Accurate closed-form eigensolutions of three-dimensional panel flutter with arbitrary homogeneous boundary conditions

Qiozhen SUN,Yufeng XING,Bo LIU,Bocheng ZHANG,Zekun WANG

a COMAC Beijing Aircraft Technology Research Institute, Beijing 102211, China

b Institute of Solid Mechanics, Beihang University, Beijing 100191, China

KEYWORDS Analytical;Closed-form;Eigenvalues;Flutter;Three-dimensional

Abstract Highly accurate closed-form eigensolutions for flutter of three-dimensional (3D) panel with arbitrary combinations of simply supported(S),glide(G),clamped(C)and free(F)boundary conditions (BCs), such as cantilever panels, are achieved according to the linear thin plate theory and the first-order piston theory as well as the complex modal analysis, and all solutions are in a simple and explicit form.The iterative Separation-of-Variable(iSOV)method proposed by the present authors is employed to obtain the highly accurate eigensolutions. The flutter mechanism is studied with the benefit of eigenvalue properties from mathematical senses.The effects of boundary conditions, chord-thickness ratios, aerodynamic damping, aspect ratios and in-plane loads on flutter properties are examined. The results are compared with those of Kantorovich method and Galerkin method, and also coincide well with analytical solutions in literature, verifying the accuracy of the present closed-form results. It is revealed that, (A) the flutter characteristics are dominated by the cross section properties of panels in the direction of stream flow; (B) two types of flutter, called coupled-mode flutter and zero-frequency flutter which includes zero-frequency single-mode flutter and buckling, are observed; (C) boundary conditions and in-plane loads can affect both flutter boundary and flutter type;(D)the flutter behavior of 3D panel is similar to that of the two-dimensional(2D)panel if the aspect ratio is up to a certain value;(E)four to six modes should be used in the Galerkin method for accurate eigensolutions,and the results converge to that of Kantorovich method which uses the same mode functions in the direction perpendicular to the stream flow.The present analysis method can be used as a reference for other stability issues characterized by complex eigenvalues, and the highly closed-form solutions are useful in parameter designs and can also be taken as benchmarks for the validation of numerical methods.

1. Introduction

Panel flutter is a self-excited oscillation of the external surface skin of a flight vehicle, resulting from the dynamic instability of the aerodynamic, inertial and elastic forces. Literature reviews on panel linear and nonlinear supersonic and hypersonic panel flutter have been presented by Mei1and Abbas et al.2. Although theoretical and experimental investigations are still focusing on high supersonic/hypersonic flow,3-4many flutter incidents in practice happened at low supersonic/transonic Mach numbers.5The panel flutter problems can be roughly classified into two categories,one is the determination of flutter boundaries,which is an eigenvalue problem based on linear models,and the other is the analysis of limit cycle oscillations or flutter responses based on nonlinear models.

The present work focuses on the solution methods of the eigenvalue problems of panel flutter, but for the completeness of the present paper, a brief review is presented on numerical and analytical methods for linear and nonlinear panel flutter.

Among the numerical methods which are applicable to both linear and nonlinear panel flutter,1,6-7the Galerkin method,8-13FEM (Finite Element Method)14-18and the Rayleigh Ritz method19-21are the most popular ones in panel flutter analysis.It was indicated that at least four to six modes are required in Galerkin method to obtain the convergent solutions of limit cycle responses. In addition, the Assumed Mode Method(AMM)combined with Ritz method has also been widely used for panel flutter analysis.22-28It was concluded that AMM might give incorrect results in flutter analysis of SCSC(Simply supported-Clamped-Simply supported-Clamped), SFSF (Simply supported-Free-Simply supported-Free) plates and plates with elastic BCs (Boundary Conditions), and is less accurate than FEM; assumed mode functions always have a great difference with real functions,so that for a complex structure system, many assumed mode functions are required to obtain relatively accurate results.

Compared with numerical methods,analytical methods are more effective and accurate for analyzing flutter problems of plates and shells.There are two main types of analytical methods, most of which are suitable to linear problems. One is the series expansion method, the other is the separation-ofvariable method which can be used to achieve closed-form eigensolutions. Zhou et al.4presented analytical solutions of flutter speed for elastic boundary panel using a modified Fourier series method. Zhou et al.29-30also developed a modified Fourier method to analyze the aero-thermal-elastic flutter boundary for supersonic plates, where both the classical and non-classical BCs can be dealt with. Muc and Flis18analyzed the optimal design of multilayed composite structures subjected to supersonic flutter constraints with semi-analytical method(specific boundary conditions),which are derived with the use of the Love-Kirchhoff hypothesis. Different from the series expansion method, the separation-of-variable method is more concise and has no truncation error.For 2D panel flutter, using both superposition method of eigensolutions31and Laplace transform method can achieve exact solutions. For 3D panel flutter, the semi-inverse method32and the Kantorovich-Krylov (KK) method are the popular ones. As is well known,if the edges parallel to the stream are not simply supported, the semi-inverse method is not applicable, but the KK method is, which assumes that mode functions satisfy BCs in the y direction perpendicular to the stream, and then 2D flutter equation is reduced to one-dimensional equation in the x direction. The KK method can be used to achieve the closed-form eigensolutions of rectangular panels with simple supports, clamped supports and ecstatically restrained against edge rotation.33-34It can be concluded that all these analytical methods can be used to cope with the eigenvalue problems of panel flutter.

Due to the difficulties in dealing with BCs in the studies of 3D panel flutter,the analytical eigensolutions are not as many as 2D case, and most research is based on SSSS plates. The exact eigensolutions of 3D SSSS panel flutter were first obtained by Hedgepeth32who introduced the 2D static aerodynamic approximation which greatly simplified the procedure of flutter analysis. In 1994, in consideration of aerodynamic damping and using the similar method of Hedgepeth,32Hassan35gave exact solutions of flutter for SSSS panels based on the first-order piston theory. However, in order to arrive at a closed-form expression for the critical Mach number,the coefficients of equations were transformed from complex number to real. As for cantilever configuration, the lack of impetus in tackling its flutter phenomenon is due to two reasons.One is the complication caused by the downstream wake flow and the other is the previous lack of a practical motivation.36Thus, most publications about panel flutter focus on CCCC or SSSS plates undergoing supersonic flow.37And most studies for cantilever plates use numerical methods.21,36,38

It can be concluded from the above literature review that,for the eigenvalue problems of 3D panel flutter,there are exact solutions only for two cases, one is the SSSS case, and the other is the case where the pair of opposite edges parallel to stream are simply supported. Although the KK method can deal with all homogeneous BCs, its usage is not convenient,and there are no explicit closed-form solutions so far. Moreover, the previous closed-form solutions for plates with two adjacent free edges are not very reasonable.39Therefore, it is of great significance to develop a unified separation-ofvariable method for the analysis of the flutter characteristics of panels with arbitrary classical BCs.

It is noteworthy that there are two types of panel flutters.One is the classical type called coupled-mode flutter featured by the interaction of two eigenmodes, and the other is the single-mode flutter. The first type has been studied relatively fully based on the piston theory, which is only valid at high Mach number and low frequencies and coincides excellently with experiments if Ma ＞1.7.40The second type which can only be studied by the exact potential flow theory or more complex theories40has been verified in theoretical analysis5,8,41-42and experimental investigations.40However, a special single-mode flutter with zero frequency was observed by the present authors in the flutter studies of 2D panel using the piston theory.43

In this context, this work uses the iterative Separation-of-Variable (iSOV) method39proposed by the present authors to achieve highly accurate closed-form eigensolutions of 3D panel flutter for arbitrary homogeneous BCs,including all possible combinations of S,G,C and F.The highly accurate solutions are compared with the results of Galerkin method and Kantorovich method. In addition, the effects of parameters such as chord-thickness ratio,aspect ratio,BCs,in-plane loads and aerodynamic damping on flutter boundary are examined.Two types of flutter are also observed for the 3D panel.

2. Basic equations

In this section, the potential energy and kinetic energy functions of a rectangular panel are given first, and then the governing equation of panel flutter is given based on the general form of the Hamilton’s variational principle.44Finally, the ODE(Ordinary Differential Equation) from which highly accurate closed-form solutions are solved is derived using separation-of-variable approach.

Consider a 3D panel(a/b ≠0)8with chord a,span b,thickness h and density ρm,and one side of the panel is exposed to a supersonic airflow and the other to still air(see Fig.1).The airflow with undisturbed velocity V and density ρapasses normally to the supported edges parallel to the x direction.Additionally, the panel is subjected to constant mid-plane compressive forces Nxand Ny.The coordinate system is established in the mid-plane and the origin O is built at the center of the panel.

In the following analysis, the Kirchhoff plate theory and the first-order piston theory are used, so the strain energy of the panel is

Fig. 1 3D panel subjected to aerodynamic loading over one surface.

To solve the flutter problem of a panel with free edges, in this work, we employ the iSOV method to reduce the partial differential equation to two ODEs (Ordinary Differential Equations), instead of solving Eq.(8) directly. In the iSOV method, w is written in the separation-of-variable form as

Then the iSOV method is performed according to Eq.(11),and for the implementation of the iSOV method,one can refer to Ref. 39. The iteration process is given in Section 3.

3. Iterative separation-of-variable method

In the iSOV method, the mode function has the form,W(η,ξ) = φ(η)ψ (ξ) (see Eq. (10)), and initially we assume the deflection of a direction to determine the deflection of another direction. Without loss of generality, first we assume the deflection in the y direction,that is ψ(ξ)=Ψ0(ξ),and then determine the deflection φ1(η) in the x direction.

3.1. Determination of φ1

where λ is the spatial eigenvalue in the x direction. From our previous study,43Eq.(19)has four eigenvalues in the following form:

where λi(i=1,2,...,4)is the complex root of Eq.(19),and for simplicity, the roots are expressed in Eq. (20), where ? is assumed as the real part of λi; α1and β1are the imaginary parts. Thus, the general form of eigenfunction φ1is

To assure nontrivial solution for A1, A2, A3and A4, the coefficient determinant of Eq. (23) must be zero, yielding the frequency equation as

Table 1 Boundary conditions along edges x = -a and a in iSOV method.

Considering Eq. (22), one can find that only ? is independent variable in Eq. (24) since Pland Rηcan be determined by the given properties of fluid and panel and the integrals in Eq.(13). Eq. (17) and Eq. (22) show that the frequency Ω included in kηdepends on ? which can be solved by Eq. (24).

By referring to the third relation in Eq. (17), the first two terms involving Ω in kηcan be written as

3.2. Determination of Ψ1

As the initial function Ψ0(ξ) is assumed and not required to satisfy BCs, the mode function in the y direction needs to be improved in the similar procedure as in Section 3.1. Here,W（η，ξ ）=φ1（η ）ψ1（ξ ）, where ψ1（ξ ）, the improvement of Ψ0(ξ),needs to be determined.For this case,Eq.(11)becomes

Similarly as in Section 3.1, to find out the two eigenvalues α2, β2and the frequency parameter k, Eq. (40) should be substituted into the BCs of clamped edge (see Table 2). Then one can obtain four homogenous algebraic equations for unknown constants B1, B2, B3and B4.

There are three unknown variables α2, β2and k in Eq.(42), but only one of them is independent, say k. Thus,α2, β2and k can be solved by Eqs. (39) and (42), and B1, B2, B3and B4are given by Eq. (43). In the iterative calculation, k defined by Eq. (26) in Section 3.1 is denoted by kφ1for the sake of simplicity. And kφ1is updated to kψ1here.

The eigenvalue equations and coefficients achieved in last two sections are for CCCC plate,and the frequency equations and coefficients for rectangular plates with other BCs are presented in Appendix A.

3.3. Iteration

Sections 3.1 and 3.2 show the procedures how to solve φ1and Ψ1in the first iteration.39In order to get more accurate closedform solutions, the iterative calculation from Section 3.1 to Section 3.2 continues until a convergent condition is satisfied.For a better understanding of the present paper, the iteration procedure is presented below.

Initialization: In the y direction, initialize the eigenfunction Ψ0. For different homogeneous BCs, Ψ0can be assumed as

where n=1，2，···, is the frequency order. Eq. (44) indicates that α2= （n+θ）π/（2b ） and β2=0, B1= B3= B4= 0,B2= 1. In general, θ is in the domain (-0.5, 0.5), but θ = 0 for simple support and θ = 0.2 for other BCs are used in our calculations.

Step 1.Using Ψ0gives an ODE in terms of φ1as in Eq.(16),and then the independent variable ?1, frequency parameter kφ1, frequency Ωφ1and φ1are determined by the BCs along edges η = -1 and 1.

Step 2. Using φ1leads to an ODE about Ψ1as in Eq. (34),and then kΨ1,ΩΨ1and Ψ1can be determined by the BCs along two edges ξ = -1 and 1.

Step 3. Check the convergence condition.

where ε=1.0×10-12is used in our work.If the condition Eq.(45) is satisfied, the iteration stops. Otherwise, go to Step 1 where Ψ0is replaced by Ψ1.In general,three or four iterations are needed, but the number of iterations depends on the selected initial values and the eigenvalue characteristics.

Table 2 Boundary conditions along edges y = -b and b in iSOV method.

Table 3 Material property of panel and gas flow.

4. Numerical results and comparison

Flutter boundary and flutter properties in terms of frequency and mode are examined in this section. The physical parameters used in numerical calculation are presented in Table 3,and the velocity of sound ac= 340 m/s. The subscripts ‘1st’and ‘2nd’ are pertaining to the first- and second-order modes in chord-wise direction respectively; the subscript ‘cr’ stands for the critical point when the first two order frequencies coincide for coupled-mode flutter and the first-order frequency becomes zero for zero-frequency flutter; the subscript ‘f’denotes the point when panel flutters. The in-plane load is not involved without particular declaration. In the comparison, the results obtained by Galerkin and Kantorovich method are achieved using the first-order mode in span-wise direction unless stated otherwise.

4.1. Flutter frequency and flutter type for different boundary conditions

By the highly accurate closed-form solutions achieved in Section 3, it follows that the flutter types of panel depend on the BCs of two edges(η=-1 and 1)which are perpendicular to the airflow direction, and are independent of the BCs of edges (ξ = -1 and 1) (see Table 4). For clarity, the detailed relationships between the flutter types and the concrete bound-ary conditions are discussed with three classifications in the following.

Table 4 Flutter characteristics regarding boundary conditions in y direction (ga = 0).

Classification 1: the BCs of two edges (η = -1 and 1) are S/C(η = -1) - S/C(η = 1)-, G(η = -1)-G(η = 1)- and F(η = -1) - F(η = 1)-. Here the cases of SSSS and CCCC are chosen to illustrate the flutter characteristics of this classification (see Table 5).

Fig.2 and Fig.3 show the relationships between ^β and Ma as well as between ^ω and Ma for SSSS panel.According to Eq.(27), gavaries nonlinearly with Ma due to the introduction of the correction factor as given in Eq.(3),which is different from that in Ref. 43. Fig. 2 shows that ^β is negative when Ma ＜Maf=4.7628,indicating that the vibration before flutter is a damping vibration; ^β reaches its minimum when Ma = Macr= 4.7407, implying that the first two frequencies coalesce, and then ^β approaches to zero as Ma = Maf. If ga=0, Macr= Maf,and ^β remains zero before Ma = Macr.The real number^β=0 implies that the vibration before flutter is harmonic.

Fig.3 indicates that ^ω1stand ^ω2ndget close to each other as Ma increases, and are equal at Macr. After this moment, they keep equal and increase as Ma rises until Ma = Maf. If Ma ＞Maf,the panel flutters and ^β ＞0.In some sense, ^β represents the vibration amplitude or the system energy with respect to time,and ^β =0 stands for the flutter state in which the energy is constant; ^β ＞0 denotes that the energy becomes more and more (divergent state); ^β ＜0 represents that the energy dissipates (steady state).43

Table 5 Eigenvalue and flutter properties for the cases with aerodynamic damping.

Fig. 2 Relation between ^β and Ma for Case SSSS.

Fig. 3 Relation between ^ω and Ma for Case SSSS.

Fig. 4 Relation between ^β and Ma for Case CCCC.

It is noteworthy that: (A) after ^ω1st= ^ω2nd, the spatial eigenroots become multiple, but ^ω1stand ^ω2ndcorrespond to the same mode, showing that the system becomes a defective system; (B) corresponding to the same Ma, the frequencies for neglecting and considering aerodynamic damping have a little difference.

Fig. 5 Relation between ^ω and Ma for Case CCCC.

The results of CCCC panel are presented in Fig. 4 and Fig. 5. Compared to the case SSSS, the very difference is that the Mafof CCCC panel is higher since the stiffness of CCCC panel is higher.In addition,the frequency equation of G_G_is the same as that of S_S_,but they have different flutter modes which are shown in the next section (see Table A1). However,unlike the 2D panel flutter,Mafof the case FSFS,as shown in Fig.6 and Fig.7,is not equal to that of CSCS,and the reason for this is that the influences of both x and y directions are considered for 3D panel in the present study.

To summarize this classification, we conclude: (A) for unstressed panels or the cases without in-plane loads, as Ma increases, ^ω1stincreases, whereas ^ω2nddecreases until they coalesce to a single frequency ^ωcr, called the nondimensional critical frequency, and the corresponding Ma to^ωcris Macr, then ^ωcrapproaches to the flutter frequency ^ωfat which the panel flutters, and the corresponding Ma to ^ωfis Maf;(B) ^β ＜0 before flutter or Ma ＜Macr,at the moment Ma=Macr,^β begins to increase until^β =0 where Ma=Maf,and ^β ＞0 when Ma ＞Maf. If ga= 0, ^β =0 before flutter,and ^ωcr= ^ωf,Macr=Maf.This type of coupled-mode flutter is denoted by the coupled-mode flutter 1 (see Table 5).

Classification 2:the BCs of two edges(η=-1 and 1)are G(η=-1)-S/C(η=1)-and F(η=-1)-S/C/G(η=1)-.Here the case GCCC(2a/h=80)is chosen to show the flutter properties (also see Table 5). It follows from Fig. 8 and Fig. 9 that^ω1stand ^ω2ndnever coincide for this classification. As Ma increases, all frequencies decrease but ^ω1stdecreases sharply.When Ma=3.6010,ω1st=0 and^β reaches its minimum,then^β =0 at Ma = Maf= 3.6103 and panel flutters. This is the zero-frequency flutter or static divergence (see Table 5). In addition,when ga=0, ^β =0 at Ma=Macr=Maf,indicating that the zero-frequency flutter is a typical instability.

Classification 3: the BCs of two edges (η = -1 and 1) are S/C(η=-1)-G/F(η=1)-and G(η=-1)-F(η=1)-.Here the cases CSFS(2a/h=120)and CFFF(2a/h=120)are chosen to show the flutter properties (also see Table 5).

It follows from Figs.10-13 that the first two frequencies get larger as Ma increases before a certain point. But when Ma is larger than that value, ^ω2ndstarts to decrease and is finally equal to ^ω1stat Macr. This type of flutter is also the coupled-mode flutter, denoted by the coupled-mode flutter 2 here,which is different from the coupled-mode flutter 1 in classification 1 where only ^ω1stincreases, and also different from the zero-frequency type where all frequencies decrease as Ma rises. Note that Fig. 4 in Ref. 43 is wrong, and the correct one is shown as Fig. 14 here.

Fig. 6 Relation between ^β and Ma for Case FSFS.

Fig. 7 Relation between ^ω and Ma for Case FSFS.

Fig. 8 Relation between ^β and Ma for Case GCCC.

To compare the cases discussed above, the flutter frequencies and Mach numbers of 11 typical cases are listed in Table 6 for the situation of aerodynamic damping ga= 0. And from the above discussions, we have the following observations:

Fig. 9 Relation between ^ω and Ma for Case GCCC.

Fig. 10 Relation between ^β and Ma for Case CSFS.

Fig. 11 Relation between ^ω and Ma for Case CSFS.

1) The variations of ^β with Ma are all the same for three classifications (see Figs. 2, 4, 6, 8, 10 and 12).

Fig. 12 Relation between ^β and Ma for Case CFFF.

Fig. 13 Relation between ^ω and Ma for Case CFFF.

Fig.14 Correct relation between β and Ma for CF panel in Ref.43.

2) Due to the asymmetry of panel stiffness caused by aerodynamic force,the flutter phenomenon may be quite different for asymmetric BCs. For example, the flutter types for GCCC and CCGC are the zero-frequency and the coupled-mode flutter 2 respectively,and the latter can hardly happen since its Maf(2a/h = 80) is 22.8122. Similarly, the GSSS panel has the zerofrequency flutter, the SSGS panel has the coupledmode flutter 2, and Maf= 2.6723 for case GSSS while Maf= 37.2634 for case SSGS (2a/h = 60).

3) The zero-frequency flutter always occurs at lower Mach number than the coupled-mode flutter. Note that these two types of flutter correspond to different BCs of edges η = -1 and 1.

Moreover, the relationships between the aerodynamic elastic coefficient pland the dimensionless frequency ^ω for five typical cases are shown in Fig. 15 for situation of ga= 0. It can be seen that, for given a/b and V, ^ω at pl= 0 (the panel is in vacuo) has two values for coupled-mode flutter and one value for zero-frequency flutter, and with the increase of pl,two types of flutter can be observed. For the coupled-mode flutter, the first two order frequencies eventually coalesce to the critical frequency at the peak point where plis called the critical aerodynamic elastic coefficient denoted by plcr. For zero-frequency flutter (GCCC), all frequencies decrease and the first frequency becomes zero when pl= plcr.

4.2. Flutter modes

The flutter frequencies and types have been elaborated in Section 4.1. For a better understanding of the panel flutter, this section presents the flutter modes.

Figs. 15-19 show the cross sections of flutter modes in gas flow direction for SSSS, CCCC, GSGS, FSFS, CSFS, CFFF,SCSF, CCCF and FCFF panels, wherein w1denotes the real part of mode functions and w2the imaginary part. As can be seen, there is a very pronounced second-mode content at flutter;the two mode shapes of S_S_and G_G_are quite different although their flutter frequencies are the same;the flutter mode shapes of S_S_ are almost the same, and so are the mode shapes of F_F_ and G_G_.

Fig. 18 and Fig. 19 illustrate the flutter modes of the cantilever panels. It also follows that the cross sections of flutter modes are dominated by the chord-wise BCs, and if the BCs of edges η=-1 and 1 are the same,the cross sections of flutter modes are almost the same (see Fig. 18), whereas the cross sections of flutter modes are quite different (see Fig. 19).Fig. 20 shows the real parts of the flutter modes from which we can see the detailed differences among nine cases.

Fig.21 and Fig.22 show the variation of w1for the first two modes of the SSSS and CSFS panels. There the numbers following ‘1st’ and ‘2nd’ are the Mach numbers, and ‘cr’ and ‘f’stand for Macrand Mafrespectively. The flutter type of the SSSS panel is the coupled-mode flutter 1, and the flutter type of the CSFS panel is the coupled-mode flutter 2. Conclusions can be drawn for the coupled-mode flutter that as Ma rises,the two modes (1,1) and (1,2) get close to each other and coalesce at Macr; if Ma ＜Macr, w2remains zero; if Ma ＞Macr,the two flutter mode functions are conjugate, and in fact the corresponding two eigenvalues are also conjugate. Note that Fig. 17 in Ref. 43 is wrong, which is the flutter mode for CF panel, and the correct one is presented in Fig. 23.

Table 6 Flutter parameters and Mach number (ga = 0, a/b = 1/3).

Fig. 15 Variation of aerodynamic elastic coefficient pl with dimensionless frequency ^ω.

Fig. 16 Flutter modes of SSSS and CCCC panels.

4.3. Flutter boundary

Flutter boundary is the boundary between stable and unstable motions. It is commonly defined as plor Ma at which panel flutter occurs.Without loss of generality,the flutter boundaries are investigated for the panels with aspect ratio b/a = 3. The effects of a/b are examined in Section 4.3.3.In addition,the inplane stress is neglected except for Section 4.3.4 where the effects of in-plane stress are discussed.

Fig. 17 Flutter modes of GSGS and FSFS panels.

Fig. 18 Flutter modes of CSFS and CFFF panels.

4.3.1.Relation between Mach number and chord-thickness ratio

The coefficient plcris important because using its definition we can determine the relationship between the critical Mach number Macrand the chord-thickness ratio a/h. Recalling the definition of plin Eq.(17), we have

Fig. 19 Flutter modes of SCSF, CCCF and FCFF panels.

Fig. 21 Variation of modes (1,1) and (2,1) for Case SSSS.

Fig. 20 Flutter modes for different boundary conditions.

Fig. 22 Variation of modes (1,1) and (2,1) for Case CSFS.

Fig. 23 Correct flutter mode of CF panel in Ref. 43.

which means that Macris proportional to (h/a)3. With the closed-form solutions achieved in Section 3, we can also find the relationship, as shown in Fig. 24, between Macrand a/h for a given a/b = 1/3. In addition, the present simulation reveals that plcris independent of a/h if ga= 0, which can be explained by the comparison in Fig.25.Another observation is that ^ωfis also independent of a/h when ga=0,but ωfdepends on a/h for all situations. Fig. 25 also shows that flutter occurs more easily if the panel is more flexible or a/h is larger. And among three cases of SSSS, CSFS and GCCC, the GCCC panel flutters easiest,indicating that the zero-frequency occurs at a smaller Ma.

4.3.2. Aerodynamic damping effects

Fig. 24 Variation of Macr with chord-thickness ratio for different boundary conditions.

Fig. 25 Variation of plf with a/h for three cases.

Fig. 26 Variation of plf with ga for three cases.

If the aerodynamic damping is considered, ^ωfand Mafdepend on both a/b and a/h. Fig. 25 and Fig. 26 show the flutter boundaries with different a/h and garespectively for three kinds of BCs. The line denoted by SSSS stands for the flutter boundary when ga≠0 while the line denoted by SSSS(ga=0) stands for the flutter boundary when ga=0, and the denotations for the other two kinds of BCs have the samemeaning. It follows that, the cases of SSSS and CSFS, or the two types of coupled-mode flutter have higher flutter boundaries than the case of GCCC, or the zero-frequency flutter;gahas a slight effect on the two types of coupled-mode flutter(see Table 7 and Table 8), while it has no effect on the zerofrequency flutter which in effect is static buckling. Therefore,gacan be ignored when determining flutter boundary for most cases,especially for the GCCC case which is independent of ga.

Table 7 Critical and flutter results of SSSS panel for different a/b when 2a/h = 150.

Table 8 Critical and flutter results of SCSF panel for several a/b when 2a/h = 150.

4.3.3. Pure aspect-ratio effects

Fig. 27 Variation of Macr with 2a/h for different a/b in the case of SSSS.

Fig. 27 shows the variation of the critical Mach number Macrwith 2a/h for different aspect ratios of SSSS panel ignoring aerodynamic damping or ga=0.Note that a/b=0 represents the 2D panel.We can see that if a/b ＜1/2,all the relationships between Macrand 2a/h are very close to each other;the details for 2a/h = 150 are presented in Table 7, where for checking the effect of gaon flutter properties, the results for nonzero gaare also included. It can be concluded that (1) the behavior of 3D panel flutter is similar to that of 2D panel flutter for all a/b considered; (2) if a/b ＜1/3, the flutter boundaries of 3D panel are quite close to those of 2D panel, so that if a/b ＜1/3, the 3D panel flutter can be simplified to the 2D panel flutter, and the differences of ^ωcrand plcrbetween 3D and 2D panel are less than 5%; (3) both ^ωcrand plcrbecome smaller for smaller a/b;(4)the effect of gaon flutter parameters is small.

Fig. 28 Flutter modes of SSSS plate for different a/b.

Table 9 Critical and flutter results of CCCC plate for different a/b when 2a/h = 150.

Furthermore, for the situation of ga≠0, Fig. 28 shows flutter modes of the SSSS plate for different a/b listed in the first column of Table 7.Compared with Fig. 16,it follows that a/b and gahave little effect on the flutter mode shapes, therefore the legends of all curves are not shown in Fig. 28.

Although the relationships between flutter boundaries (^ωf,plf,Maf)and aspect ratio b/a are similar for different BCs,such as for the SSSS,SCSF and CCCC panels,as shown in Table 7,Table 8 and Table 9 respectively, the critical aspect ratios(b/a)(cr)at which the 3D panel flutter can be simplified as 2D panel flutter are different. (b/a)(cr)is larger than or equal to 3 for the coupled-mode flutter, and (b/a)(cr)is larger than or equal to 8 for the zero-frequency flutter (see Table 10).

In short, the flutter characteristics of the 3D panel flutter are qualitatively similar to those of the 2D panel,and for a certain aspect ratio b/a,the coupled-mode flutter of the 3D panels can be simplified to that of the 2D panels. However, the 3D panel model is preferable for the investigation of the zerofrequency flutter.

4.3.4. Flutter boundary of rectangular panels subjected to inplane load

As in-plane load can change the transverse stiffness of panels,the effect of in-plane load on panel flutter boundary is not negligible and discussed in this section. The variation of plcrwith Rηfor SSSS panel is shown in Fig.29,where the in-plane load Ny= 0 and ga= 0. As can be seen, plcrdecreases monotonically with the increase of Rηsince the transverse stiffness of the panel is monotonically reduced by the compressive in-plane load Nx. For a fixed value Rη, numerical results reveal that plcris independent of a/b.However,the panel cannot experience coupled-mode and single-mode flutter but can experience static instability or buckling if plis small while Nxis big enough.For example, a rectangular panel with a/b = 1/3 buckles when pl＜14.23 while a square panel buckles when pl＜10.66.For this reason, Fig. 29 is only valid for Rη＜ 3.85 if a/b=1/3 and valid for Rη＜4.40 if a/b=1.Thus,the validity range of Fig.29 depends on the buckling characteristics of the panel.For showing more clearly,the variations of plcrand ^ωcrwith Rηfor a panel(a/b=1/3)are presented in Fig.30.It can be observed that ^ωcrdecreases as Rηincreases; when pl＜14.23, the panel buckles first, and frequency becomes zero. But plcris finite even though the ‘‘effective stiffness” of the panel is zero, which is measured by the lowest in-vacuo natural frequency.

If Ny＞0,all frequencies decrease.However,as long as the panel does not buckle due to Ny, plcris not influenced, that is,the flutter speed is independent of Ny.

Table 10 Critical aspect ratios for different boundary conditions.

Fig. 29 Variation of plcr with Rη for SSSS panel (Ny = 0,ga = 0).

Fig. 30 Variation of both plcr and ^ωcr with Rη for SSSS panel(Ny = 0, ga = 0, a/b = 1/3).

4.4. Validation

The above discussion is based on the highly accurate closedform eigensolutions in Table A1 and Table A2. In fact, the Galerkin and Kantorovich methods are widely used in flutter analysis. In order to verify the present closed-form solutions,we compare the present solutions with the Galerkin solutions and Kantorovich solutions as well as some results in literature.

4.4.1. Galerkin method

In this method, the deflection has the following form:

where M and N are the number of terms adopted in x and y direction respectively;Amnis the coefficient of each order mode function.

In view of the fact that flutter is caused by aerodynamic forces and the first span-wise mode is dominant in linear theory for un-buckled panel,10only the first span-wise mode is retained here, that is, n = 1. Henceforth, the subscript n is dropped somewhere for clarity.

(1) A simply supported rectangular panel

For a SSSS panel, the eigenfunctions are

4.4.3. Comparison

In order to validate the accuracy of the present closed-form solutions, here the present results are compared with the results of the Galerkin method and the Kantorovich method and the results in literature. We use the relative deviation defined by (iSOV - other)/iSOV × 100%, denoted by Difference in the following tables, to assess the accuracy of an approximate method.For the SSSS panel,Table 11 shows that the Kantorovich method and the iSOV method are all exact.However, as an approximate method, the accuracy of the Galerkin method depends on the number of modes adopted.For a SSSS panel,one should use at least four modes to obtain accurate results. It can be seen that, (A) using odd number ofmodes predicts larger solutions compared with the exact,whereas using even number of modes predicts lower solutions;(B)for a certain Mach number, ^β0is accurate for all methods.

Table 11 Flutter results of SSSS panel (a/b = 1/3, a/h = 150).

Table 12 Flutter results of CCCC panel (a/b = 1/3, a/h = 150).

Table 13 Flutter results of SCSF panel.

For CCCC panel,the exact eigensolutions in separation-ofvariable form do not exist, but here we achieve the accurate closed-form eigensolutions in separation-of-variable form using the iSOV method, in which both the span-wise and chord-wise mode function are related to the panel but not beam.From Table 12,we can see that the results of the Galerkin method converge to the results of the Kantorovich1. One can also observe that, for the Galerkin method, (A) using odd number of modes predicts larger solutions while using even number of modes predicts lower; (B) at least four modes need to be used to yield accurate solutions; (C) more accurate results can be obtained using more modes, and no meaningful results can be achieved through three modes.

The results are also compared with the results in Ref. 48 where the Kantorovich method and beam mode functions were used. Table 13 and Table 14 show the comparison for SCSF and CCCF panels with different a/b. It can be seen that the present results are more accurate than the results in Ref. 48 since the literature results are larger than the present ones,and the differences between them decrease as a/b decreases.This is because the panel is more and more shaped like a beam as a/b decreases, leading to the increase in the accuracy of the results in Ref. 48.

5. Conclusions

In the present paper,the governing equations for the flutter of three-dimensional (3D) panels were derived based on theHamilton principle, the classical thin-panel theory and the first-order piston theory. The highly accurate closed-form eigensolutions for rectangular panels with arbitrary homogeneous boundary conditions were achieved using the iSOV method for the first time, and all mode functions and frequency equations were presented in an elegant, explicit and closed form. If the edges in the direction perpendicular to the flow direction are simply supported, the solutions of the iSOV method are exact, which are the Navier and Levy types of exact solutions, otherwise they are highly accurate.

Table 14 Flutter results of CCCF panel.

The present investigation provided insight into the flutter characteristics of the 3D panels with regard to structural and aerodynamic factors such as the aspect ratio, the thicknesschord ratio, the boundary conditions and the aerodynamic damping. Considering these factors, both static and dynamic boundary conditions have been achieved. There were the following observations: (A) although the aerodynamic damping has significant effect on limit cycle deflections,it has little effect on flutter boundaries; (B) among these factors, only the boundary conditions and in-plane loads can affect the flutter types, and two types of flutter called the coupled-mode flutter and the zero-frequency flutter which includes zero-frequency single-mode flutter and buckling have been discovered; (C)the flutter characteristics which are similar with those of the 2D panels are dominated by the properties of panels in the direction of the stream flow, and the 3D panel problem can be reduced to 2D problem if the aspect ratio is high enough for the coupled-mode flutter; (D) the flutter speed is independent of in-plane loads in the direction perpendicular to the flow direction,and if the in-plane loads are large enough,panels experience buckling and there is no occurrence of flutter.

The mechanism of the panel flutter was also revealed.Once the panels are subjected to the aerodynamic pressure,the panel stiffness becomes asymmetrical due to effects of aero-elasticity on it. And at the critical Mach number, the first two modes degenerate into one new mode or the first frequency is zero.In general,the coupled-mode flutter occurs at high Mach number while zero-frequency flutter or buckling happens at low Mach number.

The newly achieved eigensolutions have been verified by comparing with those of the Galerkin method and the Kantorovich method. The solutions can serve as benchmarks for the validation of numerical methods, and the basis of structural parametric design in engineering.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Nos. 11872090, 11672019 and 11472035).

Appendix

Table A1 Eigensolutions for two edges at η=±1.

Table A1 (continued)

Table A1 (continued)

Table A2 Eigensolutions for two edges at ξ=±1.