Analysis Of Nonlinear Panel Flutter In Supersonic Flow Based On Reduced-order Model

Panel flutter is a self-excited oscillation induced by the interaction between initial, elasticand aerodynamic forces for the skin or other thin structures of a wing. Panel flutter is not alwaysresulting in a fatal structural failure like wing flutter, but there may exist fatigue failures. Thestudy of panel flutter is of importance in the practical engineering background since so manydisasters were due to the panel flutter. On the other hand, panel flutter is a typical fluid-structureinteraction problem, which is so common in many felds. Hence, panel flutter analysis has thesignifcance of theoretical academic value. However, there are still some problems unsolvedfor panel flutter:1) The classical methods such as the Galerkin and Rayleigh-Ritz methodsare semi-analytical methods, which are high-precision but limited to the simple geometries andboundary conditions. By contrast, to circumvent this problem the fnite element method isapplied, which however, results in dramatic computational cost.2) There are few studies oncantilever plate, and they are limited to the simple limit cycle oscillations. Complex dynamicbehaviors such as chaos have not been investigated. The algorithms for fatigue life predictionare different for complex responses, and hence, the fatigue life is uncontinuous, which is veryimportant for prediction of fatigue life.For solving the two problems aforementioned, this paper performs studies as follows:1)POD-based reduced-order models are constructed to reduce the dimension of the system andimprove the computational effciency in analysis of nonlinear panel flutter;2) The complexdynamics of systems such as chaos are investigated based on the bifurcation theory to providethe important references for the predication of fatigue life. Additionally, the study of chaoscan be used in analysis of structural damage detection. The main contents, conclusions andinnovations of this paper are summarized as follows:1. The nonlinear aeroelastic equations of a rectangular and simply-supported flutteringplate in both two and three dimensions are constructed using von Karman’s large deflectionplate theory and frst-order piston aerodynamic theory. The Galerkin and POD methods areused to discretize the aeroelastic equations, transforming the partial differential equations intoordinary differential equations, which are then solved by the numerical fourth-order Runge-Kutta method. Numerical simulations demonstrate that the POD modes extracted from thechaotic responses are the optimal and global, which need to be calculated once and can be usedto analyze dynamic responses under variable system parameters. This paper frst proposes thatthe POD modes are actually the nonlinear aeroelastic modes of the fluttering panel, and thusthey can reconstruct the system with the lowest dimension. Consequently, the POD method can reduce the computational cost dramatically and hence improve the effciency with goodaccuracy. In use of the POD method, there is no limitation for the geometry or boundarycondition of the fluttering plate, and the only input necessary for the POD method is a set oftime responses of the system. Hence, the POD method is convenient for further study of panelflutter systematically and fundamentally.2. The governing equations are constructed based on the Lagrange equation. TheRayleigh-Ritz approach is adopted to discretize the aeroelastic equations, and the resultingODEs are solved by the fourth-order Runge-Kutta method. We frst investigate the chaos andthe route to chaos for a cantilever plate in supersonic flow. Numerical simulations demonstratethat the route to chaos depends on the panel’s length-to-width ratio a/b. For the square pan-el, a period-doubling of periodic motion occurs before transition to chaos. Another route tochaos is via direct quasi-periodic response for the panel of a/b=2. The panel of a/b=0.5oscillates as chaotic and periodic motions in turn, and with the increase of dynamic pressurebeyond a certain value, the panel stabilizes into a buckling state. Bifurcation diagrams pro-vide the boundaries between various responses, including LCO, buckling, quasi-periodic andchaotic motions, and these are important references for prediction of fatigue life. Referring tothe practical application, chaotic responses of the cantilever plate are the optimal snapshots toconstruct the POD based reduced-order model.3. A fast and effcient POD method is proposed to construct the ROM in analysis of non-linear panel flutter for a cantilever plate in supersonic flow. The fast POD method in this papercalculates the derivatives of the POD modes numerically to avoid the projection from the POMsto the Rayleigh-Ritz modes in the traditional way, which omits the complex math derivations,simplifes the equations construction and improves the computational effciency by two magni-tudes compared to the traditional POD method. We frst apply the POD method to analyze thenonlinear panel flutter of a cantilever plate and prove that the two-dimensional POD modes arethe nonlinear aeroelastic modes of the cantilever plate.4. Considering a simply-supported plate subjected to uniform thermal loads, the aerother-moelastic equations are constructed using the von Karman plate theory, frst-order piston theoryand quasi-steady thermal stress theory. The governing equations are discretized using the PODmethod. This study frst applies the POD method for analysis of complex dynamics such aschaos, and we fnd chaotic transients in the system of fluttering plate. The chaotic transient iseasily mistaken as a chaotic motion for the long period of chaotic motion. However, the PODmethod can identify the chaotic transient phenomenon more accurately and more effciently,since that the POD method converges much faster than the Galerkin method.