This paper investigates the parametric instability of a panel (beam) under high speed air flows and axial excitations. The idea is to affect out-of-plane vibrations and aerodynamic loads by in-plane excitations. The periodic axial excitation introduces time-varying items into the panel system. The numerical method based on Floquet theory and the perturbation method are utilized to solve the Mathieu-Hill equations. The system stability with respect to air/panel density ratio, dynamic pressure ratio, and excitation frequency are explored. The results indicate that panel flutter can be suppressed by the axial excitations with proper parameter combinations.

Panel (beam) flutter usually occurs when high speed objects move in the atmosphere, such as flight wings [

Dowell [

Panel is usually excited by in-plane loads resulting from the vibrations generated and/or transmitted through the attached structures and dynamics components when experiencing aerodynamic loads. If the in-plane load is time dependent, the system becomes time-varying. The topic of dynamic stability of time-varying systems attracts many attentions. Iwatsubo et al. [

Nevertheless, the published investigations on the parametric stability of the flutter panel (beam) with the periodically time-varying system stiffness due to axial excitations are scarce. This paper is to explore the coactions of time-varying axial excitations and aerodynamic loads on panel (beam) and conduct parameter studies. The stability analysis is executed first by Floquet theory numerically and then by Hsu’s method analytically for approximations.

The configuration considered herein is an isotropic thin panel (beam) with constant thickness and cross section. As shown in Figure

Simply supported panel (beam) subjected to an air flow and an axial excitation.

The total kinetic energy of the penal due to lateral displacements is

The total potential energy of the panel due to lateral displacements is

For material viscous damping, a Rayleigh dissipation function is defined as

The aerodynamic load is expressed by using the classic quasisteady first-order piston theory [

The flow goes against the lateral vibrations of the panel, so the nonconservative virtual work from aerodynamic load is negative and its expression is

For a simply supported panel, the modal expansion of

The stiffness matrix

The elements of the N.D. coefficient matrices in (

Here,

Due to the periodic axial excitations, (

To implement Hsu’s perturbation method, the time-invariant part of the system stiffness matrix,

The damping matrix and the time-varying stiffness matrix resulting from the axial excitation are assumed to be small quantities relative to the time-invariant system stiffness for better predictability through Hsu’s method. The standard form in Hsu’s method is obtained by separating the regular and perturbed items in (

For the simple demonstrations of the model and the solving process developed above, only the first two modes are considered so that the closed-form stability solutions can be obtained. The axial excitation force considered here is a single frequency cosine function:

The N.D. natural frequencies due to materials and aerodynamic loads are plotted in Figure

N.D. natural frequency variations with respect to dynamic pressure ratio: Ma = 2, ^{−6}.

The system stability with respect to the axial excitation frequency and the dynamic pressure ratio is shown in Figure

Stability plot for the flutter panel with respect to axial excitation frequency and dynamic pressure ratio: Ma = 2, ^{−6}, ^{−5}, and

The system stability with respect to the axial excitation frequency and the air/panel density ratio is shown in Figure

Stability plot for the flutter panel with respect to axial excitation frequency and air/panel density ratio: Ma = 2, ^{−6}, ^{−6}, and

It can be observed in Figures

Stability plot for the flutter panel with respect to air/panel density ratio and dynamic pressure ratio: Ma = 2, ^{−6}, and

This paper investigates the parametric stability of the panel (beam) under both aerodynamic loads and axial excitations. The dimensionless equation-of-motion is derived, including material viscous damping, axial excitation, and aerodynamic load. The eigen-analyses based on the first two modes are taken as examples to explore the stability properties of the flutter panel system with an axial single frequency cosine excitation. Both numerical FTM method and analytical perturbation method solve the problem and their results match each other very well.

The panel may flutter under high-speed air flows when its out-of-plane dynamics couples with the aerodynamic loads. The parameter study was conducted for the system instability zones with respect to axial excitation frequency, air/panel density ratio, and air/panel dynamic pressure ratio. Different from the static axial force, this paper introduces a periodic axial excitation that brings the system into the time-varying domain. The axial excitation force could increase the panel stiffness locally to overcome aerodynamic loads when interacting with the out-of-plane vibrations. The study results in this paper indicate that the system is stable under the combinations of the proper excitation frequency and certain air/panel density ratio and dynamic pressure ratio.

The perturbation method developed in this paper saves lots of computations, which can help understand the flutter phenomenon of the panel with axial excitations more efficiently.

The authors declare that they have no competing interests.