Static Stability Analysis of a Thin Plate with a Fixed Trailing Edge in Axial Subsonic Flow: Possio Integral Equation Approach

TL;DR

This paper develops a new integral equation approach to analyze the static stability of thin plates with fixed trailing edges in axial subsonic flow, deriving explicit formulas and stability conditions, including effects of piezoelectric coupling.

Contribution

It introduces a novel derivation of the Possio integral equation for stability analysis and provides explicit formulas for divergence speeds of plates, including piezoelectric effects.

Findings

Free-pinned plates are statically unstable.

Explicit divergence speed formulas for free-clamped plates.

Piezoelectric plates exhibit enhanced static stability.

Abstract

In this work, the static stability of plates with fixed trailing edges in axial airflow is studied using the framework of Possio integral equation. First, we introduce a new derivation of a Possio integral equation that relates the pressure jump along thin plates to their downwash based on the linearization of the governing equations of an ideal compressible fluid. The steady state solution to the Possio equation is used to account for the aerodynamic forces in the steady state plate governing equation resulting in a singular differential-integral equation which is transformed to an integral equation. Next, we verify the solvability of the integral equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. Then, we derive explicit formulas for the characteristic equations of free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. After that, we move to derive analytically flow speed intervals that correspond to static stability regions for free-clamped plates. We also resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.