Nonlinear Vibration and Dynamic Bifurcation of Axially Moving Plates Under Subsonic Airflow in a Narrow Space

Abstract

In this paper, the nonlinear vibration and dynamic bifurcation of axially moving plates under subsonic airflow in a narrow space concerning the background of the mining industry are investigated. The nonlinear dynamic equations interacting with narrow space airflow are established using Hamilton’s principle and linear potential flow theory. The dynamic bifurcation of vibration characteristics of axially moving plates caused by airflow is studied. The displacement–time diagrams, phase diagrams, and Poincare maps are plotted to distinguish the motion behaviors. The incremental harmonic balance method is used to study nonlinear vibration. The effects of airflow velocity, axial velocity and the narrow gap height on stability and nonlinear vibration characteristics are discussed. With the increase of axial velocity and air velocity and the decrease of narrow gap height, the resonance frequency of the plate decreases and the vibration peak increases. A smaller narrow gap height magnifies the effect of airflow on stability and nonlinear vibration, and a larger narrow gap height makes the magnification disappear. The findings in this paper provide valuable insights into the nonlinear vibration of axially moving thin plates interacting with subsonic airflow in a narrow space, and improve the understanding of the stability, controllability, and predictability of this system in future design works.

Appendix

Based on Eq. (27), the state equations of four generalized coordinates are obtained

$$\begin{aligned} & m_{11} \ddot{q}_{11} + c_{11} \dot{q}_{11} + k_{11} q_{11} + R_{1} (q) = 0, \\ & m_{12} \ddot{q}_{12} + c_{12} \dot{q}_{12} + k_{12} q_{12} + R_{2} (q) = 0, \\ & m_{21} \ddot{q}_{21} + c_{21} \dot{q}_{21} + k_{21} q_{21} + R_{3} (q) = 0, \\ & m_{22} \ddot{q}_{22} + c_{22} \dot{q}_{22} + k_{22} q_{22} + R_{4} (q) = 0, \\ \end{aligned}$$

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in which R₁(q), R₂(q), R₃(q), and R₄(q) are nonlinear terms.

The nonlinear matrix K⁽³⁾ is extracted from the Jacobian matrix

$${\mathbf{K}}^{(3)} = \frac{1}{3}\left[ {\begin{array}{*{20}c} {\frac{{\partial R_{1} (q)}}{{\partial q_{11} }}} & {\frac{{\partial R_{1} (q)}}{{\partial q_{12} }}} & {\frac{{\partial R_{1} (q)}}{{\partial q_{21} }}} & {\frac{{\partial R_{1} (q)}}{{\partial q_{22} }}} \\ {\frac{{\partial R_{2} (q)}}{{\partial q_{11} }}} & {\frac{{\partial R_{2} (q)}}{{\partial q_{12} }}} & {\frac{{\partial R_{2} (q)}}{{\partial q_{21} }}} & {\frac{{\partial R_{4} (q)}}{{\partial q_{22} }}} \\ {\frac{{\partial R_{3} (q)}}{{\partial q_{11} }}} & {\frac{{\partial R_{3} (q)}}{{\partial q_{12} }}} & {\frac{{\partial R_{3} (q)}}{{\partial q_{21} }}} & {\frac{{\partial R_{4} (q)}}{{\partial q_{22} }}} \\ {\frac{{\partial R_{4} (q)}}{{\partial q_{11} }}} & {\frac{{\partial R_{4} (q)}}{{\partial q_{12} }}} & {\frac{{\partial R_{4} (q)}}{{\partial q_{21} }}} & {\frac{{\partial R_{4} (q)}}{{\partial q_{22} }}} \\ \end{array} } \right]$$

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