Aeroelastic eff ect on aerothermoacoustic response of metallic panels in supersonic flow

Wng Xiochen,Yng Zhichun,*,Zhou Jin,Hu Wei

aSchool of Aeronautics,Northwestern Polytechnical University,Xi’an 710072,China

bXi’an Aerospace Propulsion Institute,Xi’an 710100,China

Aeroelastic eff ect on aerothermoacoustic response of metallic panels in supersonic flow

Wang Xiaochena,Yang Zhichuna,*,Zhou Jianb,Hu Weia

aSchool of Aeronautics,Northwestern Polytechnical University,Xi’an 710072,China

bXi’an Aerospace Propulsion Institute,Xi’an 710100,China

A finite element formulation is presented for the analysis of the aeroelastic effect on the aerothermoacoustic response of metallic panels in supersonic flow.The first-order shear deformation theory(FSDT)and the von Karman nonlinear strain-displacement relationships are employed to consider the geometric nonlinearity induced by large deflections.The piston theory and the Gaussian white noise are used to simulate the mean flow aerodynamics and the turbulence from the boundary layer.The thermal loading is assumed to be steady and uniformly distributed,and the material properties are assumed to be temperature independent.The governing equations of motion are firstly formulated in structural node degrees offreedom by using the principle of virtual work,and then transformed and reduced to a set of coupled nonlinear Duffing oscillators in modal coordinates.The dynamic response of a panel is obtained by the Runge-Kutta integration method.The results indicate that the increasing aeroelastic effect can lead the panel vibration from a random motion to a highly ordered motion in the fashion of diffused limit cycle oscillations(LCOs),and remarkably alter the stochastic bifurcation and the spectrum of the aerothermoacoustic response.On the other hand there exists a counterbalance mechanism between the external random loading and the aeroelastic effect,which mainly functions through the nonlinear frequency-amplitude response.It is surmised that the aeroelastic effect must be considered in sonic fatigue analysis for panel structures in supersonic flow.

1.Introduction

Skin panels of advanced high-speed aircraft are simultaneously subjected to mean flow aerodynamic loading,thermal loading,boundary layer turbulent loading,and even shock impinging.1Under such extreme combined loading,heated panels may experience a co-existing behavior associated with panel flutter,thermal buckling snap-through,and random motion with large amplitudes.This co-existing behavior can bring a high alternating stress and thus significantly accelerate the panel fatigue damage.Based on this consideration,an accurate prediction of the nonlinear dynamic response and an analysis of the interaction mechanism(especially the aeroelastic effect)within a multi-physical system are required.However,there have been few studies to investigate the aerothermoacoustic responses of skin panels.For a long time,this multi-physical dynamic problem of heated panel structures in supersonic flow has always been divided into two separate classes,namely,(1)thermo-acoustic fatigue and(2)aeroelastic instability(panel flutter).2

The existing studies on thermo-acoustic fatigue prediction are mainly based on the response of the single degree offreedom(SDOF)model of a structure,in which a single-mode Fokker-Planck equation is employed to simulate the displacements and strain histograms of a target panel under combined thermo-acoustic loading.3,4An excellent review of sonic fatigue given by Clarkson3listed different sources of the random loading for aircraft skin panels;another detailed review given by Wolfe et al.4presented design guides,finite element methods,and validation technologies for panel sonic fatigue response prediction.To predict the panel stochastic response,a nonlinear finite element technique,which is applicable in the low and medium frequency ranges for the nonlinear response prediction of stiffened panels,was proposed by McE-wan et al.5The Galerkin and Monte Carlo(MC)methods for isotropic/composite panel structures under thermo-acoustic loading were developed by Vaicaitis.1Meanwhile,a wellcharacterized experiment conducted by Ng and Clevenson6investigated the snap-through phenomenon in detail for heated panels under thermo-acoustic loading.Recently,a review of indirect/non-intrusive reduced-order methods that are capable of dealing with geometric nonlinearity was given by Mignolet et al.7,8To determine the suitability of various high-cycle fatigue models for metallic panel structures under thermalacoustic loading,a methodology was developed by Przekop and Rizzi,9,10and the influences of different basis selections were also investigated.

Panel flutter is an aeroelastic instability phenomenon due to the frequency coalescence of corresponding coupled modes.Most flutter analysis can be grouped into four categories based on linear/nonlinear structure theories and different aerodynamic theories.The main task offlutter analysis was to determine the critical dynamic pressure,the mode shape,and the frequency at the onset offlutter.11,12An excellent investigation on nonlinear oscillations offluttering plates was given by Dowell,13in which the results indicated that four to six modes should be used for panel flutter analysis,and small static pressures could alter the flutter boundary.From the amplitude statistics perspective,the nonlinear response of composite/metallic panels under combined acoustic and aerodynamic loading was studied by Abdel-Motagaly et al.,14and the results showed that the aeroelastic effect must be considered at a high dynamic pressure level.Similar conclusions for clamped functionally graded material(FGM)panels were confirmed by Ibrahim et al.,15in which thermal loading was taken into account,and the thermal buckling snap-through and the post-buckling motion were examined based on deflection time histories.Then,atwo-waycouplingaerothermoacoustic numerical model for panels at high-speed flow was developed by Miller et al.,16and the results showed that the inclusion offorced pressure loading could reduce the onset time of panel flutter.In order to validate the numerical model,researchers at the Air Force Research Laboratory(AFRL),USA,have conducted a series of experiments to investigate the co-existing behavior for panel structures by using a novel continuous flow wind tunnel,RC-19,in which both the turbulent boundary layer and the mean airflow can be simulated,and the experimental results have confirmed the necessity of an integrated aerothermoacoustic analysis for sonic fatigue ofskin panels.17,18

Considering that the governing equations of a multiphysical dynamic system can be treated as a set of coupled Duffing oscillators driven by a random excitation in nature,a single Duffing oscillator can be employed as a classical paradigm to illustrate some nonlinear phenomena of the co-existing behaviors associated with stochastic bifurcations and jumps.For a single Duffing oscillator,it is well known that there exist some unstable specific frequency regions in which the vibration amplitude suddenly jumps up/down,19and thus the response amplitudes can be multi-valued.20–24Recently,based on the SDOF Duffing model of a buckled beam,Wiebe and Spottswood25investigated the effects of damping and excitation intensity on the co-existing response numerically and experimentally,and it is shown that the co-existing behaviors are sensitive to both the excitation frequency and intensity.

To the best of the authors’knowledge,most conclusions of the previous aerothermoacoustic investigations are obtained from the perspective of the response time history or its statistic characteristics.However,it is also important to study the spectrum characteristics of the co-existing behavior.Motivated by the need for an expanded investigation on the integrated aerothermoacoustic response,especially the aeroelastic effect on the dynamic responses of panel structures in supersonic flow,a nonlinear finite element formulation is presented to investigate the aeroelastic effect in this paper.The nonlinear equations of motion are firstly formulated in the structure node degrees offreedom,and then the model order is reduced by using a modal transformation approach.The Runge-Kutta integration method is employed to obtain the dynamic response.Finally,numerical examples are presented to illustrate the aeroelastic effect on the aerothermoacoustic response.

2.Finite element formulation

In this section,the formulation of the finite element governing equations is deduced based on the following assumptions or theories:(1)the cavity effect of the air entrapped under the skin panel is neglected,(2)the first-order piston theory is used to simulate the supersonic mean flow aerodynamic loading,(3)for convenience,the turbulent loading from the boundary layer is assumed to be a band-limited Gaussian white noise in time domain and uniformly distributed over the panel surface,(4)the thermal loading is assumed to be steady and uniformly distributed in the panel,and(5)the panel parameters are independent on temperature.

The corresponding model can be schematically depicted in Fig.1,which consists of three components:(1)the aerodynamic loading,which is coupled with the panel vibration,(2)the transient deformation of the heated panel,which is excited by both the disturbance loading and the self-induced aerodynamic loading,and(3)the disturbance loading caused by the boundary layer/engine noise.The first two components can build a traditional aeroelastic self-sustained system,while the uncoupled disturbance loading can be treated as a pure external excitation.

Fig.1 Schematic of an aerothermoacoustic model for skin panels.

2.1.Nonlinear strain-displacement relations

Based on the first-order shear deformation panel theory,which assumes that the transverse shear deformation is constant through the panel thickness,the displacement fields of the panel can be expressed as

where u（x，y，z）,v（x，y，z）,and w（x，y，z）are the displacements of the panel in the x,y,and z directions,respectively,and z is the transverse coordinate.u0（x，y）,v0（x，y）,and w0（x，y）are three unknown displacement components of the middle plane in the x,y,and z directions,respectively. φxand φyare the rotations of the normal to the mid-surface with respect to the x and y axes,respectively.The nodal degrees offreedom vector of a three-node triangular Mindlin plate element(MIN3)can be written as

where wbis the transverse displacement vector of the middle plane.The relation between the element displacements and the nodal displacements can be presented in terms of interpolation shape function matrices as

where Huand Hvdenote the in-plane displacement interpolation shape function matrices,Hwdenotes the transverse displacement interpolation shape function matrix,while Hwφ,Hφx,and Hφydenote the rotation displacement interpolation shape function matrices.26These six interpolation shape functions are formulated in details in Appendix A.Based on the von Karman deflection theory,the in-plane strains and curvatures can be expressed as

where εm, εmb,and zk are the linear membrane strain vector,the nonlinear membrane strain vector,and the bending strain vector,respectively.Using the interpolation shape function matrices in Eq.(3),each term in Eq.(4)can be written as

2.2.Stress-strain relationship

The stress-strain relations of isotropic metallic panels subjected to a temperature elevation of ΔT（x，y）can be given by

where υ,E,and G are the material properties of the metallic panel,and α is the thermal expansion coefficient.In the current study,the material properties are assumed to be temperature independent.Based on the Reissner-Mindlin theory that assumes that the constitutive relations between the transverse shear stress resultants and the shear strains are satisfied only in an average-corrected form,the constitutive equations for the isotropic metallic panel can be written as

where N,M,and R denote the in-plane force vector,the bending moment vector,and the transverse shear force vector,respectively.A,B,D,and Asare the in-plane stretching stiffness matrices,the bending-stretching coupling stiffness matrices,the bending stiffness matrices,and the shear stiffness matrices,respectively.Meanwhile,NTand MTare the thermal in-plane force resultant and the thermal moment resultant vectors.

2.3.Aerodynamic loading

For most studies of supersonic panel flutter,the aerodynamic load on the panel outside surface can be approximated by the quasi steady first-order piston theory,which is valid forand showsgood accuracy in therangeofand the aerodynamic loading with zero yaw angle air flow parallel to the panel surface can be expressed

where qa=12ρaV2∞is the mean flow dynamic pressure,V∞is the mean flow velocity,ρais the air density,Ma is the Mach number,D110is the first element D(1,1)of the bending stiffness matrix D in Eq.(10),is the non-dimensional dynam-the stream wise direction.The aerodynamic damping coeff i-For Ma≫1,RM≈μ/Ma,and in this study,the value RM=0.1 is selected.Applying the interpolation shape function matrices described in Eq.(3),the aerodynamic loading can be written as

2.4.Acoustic loading

For the nonlinear finite element approach presented in this study,the input acoustic excitation is assumed to be a stationary,band-limited Gaussian random noise and uniformly distributed over the panel surface.The power spectrum density function can be specified as27–29

where Pref=20 μPa is the reference pressure,fcis the selected cutofffrequency in Hertz,and the sound pressure level(SPL)is the sound spectrum level in decibels(dB).

2.5.Governing equations and solution procedures

To obtain motion equations for heated skin panels subjected to combined aerodynamic and random acoustic loads,the principle of virtual work,which states that for an equilibrium system,the total work done by internal and external forces with an virtual displacement is zero,can be expressed as

where k,ks,and kTare the elementary linear,shear correction,and thermal stiffness matrices,respectively,while k1and k2are the elementary first-and second-order nonlinear stiffness matrices,respectively.In addition,pφTand pmTare thermal induced load on the rotational and membrane displacements,respectively.αsis the improved shear correction factor,whose value can be defined as

where khsis the thickness correction factor,and in Reissner plate theory,khs=5/6.kps,the in-plane correction factor obtained from ksand k in Eq.(17),can be written as14,26

The elementary external virtual work,without considering inertial coupling between the in-panel and transverse deflections,can be expressed as

where m,Ca,and kaare the elementary mass matrix,aerodynamic damping matrix,and aerodynamic stiffness matrix,respectively,while pbband pbφare the external random load vectors on the transverse and rotational displacements,respectively.Assembling each elementary virtual work described in Eqs.(17)and(20),the system governing equations in nodal degrees offreedom can be obtained as

where subscripts ‘B”, ‘m”, ‘A”,and ‘T” stand for the bending-stretching,membrane,aerodynamic,and thermal components,respectively,as different combinations of subscripts ‘Bm” (‘mB”),‘BT” and ‘mT” denote that the corresponding matrices/vectors are related to the bending-stretching displacement wB=[wbwφ]Twith the membrane displacement wm,as well as the thermal induced forces on the bending-stretching and membrane displacements.The terms with superscript ‘—” denote the system displacement and force vectors,and separating the membrane and transverse displacement equations in Eq.(21)results in30

Because the thermal force vectors pmTare assumed to be steady,thus the in-plane inertia termcan be neglected,and the in-plane displacement can be expressed as

Thus the system equations of motion can be stated as a function of wBas

The bending displacement in Eq.(25)can be expressed as a linear combination of normal transverse modes as

where Q=[Q1Q2...QN]is the selected natural mode shapes of the panel in the absence of aerodynamic stiffness.These normal transverse modes can be obtained from a derived eigen problem without the thermal effect of Eq.(25)as

Accordingly,the system governing equations in Eq.(25)can be transformed into the modal coordinates as

where the terms with superscript ‘—”denote the corresponding matrices in the modal system,and

Considering the structural damping effect,a structural modal damping matrix 2ξrfrI MBhas been added to Eq.(29).The coefficient ξr=0.01 is the modal damping ratio of the rth mode,and fris the corresponding natural frequency.Finally,the fourth-order Runge-Kutta integration method is employed to obtain the dynamic response.Eq.(28)can be rewritten in the state space,and the fourth-order Runge-Kutta numerical integration method with a fixed time step of 1/5000 s is adopted to solve the dynamic differential equations,while a random acoustic loading is generated with the same time step.15

Table 1 Geometric and material properties of titanium alloy panel.

Table 2 Modal convergence for composite panel.

Fig.2 Comparison of limit-cycle oscillation amplitudes.

3.Numerical results and discussion

The aeroelastic effect on the nonlinear dynamic response of heated square isotropic panels under combined acoustic and aerodynamic loading is investigated in this section.Meanwhile,the influences of three parameters:temperature elevation,sound pressurelevel(SPL),and non-dimensional dynamic pressure are examined.In the example studies,the dimension of the square titanium alloy panel is 0.3 m×0.3 m×0.001 m with fully clamped edges,and the panel is modeled with a 20×20×2 mesh with MIN3 elements.The reference temperature is assumed to be 24°C,and the first-order critical buckling temperature elevation is ΔTcr=4.626 °C.A thermal loading with a specific temperature elevation of ΔT/ΔTcr=1.2 is uniformly applied to the panel,and thus there exists the thermal induced in-plane stress which can make the panel prone to static buckling or dynamic snap-through.The panel’s geometric and material properties are presented in Table 1.

3.1.Validation offormulation

Before conducting detailed studies,a converged study is firstly investigated with SPL=120 dB, ΔT/ΔTcr=1.0, and λ=1000,as shown in Table 2.Thus the first twenty-six modes are selected for the following validation and examples analysis with a convergent solution.

In addition,the proposed FE method for flutter analysis is validated at ΔT=0.It can be seen from Fig.2 that the limitcycle oscillation non-dimensional amplitudes calculated using the proposed method are in good agreement with those of Dowell.13

Additionally,the first ten eigen frequencies of the panel at room temperature were calculated by the proposed method.It can be observed from Table 3 that the obtained eigen frequencies are in good agreement with those obtained from MSC/NASTRAN. Moreover, the triangular meshes(20×20×2)with MIN3 elements employed here can be adequate to model the metallic panel for further dynamic and flutter analysis.

3.2.Example analysis

The corresponding nonlinear dynamic responses of the panel under different combined acoustic and aerodynamic loadings are shown in Figs.3–10,respectively.At each fixed SPL of the random acoustic loading,four different non-dimensional aerodynamic pressures(λ=0,300,400,and 700)are chosen to investigate the aeroelastic effect on the nonlinear structural dynamic behaviors.The results include the root mean square(RMS)value,non-dimensional deflection time histories,phase plots,power spectrum density(PSD),and non-dimensionaldisplacement probability density distributions of the stationary response.

Table 3 Natural frequencies(Hz)of a fully clamped titanium plate.

For a fixed SPL(120 dB),the responses of the heated panel(ΔT/ΔTcr=1.2)at different non-dimensional aerodynamic pressures:0,300,400,and 700 are presented in Figs.3–6,respectively.Atacombined loadingofSPL=120 dB,ΔT/ΔTcr=1.2,and λ =0,the panel mainly experiences a small-amplitude linear vibration about one of the two thermal buckled equilibrium positions shown in the non-dimensional deflection time history and phase plots(Figs.3(a)and(b)).The static instability phenomenon,thermal buckling motion can be shown as a peak at the zero frequency as shown in the spectrum response(Fig.3(c)),which indicates the stiffness loss of the heated panel.Thus this dynamic post-buckling motion is dominated by the second peak below 80 Hz,which indicates the first resonant frequency of the post-buckled panel.In the non-dimensional deflection probability density distribution plot(Fig.3(d)),this post-buckling motion can be directly demonstrated as one peak with a Gaussian distribution around the nonzero post buckled equilibrium position.

To investigate the aeroelastic effect on the structural dynamic behavior of the acoustically excited heated panel in supersonic flow,the non-dimensional aerodynamic pressure is increased to λ=300.The panel mainly experiences a small-amplitude linear vibration around the zero amplitude position shown in the non-dimensional deflection time history and phase plots(Figs.4(a)and(b)).Compared with its counterpart case with null aerodynamic loading(Fig.3),the spectrum response can be significantly changed.Firstly,the initial peak at the zero frequency indicating that the thermal induced static buckling motion disappears.This change indicates that the stability of the thermal induced buckled equilibrium positions can be changed by the increasing aeroelastic effect.Secondly,the initial dominant single resonance peak(Fig.3(c))is replaced with two nonzero peaks,which indicates that there exist more modes induced by the spatially correlated aerodynamic loading due to the increased aeroelastic effect.The non-dimensional deflection probability density distribution is also Gaussian,but around a newly established equilibrium position,zero deflection(Fig.4(d)).As pointed out by Zhu et al.,21,22the stochastic bifurcation means the changes in the number,location,shape,and magnitude of the peaks of the amplitude probability density distribution of the stationary dynamic response.Thus the panel experiences a phenomenon bifurcation due to the change of the equilibrium position.

At λ=400,the heated panel mainly experiences diffused limit cycle oscillations(LCOs)(Figs.5(a)and(b)).The initial two dominant resonance peaks merge as one single peak due to the increasing aeroelastic effect shown in the power spectrum density plot(Fig.5(c)).In this case,the response is mainly dominated by the flutter motion.The non-dimensional deflections probability density distribution present two symmetric peaks indicating the limit cycle oscillation amplitudes,and thus the panel experiences dynamic-bifurcation due to panel flutter.

As the non-dimensional aerodynamic pressure is increased to λ=700,the random response of the heated panel can be much less disturbed by the external random loading,and with such a high aerodynamic loading,the RMS of the stationary solution can be increased to the panel thickness level.Thus the cubic nonlinear stiffness induced by large limit cycle oscillation amplitudes can affect the panel response.Firstly,the symmetry of the initial phase plot(Fig.5(b))can be changed(Fig.6(b)).Secondly,the spectrum response,in addition to the shifted frequency due to increased λ is enriched by its ultra-harmonic motions(3,5,and 7 times of the fundamental resonant frequency)(Fig.6(d)).

In these three cases with a fixed SPL(120 dB),it is concluded that increasing λ can break down the thermal induced static buckling bifurcation,and thus establish a new equilibrium position.Additionally,the spectrum response can be significantly altered,as the spatially correlated aerodynamic loading can be excited by the aeroelastic coupling mechanism.Thus the increasing aeroelastic effect can lead the random aerothermoacoustic response into a highly ordered motion(LCOs)compared with the pure thermo-acoustic random motion.

For another fixed SPL=140 dB,the responses of the heated panel at different non-dimensional aerodynamic pressures λ =0,300,400,and 700 are presented in Figs.7–10,respectively.It can be seen that the panel exhibits totally different behaviors.

Comparing the case of SPL=140 dB(Fig.7)and λ=0 with the case of SPL=120 dB and λ=0(Fig.3),the panel basically experiences an intermittent snap-through motion,as the random acoustic loading with a higher intensity can easily drive the panel from one thermal buckled equilibrium position to the other symmetric one as shown in the time history plot(Fig.7(a))and the phase plot(Fig.7(b)).As the snapthrough behavior is characterized with zero eigen frequency,and the spectrum response of the post-buckling motion is related to the additive stiffness due to diffused buckled deflections,so there exists a broad spectrum plateau near the zero frequency(Fig.7(c)).Meanwhile,the sub-motions including snap-through and post-buckling motion can be directly seen in the phase plot(Fig.7(b))and the non-dimensional deflections probability density distribution plot(Fig.7(d)).In the latter plot,there exist two distribution plateaus indicating the post-buckling motion affected by the counterbalance mechanism between random excitation and additive stiffness due to diffused post-buckled deflections.

As the non-dimensional aerodynamic pressure is increased to λ=300,the panel also experiences a small linear vibration around the zero amplitude position as shown in the nondimensional deflection time history(Fig.8(a)).Compared with the counterpart cases(Figs.3 and 4)with SPL=120 dB,the RMS can be less reduced by the increasing aerodynamic loading than that of the case with null aerodynamic loading.Additionally,the spectrum response of the heated panel can be largely affected by the increased SPL(140 dB)in this case.The initial two dominant resonance peaks can be broadened and shifted to a higher frequency due to larger deflections(Fig.8(c)).Correspondingly,the initial distribution peak(Fig.4(d))is replaced with a distribution plateau around zero position in the non-dimensional deflections probability density distribution plot(Fig.8(d)).

Fig.3 Response of heated(ΔT/ΔTcr=1.2)panel at 120 dB,λ =0,and RMS=0.3977.

Fig.4 Response of heated(ΔT/ΔTcr=1.2)panel at 120 dB,λ =300,and RMS=0.1374.

Fig.5 Response of heated(ΔT/ΔTcr=1.2)panel at 120 dB,λ =400,and RMS=0.4403.

Fig.6 Response of heated(ΔT/ΔTcr=1.2)panel at 120 dB,λ =700，and RMS=0.9172.

Fig.7 Response of heated(ΔT/ΔTcr=1.2)panel at 140 dB,λ =0,and RMS=0.4189.

Fig.8 Response of heated(ΔT/ΔTcr=1.2)panel at 140 dB,λ =300,and RMS=0.4627.

Fig.9 Response of heated(ΔT/ΔTcr=1.2)panel at 140 dB,λ =400,and RMS=0.5415.

Fig.10 Response of heated(ΔT/ΔTcr=1.2)panel at 140 dB,λ =700，and RMS=0.9205.

As the non-dimensional aerodynamic pressure is increased to λ=400,compared with itscounterpartcase with SPL=120 dB,the non-dimensional deflection time history(Fig.9(a))does not present a typical flutter motion,i.e.,limit cycle oscillations,but a random motion.In the phase plot(Fig.9(b)),the ellipse orbits of diffused LCOs can be largely disturbed by the external random loading,and thus the ampli-tudes can be diffused in a much wider range.Correspondingly there exists a distribution plateau in the non-dimensional deflection probability density distribution plot(Fig.9(d)).All these random phenomena can be attributed to the particular spectrum response(Fig.9(c)),where the initial resonance peaks become much broader.Compared with the counterpart case(Fig.5(c)),the neighboring frequency regions of the first resonant peak are elevated to a higher level due to large deflections with the panel thickness level,which can result in a nonlinear frequency-amplitude response and multi-value frequency regions.

Ultimately,as the non-dimensional aerodynamic pressure is increased to λ=700,the panel mainly experiences diffused LCOs shown in the non-dimensional deflection time history(Fig.10(a))and the phase plot(Fig.10(b)).Compared with the case of SPL=120 dB(Fig.6),the heated panel can be easily disturbed by the increased external random acoustic(SPL=140 dB).As the RMS(=0.9205)is near to the panel thickness level,thus the response can be complicated by the nonlinear frequency-amplitude response.Thus in the spectrum response(Fig.10(c)),there are broader resonant peaks excited compared with its counterpart case(Fig.6(c)).Correspondingly,the LCO amplitudes can be kept in a wider range as shown in the phase plot,which can be alternatively demonstrated as out rings beyond two symmetric LCO amplitude peaks in the non-dimensional deflection probability density distribution(Fig.10(d)).In this case,the response can be considered as a compromise motion due to the balance between the random vibration and the LCO.

To comprehensively study the counterbalance relationship between random excitation and aeroelastic effect,the fluttering panel can be reasonably treated as a linear filtering31to the wind band random excitation(Fig.6(c)).On the other hand the spectrum response of the heated panel can be changed by external random excitation through the nonlinear frequencyamplitude response due to nonlinear stiffness,which can be excited by the large deflections of the panel thickness level under high-intensity random excitation.

4.Conclusions

In the present study,a time domain finite element method was presented for the nonlinear aerothermoacoustic response of heated metallic panels subjected to combined acoustic and aerodynamic loading.The aeroelastic effect on the nonlinear dynamic response was investigated from the perspectives of the statistics and the spectrum of the stationary response.The following conclusions can be obtained:

(1)The aerothermoacoustic response can be remarkably affected by the increasing aeroelastic effect.For a relatively low-level random acoustic excitation,the increasing aeroelastic effect can change the equilibrium position of the heated panel from two symmetric thermal buckled positions to one single equilibrium position,the flat position.Thus the stochastic bifurcations can be obviously altered.

(2)Additionally,the increasing aeroelastic effect can induce a specific spatially correlated aerodynamic load.Thus the spectrum response of the heated panel under spatially uniform,random acoustic loading can be signif icantly affected by the aeroelastic effect.This effect can be characterized by the frequencies coalescence mechanism,which is the reason for panel flutter.With such a highly ordered spectrum,the fluttering panel can be treated as a linear filtering.Thus the increasing aeroelastic effect can lead the random thermo-acoustic response to a highly ordered motion,diffused limit oscillation.All these changes can be accompanied with specific stochastic bifurcations.

(3)As for a higher-level random acoustic excitation,the counterbalance mechanism between the random acoustic excitation and the aeroelastic effect functions is like this.Under high-intensity random loading,the nonlinear stiffness,induced by large deflections with the panel thickness level,can result in a nonlinear frequencyamplitude response,which indicates some unstable frequency regions with multi-valued amplitudes.Thus the highly ordered spectrum response due to a flutter motion can be altered,and the panel presents a random thermo-acoustic response again.On the other hand under high aerodynamic pressure,this nonlinear stiffness can result in an ultra-harmonic motion.It is concluded that the aeroelastic effect has to be considered for an accurate fatigue life prediction ofpanel structures.

Acknowledgments

The work was supported by the National Natural Science Foundation of China(No.11472216).The first author would like to acknowledge the support from China Scholarship Council(CSC)and German Aerospace Center(DLR).

Appendix A.Interpolation shape function of MIN3 element

In the derivation of the interpolation shape function,it is necessary to present the parametric coordinates expressed in terms of sub-triangle areas(refer to Fig.A1)using Cartesian coordinates of the three nodes 1（x1，y1）,2（x2，y2）,and 3（x3，y3）as26

Fig.A1 Triangular element coordinate description.26

Introducing the following parameters

Thus the parametric coordinate is

Based on the parametric coordinate,the relationships between these two coordinates are

The nodal displacement vectors of an MIN3 element(Mindlin-type,three nodes with each five degrees),i.e.,

can be rewritten using the interpolation shape function matrices as

The interpolation shape function matrices can be expressed in terms of parametric coordinates as

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Wang Xiaochen is a Ph.D.student in the School of Aeronautics at Northwestern Polytechnical University.His research area includes fluid-solid interaction(FSI,aerothermoelasticity)and adaptive structure system(nonlinear energy harvesting)design.

Yang Zhichun is a professor(Ph.D.advisor)in the School of Aeronautics at Northwestern Polytechnical University.His main current research interests are aeroelasticity,structural health monitoring,and smart materials.

7 September 2015;revised 18 April 2016;accepted 27 August 2016

Available online 17 October 2016

Aeroelastic effect;

Aerothermoacoustic response;

Metallic panels;

Nonlinear frequencyamplitude response;

Sonic fatigue

Ⓒ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.Tel.:+86 29 88460461.

E-mail addresses:wxc_npu@163.com(X.Wang),yangzc@nwpu.edu.cn(Z.Yang).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.10.003

1000-9361Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).