Stability and nonlinear vibration of an axially moving plate interacting with magnetic field and subsonic airflow in a narrow gap

Abstract

In this paper, the stability and nonlinear vibration properties of an axially moving plate interacting with magnetic field and subsonic airflow in a narrow gap on the background of the maglev high-speed train chassis are investigated. The effect of the ground is simulated by a rigid wall influencing the subsonic aerodynamic pressure. The motion equation is established by using the generalized extended Hamilton principle based on the Burger nonlinear plate theory and solved by the incremental harmonic balance (IHB) method. The aerodynamic pressure in a narrow gap is built by using the linear Bernoulli equation based on the assumed mode method. The stability of the plate is studied by analyzing the natural frequencies of the system. The nonlinear vibration properties are investigated by analyzing the amplitude–frequency responses of the first four generalized coordinates. The effects of the gap height, the magnetic field parameters, the axially moving velocity, the flow velocity and the external excitation on the nonlinear vibration properties are discussed. From the study, rich nonlinear dynamic phenomena are observed. The frequency responses of the first generalized coordinate show hardening type of nonlinearity, and the second to the fourth generalized coordinates exhibit complex nonlinear behaviors. The gap height has significant effects on the stability of the plate when the gap is narrow. The stability of the vibration becomes worse with the flow velocity, the axially moving velocity, the magnetic induction density and the excitation amplitude increasing. The present research can provide some enlightenments for the stability design and vibration reduction of the maglev high-speed trains.

Appendix

As shown in Fig. 

Fig. 17

Schematic diagram of the plate unit at different times

17, to derive Eq. 13, a unit of the airflow can be taken as the reference. The plate unit has a velocity U_∞ relative to the flow unit. The coordinates of the points A – D are w(x, y, t), w(x + dx, y, t), w(x + U_∞dt, y, t + dt), and w(x + U_∞dt + dx, y, t + dt), respectively.

The displacement of the center of the plate unit in z direction can be expressed as.

$$ {\text{d}}w = \frac{1}{2}[w(x + U_{\infty } {\text{d}}t,y,t + {\text{d}}t) + w(x + U_{\infty } {\text{d}}t + {\text{d}}x,y,t + {\text{d}}t)] - \frac{1}{2}[w(x,y,t) + w(x + {\text{d}}x,y,t)] $$

(73)

Using the second order Taylor series, and omitting higher order infinitesimals, Eq. (73) is changed into.

$$ \begin{gathered} {\text{d}}w = \frac{1}{2}\left[ \begin{gathered} w(x,y,t) + \frac{\partial w(x,y,t)}{{\partial x}}U_{\infty } {\text{d}}t + \frac{\partial w(x,y,t)}{{\partial t}}{\text{d}}t \hfill \\ + w(x,y,t) + \frac{\partial w(x,y,t)}{{\partial x}}(U_{\infty } {\text{d}}t + {\text{d}}x) + \frac{\partial w(x,y,t)}{{\partial t}}{\text{d}}t \hfill \\ \end{gathered} \right] \hfill \\ \, - \frac{1}{2}\left[ {2w(x,y,t) + \frac{\partial w(x,y,t)}{{\partial x}}{\text{d}}x} \right] \hfill \\ \, = \frac{\partial w(x,y,t)}{{\partial x}}U_{\infty } {\text{d}}t + \frac{\partial w(x,y,t)}{{\partial t}}{\text{d}}t \hfill \\ \end{gathered} $$

(74)

So the normal velocity of the plate unit interacting with the airflow can be expressed as.

$$ \frac{{{\text{d}}w}}{{{\text{d}}t}} = \frac{\partial w}{{\partial t}} + U_{\infty } \frac{\partial w}{{\partial x}} $$

(75)

and the boundary condition of the velocity potential on the lower surface of the plate is expressed as.

$$ \left. {\frac{\partial \Phi }{{\partial z}}} \right|_{{z = \frac{h}{2}}} = \frac{{{\text{d}}w}}{{{\text{d}}t}} = \frac{\partial w}{{\partial t}} + U_{\infty } \frac{\partial w}{{\partial x}} $$

(76)