Applied Mathematics and Mechanics >
Nonlinear vibrations of axially transporting viscoelastic plates immersed in liquids
Received date: 2025-07-23
Revised date: 2025-09-30
Online published: 2025-11-28
Supported by
Project supported by the Science and Technology Support Plan for Youth Innovation of Colleges and Universities of Shandong Province of China (No. 2023KJ215), the National Natural Science Foundation of China (Nos. 12002142 and 52405274), and the National Natural Science Foundation of Shanghai of China (No. ZR2023QE100)
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In industrial applications, plate-like structures such as steel strips in continuous hot-dip galvanizing and papers under fan action are ubiquitous. The vibration issues that arise when these structures are in axial motion, and are influenced by fluids and thermal fields, have attracted significant attention from the academic community. This study focuses on the nonlinear dynamic behavior of axially transporting immersed viscoelastic plates with particular emphasis on internal resonance and speed-dependent tension. The governing equation and the related boundary conditions for the axially transporting viscoelastic immersed plate are derived with Hamilton’s principle, prioritizing the impact of time-varying tension induced by speed perturbations. Based on the second-order Galerkin truncation, the governing equation is discretized into a system of second-order ordinary differential equations. The multi-scale method is used to analyze the stable steady-state response of the immersed viscoelastic plate. The conditions for achieving a 3 : 1 frequency ratio between the first two orders of the system are analytically deduced. Notably, when the viscoelastic coefficient diminishes, the stability boundaries exhibit increased complexity, manifesting as the irregular W-shaped contours in the parameter space. Numerical examples comprehensively investigate the effects of viscoelasticity on both the stability region and the steady-state response under internal resonance conditions. Finally, the accuracy of the obtained results is validated through numerical computation.
Dengbo ZHANG , Qingke ZHOU , Xiangfei JI , Youqi TANG . Nonlinear vibrations of axially transporting viscoelastic plates immersed in liquids[J]. Applied Mathematics and Mechanics, 2025 , 46(12) : 2341 -2360 . DOI: 10.1007/s10483-025-3329-8
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