	1.432E-1			6 864.657

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参考文献(18)

[1]	周博, 郑雪瑶, 康泽天, 等. 基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究[J]. 应用数学和力学, 2019, 40(12): 1321-1334. doi: 10.21656/1000-0887.400056 ZHOU Bo, ZHENG Xueyao, KANG Zetian, et al. A Timoshenko micro-beam model and its size effects based on the modified couple stress theory[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1321-1334. (in Chinese) doi: 10.21656/1000-0887.400056
[2]	关玉铭, 戈新生. 基于非约束模态的中心刚体-Timoshenko梁动力学建模与分析[J]. 应用数学和力学, 2022, 43(2): 156-165. doi: 10.21656/1000-0887.420089 GUAN Yuming, GE Xinsheng. Dynamic modeling and analysis of the central rigid body-Timoshenko beam model based on unconstrained modes[J]. Applied Mathematics and Mechanics, 2022, 43(2): 156-165. (in Chinese) doi: 10.21656/1000-0887.420089
[3]	LOVE A E. A Treatise on the Mathematical Theory of Elasticity[M]. New York: Dover, 1927: 314-331.
[4]	TRAILL-NASH R W, COLLAR A R. The effects of shear flexibility and rotatory inertia on the bending vibrations of beams[J]. The Quarterly Journal of Mechanics and Applied Mathematics, 1953, 6(2): 186-222. doi: 10.1093/qjmam/6.2.186
[5]	LUBLINER E, ELISHAKOFF I. Random vibration of system with finitely many degrees of freedom and several coalescent natural frequencies[J]. International Journal of Engineering Science, 1986, 24(4): 461-470. doi: 10.1016/0020-7225(86)90038-8
[6]	ELISHAKOFF I. Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories[M]. New York: Isaac Elishakoff, 2019: 107-138.
[7]	余云燕, 孔嘉乐, 陈进浩, 等. 变截面修正Timoshenko梁自振频率的回传射线矩阵法分析[J]. 地震工程学报, 2022, 44(4): 751-758. YU Yunyan, KONG Jiale, CHEN Jinhao, et al. Natural frequency of a modified Timoshenko beam with variable cross-section with the method of reverberation ray matrix[J]. China Earthquake Engineering Journal, 2022, 44(4): 751-758. (in Chinese)
[8]	LI M L, WEI P J, ZHOU X L. Wave propagation and free vibration of a Timoshenko beam mounted on the viscoelastic Pasternak foundation modeled by fraction-order derivatives[J]. Mechanics of Time-Dependent Materials, 2023, 27(4): 1209-1223. doi: 10.1007/s11043-022-09541-4
[9]	吴晓, 罗佑新. 用Timoshenko梁修正理论研究功能梯度材料梁的动力响应[J]. 振动与冲击, 2011, 30(10): 245-248. WU Xiao, LUO Youxin. Dynamic responses of a beam with functionally graded materials with Timoshenko beam correction theory[J]. Journal of Vibration and Shock, 2011, 30(10): 245-248. (in Chinese)
[10]	FALSONE G, SETTINERI D, ELISHAKOFF I. A new class of interdependent shape polynomials for the FE dynamic analysis of Mindlin plate Timoshenko beam[J]. Meccanica, 2015, 50(3): 767-780. doi: 10.1007/s11012-014-0032-9
[11]	ELISHAKOFF I, TONZANI G M, MARZANI A. Effect of boundary conditions in three alternative models of Timoshenko-Ehrenfest beams on Winkler elastic foundation[J]. Acta Mechanica, 2018, 229(4): 1649-1686. doi: 10.1007/s00707-017-2034-x
[12]	ELISHAKOFF I, TONZANI G M, MARZANI A. Three alternative versions of Bresse-Timoshenko theory for beam on pure Pasternak foundation[J]. International Journal of Mechanical Sciences, 2018, 149: 402-412. doi: 10.1016/j.ijmecsci.2017.10.043
[13]	KARNOPP B H. Duality relations in the analysis of beam oscillations[J]. Zeitschrift für Angewandte Mathematik und Physik ZAMP, 1967, 18(4): 575-580.
[14]	CHEN Q, ZHU D. Vibrational analysis theory and application to elastic-viscoelastic composite structures[J]. Computers & Structures, 1990, 37(4): 585-595.
[15]	RAM Y M, ELHAY S. Dualities in vibrating rods and beams: continuous and discrete models[J]. Journal of Sound and Vibration, 1995, 184(5): 759-766. doi: 10.1006/jsvi.1995.0345
[16]	LI X, XU F, ZHANG Z. Symplectic method for natural modes of beams resting on elastic foundations[J]. Journal of Engineering Mechanics, 2018, 144(4): 04018009. doi: 10.1061/(ASCE)EM.1943-7889.0001427
[17]	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. HU Haiyan. Duality relations of beams in natural vibrations[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(1): 139-149. (in Chinese)
[18]	ELISHAKOFF I, AMATO M. Flutter of a beam in supersonic flow: truncated version of Timoshenko-Ehrenfest equation is sufficient[J]. International Journal of Mechanics and Materials in Design, 2021, 17(4): 783-799. doi: 10.1007/s10999-021-09537-x