Flutter of sandwich panels at supersonic speeds

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Virginia Polytechnic Institute


Panel flutter is an important design consideration for vehicles traveling at supersonic speeds. Most theoretical analyses of panel flutter consider the motion of the panel to be described adequately by classical thin plate theory. In such a theory, transverse shear deformations are neglected which is a reasonable assumption for solid plates. For a sandwich panel, neglect of transverse shear deformations may not be a good assumption in flutter analysis inasmuch as studies have indicated that the vibration and buckling behavior of such panels can be affected significantly by shear deformations. An analysis which considers transverse shear deformations is presented in order to determine the effect of finite transverse shear stiffness on the flutter behavior of sandwich plates.

The sandwich theory used is due to Libove and Batdorf. The essential feature of this theory is that straight line elements perpendicular to the undeformed middle surface remain straight and of the same length but are not necessarily perpendicular to the deformed middle surface. The aerodynamic loading on the panel is given by two-dimensional static aerodynamics. The adequacy of such an approximation has been demonstrated for panels rigid in shear and the mathematical simplicity allows closed-form solutions to be found.

The analysis proceeds from consideration of the equilibrium of an infinitesimal element. If equations are written in terms of the deflection and two shear deformations for equilibrium of forces in the z direction and equilibrium. of moments about the x and y axis, three differential equations involving the three unknown displacements are obtained. This system of equations is of sixth order with constant coefficients, but for simple support boundary conditions on the streamwise edges an exact solution can be obtained. The associated characteristics equation can be factored into a fourth degree equation and a second degree equation; thus an analytical expression can be obtained for the characteristic roots.

The solution just described is a general solution for the motion of a sandwich panel simply supported along streamwise edges and subject to inertia loading and aerodynamic forces given by two-dimensional static aerodynamics. Any combination of boundary conditions consistent with the sandwich plate theory used can be applied at the leading and trailing edges. Two cases are considered: simply supported leading and trailing edges and clamped leading and trailing edges. With the use of either set of boundary condition, a transcendental equation is obtained which is satisfied by various combinations of frequency and dynamic pressure. The dynamic pressure necessary to cause the frequency to become complex corresponds to divergent oscillatory motion or flutter.

Values of the flutter dynamic pressure have been calculated as a function of length-width ratio for a large range of shear stiffness. For infinite shear stiffness the results agree with those established by previous investigators. As shear stiffness decreases, the flutter dynamic pressure usually decreases also. An unusual result of the analysis is that at low length-width ratios, a clamped panel has a lower flutter dynamic pressure than a simply supported panel even though the vibration frequencies are higher for the clamped panel. Results are not presented for panels with normal inplane loadings but they can be obtained from the equations given. The analysis shows that flutter is independent of normal inplane loadings perpendicular to the flow direction just as was found for panels rigid in shear.

An approximate two-mode Galerkin solution to the problem has been obtained by a previous investigator. Comparison of the exact solution to the approximate solution shows the approximate analysis to be in increasing error as length-width ratio increases or shear stiffness decreases.