Shokrollahi Saeed,Shafaghat Salman

Department of Aerospace Engineering,Malek Ashtar University of Technology,Tehran,Iran

Flutter analysis of hybrid metal-composite low aspect ratio trapezoidal wings in supersonic flow

Shokrollahi Saeed*,Shafaghat Salman

Department of Aerospace Engineering,Malek Ashtar University of Technology,Tehran,Iran

Assumed mode method;Flutter;Hybrid trapezoidal plate;Mach box method;Supersonic flow

An effective 3D supersonic Mach box approach in combination with non-classical hybrid metal-composite plate theory has been used to investigate flutter boundaries of trapezoidal low aspect ratio wings.The wing structure is composed of two main components including aluminum material(in-board section)and laminated composite material(out-board section).A global Ritz method is used with simple polynomials being employed as the trial functions.The most important objective of the present research is to study the effect of composite to metal proportion of hybrid wing structure on flutter boundaries in low supersonic regime.In addition,the effect of some important geometrical parameters such as sweep angle,taper ratio and aspect ratio on flutter boundaries were studied.The results obtained by present approach for special cases like pure metallic wings and results for high supersonic regime based on piston theory show a good agreement with those obtained by other investigators.

1.Introduction

Aeroelastic analyses of lifting surfaces are an important step in aircraft structural design procedures and a necessary issue in getting flight certification,the so-called flutter clearance.In the early design stages,it is a common practice to use an equivalent plate model for low aspect ratio wings because they are likely to behave more as a plate than as a beam.1–3On the other hand,nowadays the tendency for employing the advanced composite structures in aircraft industry is increasing due to their novel properties such as high strength to weight and stiffness to weight ratios.Despite of these properties,there are some problems with these materials that limit their usage in all parts of an air vehicle structure.For instance,one of the most critical problems in composite materials is failure potential in joint region,where two structural elements are joining to each other.Among all structural joining parts in aircraft,the wing-fuselage interface joints are the most critical points due to high shear forces and bending moments in these areas.One solution to overcome this problem would be employing hybrid metal-composite structures which can benefit from both components’advantages.Here,as shown in Fig.1,a hybrid structure commonly consists of two main parts including in-board metallic material,mostly an aluminum alloy and out-board laminated composite material like carbon or glass fibers.4

Although theoretical modeling of such a hybrid structure needs some considerations in metal-composite interface region because of different thermal expansion coefficients,multiplicity of failure modes,damage tolerance and buckling instabilities,the main purpose of present research is to investigate the structural dynamics and aeroelastic behavior of the hybrid wing by neglecting the above mentioned considerations.In fact,it is assumed that the wing deformation is continuous in the metal-composite interface region,which may be not an exact assumption in reality.Supersonic aircraft,on the other hand,usually have trapezoidal or delta wing planforms which essentially arise from aerodynamic considerations and aircraft performance requirements.As noted above,structural dynamics modeling of such wings can be performed using classical(thin plate)or non-classical plate theories.There exists a considerable body of work on the dynamic behavior of all kinds of plates.A thorough description of literature on the study of thin,thick and laminated composite plates of trapezoidal shapes was given by Lovejoy and Kapania5and other investigators.6–22One of the most critical points in the analysis of non-rectangular plates is difficulty of satisfying boundary conditions.This problem can be solved by using a suitable mapping approach through which the initial non-rectangular domain of the problem can be transformed into a rectangular one.Leissa has published a fairly complete vibration formulation of non-rectangular plates,by mapping procedure,mostly based on thin plate theory.23Shokrollahi and Bakhtiari-Nejad used a similar mapping approach to investigate the flutter boundaries and limit cycle oscillations of low aspect ratio swept back trapezoidal wings in low subsonic flow based on linear and nonlinear thin plate theories combined with unsteady vortex lattice method.24

Shokrollahi and Shafaghat25carried out the free vibration analysis of trapezoidal hybrid metal-composite thick plates based on non-classical first-order shear deformation theory.They studied the effect of composite to metal proportion as well as some geometrical parameters on natural frequencies of the plate using proper polynomials as trial functions.

Nakai et al.29have done an experimental investigation of the supersonic flutter characteristics of thin cantilever plate surfaces having a plate aspect-ratio of 1.0 and a taper ratio of 0.63,in the NASA l m×l rri supersonic blow-down wind tunnel at Mach numbers from 1.519 to 4.140.Each model was cantilevered on the side wall of test section of the wind tunnel with a model injection-rejection rig.

They have compared the experimental results with the calculated results by employing piston theory and quasi-steady second-order theory as the required oscillatory aerodynamic forces and utilizing the first three normal coupled modes which include cambering deflections.Both theories were unconservative for the configuration of the models experimented and the range of Mach numbers investigated.29

In present paper,some important aspects of new aircraft structural material tendencies,so called hybrid compositemetal construction,are used in combination with an effective supersonic approach named Mach box method to investigate the flutter boundaries of trapezoidal wings in a vast flow field regime from low to moderate supersonic Mach numbers.The most important issue in present research is studying the effect of composite to metal proportion of the hybrid wing structure on flutter Mach number and frequencies.In addition,the effects of some important wing geometrical parameters like sweep angle,taper ratio and aspect ratio on flutter boundaries are studied.

2.Theoretical development

2.1.Aerodynamics modeling—supersonic Mach box method

The airflow about the wing is assumed to be a 3D potential supersonic flow where the Mach number ranges from low to moderate supersonic regimes.For these situations,it has been shown that the Mach box method is an efficient and accurate approach.In contrast,the well-known piston theory as a simple method in approximating aerodynamic forces is not satisfactory for this range of Mach numbers,especially for low supersonic velocities.

The linearized disturbance velocity potential equation and principal boundary conditions for a 3D unsteady supersonic flow are

where φ,Maanda∞r(nóng)epresent the disturbance velocity potential,Mach number and speed of sound,respectively;ˉwis the downwash velocity on the wing;wthe displacement inzdirection;Uthe air flow velocity;x，y，zare the plate coordinate in stream,cross-stream and thickness direction,respectively.

The solution of Eq.(1)for a planer surface undergoing a small harmonic oscillation with reduced frequencyˉω can be obtained by the following relation30:

where ω is natural frequency,

where η，ξ are the dummy variable of integration foryandxdirections;β=(Ma2-1)1/2.

Using the linearized expression for the local differential pressure coefficient ΔCp,the complex amplitude of the pressure coefficient can be expressed as

Using Eq.(4),the pressure can be determined at any point in the plane of the planform once the downwash is specified overS.

In present study,according to basic approach of Pines et al.,31the trapezoidal wing planform is divided into small rectangular boxes with a sufficiently fine grid so that the downwash over any one box can be taken as uniformly distributed at any instant and the resulting pressure disturbance at the center of each box is a sufficiently accurate average of the pressure distribution over that box.A typical planer box mesh for a 3D flow is shown in Fig.2,wherexi，yjarexandyat center of boxi,j.

With this simplification,Eq.(4)becomes

or

Eq.(6)is a complex variable equation which can be written for all boxes over the wing planform,resulting in a matrix form:

When the Mach box method is used,there are two distinguishable cases associated with the wing edge conditions.If all the edges are supersonic,it can be assumed that the wing upper and lower surfaces have no aerodynamic interactions,since only for the supersonic edges no information can be transformed from one surface to another.On the other hand,if the edges are subsonic,the situation is different.Evvard32named the region between a subsonic leading edge and Mach cone the ‘diaphragm” and recognized that the required extra relationship is that the differential pressure in this region must be identically zero.Fig.3 shows the concept of diaphragm for a trapezoidal wing.

wherepww，pwd,pdwandpddare influence coefficient matrices.The diaphragm downwash is now determined using the second line of Eq.(8):

Substitute the result into Eq.(8),and the desired lifting pressure coefficient is obtained:

This equation will be coupled to structural dynamics equations in order to generate an aeroelastic model in next sections.

2.2.Wing structural modeling—assumptions and formulations

As mentioned previously,in conceptual or early preliminary design stages,the low aspect ratio wing structural dynamics can be modeled based on plate theories.In present research,the Reissner–Mindlin plate theory is used to investigate the aeroelastic stability of low aspect ratio trapezoidal wings having a hybrid metal-composite structure.For this purpose,a general trapezoidal hybrid plate with uniform thicknesshand cantilever boundary conditions is considered.The Cartesian coordinate systemxOypassing through the mid-surface of the plate and the geometry of the plate with side lengthsa,b,c,dand internal angles α and λ is shown in Fig.4.The lamination scheme is symmetric with respect to the midplane for composite part.The angle of fiber orientation is denoted by θ,which is measured fromx-axis to fiber direction,Ymcis the coordinates of composite to metal connection inydirection(Fig.4).

2.2.1.Structural dynamics equations

In order to drive the structural dynamics equations of a hybrid wing,an energy based approach according to Lagrange’s equations is used.The components of the displacement field based on the first-order shear deformation plate theory(FSDT)inx,yandzdirections at any timetare given by

wherew0(x，y，t)is the deflections of the mid-plane points inz-directions; θx(x，y，t)and θy(x，y，t)are the rotations of the cross sections about the coordinatesxandy,respectively.The strain-displacement relations for a transverse shear deformable plate can be written as follows:

The strain energy,now,can be calculated as follows:

whereVia the potential energy,VmandVcare referred to metallic and composite parts of structural volume of hybrid wing respectively and ｛σ｝ is the stress vector whose components depend on the material type.

The kinetic energyTof the hybrid wing can be written as

where ｛v｝ is the velocity vector whose components are

For the convenience of calculations,a transformation from(x,y)to(u,v)is performed.By this transformation,the actual trapezoidal plate in thexOyphysical domain is mapped into a computational square planform in theu，vdomain(-1≤u≤ 1，-1≤v≤ 1)(Fig.5).The transformation can be formulated as

2.2.2.Structural dynamics eigenanalysis—modal properties extraction

Before any aeroelastic analysis,it is convenient,in general,to determine the natural frequencies and mode shapes of the structure under consideration.For this purpose,the displacement and rotations are given by harmonic functions of the time,i.e.,

2.2.3.Application of Ritz method

Application of Ritz method requires the minimization of the following energy functional:

Table 1 The first four natural frequencies(convergence study)of boron-epoxy[±45]2sand aluminum alloy for hybrid trapezoidal wings(υ=0.3，k=0.833，α = β =10°).

whereVmaxandTmaxare given by Eqs.(13)and(14)respectively.Minimization of the functional leads to the following eigenvalue equations:

A complete eigenanalysis of the trapezoidal hybrid wings with various metal to composite proportions and different geometric conditions has been carried out by current paper’s authors in Ref.25.Table 1 shows the results of an eigenanalysis of a typical trapezoidal hybrid wing with α = β =10°.In this table,the convergence of the solution is shown too.

2.3.Aeroelastic model

In this stage,the wing structural dynamics response can be combined with aerodynamic equations to obtain an aeroelastic model.For this purpose,we use Lagrange’s equations according to expansion forms of structural response in terms of generalized coordinates in computational domain:

Based on these generalized coordinates,the Lagrange’s equations can be written as

whereQa,QbandQcare non-conservative forces corresponding to generalized coordinatesw,θxand θyrespectively,which,in this case,are aerodynamic in nature.In present case,the aerodynamic forcing term is accordingly induced inz-direction by supersonic pressure-downwash relation(Eq.(7))and is presented as a non-conservative force in the first equation of Eq.(26).

The transformed generalized force in computational(u,v)plane can be written as follows:

Other generalized forces are vanished in Eq.(26).Substituting expansions Eq.(25)into Lagrange’s Eq.(26)and using Eq.(7)as a generalized force term,we obtain aeroelastic equations in matrix form as follows:

where A(q，˙q)stands for aerodynamic forces obtained from Mach box method(Eq.(10))in terms of generalized coordinates.We refer to Eq.(28)as the complete fluid/structure model.The eigenvalues of Eq.(28)determine the stability of the aeroelastic system.

Table 2 Comparison of results obtained by present method with results of Ref.29

2.4.Numerical results

Various types of trapezoidal hybrid wing models with varying aspect ratio,sweep angle and metal to composite proportion were considered.The hybrid wing models are taken to be cantilever trapezoidal plates composed of an inboard aluminum alloy part and an outboard laminated composite part with various fiber orientation and stacking sequence.

2.4.1.Stability of aeroeleastic model

The aeroelastic eigenvalue solution of the linear model(Eq.(28))determines the stability of the system.The general solution of Eq.(28)may be written as follows:

whereQis in general a complex constant vector(eigenvectors)andpa complex eigenvalue in the form:

where σ′and ω′are real and imaginary parts of the solution,respectively.When the real part of any eigenvalue σ becomes positive,the entire system becomes unstable.In order to investigate the aeroelastic behavior of hybrid trapezoidal wings at supersonic regimes,the results of a selected flutter analysis case in the form of frequency versus Mach number and damping ratio versus Mach number diagrams are given in Fig.6.As it can be seen from Fig.6,the flutter Mach number is nearly equal to 3.

2.4.2.Effect of metal to composite length ratio on flutter boundaries

In this section,the effects of metal to composite length ratio on flutter Mach number of wings with various taper ratios and sweep angles are considered.Three cases are shown here for taper ratios 0.50,0.75 and 1.00 in Fig.7.In each case,the diagrams are given for three sweep angles β =0°,15°,30°,respectively.

2.4.3.Fiber orientation effect on flutter boundaries

The effect of composite part fiber orientation of hybrid wing on natural frequencies was studied in Ref.25In this section,we have studied this effect on flutter boundaries of the wing for various taper ratios and sweep angles as shown in Fig.8.Taper ratios are 1.00,0.75 and 0.50 and sweep angles are 0°,15°and 30°.In all of these investigations,the spanwise position of the composite-metal interface,Ymc,is in wing midspan station.

As it can be seen from Fig.8,the proper fiber orientation angles,i.e.,maximum flutter Mach number,depend on wing taper ratio.For example,for TR=1.00 the optimum orientation angle is 30°,whereas for TR=0.50 the optimum orientation angle is 0°.In contrast,the figures show that the sweep angle has less effect on maximum flutter Mach number.

2.4.4.Validation of results

In order to validate the results of present study,a comparison with the results obtained by Nakai et al.,29was performed.

These investigators studied the flutter boundaries of trapezoidal wings at supersonic flow theoretically and experimentally.They employed the piston theory as a quasi-steady aerodynamic method in their theoretical approach.To verify the validation of present work,five scaled models were used.These models had been used by Nakai et al.previously.The results are given in Table 2 for comparison.As it can be seen from Table 2 that the results obtained from present study are in better agreement with the test results in comparison with the results of piston theory.

3.Conclusion

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4 January 2016;revised 5 July 2016;accepted 1 August 2016

Available online 22 December 2016

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail address:s_shokrollahi@mut.ac.ir(S.Saeed).

Peer review under responsibility of Editorial Committee of CJA.