Critical Dynamic Pressure Research Articles

Flutter stabilization of cantilevered plates using a bonded patch

In recent years, many researchers have studied active vibration suppression of fluttering plates using piezoelectric actuators. Lots of these researchers have focused on optimal placement of piezoelectric patches to obtain maximum controllability of the plate. Although mass and stiffness characteristics of bonded patches can alter the aeroelastic behavior of fluttering plates, few of the investigators have considered the effect of the mentioned parameters in the optimization process. This paper investigates the effect of a bonded patch on the aeroelastic behavior of cantilevered plates in supersonic flow and examines the optimal location of the patch for the best controllability performance. For mathematical simulation of the structure, linear von Karman plate theory along with first-order piston theory is employed. The results obtained through this study reveal that a bonded patch with a small mass ratio can change the system critical dynamic pressure significantly. The maximum raise of the critical dynamic pressure is acquired when the bonded patch is placed on the leading edge of the plate. A variation of the system’s aerodynamic characteristics, subsequently, influences the control performance of the bonded patch and alters the optimal patch location.

Nonlinear Model Reduction of von Kármán Plates Under Quasi-Steady Fluid Flow

A reduced-order model for von Karman plates, using a method of quadratic components, is presented in this paper. The method of quadratic components postulates the full kinematics of the plate as consisting of a combination of linear and quadratic components. This reduced model is then discretized in space with a Galerkin approximation and in time with an implicit integration scheme. The plate is coupled with a fluid flowing over it at a supersonic speed using a quasi-steady pressure model commonly referred to as piston theory. A static loading example is used to validate the model reduction of the plate with respect to other numerical and approximate solutions. The limit cycles of the plate coupled with piston theory are calculated via a cyclic method, and the results from parameter studies are compared to classical results. A previously unexplored regime of the limit cycle amplitudes is investigated; a second, coexisting, limit cycle is found that yields a discontinuity in the limit cycle amplitudes at a critical dynamic pressure.

Nonlinear flutter analysis of stiffened composite panels in supersonic flow

The flutter instability of stiffened composite panels subjected to aerodynamic forces in the supersonic flow is investigated. Based on Hamilton’s principle, the aeroelastic model of the composite panel is established by using the von Karman large deflection plate theory, piston theory aerodynamics and the quasi-steady thermal stress theory. Then, using the finite element method along with Bogner-Fox-Schmit elements and three-dimensional beam elements, the nonlinear equations of motion are derived. The effect of stiffening scheme on the flutter critical dynamic pressure is demonstrated through the numerical example, and the nonlinear flutter characteristics of stiffened composite panels are also analyzed in the time domain. This will lay the foundation for design of panel structures employed in aerospace vehicles.

Effect of a Bonded Patch on Aeroelastic Behavior of Cantilevered Plates

In recent years, many researchers have studied vibration suppression of fluttering plates using piezoelectric actuators. Lots of these researchers have focused on optimal placement of piezoelectric patches to obtain maximum controllability. Although mass and stiffness characteristics of bonded patches can alter aeroelastic behavior of fluttering plates, few of them considered the effect of the mentioned parameters in optimization process. This paper investigates effect of a bonded patch on aeroelastic behavior of cantilevered plates in supersonic flow. For this purpose, critical dynamic pressure and limit‐cycle oscillations of the system are studied. Von Karman plate theory along with first order piston theory is employed for mathematical simulation of the system. Obtained results reveal that a bonded patch with a small mass ratio can change the system critical dynamic pressure significantly, where the main part of the variations is resulted from the added mass of the bonded patch. The maximum raise of dynamic pressure is acquired when the patch is placed on the plate’s leading edge. The results show that mass and stiffness characteristics of bonded piezoelectric patches can have a great impact on aeroelastic performance of fluttering plates. Therefore, these parameters must be considered as effective factors for optimal placement of piezo‐actuators.

Limit-Cycle Oscillation of Shape Memory Alloy Hybrid Composite Plates at Elevated Temperatures

A traditional composite plate impregnated with pre-strained shape memory alloy fibers and subject to combined thermal and aerodynamic loads is investigated to demonstrate the effectiveness of using the SMA fiber embeddings in improving the static and dynamic response of composite plates. The problems investigated can be categorized into: thermal buckling subject to aerodynamic loading, linear flutter boundary at elevated temperatures, nonlinear flutter limit-cycle, and chaotic oscillations at elevated temperatures. A nonlinear finite element model based on the von Karman strain displacement relations and first-order shear deformable plate theory is derived. Aerodynamic pressure is modeled using the quasi-steady first-order piston theory. The governing equations are obtained using the principle of virtual work based on thermal strain being a cumulative physical quantity. Newton-Raphson iteration is employed to obtain the static aero-thermal large deflection at each temperature step and the dynamic response at each time step of the Newmark numerical integration scheme. A frequency domain solution is presented for predicting the flutter boundary at elevated temperatures, while the time domain method along with modal transformation is applied to numerically investigate periodic, non-periodic, and chaotic limit-cycle oscillations. The results show that the critical buckling temperature of the plate is greatly increased, and hence the thermal post-buckling deflection is suppressed by using SMA fiber embeddings. The SMA fiber embeddings caused an increase in the critical dynamic pressure at elevated temperatures, and enlargement of the static flat and dynamically stable region of the panel.

Thermal Buckling and Flutter Behavior of Shape Memory Alloy Hybrid Composite Shells

A new nonlinear finite element formulation is presented to predict the thermal buckling and flutter boundaries of shape memory alloy hybrid composite cylindrical panels at elevated temperatures. The governing equations are obtained using Marguerre curved-plate theory and the principle of virtual work. The effect of large deflection is included in the formulation through the von Karman nonlinear strain-displacement relations. To account for the temperature dependence of material properties, the thermal strain is stated as an integral quantity of the thermal expansion coefficient with respect to temperature. The aerodynamic pressure is modeled using the quasi-steady first-order piston theory. The Newton―Raphson iteration method is employed to obtain the nonlinear thermal postbuckling deflections, and a frequency-domain solution is presented to predict the critical dynamic pressure at elevated temperatures. Numerical results are presented to illustrate the effect of shape memory alloy fiber embeddings, temperature rise, height-to-thickness ratios, and boundary conditions on the panel response.

Transonic Limit Cycle Flutter of High-Aspect-Ratio Swept Wings

This paper is motivated by the observed transonic limit cycle flutter of a high-aspect-ratio swept wing tested at DLR, German Aerospace Center in Gottingen. We show that the aeroelastic mechanism responsible for the limit cycle flutter behavior can be explained in terms of the structural washout effect, which is strongly stabilizing at transonic Mach numbers. This limits the energy transfer rate to the wing, resulting in a limit cycle flutter mode in which bending and torsion are almost perfectly in phase, and the streamwise wing-section motion resembles a single-degree-of-freedom torsion mode with an axis of rotation forward of the leading edge. For swept wings, the apparent pitching motion of the wing sections arises naturally through the structural washout and is present in the first bending mode that enters limit cycle flutter. The aeroelastic washout effect may produce counterintuitive results, and increasing the dynamic pressure may actually be stabilizing. In such a case, decreasing the dynamic pressure would increase the limit cycle amplitude, and high-altitude transonic flutter becomes a real possibility. The results demonstrate the importance of using a nonlinear structural model in these calculations. If a linear structural model is used, the critical dynamic pressure at limit cycle flutter onset is overpredicted by a factor of nearly 3, and the predicted flutter mode is at a frequency much higher than was observed during wind-tunnel tests.

Thermal Buckling and Nonlinear Flutter Behavior of Functionally Graded Material Panels

The nonlinear flutter and thermal buckling of an functionally gradient material panel under the combined effect of elevated temperature conditions and aerodynamic loading is studied. A nonlinear finite element model based on the first-order shear deformable plate theory and von Karman strain-displacement relations is adopted. The governing nonlinear equations are obtained using the principal of virtual work, adopting an approach based on the thermal strain being a cumulative physical quantity to account for temperature-dependent material properties. The aerodynamic pressure is modeled using the quasi-steady first-order piston theory. This system of nonlinear equations is solved by the Newton-Raphson numerical technique. It is found that the temperature increase has an adverse effect on the functionally gradient material panel flutter characteristics through decreasing the critical dynamic pressure. Decreasing the volume fraction enhances flutter characteristics, but this is limited by structural integrity aspect. The presence of aerodynamic flow results in postponing the buckling temperature and in suppressing the postbuckling deflection, and the temperature increase gives way for higher limit-cycle amplitude.

Non-linear panel flutter for temperature-dependent functionally graded material panels

The non-linear flutter and thermal buckling of an FGM panel under the combined effect of elevated temperature conditions and aerodynamic loading is investigated using a finite element model based on the thin plate theory and von Karman strain-displacement relations to account for moderately large deflection. The aerodynamic pressure is modeled using the quasi-steady first order piston theory. The governing non-linear equations are obtained using the principal of virtual work adopting an approach based on the thermal strain being a cumulative physical quantity to account for temperature dependent material properties. This system of non-linear equations is solved by Newton–Raphson numerical technique. It is found that the temperature increase has an adverse effect on the FGM panel flutter characteristics through decreasing the critical dynamic pressure. Decreasing the volume fraction enhances flutter characteristics but this is limited by structural integrity aspect. The presence of aerodynamic flow results in postponing the buckling temperature and in suppressing the post buckling deflection while the temperature increase gives way for higher limit cycle amplitude.

曲面パネルの流体構造連成計算法によるフラッタ解析

A computational program using a CFD-CSD coupled method has been developed to study the problem of low supersonic curved panel flutter. So far the supersonic aerodynamic model has been commonly employed to compute flutter boundaries. However, the present CFD-CSD coupled method makes it possible to analyze flutter boundaries at all speeds including transonic speeds. The computed flutter dynamic pressure for M∞=3 agrees with others’ results. A curved panel flutter were analyzed at a low supersonic speed, M∞=1.2. The flutter for small curvature panels is first-mode flutter, which is similar to flat panel flutter, whereas it becomes higher-mode flutter for larger curvature plates. In this study, flutter boundaries for an access panel have also been simulated. Its critical dynamic pressure becomes significantly lower compared with the case of a panel with its all edges simply supported.

Flutter Suppression of Cantilevered Plate Wing using Piezoelectric Materials

The supersonic flutter suppression of a cantilevered plate wing is studied with the finite element method and the quasi-steady aerodynamic theory. We suppress wing flutter by using piezoelectric materials and electric devices. Two approaches to flutter suppression using piezoelectric materials are presented; an energy-recycling semi-active approach and a negative capacitance approach. To assess their flutter suppression performances, we simulate flutter dynamics of the plate wing to which piezoelectric patches are attached. The critical dynamic pressure drastically increases with our flutter control using a negative capacitor.

Robust structural optimization of plate wing corresponding to bifurcation in higher mode flutter

We propose a robust structural optimization of a plate wing that considers the bifurcation in higher mode flutter, which affects the robustness of an optimum design and the convergence of an optimization. In the optimization, the thickness distribution of a delta wing is considered to improve the critical dynamic pressure of supersonic flutter under the constant total mass. The convergence is improved by detaching the distance of eigenvalues between adjacent modes by the positive constant value in the optimization. Consequently, the critical dynamic pressure of the optimum design is six times larger than that of the uniform design. Moreover, to improve the robustness, the constraints are proposed to keep eigenvalues between adjacent modes apart by the specified distance which is composed of the sensitivity of the distance of eigenvalues. Numerical results indicate that the robustness of the optimum design is quantitatively obtained with the constraints.

Thermal post-buckling and flutter characteristics of composite plates embedded with shape memory alloy fibers

It is investigated that the composite plate embedded with shape memory alloy (SMA) fibers is subject to the aerodynamic and thermal loading in the supersonic region. The nonlinear finite element equations based on the first-order shear deformation plate theory (FSDT) are formulated for the laminated composite plate embedded with SMA fibers (SMA composite plate). The von Karman strain–displacement relation is used to account for the large deflection. The incremental method considering the influence of the initial deflections and initial stresses is adopted for the temperature-dependent material properties of SMA fibers and composite matrix. The first-order piston theory is used for modeling aerodynamic loads. This study shows the effect of the SMA on the critical temperature, thermal post-buckling deflection, natural frequency and critical dynamic pressure of the SMA composite plate.

Effects of hysteretic and aerodynamic damping on supersonic panel flutter of composite plates

The governing equation for the finite element analysis of the panel flutter of composite plates including structural damping is derived from Hamilton's principle. The first order shear deformable plate theory has been applied to structural modelling so as to obtain the finite element eigenvalue equation. The unsteady aerodynamic load in a supersonic flow is computed by using the linear piston theory. The critical dynamic pressures for composite plates have been calculated to investigate the effects of structural damping on flutter boundaries. The effects are dependent on fiber orientation because flutter mode can be weak or strong in the fiber orientation of composite plates. Structural damping plays an important role in flutter stability with low aerodynamic damping but would not affect the flutter boundary with high aerodynamic damping.

117 理論感度解析を用いた超音速フラッタ限界動圧向上のための平板翼の構造最適化

This paper investigates the structural optimization of plate wing for supersonic flutter. The analytical sensitivity of the critical dynamic pressure on supersonic flutter is formulated by the theoretical sensitivity analysis. The sensitivity is implemented in the steepest descent method to investigated the performance of the optimization. A delta wing model is considered for a numerical example. The numerical results indicate that the computational cost is reduced and the critical dynamic pressure is improved with respect to the results with the evolutionary structural optimization.

225 超音速フラッタ特性と構造質量を同時に考慮した平板翼の多目的複合領域構造最適化

This paper investigates the properties of the optimum structural design for supersonic flutter of the plate wing. The optimization method is based on the Evolutionary Structural Optimization method. In order to consider the properties of the optimum structural design, the optimum mass distribution, the optimum stiffness distribution and the optimum equivalent thickness distribution are compared in the numerical experiments. Numerical results show that the equivalent thickness distribution is the most suitable for the design variable of the structural optimization for the supersonic flutter. Numerical results also show that the optimum stiffness distribution has a relationship with the strain energy density of the 2nd flutter mode shape near the critical dynamic pressure.

Methodology for Supersonic Panel Flutter Analysis of Thermal Protection System Seals

A methodology is presented for the e utter analysis of the seal of the thermal protection system panel of the X-33 Advanced Technology Demonstrator test vehicle. The seal is simulated as a two-dimensional cantilevered panel with an elastic stopper, which is modeled as an equivalent spring. This cantilever beam ‐spring model under the aerodynamic pressure at supersonic speeds becomes an impact nonlinear dynamic system. The e utter analysis of the seal is, thus, carried out using time-domain numerical simulation with a displacement stability criterion. The e utter boundary of the seal is further verie ed with a family of three traditional and one nontraditional panel e utter models. The frequency-domain method that applies eigenanalysis on the traditional panel e utter problem was used. The results showed that the critical dynamic pressure could be more than doubled with the properly chosen materialforthebase stopper. Theproposed methodology can beeasily extended to three-dimensional panel seals with e ow angularity. Nomenclature [aa], [Aa] = element and system aerodynamic ine uence matrices b = beam width Ca = aerodynamic damping coefe cient fdg = element nodal displacement vector E = Young’ s modulus Fe

209 平板翼における超音速フラッター特性/構造の創発的最適化

This paper investigates the structural optimization of plate wing for supersonic flutter. The optimization is performed to improve the critical dynamic pressure of the wing model by extending the Evolutionary Structural Optimization method. A delta wing with elastic plate is considered for the analysis model. The objective function is the difference between the variation of the first and second eigenfrequency in the vicinity of the critical dynamic pressure. Numerical results show that the critical dynamic pressure has been improved by the proposed optimization procedure.

High-Supersonic/Hypersonic Flutter of Prismatic Composite Plate/Shell Panels

A spline e nite strip modal method is developed for e utter instability analysis of prismatic composite plate/shell panels exposed to high-supersonic/hypersonic e ow. The structural model is formulated within the context of e rstorder shear deformation shell theory, and the aerodynamic pressure is evaluated using the two-dimensional static aerodynamicapproximation.Thethermaleffectduetoaerodynamicheatingisnotconsidered.Theanalysisconsists of two stages. In the e rst stage, the free-vibration problem of the panel is performed by using a superstrip concept and a multilevel substructuring technique. The natural frequencies and vibration modes are, thus, obtained by employing the highly reliable bisection Sturm sequenceapproach with great accuracy and efe ciency. In thesecond stage, the obtained natural frequencies and the corresponding mode shapes are utilized to reduce considerably the size of the original aeroelastic eigenvalue problem, which is then solved to determine the critical dynamic pressure and the corresponding e utter frequency. A variety of numerical experiments, which are conducted to validate and test the performance of the formulation, demonstrates the reliability, efe ciency, accuracy, and versatility of the method.

ANALYTIC SENSITIVITIES IN COMPOSITE WING-SHAPE YNTHESIS PART TWO: PANEL FLUTTER CONSTRAINTS

ABSTRACTThe equivalent plate modeling formulation of composite wings, described in Part One, is extended in this paper to include the explicit analytic expressions for terms of the mass and aerodynamic matrices. Analytic sensitivities of the mass and aerodynamic matrices with respect to panel shape, thickness and fiber orientations are derived. Before proceeding in the study of analytic sensitivities using the derived expressions, the accuracy of its panel flutter analysis was numerically tested for simply supported panels of trapezoidal and skewed shape, of isotropic and composite materials and variable fiber orientations. The nondimensional critical dynamic pressure and critical frequency of flutter are compared with the results of finite element flutter analysis of composite skew panels by Chowdary, T.V.R. and al given in Computers and Structures, (1996). The correlation of the results is good for moderate in-plane loading. Accuracy of the approximations is studied for the integration with nonlinear programming/approximation concept aeroelastic design synthesis methodology.