Flutter Of Plates Research Articles

Effect of Yaw Angle on Flutter of Rectangular Plates at Low Supersonic Speeds

The stability of the infinite series of thin elastic rectangular plates simply supported along all edges and exposed to a supersonic flow is investigated. The nonzero flow yaw angle with supersonic leading edge is considered. To derive expression for the unsteady aerodynamic pressure distribution over the oscillating plate, potential flow theory was used, and an integrodifferential eigenvalue problem for finding complex eigenvalues was obtained. Flutter boundaries for the first and second modes are computed. Changing of the single-mode and coupled-mode flutter boundaries with the change of the yaw angle was shown. While they are smooth at zero angle, the boundaries become irregular, and additional isolated regions of stability and instability appear when increasing the yaw angle. This irregularity occurs due to the interaction of multiple spanwise modes, which takes place at nonzero yaw angle due to their aerodynamic coupling.

The influence of gap- and chord-widths for multi-box girders: Superposition of flat plate flutter derivatives and section model tests

A two dimensional model based on the superposition of Theodorsen’s flat plate theory generalized to approximate the aeroelastic effects on an arbitrary multi-box bridge girder is developed. The model is compared with wind tunnel test results of a single box girder and two triple-box girders having different gap-widths between the center box and the external boxes. Reasonable agreement between the flutter derivatives obtained from single degree of freedom harmonic forced motion tests and superposition of flat plate flutter derivatives is obtained. For vertical motion, the aerodynamic damping is correlated with the sum of chord-widths of the interconnected boxes. The aerodynamic damping during harmonic torsional motion is correlated with the distance from the elastic axis of the section to the mid-chords of the external boxes. The torsional aerodynamic damping agrees with the superposition of flat plate derivatives while the torsional aerodynamic stiffness disagrees. This means that the flat plate model overestimates the critical flutter wind speed for the present sections, but it can be used to approximate the flutter derivatives: H1, H2, H3 and A2. Empirical results are used to tune the superposed flat plate model, which is then used to approximate the critical flutter wind speed for a 2200 m single span suspension bridge featuring a light-weight triple-box girder. The wind tunnel tests indicate that the light-weight triple-box girder is aerodynamically stable up to at least 70 m/s even though the torsional frequency is lower than the vertical.

Plate Flutter Problem with Mixed Boundary Conditions

Plate Flutter Problem with Mixed Boundary Conditions

Controlling flutter of a cable-stayed bridge with output feedback driven winglets

Active control of flutter of a long span cable-stayed bridge is considered in this study. The control action is provided by winglets actuated using constant/variable gain output feedback controllers. Two FE modeling schemes are adopted, namely one element cables, and multi-element cables with each stay-cable discretized into multiple elements to consider the cable vibration effect. Cable discretization results in additional closely spaced modes. Rational function approximation of flat plate flutter derivatives is considered to model aeroelastic forces on deck and winglets. The output feedback controller yields the desired winglet rotation by using vertical and torsional displacements of the deck. Controllability Gramian and response indices are used to obtain the optimal location of winglets, while the best sensor location is obtained based on a combined index. This yields around 40% increase in the stability limit with winglets covering 12% of the central span, and eight measured outputs for feedback. Response indices reduce substantially if a shift from constant to variable gain is performed at a certain wind speed. Controller design based on the one-element model, when applied to the more realistic multi-element model, results in substantially lesser response attenuation. Hence, cable vibration needs to be consider for the controller design. Keywords: Flutter, Cable-stayed bridge, Winglet, Rational function approximation, Variable gain output feedback, Cable vibration.

Nonlinear dynamics and flutter of plate and cavity in response to supersonic wind tunnel start

The transient response of a plate and a cavity is investigated in a supersonic wind tunnel start experiment where the freestream flow inside the test section reaches turbulent flow at Mach 2. Experimentally measured plate displacement time history shows flutter onset, transition to limit cycle oscillation, and stabilization at a static deformed state during the 30 s run. To analyze and interpret the measured plate response, a fully coupled aero-thermal-acousto-elastic analysis is carried out. A theoretical–computational model is formulated with a nonlinear structural plate model, acoustic pressure equation for the stationary fluid in a cavity, and the first-order Piston Theory aerodynamics. A linear stability analysis is performed that includes the nonlinear added stiffness due to an initial deformation to investigate the combined effects of freestream coupling and temperature differential on system stability. Also, direct time integration of the nonlinear fluid structural equations of motion is performed using experimentally measured flow parameters as inputs. All stability transitions are captured using the theoretical model with good agreement with experiment for transitions from no flutter to flutter/limit cycle oscillations (LCO) although the theoretical LCO amplitude is approximately $$50\%$$ larger than measured. The system’s sensitivity to cavity coupling, temperature differential, thickness calibration, static pressure differential, and turbulent pressure fluctuations are investigated. Lastly, snap-through buckling analyses in response to periodic and quasi-static excitations are conducted.

Computational Experiments to Evaluate the Approaches to the Modeling of Viscoelastic Plates Motion Based on Various Theories

Mathematical and computer modeling of the flutter of elements and units of the aircraft design is an actual scientific problem; its study is stimulated by the failure of aircraft elements, parts of space and jet engines. In view of the complexity of the flutter phenomenon of aircraft elements, simplifying assumptions are used in many studies. However, these assumptions, as a rule, turn out to be so restrictive that the mathematical model ceases to reflect the real conditions with sufficient accuracy. Therefore, results of theoretical and experimental studies are in bad agreement.At present, the problem of panel flutter is very relevant. Improvement of characteristics of military and civil aircraft inevitably requires reducing their weight, and consequently, the rigidity of paneling, which increases the possibility of a panel flutter. The concept of creating the aircraft with a variable shape, which would inevitably lead to a reduction in paneling thickness are actively discussed. Finally, the use of new materials and, in particular, composites, changes physical properties of the panels and can also lead to a flutter.The above-mentioned scientific problem gives grounds to assert that the development of adequate mathematical models, numerical methods and algorithms for solving nonlinear integral-differential equations of dynamic problems of the hereditary theory of viscoelasticity is actual.In connection with this, the development of mathematical models of individual elements of aircraft made of composite material is becoming very important.Generalized mathematical models of non-linear problems of the flutter of viscoelastic isotropic plates, streamlined by a supersonic gas flow, are constructed in the paper on the basis of integral models. To study oscillation processes in plates, a numerical algorithm is proposed for solving nonlinear integro-differential equations with singular kernels. Based on the developed computational algorithm, a package of applied programs is created. The effect of the singularity parameter in heredity kernels on the vibrations of structures with viscoelastic properties is numerically investigated. In a wide range of changes in plate parameters, critical flutter velocities are determined. Numerical solutions of the problem of viscoelastic plate flutter are compared for different models. It is shown that the most adequate theory for investigating a wide class of problems of the hereditary theory of viscoelasticity is the geometric nonlinear Kirchhoff-Love theory with consideration of elastic waves propagation. It is established that an account of viscoelastic properties of plate material leads to 40-60% decrease in the critical flutter velocity.

Supersonic Flutter and Buckling Optimization of Tow-Steered Composite Plates

The supersonic aeroelastic stability of tow-steered carbon reinforced composite panels, in each layer of which the fibers follow curvilinear paths, is assessed. A structural model based on the Rayleigh–Ritz method, combined with the aerodynamic piston theory, is derived to represent the aeroelastic behavior of rectangular plates under different boundary conditions. In this model, the classical lamination theory, considering a symmetric stacking sequence and fiber trajectories described by Lagrange polynomials of different orders, is used. In addition, manufacturing constraints, which impose limitations to the feasible fiber trajectories, and the effect of in-plane loads are considered in the model. Using a multicriteria differential evolution algorithm, numerical optimization is performed for a variety of scenarios and aimed at increasing the flutter and linear buckling stability margins of tow-steered plates, considering the geometrical parameters defining the fiber trajectories on the layers as design variables. The results obtained for the different optimization scenarios are compared, having a composite plate with unidirectional fibers as the baseline and aimed at evaluating the benefits achieved by the optimum tow-steered plates. The results enable quantification of the stability improvements by exploring fiber steering, which has been shown to be beneficial, even in situations in which manufacturing constraints are accounted for.

Non-linear limit cycle flutter of a plate with Hertzian contact in axial flow

This paper is aimed at the nonlinear flutter of a cantilevered plate with Hertzian contact in axial flow. The contact effect is modeled as a nonlinear spring force with both square and cubic nonlinearity. The fluid force is considered as the sum of two parts, one is the reactive fluid force due to plate motion and the other is the resistive fluid force independent on plate motion. The reactive fluid force is derived by solving the bound and wake vorticity with the help of Glauert’s expansions, and the resistive force is evaluated in terms of drag coefficient. The governing nonlinear partial differential equation of the system is discretized in space and time domains by using the Galerkin method. Results show that the plate loses its stability by flutter and then undergoes limit cycle motions due to the contact nonlinearity after instability. The present fluid model is reliable and shows good agreement with other theories archived. A heuristic analysis scheme based on the equivalent linearization method is developed for the analysis of bifurcations and limit cycles. The Hopf bifurcation is either supercritical or subcritical, which is closely dependent on the contact location. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. When the plate experiences limit cycles, with the increasing dynamic pressure there firstly appear the lock-in motions; and then the quasi-periodic motions show up as a breaking of limit cycle by inclusion of a secondary significant frequency with an irrational value of 14π of the dominant limit cycle frequency. Finally the plate undergoes dynamic buckling characterized by quasi-periodic divergence when the dynamic pressure is relatively large.

An analytical solution to the aeroelastic response of a two-dimensional elastic plate in axial potential flow

This paper presents a novel analytical model that predicts the two-way coupled aeroelastic response of a linear elastic plate in axial potential flow, including the effects of plate curvature. The plate deforms in dynamic balance of the inertia, elastic, and aerodynamic forces. Analytical solutions are obtained by deriving the generalized aerodynamic force with respect to the beam eigenfunctions, which are expressed in a Chebyshev polynomial expansion. Exact expressions are derived for the generated lift, thrust and required input power. The derived solution agrees well with the results reported in the literature for plate flutter and flapping wings.

Aero-thermo-elastic flutter analysis of supersonic moderately thick orthotropic plates with general boundary conditions

In this paper, a unified solution is proposed to evaluate the aero-thermo-elastic flutter of supersonic plates with general boundary conditions, in which the classical and non-classical boundary conditions can be dealt with. The Mindlin plate theory and supersonic piston theory are employed to formulate the strain energy, kinetic energy and external work functions of the system. The yawed flow angle effect and temperature dependent properties of materials are also considered in the proposed formulation. The motion equations of the supersonic plate are derived by using the Hamilton's principle and the displacement components of the supersonic plate are expanded by the Fourier series combined with auxiliary functions, which can readily satisfy the classical and elastic boundary conditions. A considerable number of numerical examples concerning the vibration and flutter of the supersonic plate are carried out to show the performance of the described method and the comparisons of the obtained results with those from the existing work and finite element method (FEM) are performed. It is found the proposed method is accurate and versatile to predict the flutter behaviors of the supersonic plate. Finally, the effects of the aerodynamic pressure, yawed flow angle, thermal loads, boundary conditions and temperature dependence of material properties on the flutter characteristics of the supersonic plate are analyzed in detail.

Bridge Deck Flutter Control Using Winglets and Static Output Feedback

Control of wind-induced flutter of a bridge deck is studied using static output feedback. Servomotor-actuated winglets provide the control forces. Deck and winglets are modeled as flat plates and their aerodynamic interaction is neglected. Self-excited wind forces acting on deck and winglets are modeled using the Scanlan–Tomko model, with flat plate flutter derivatives (FDs) obtained from Theodorsen functions. Rogers rational function approximation (RFA) is used for time domain representation of wind forces in order to simplify the stability and control analyses. Control input to servomotors is based on direct feedback of vertical and torsional displacements of deck. Feedback gains that are constant, or varying with wind speed, are considered. Winglet rotations being restricted, flutter and divergence behavior is studied using system eigenvalues as well as responses. Results show that variable gain output feedback (VGOF) control using servomotor driven winglets is very effective. It provides the maximum increase in critical speed and maximum attenuation of response, followed by control with gain scheduling, with the former requiring less input power. Control with constant gain is least effective. Control of deck rotation generally appears to improve with wind speed.

Flutter Analysis of Laminated Curvilinear-Stiffened Plates

This paper proposes the use of curvilinear stiffeners as a mechanism to control supersonic panel flutter. To account for transverse shear deformation, the plate and stiffeners are modeled according to the first-order shear deformation theory and Timoshenko beam theory, respectively. The Chebyshev polynomials are the basis of the deflection and rotation functions in the Ritz method. The aeroelastic load is formulated according to the first-order high-Mach-number approximation to linear potential flow theory. The minimum potential energy and Hamilton’s principle are used to solve the problem. Plots of frequency versus aerodynamic pressure and damping versus aerodynamic pressure are used to determine the critical aerodynamic pressure, and hence to predict flutter of various isotropic and composite, stiffened, and unstiffened plates. The results for the flutter of straight-stiffened plates are validated through comparison with published papers. Several numerical examples are discussed, for which parametric studies for the shape and number of stiffeners and fiber orientation are performed. The flutter mode shapes are also presented. The present study attests that the critical dynamic pressure can be enhanced, and flutter can be successfully suppressed by changing the stiffener’s shape and fiber orientation.

Flutter of clamped orthotropic rectangular plate

Flutter of clamped orthotropic rectangular plate

Flutter suppression of plates using passive constrained viscoelastic layers

Flutter in aeronautical panels is a self-excited aeroelastic phenomenon which occurs during supersonic flights due to dynamic instability of inertia, elastic and aerodynamic forces of the system. In the flutter condition, when the critical aerodynamic pressure is reached, the vibration amplitudes of the panel become dynamically unstable and increase exponentially with time, significantly affecting the fatigue life of the existing aeronautical components. Thus, in this paper, the interest is to investigate the possibility reducing the effects of the supersonic aeroelastic instability of rectangular plates by applying passive constrained viscoelastic layers. The rationale for such study is the fact that as the addition of viscoelastic materials provides decreased vibration amplitudes it becomes important to quantify the suppression of plate flutter coalescence modes that can be obtained. Moreover, despite the fact that much research on the suppression of panel flutter has been carried out by using passive, semi-active and active control techniques, few works have been proposed to deal with the problem of predicting the flutter boundary of aeroviscoelastic systems, since they must conveniently account for the frequency- and temperature-dependent behavior of the viscoelastic material. After the presentation of the theoretical foundations of the methodology, the description of a numerical study on the flutter analysis of a three-layer sandwich plate is addressed.

Флаттер упругой прямоугольной пластинки при сверхзвуковом обтекании и наличии инерционных моментов на кромках.

Флаттер упругой прямоугольной пластинки при сверхзвуковом обтекании и наличии инерционных моментов на кромках.

Damping effect on supersonic panel flutter of composite plate with viscoelastic mid-layer

The panel flutter of composite plate with viscoelastic mid-layer in supersonic airflow is investigated in this study. The hysteric damping model is used to describe the viscoelastic property of the mid-layer material and the first piston theory to model the aerodynamic forces. Hamilton’s principle is employed to derive the partial differential equations governing the vibrations of the laminated composite plate. By Galerkin method the governing partial differential equations are truncated into a set of ordinary differential equations. The critical dynamic pressure for panel flutter has been studied by considering the eigenvalue problem of the set of ODEs. It was concluded that the introduction of aerodynamic damping postpones the threshold of flutter, while the viscoelastic damping of the soft mid-layer presents a phenomenon of dual effect. When the viscoelastic damping value is low, the internal material damping shows detrimental effect to the flutter depression. If the viscoelastic damping is increased further, the flutter resistance of the composite plate will be enhanced. The convergence of the Galerkin method is also discussed in this study. The influences of some parameters to the convergence have been investigated in details.

Flutter of spring-mounted flexible plates in uniform flow

A fluid–structure interaction (FSI) system is studied wherein a cantilevered flexible plate aligned with a uniform flow has its upstream end attached to a spring mounting. This allows the entire system to oscillate in a direction perpendicular to that of the flow as a result of the mounting׳s dynamic interaction with the flow-induced oscillations, or flutter, of the flexible plate. We also study a hinged-free rotational-spring attachment as a comparison for the heaving system. This variation on classical plate flutter is motivated by its potential as an energy-harvesting system in which the reciprocating motion of the support system would be tapped for energy production. We formulate and deploy a hybrid of theoretical and computational modelling for the two systems and comprehensively map out their linear-stability characteristics at low mass ratio. Relative to a fixed cantilever, the introduction of the dynamic support in both systems yields lower flutter-onset flow speeds; this is desirable for energy-harvesting applications. We further study the effect of adding an inlet surface upstream of the mount as a means of changing the destabilising mechanism from single-mode flutter to modal-coalescence flutter which is a more powerful instability more suited to energy harvesting. This strategy is seen to be effective in the heaving system. However, divergence occurs in the rotational system for low spring natural frequencies and this would lead to its failure for energy production. Finally, we determine the power-output characteristics for both systems by introducing dashpot damping at the mount. The introduction of damping increases the critical speeds and its variation permits optimal values to be found that maximise the power output for each system. The addition of an inlet surface is then shown to increase significantly the power output of the heaving system whereas this design strategy is not equally beneficial for the rotational system.

A comparison of numerical and semi-analytical proper orthogonal decomposition methods for a fluttering plate

Both a simple POD (S-POD) method calculating derivatives of the POD modes (POMs) numerically and a semi-analytical POD (P-POD) method with a projection from the POMs to the Rayleigh–Ritz modes are used to investigate a supersonic fluttering cantilever plate. Numerical simulations demonstrate that the S-POD method avoids the complex mathematical derivations due to the mode projection in the P-POD method, and the computational cost is remarkably reduced. In particular, the S-POD method is approximately $$50$$ times faster than the P-POD method. In addition, the simply supported plate is briefly discussed using the P-POD method, which is much more efficient compared with the Galerkin method, especially for the panel with large a/b. The present study will give the researchers crucial suggestions on how to choose the S-POD method or the P-POD method for the panel with different length-to-width ratios or support conditions.

Control of aeroelastic characteristics of cantilever wing with smart materials using state-dependent Riccati equation method

In this work, aeroelastic control of a cantilever wing with embedded shape memory alloy wires and piezoelectric actuators is carried out. Shape memory alloy wires increase the total stiffness of the system by generating an in-plane load, and they are used as passive controllers. Piezoelectric actuators are used for active control of a fluttering plate using the state-dependent Riccati equation method. The effect of shape memory alloy wires and piezoelectric actuators on the aeroelastic flutter condition and limit cycle oscillation amplitude of a rectangular cantilever wing are investigated. Wings are modeled according to the first-order shear deformation plate theory and large deflection Von Karman plate theory. Simulation results are presented, which show that the designed state feedback control law can be used for the stabilization of an aeroelastic system with structural nonlinearities.

INVESTIGATION OF THE NONLINEAR FLUTTER OF VISCOELASTIC PLATES IN A SUPERSONIC FLOW

In this paper the flutter of nonlinear viscoelastic plates in a supersonic flow is investigated. The basic direction of work is consisted in taking into account of viscoelastic material’s properties at supersonic speeds. Quasi-steady aerodynamic panel loadings are determined using piston theory. The vibration equations relatively of deflection are described by Integrо-differential equations in partial derivatives. The plate nonlinear partial integro-differential equation is transformed info a set of nonlinear ordinary IDE through a Bubnov-Galerkin’s approach. The resulting system of IDE is solved through the Badalov-Eshmatov integration method. Critical speeds for plate flutter are defined.