Piston Theory Research Articles

Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink

Abstract In this paper, a nonlinear energy sink is used to improve the dynamic behavior of a square plate in supersonic flow. The plate is examined with different boundary conditions. The effect of changing the parameters and installation location of nonlinear energy sink on improving the dynamic behavior of the plate has been studied. The governing equations are first obtained using Von Kármán’s plate theory and piston theory, and then discretized using the Rayleigh–Ritz process. These equations are then solved using the fourth-order Runge–Kutta method. Time history curves, phase portraits, Poincaré maps, and bifurcation diagrams were used to investigate the dynamic behavior and impact of nonlinear energy sink. The results show that the dynamic behavior of the plate is very complex in some cases, but with proper use of nonlinear energy sinks, this behavior can be improved.

Optimization of Free Vibration and Flutter Analysis of Composite Plates Using a Coupled Method of Genetic Algorithm and Generalized Differential Quadrature

In this study, the optimization of free vibration and flutter analysis of a composite plate with the implementation of different boundary conditions is investigated. The governing equations of a composite plate are obtained using the first shear deformation theory (FSDT) of plates. The first-order piston theory is utilized to simulate aerodynamic loads, including arbitrary yaw angle over the panel area. The coupled method of genetic algorithm with well-known generalized differential quadrature (GDQ) method is then developed to obtain optimal layup, which led to maximum fundamental frequency and optimized critical flutter aerodynamic pressure. To achieve this aim, the free vibration and flutter analysis of a composite plate will be conducted via the obtained eigenvalue problem by establishing the discretized governing equations and applying boundary conditions. The obtained results are compared and validated with the available results in the literature, which show high accuracy and a fast rate of convergence. Also, the influence of changing in the yaw angle, number of layers, thickness-to-length ratio, boundary conditions and turning symmetric layup to asymmetric layup is studied.

Complex damping influences on the oscillatory/static instability characteristics of heated panels in supersonic airflow

The aim of this study is to investigate the complex damping influences on the oscillatory/static instability characteristics of heated panels in supersonic airflow. Firstly, by assuming a constant, uniform thermal loading and adopting the piston theory, the panel aeroelastic governing equation is obtained. After deriving the panel buckling and vibration modes, the reduced order model can be built and adopted to investigate the system primary instability in the modal coordinates. Then, introducing the modal damping coefficients ratio η>0, the critical parameters of the panel flutter oscillation are theoretically evaluated based on the non-conservative energy balance principle, thus the system instability characteristics can be investigated. The results show that the system oscillatory and static buckling instability characteristics are significantly regulated by the thermal loading and modal damping. For the oscillatory instability, there exists the damping paradox, which is associated with the system energy dissipation efficiency, and can be quantitatively evaluated by the ratio 2η/1+η. The system static buckling instability characteristics is also affected by the modal damping, and this damping destabilization is clarified based on Hamiltonian energy conservation law. The results agree well with that obtained by Routh-Hurwitz criteria, and lead an in-depth understanding of the complex role played by the damping within the non-conservative dissipative systems.

Meshless numerical approach to flutter analysis of rotating pre-twisted nanocomposite blades subjected to supersonic airflow

In this study, the flutter analysis of a rotating pre-twisted functionally graded graphene nanoplatelets reinforced composite (FG GPLRC) blade under supersonic airflow is investigated. The pre-twisted blade is multi-layered and reinforced with graphene nanoplatelets (GPLs) evenly distributed in each layer while the GPL weight fraction changes from layer to layer through the thickness direction. The effective Young's modulus is determined by Halpin-Tsai micromechanical model while the Poisson's ratio and mass density are predicted by Voigt's rule for GPLRC layers. According to assumptions of the first-order shear deformation theory (FSDT), the first-order piston theory and shell theory, the dynamic model of rotating pre-twisted GPLRC blades subjected to supersonic flow is developed. Meshless the improved moving least-square Ritz method (IMLS-Ritz) is used to derive the discrete dynamic equations of rotating pre-twisted GPLRC blades under aerodynamic load. The accuracy of the IMLS-Ritz method for this problem is validated by comprehensive convergence studies and careful comparison studies. A detailed parameter investigation of the effects of GPL distribution configuration, rotation velocity and geometrical parameters on flutter behavior characteristics of GPLRC blades is systematically conducted.

Tailoring the panel inertial and elastic forces for the flutter and stability characteristics enhancement using copper patches

This paper studies the effect of tailoring the inertial and elastic forces on the flutter and the stability characteristics of the aircraft panels using copper patches that are placed at effective locations. The proposed panel is modeled as a multilayer thin composite panel. The equations of panel motion are derived based on Hamilton’s principle. The first-order piston theory is used to calculate the aerodynamic loading. The Bogner Fox-Schmit rectangular plate element is considered in the formulation of the finite element equations. The modal transformation based on the most dominant modes is applied. The validation and consistency studies are conducted using the COMSOL Multiphysics commercial code and some previous analytical and experimental studies. The critical aerodynamic pressures of the tailored panels and the pristine ones are calculated using the eigenvalues and the values of the damping ratios. The effective placement of the copper patches is determined using extensive search methods that are constrained with the weight necessities and the recommended searching focuses. The results highlight the significant effect of the proposed method as the critical aerodynamic pressure of the copper-hosted panel has increased by 120.9%, which corresponds to a maximum increase in weight of 20.8%.

Linear and Nonlinear Free Vibration of a Two-Dimensional Multiferroic Composite Plate Subjected to Magneto-Electro-Thermo-Aerodynamic Loading

Vibration response of a two-dimensional magneto-electro-elastic plate is investigated in this paper. The considered multi-phase plate is rectangular and simply-supported resting on an elastic foundation. The plate is under aerodynamic pressure and subjected to temperature change. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. The nonlinear vibrational analysis of the described plate is considered as an innovation of the present paper, which had not been done before. To model this problem, third-order shear deformation theory along with Gauss’s laws for electrostatics and magnetostatics, first-order piston theory, and Galerkin and multiple times scale methods are used. After validating the presented method, effects of several parameters on the natural frequency, time history, backbone curve, and phase plane diagram of this smart composite plate are obtained. It is found that for plates with constant a/h ratio, electric and magnetic potentials have noticeable effects on the time histories, phase plane diagrams and backbone curves of the plates with smaller thicknesses. In addition, the numerical results of this research indicate that some parameters have considerable effect on the vibration behavior of presented plate. Elastic parameters of the foundation, applied electric and magnetic potentials, and environment temperature are important parameters in this analysis.

Aeroelastic stability and nonlinear flutter analysis of viscoelastic heated panel in shock-dominated flows

In recent years, new composite materials, functional materials and intelligent materials have been widely used in the structural design of supersonic vehicle panels, and these new materials generally have viscoelastic properties. This paper analyzes aeroelastic stability and nonlinear flutter of a viscoelastic heated panel in shock-dominated flows by using theoretical analysis methods and numerical computations, respectively. The governing equations, based on von-Kármán large deflection theory of isotropic flat plates, are constructed with viscoelastic structural damping of Kelvin's model included. Local first-order piston theory is employed in the region before and after shock to estimate the aerodynamic loadings. The results show that the introduction of viscoelastic structural damping reduces the aeroelastic stability of the panel in shock-dominated flows. The presence of the oblique shock reduces the aeroelastic stability of the viscoelastic panel. Generally, as the viscoelastic damping or the shock strength increases, the critical flutter dynamic pressure decreases. However, at the strong shock strength, the critical flutter dynamic pressures and stability boundaries remain unchanged as the viscoelastic damping changes in a range of large viscoelastic damping coefficients. It is also found that for a flexible panel in shock-dominated flows, chaotic motion can be effectively regulated as periodic motions by the additive structural viscoelastic damping. This investigation has important theoretical and practical values in the aeroelastic design of thin panel structures of supersonic and hypersonic vehicles.

Fully Coupled Nonlinear Aerothermoelastic Computational Model of a Plate in Hypersonic Flow

A recent structural model of a plate is extended to include a nonuniform temperature differential that varies in time using a modal expansion. The nonlinear structural plate model with first-order piston theory aerodynamics and cavity dynamics is coupled with the heat equation to form a single system of dynamical equations for the fluid-structural-thermal system. The heat flux from the boundary layer to the plate is modeled using the temperature reference method. A computational scheme is formulated by linearizing the heat flux in terms of two variables: the local aerodynamic downwash due to plate motion and the local wall temperature. The proposed linearization produces a local force equivalent for the pointwise heat flux and substantially simplifies the time-marching scheme of the coupled equations. The model is correlated with results from a recent supersonic wind-tunnel experiment conducted at Air Force Research Laboratory (AFRL). It is found that when a nonlinear response is expected in an experiment (for example, flutter that reaches limit cycle oscillation or a large static buckled deformation), the in-plane boundary stiffness should be measured before the wind-tunnel experiment to fully characterize the experimental model. The heat flux linearization error is quantified, and the modal convergence of the temperature differential is investigated.

A lattice sandwich structure with the active variable stiffness device under aerodynamical condition

Herein, a novel pyramid lattice sandwich structure with the active variable stiffness (AVS) is proposed, which consists of a proportion integration (PI) active controller, a piezoelectric actuator, and a variable stiffness device. The principle of AVS is to process the real-time vibration feedback signal of the lattice sandwich beam by using the PI controller and the piezoelectric actuator to change structural stiffness continuously. The structure has excellent self-adjusting and active vibration reduction effect. Based on the supersonic piston theory, Hamilton's principle, and the assumed modes method, the dynamics equations of the whole structure are established. The vibration reduction and energy dissipation of lattice sandwich beams with AVS and without AVS are analyzed by using the Runge-Kutta algorithm. Moreover, the relevant parameters of PI controller, positive stiffness, negative stiffness, incoming flow velocity, damping coefficient, and installation position are compared and analyzed, respectively. The results indicate that the novel lattice sandwich structure with AVS can achieve ideal vibration reduction effect, and the efficiency can be improved by adjusting the parameters appropriately.

Explicit finite element method for nonlinear flutter analysis of composite panels

This work presents an efficient explicit finite element model for predicting the nonlinear aeroelastic behavior of composite panels in the supersonic regime. The first-order shear deformation plate theory in conjunction with the von Kármán nonlinear strains is used for structural modeling and the linear piston theory is used to model the aerodynamic loads. In order to reduce the computational cost of the simulations, a lumping procedure is employed in the mass and aerodynamic damping matrices of the finite element model. No modal reduction is performed and the central difference method is used for the numerical direct integration in time of the nonlinear equations. The model is verified using results from the literature and it is demonstrated that the lumping procedure drastically reduces the computational cost of the simulations.

Influence of vibration characteristics on the nonlinearity of piston theory

Influence of vibration characteristics on the nonlinearity of piston theory

A Combined Energy Method for Flutter Instability Analysis of Weakly Damped Panels in Supersonic Airflow

A combined energy method is proposed to investigate the flutter instability characteristics of weakly damped panels in the supersonic airflow. Based on the small damping assumption, the motion governing partial differential equation (PDE) of the panel aeroelastic system, is built by adopting the first-order piston theory and von Karman large deflection plate theory. Then by applying the Galerkin procedure, the PDE is discretized into a set of coupled ordinary differential equations, and the system reduced order model (ROM) with two degrees of freedom is obtained. Considering that the panel aeroelastic system is non-conservative in the physical nature, and assuming that the panel exhibits a single period oscillation on the flutter occurrence, the non-conservative energy balance principle is applied to the linearized ROM within one single oscillation period. The obtained result shows that the ROM modal coordinate amplitudes ratio is regulated by the modal damping coefficients ratio, though each modal damping coefficient is small. Furthermore, as the total damping dissipation energy can be eliminated due to its smallness, the He’s energy balance method is applied to the undamped ROM, therefore the critical non-dimensional dynamic pressure on the flutter instability occurrence, and the oscillation circular frequency amplitude relationship (linear and nonlinear form) are derived. In addition, the damping destabilization paradoxical influence on the system flutter instability is investigated. The accuracy and efficiency of the proposed method are validated by comparing the results with that obtained by using Routh Hurwitz criteria.

Nonlinear supersonic flutter of multibay cross-stream shallow shells

This work investigates the interaction between adjacent bays and the effect of the curvature ratio in the nonlinear response of cross-stream curved panels. The study presents a numerical investigation of shallow shells’ linear and nonlinear dynamic behavior exposed to supersonic flutter. The finite element model embedding the aerodynamic piston theory and the Newmark method for direct time-domain integration of the nonlinear aeroelastic equations are used. Results show that the downwind bay of a two-bay cross-stream curved panel oscillates with much higher amplitudes than the equivalent single panel approach for all curvature ratios under consideration. Jump discontinuities in the LCO amplitudes were observed for a range of post-flutter dynamic pressure. The mutual existence of LCO branches in the amplitude envelope was checked through the sensitivity to initial conditions. No sudden jump in the LCO amplitudes was found for the higher curvatures, whereas a more complex behavior is encountered, especially for the downwind bay. Amplitude envelopes show that the nonlinear coupling between adjacent bays severely influences the dynamic behavior of curved panels subjected to cross-stream supersonic flow. Consequently, relying on single panel modeling may be unsafe for structural design and fatigue-life prediction for supersonic panel flutter suppression.

Flutter of variable stiffness composite laminates in supersonic flow with temperature effects

The nonlinear thermal flutter behavior of variable stiffness composite laminates (VSCL) with curvilinear fibers in high supersonic flow is investigated. The first order shear deformation theory (FSDT) combining von Karman large-deflection strain-displacement relations, quasi-steady first-order piston theory aerodynamics and quasi-steady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion. The fiber orientation within a layer is assumed to vary linearly from [Formula: see text] at the center to [Formula: see text] at the vertical edges of the rectangular lamina. The flutter characteristics of variable stiffness composite laminates with different temperature distributions are then studied. The results show that the critical dynamic pressure decreases as [Formula: see text] or [Formula: see text] increases, whereas the limit cycle amplitude increases as [Formula: see text] or [Formula: see text] increases for the same dynamic pressure. The critical dynamic pressure and limit cycle amplitude both increase when the temperature gradient along panel thickness increases. Simple harmonic motions, unharmonic but periodic motions, and chaotic motions can be observed on VSCL under different temperatures. It also turns out that temperature distribution has similar influence on both the critical dynamic pressure and limit cycle amplitude of VSCL.

Nonlinear dynamics of the rotating beam with time-varying speed under aerodynamic loads

In this research, the bifurcation analysis is inspected for a rotatory pre-twisted beam which is subjected to the time varying angular velocity and aerodynamic forces. Aerodynamic loads are obtained by the Piston and Whitehead theories. It is assumed that a time-dependent periodic component is superposed on the angular velocity. The extended Hamilton's principle is utilized to derive the dynamic equations of motion in flapwise and chordwise directions. Then a combination of the Galerkin and multiple scales methods is established to obtain the modulation equations describing amplitude and phases of the interacting modes. Eigen value analysis is performed on the linear part of the equations and the possibility of 1:1 internal resonance condition is examined. The evolution of amplitudes is obtained versus the amplitude and frequency of the periodic angular velocity function. Primary and subharmonic resonance conditions are tuned between the angular velocity frequency and natural frequencies of the beam. Numerical simulations indicate that the nonlinear phenomena such as jump, saturation, and double jumping appear in nonlinear vibration of rotating beam with varying speed.

Finite element model to investigate the dynamic instability of rectangular plates subjected to supersonic airflow

In this paper, a numerical method is presented to study the dynamic behaviour of an isotropic rectangular plate subjected to aerodynamic loads induced by parallel supersonic airflow. A finite element model, based on bi-dimensional polynomial displacement functions and linear Piston theory, is used to study the dynamic behaviour of the solid plate coupled with aerodynamic loads. The developed approach is able to model flat plates and shallow shells in which the fluid–structure coupling at the interface is applied synchronously using a monolithic method. The solid model is based on Sanders’ shell theory. The stiffness and damping matrices that result from application of the aerodynamic load are coupled with those obtained from a structural model and are calculated using analytical integration. By assembling the matrices, the global mass, damping and stiffness matrices for the plate are obtained and then dynamic equations of the governing problem are derived. The eigenvalues of the system are calculated using the equation reduction technique. The critical non-dimensional aerodynamic pressure of the airflow that induces flutter of the structure is determined for various boundary conditions and geometries. The obtained results are compared with other published research works and very good agreement is observed.

Vibration and flutter characteristics of GPL-reinforced functionally graded porous cylindrical panels subjected to supersonic flow

In this study, the vibration and flutter characteristics of the graphene platelets reinforced functionally graded porous cylindrical panels subjected to the supersonic flow are investigated for the first time. The considered panels are constructed of the nanocomposites that made of matrix of open-cell metal foams with graded pore distributions and are reinforced uniformly by the graphene platelets. The typical mechanical properties of porous matrix are determined using open-cell metal foam model, which provides a typical mechanical feature to determine the relationship between coefficients of density and porosity. The effective elasticity modulus of the nanocomposite is determined based on a modified Halpin-Tsai micromechanics model, and the rule of mixture is used to calculate the Poisson's ratio and mass density. A high order shear deformable computational framework is developed with inclusion of the trapezoidal shape factor to eliminate the errors introduced by the curvature of the panels. The standard Lagrange processing in conjunction with the Navier-solution technique, with the help of the first-order piston theory, is applied to derive the equations of motion related to the vibration and flutter characteristics of the cylindrical panel with simple-supported boundary condition. Comparisons of the natural frequencies with the previous works are performed to validate the proposed model. Parameter studies are carried out, and the results in the form of graphs or tables reveal the effects of pore, graphene platelets on the supersonic flutter characteristics of the nanocomposite cylindrical panels subjected to the aerodynamic flow.

Linear aeroelastic analysis of cantilever hybrid composite laminated plates with curvilinear fibres and carbon nanotubes

In this paper, cantilever laminated composite plates, subjected to supersonic airflow, are studied in the linear elastic regime. The considered material is a multiscale three-phase composite, constituted by epoxy resin reinforced with multi-walled carbon nanotubes and curvilinear carbon fibres. The orientation of the fibres is defined using three different functions. A model based on a third-order shear deformation theory and a p-version finite element is applied. Furthermore, a hierarchical approach is applied to characterise the mechanical properties of the composite material. A combination of adequate micromechanics based models, including a modified version of the Halpin-Tsai model, the Rule of Mixtures, a unit cell-based model, and the Chamis model, is implemented. The aeroelastic analysis is performed considering the linear piston theory to evaluate flutter (dynamic instability) and divergence (static instability) in such structures. The improvements achieved through the combination of carbon nanotubes and curvilinear fibres on the instabilities are explored.

Investigation on flutter instability of magnetic-electric-thermo-elastic functionally graded plates in the supersonic airflow with any yawed angle

This paper proposes a general approach to deal with flutter instability of the magnetic-electric-thermo-elastic functionally graded plates (METE-FGPPs) subject to supersonic airflow with any yawed angle. The unified method is established on the basis of the first order shear deformation theory and the supersonic piston theory. Various boundary conditions, two porosity distributions and three temperature distributions are taken into consideration. The material properties and temperature gradient distribution changing continuously along the thickness are evaluated according to Voigt model and the formulations of motion equations of METE-FGPPs can be derived by adopting Rayleigh-Ritz method. In order to satisfy various boundary conditions easily, the physical spring technology is used. The displacement functions can be expressed by using the characteristic orthogonal polynomials to improve the convergence of the numerical calculations. Firstly, the convergence of the present method is confirmed by verifying the convergence of the physical spring and truncated numbers. Then, the versatility, stability and accuracy are verified by some numerical examples and experiments. Finally, the influences of the key parameters, including power law indexes, boundary conditions, magnetic potential, electric voltage, temperature and yawed flow angles etc, on the flutter instability of the METE-FGPPs are discussed in detail.

Nonlinear flutter of tapered and skewed cantilevered plates with curvilinear fiber paths

Aeroelastic stability of tapered/skew variable stiffness composite cantilevered plates are considered in the current study at flutter and post-flutter regions. The variable stiffness behavior is obtained by altering the fiber angles continuously according to a selected curvilinear fiber path function in the composite laminates. Flutter speed, limit cycle oscillations (LCOs), and bifurcation diagrams of tapered/skewed plates are obtained at two different fiber path functions. Nonlinear structural model is utilized based on the virtual work principle. Fully nonlinear Green’s kinematic strain relations are used to account for the geometric nonlinearities and the first order shear deformation theory (FSDT) is employed to generalize the formulation for the case of moderately thick plates including transverse shear effects. One prominent target of the present study is to determine how the variable stiffness parameters affect the nonlinear behavior. Consequently, one may find the best fiber path with improved flutter and post-flutter characteristics for tapered/skew plates in supersonic flow. First order linear piston theory is used to model the aerodynamic loading. In order to get a reliable solution, the Generalized Differential Quadrature (GDQ) method is employed. Moreover, time integration of the equations of motion is carried out using the Newmark’s average acceleration technique. It will be shown that taperness/skewness as well as variable stiffness lamination parameters have significant effects on the aeroelastic stability margins. In addition, the post-critical behavior is found to be periodic or quasi-periodic at all the presented simulations with no specific route to chaos.