Nonlinear Dynamic Responses of Shear-Deformable Composite Panels under Combined Supersonic Aerodynamic, Thermal, and Random Acoustic Loads

Abstract

The skin panel structures of vehicles in supersonic flights are subjected to combined thermal, acoustic, and aerodynamic loads. These combined loads may cause a nonlinear dynamic response of the high-speed flight vehicle’s panel structures. Among these nonlinear dynamic responses, the snapthrough and limit cycle oscillation response seriously affect the fatigue failure of the panel structures. This work investigates the nonlinear dynamic responses using the numerical method, when combined supersonic aerodynamic, thermal, and random acoustic loads are applied to the panel structures simultaneously. To consider the thin and thick composite panels, the first-order shear deformation plate theory (FSDT) and the von Karman nonlinear displacement–strain relationship are applied. The aerodynamic load in the supersonic flow is modeled using the first-order piston theory. The thermal load distribution is assumed constant in the thickness direction of the composite panel. The random acoustic load is represented as stationary white-Gaussian random pressure with zero mean and uniform magnitude over the panels. The nonlinear equations of motion of the composite panel under combined loads are derived using the principle of virtual work and the finite element method. The static displacement, which is the solution of the nonlinear static equation, is calculated using the Newton–Raphson method, and the nonlinear dynamic equation is solved using the Newmark-β time integration method. Using these numerical methods, the nonlinear dynamic analyses are conducted under various loading conditions such as thermal–random acoustic loads, thermal–supersonic aerodynamic loads, and supersonic aerodynamic–thermal–random acoustic loads. Numerical results show the nonlinear dynamic response of the composite thin and thick panels such as snapthrough and limit cycle oscillation responses. Particularly, the snapthrough response is caused when the random acoustic load is applied appropriately to the thermally buckled composite plate when the aerodynamic load is not applied or applied with the relatively small magnitude of the dynamic pressure.

Appendix A

The coefficients, matrices, and vectors for the Newmark-β time integration are defined as follows:

$$ \begin{aligned} a_{1} & = \alpha \Delta t,\,\,\,\,a_{2} = (1 - \alpha )\Delta t,\,\,\,\,a_{3} = \frac{1}{{\beta (\Delta t)^{2} }},\,\,\,\,a_{4} = a_{3} \Delta t, \\ a_{5} & = \frac{1}{\gamma } - 1,\,\,\,\,a_{6} = \frac{\alpha }{\beta \Delta t},\,\,\,\,a_{7} = \frac{\alpha }{\beta } - 1,\,\,\,\,a_{8} = \left( {\frac{\alpha }{\gamma } - 1} \right)\Delta t \\ \alpha & = \frac{1}{2},\,\,\,\,\beta = \frac{1}{4},\,\,\,\,\gamma = 2\beta, \\ \end{aligned}, $$

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$$ {\hat{\mathbf{F}}}^{i + 1} = {\mathbf{F}}_{eff}^{i + 1} + {\mathbf{M}}^{i + 1} (a_{3} {\mathbf{d}}_{t}^{i} + a_{4} {\dot{\mathbf{d}}}_{t}^{i} + a_{5} {\mathbf{\ddot{d}}}_{t}^{i} ), $$

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$$ {\mathbf{F}}_{eff}^{i + 1} = {\mathbf{F}}^{i + 1} + {\mathbf{M}}^{i + 1} \left( {a_{3} {\mathbf{d}}_{t} + a_{4} {\dot{\mathbf{d}}}_{t} + a_{5} {\mathbf{\ddot{d}}}_{t} } \right) + {\mathbf{C}}^{i + 1} \left( {a_{6} {\mathbf{d}}_{t} + a_{7} {\dot{\mathbf{d}}}_{t} + a_{8} {\mathbf{\ddot{d}}}_{t} } \right), $$

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$$ {\mathbf{\ddot{d}}}_{t}^{i + 1} = a_{3} \left( {{\mathbf{d}}_{t}^{i + 1} - {\mathbf{d}}_{t}^{i} } \right) - a_{4} {\dot{\mathbf{d}}}_{t}^{i} - a_{5} {\mathbf{\ddot{d}}}_{t}^{i}, $$

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$$ {\dot{\mathbf{d}}}_{t}^{i + 1} = {\dot{\mathbf{d}}}_{t}^{i} + a_{2} {\mathbf{\ddot{d}}}_{t}^{i} + a_{1} {\mathbf{\ddot{d}}}_{t}^{i + 1} . $$

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