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Aeroelastic stability analysis of a flexible panel subjected to an oblique shock based on an analytical model

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Abstract

For a deeper understanding of the physical phenomenology of shock-induced panel flutter, a theoretical model for analyzing aeroelastic stability of flexible panels subjected to an oblique shock has been developed. The von Kármán large deflection plate theory is used to account for the geometrical nonlinearity, and local first-order piston theory is employed to predict unsteady aerodynamic loading in shock-dominated flows. In order to consider the nonuniform static pressure differentials induced by the shock, we regard the final total displacement of the panel as the superposition of static deformation and dynamic displacement, which is in accord with the actual situation of physicality. The static deformation is obtained by solving the static aeroelastic equation, and then it is introduced into the dynamic aeroelastic equations in the form of the stiffness by the nonlinear induced loading. According to Lyapunov indirect method and the Routh–Hurwitz criterion, a theoretical solution for the aeroelastic stability boundaries of the flexible panel subjected to an oblique shock is derived. The results show that the presence of an impinging shock wave is found to produce panel flutter that is characteristically different from that with the shock-free condition. For a complex aeroelastic system in shock-dominated flows, there exists a game between the static pressure differential and the unsteady dynamic pressure. When the dynamic pressure gains the upper hand, the presence of shock reduces the aeroelastic stability of the panel. In contrast, when the static pressure difference has the upper hand, the presence of shock will enhance stability of the panel. The dimensionless aerodynamic parameter, which is the ratio of the non-dimensional static pressure to the non-dimensional dynamic pressure of the incoming flow, plays a significant role in aeroelastic stability of panels in shock-dominated flows. For different dimensionless aerodynamic parameters, the flutter boundaries will present different characteristics. As this dimensionless aerodynamic parameter increases, the non-dimensional critical flutter dynamic pressure will increase monotonously.

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Abbreviations

\(a\) :

Speed of sound

\(a_{\infty }\) :

Free stream sound velocity

\(D\) :

Plate stiffness

\(E\) :

Elasticity modulus

\(h\) :

Plate thickness

\(l\) :

Plate length

\({\text{Ma}}\) :

Mach number

\(M_{{\text{R}}}\) :

Dimensionless aerodynamic parameter

\(N\) :

Mode number

\(N_{x}\) :

In-plane force

\(q = \rho _{\infty } U_{\infty }^{2} /2\) :

Dynamic pressure

\(R_{x}\) :

Non-dimensional in-plane force

\(t\) :

Physical time

\(U\) :

Flow velocity

\(W\) :

Non-dimensional plate delection

\(W_{{\text{S}}}\) :

Non-dimensional static deformation

\(W_{{\text{D}}}\) :

Non-dimensional dynamic displacement

\(w\) :

Plate deflection

\(x\) :

Streamwise coordinate

\(\gamma\) :

Air specific heat ratio

\(\lambda\) :

Non-dimensional dynamic pressure, \(2ql^{3} /{\text{Ma}}D\)

\(\upsilon\) :

Poisson’s ratio

\(\rho _{{\text{m}}}\) :

Density of plate

\(\rho\) :

Air density

\(\rho _{\infty } ,U_{\infty } ,p_{\infty } ,M_{\infty }\) :

Free stream air density, velocity, pressure, Mach number

\(\sigma\) :

Shock angle

u:

Upper surface of the panel

d:

Lower surface of the panel

l:

Left side of the shock

r:

Right side of the shock

loc:

Local condition

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants No. 11732013) and the Project (NNW2019ZT3-A15).

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  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Liuqing Ye, Zhengyin Ye, Kun Ye & Jie Wu

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Ye, L., Ye, Z., Ye, K. et al. Aeroelastic stability analysis of a flexible panel subjected to an oblique shock based on an analytical model. Acta Mech 232, 3539–3564 (2021). https://doi.org/10.1007/s00707-021-03023-3

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