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Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam

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Abstract

The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.

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Acknowledgements

This study was funded by the National Natural Science Foundation of China (Nos. 11772181, 11422214,) the “Dawn” Program of Shanghai Education Commission, (No. 17SG38), and Innovation Program of Shanghai Municipal Education Commission (No. 2017-01-07-00-09-E00019).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 149 Yan Chang Road, Shanghai, 200072, China

    Hu Ding, Yi Li & Li-Qun Chen

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai, 200072, China

    Hu Ding & Li-Qun Chen

  3. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Li-Qun Chen

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  1. Hu Ding
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  2. Yi Li
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  3. Li-Qun Chen
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Correspondence to Hu Ding.

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Ding, H., Li, Y. & Chen, LQ. Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam. Meccanica 53, 3233–3249 (2018). https://doi.org/10.1007/s11012-018-0891-6

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  • DOI: https://doi.org/10.1007/s11012-018-0891-6

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