Normalization and Duality Relations of Modified Timoshenko Beams

CAI Wenxiu; ZHENG Gang; TANG Yu; SUN Ceshi; YE Nianyu; XUE Wenqi

doi:10.21656/1000-0887.450292

Volume 46 Issue 12

Dec. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(12): 1540-1549

CAI Wenxiu, ZHENG Gang, TANG Yu, SUN Ceshi, YE Nianyu, XUE Wenqi. Normalization and Duality Relations of Modified Timoshenko Beams[J]. Applied Mathematics and Mechanics, 2025, 46(12): 1540-1549. doi: 10.21656/1000-0887.450292

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Normalization and Duality Relations of Modified Timoshenko Beams

doi: 10.21656/1000-0887.450292

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

Funds:

The National Science Foundation of China(52378284)

Received Date: 2024-10-28
Rev Recd Date: 2025-11-17

Available Online: 2025-12-31

Abstract

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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References(18)

References

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