Xiojie ZHANG, Ynrong WANG,b, Weiyu CHEN, Xinghu JIANG,b,*

a School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Jiangxi Research Institute Beihang University, Nanchang 330096, China

KEYWORDS Aerodynamic damping;Flutter;Hollow fan blade;Hollow structure;Orthogonal experimental design

Abstract In recent years,the hollow fan blades have been widely used to meet the demand for light weight and good performance of the aero-engine. However, the relationship between the hollow structure and the aeroelastic stability has not been studied yet in the open literature. In this paper,it has been investigated for an H-shaped hollow fan blade.Before studying the flutter behavior,the methods of parametric modeling and auto-generation of Finite Element Model (FEM) are presented. The influence of the feature parameters on the vibration frequency and mode shape (as the input of flutter calculation)of the first three modes are analyzed by the Orthogonal Experimental Design(OED)method.The results show that the parameters have a more remarkable impact on the first torsional mode and thus it is concerned in the flutter sensitivity analysis.Compared with the solid blade, the minimum aerodynamic damping of the hollow blade decreases, indicating that the hollow structure makes the aeroelastic stability worse. For the parameters describing the hollow section, the rib number N has the greatest influence on the minimum aerodynamic damping, followed by the wall thickness W5. For the parameters in the height of hollow segment, the aerodynamic damping increases with the increase of parameters M1 and M2. This means that reducing the height of the hollow segment is helpful to improve the aeroelastic stability. Compared with the impact of parameters in hollow section, the variation of aerodynamic damping caused by the height of the hollow segment is small.

1. Introduction

Modern aircraft engine pursues higher thrust-to-weight ratios,better aerodynamic performance and structural strength,which put forward new requirements for the design of aeroengine in many aspects, such as material, aeromechanical design, and fabrication technology. Utilizations of many new structures and technologies contribute to such advanced aero-engine, and the shroudless, hollow fan blade has secured much attention.

In the 1960s, the great majority of fan blades were solid structure with large aspect ratio. A major challenge of this kind of fan blade is prominent vibration problem. Therefore,it often requires the installation of part-span shroud to improve the blade rigidity and anti-vibration ability.However,the existence of shroud has certain problems, which are not allowed by high-performance fans. On the one hand, the shroud leads to the significant increase in the component weight and the difficulty of manufacturing.On the other hand,it offers a substantial increase in aerodynamic losses which is directly translatable into a less efficient component and higher fuel consumption.1

In order to overcome the disadvantages of the conventional shrouded,solid fan blades,people began to investigate the possibility of the shroudless fan blade. The aeroelastic instability caused by the removal of shroud is mitigated by reducing the aspect ratio. Meanwhile, the hollow structure is applied to lighten the weight increment produced by the large chord of the blade.With the complex hollow structure inside the blade,it brings new challenges for designers. Several areas are required further research,such as fabrication feasibility,structural integrity,sensitivity to foreign object damage,and repairability.2

In the beginning of the hollow fan blade research,it mainly focused on the exploration of fabrication technology. The Superplastic Forming combined with Diffusion Bonding(SPF/DB) process was extensive evaluated from diamond shaped specimens to full-size hollow prototype blades.1,3After years of research, the SPF/DB process of titanium alloy used in the fan blades has been relatively mature. It has become the main fabrication technology of the hollow fan blade due to the advantage of apparent weight and cost savings.4–5

There are three typical hollow fan blades with different configurations of reinforcing ribs, which are already used in the aero-engines.The first-generation hollow fan blade was honeycomb structure designed by Rolls-Royce, as shown in Fig. 1(a). It was successfully applied to RB211-535E4 engine for the first time in 1984. However, it was gradually disused because of its poor weight reduction and strength performance. Based on the SPF/DB process, the second-generation hollow fan blade was developed with a rib configuration similar to the triangular truss, which can be called W-shaped hollow fan blade,as shown in Fig.1(b).It was successfully applied to TRENT700 engine in 1994. Compared with the first generation, this kind of hollow fan blade is lighter in weight, with about 15%reduction of each blade.Since 1980s,Pratt&Whitney Aircraft started to develop the hollow fan blade, and successfully applied to PW4084 engine in 1994. The hollow configuration is H-shaped, as shown in Fig. 1(c), which is different from the above two blades designed by Rolls-Royce.6

Fig. 1 Three typical hollow fan blades.

In the early 21st century, composite materials were used to replace the internal metal rib structure.In 2002,NASA Glenn laboratory began to develop a lightweight and low-noise composite hollow fan blade.7The simulation and experimental results showed that the blade has potential aeroelastic instability, so that the research has centered on the arrangement of damping composite materials to obtain the best damping effect.8Some researchers are also committed to the design of hollow fan blade from the perspective of aerodynamic efficiency.For example,the airflow passage is designed in the hollow structure to reduce the flow separation.9In addition,many researchers have been putting a lot of efforts to design the novel rib structure.10–13But so far, the W-shaped of Rolls-Royce and H-shaped of Pratt&Whitney Aircraft are the most widely used hollow fan blades due to the mature level of design and manufacture. These two types of hollow fan blades show good performance in both test and practical application.

In terms of structural strength, Ogawa et al.14investigated the location and number of H-shaped ribs in relation to the strength and torsional rigidity using the two-dimensional torsional stress analysis. Kajbyshev et al.15further explored the influence of rib types (cell, corrugation and truss) on the strength and stiffness using shell elements. After clarifying the influence of ribs on strength, many researchers optimize the distribution of ribs based on different targets.Meng et al.16presented a new Fork-shaped rib structure based on the target of static strength, frequency and stress level under bird-strike condition. However, this Fork-shaped blade cannot be manufactured with SPF/DB process and will not be employed in practice in a short time.Ma et al.17focused on the weight optimization through changing the thickness and distribution of the W-shaped ribs under the constraints of stress, displacement, and frequencies. In addition, Audic18and Hou19et al.studied the transient response of hollow fan blade under bird strike by numerical simulation and the analysis method was discussed in detail. Kielb20used a simplified cross section model to investigate the effect of mass balancing on flutter alleviation of the H-shaped hollow fan blade. However, this simplified model and analytical method are no longer applicable to the present complex hollow fan blades.

From the significant advantages of hollow fan blade in practical application, its structural design has achieved outstanding results.However,the detailed design theory and analysis of hollow fan blade are rarely seen in the open literature.There is no clear description for the key structural parameters of the hollow fan blade, especially the internal hollow structure. What’s more, the relationship between the hollow structure and the flutter characteristics at different Nodal Diameters(ND)has not been studied yet in the open literature,which has hindered the development of flutter suppression methods of the hollow fan blade.

In this paper, the influence of several important feature parameters (including rib thickness, rib number, blade thickness at different blade span) of a typical H-shaped hollow fan blade on the natural vibration characteristics and flutter characteristics are investigated. The aeroelastic eigenvalue method, which was proposed by Hanamura et al.21are performed to calculate the aerodynamic damping for all NDs.The effect of feature parameters on the minimum aerodynamic damping is discussed in detail using the energy method.

The first part of this paper introduces the test case and numerical methods. Then, a set of feature parameters that can describe the geometric structure of the hollow fan blade are proposed. On this basis, the auto-generation method of the Finite Element Model (FEM) is presented, which is the premise of subsequent modal analysis and flutter calculation.Next,the flow structure and characteristics at the design point are analyzed. In addition, the differences in the vibration frequencies and mode shapes of the first three modes between different hollow structures are compared. The parameters which have a greater impact on the first torsion mode are selected by the Orthogonal Experimental Design (OED) method. Finally,the effect of key parameters of hollow fan blade on flutter characteristics is discussed in detail.

2. Numerical methodology

2.1. Aerodynamic flow model

The commercial software CFX is used to solve the threedimensional compressible Reynolds averaged Navier-Stokes equations using the k-ε turbulence model. The highresolution scheme is used to calculate the advection term and turbulence numerics, while the second-order backward Euler scheme is used for the transient term. All simulations are solved using double precision.For steady simulations,the total pressure, total temperature, and inlet flow angle are set at the inlet boundary, and the static pressure is given at the outlet.The boundary conditions are the same for the flutter simulations, and the steady results are used as the initial field for the flutter simulations.

It is known that the position of inlet and outlet will affect the flutter characteristics due to the reflected pressure wave.22–23The purpose of this work is to associate the hollow structure with the aerodynamic damping, and the effect of the reflected pressure wave is beyond the scope of this work.Thus,the meshes of the upstream and downstream are extended and coarsened to eliminate the effect of pressure wave reflection,as shown in Fig. 2.

The mesh sensitivity study is performed using three different meshes with approximately 310000(coarse), 640000 (medium), and 1060000 (fine) nodes per passage. The steady results of aerodynamic performance on the 100% speed line are shown in Fig. 3. It can be seen that medium and fine meshes give essentially identical predictions at the design point, but little difference appears when moves towards the stall condition. Considering the computer resources and the reliability of results,the final mesh used for flutter simulations has 0.64 million nodes for each passage,which can well-predict the flow structures at the design point.

Fig. 2 Mesh overview with the extended coarse inlet and outlet.

Fig. 3 Aerodynamic performance for three mesh densities.

2.2. Mechanical model

As the blade profile shape under the operating condition is determined by aerodynamic design in advance. Thus, an untwist analysis is needed to establish the required correction from a given blade operating shape (‘‘Hot” blade) to get the manufacture blade shape(‘‘Cold”blade).A larger than normal correction is expected due to the large size and low rigidity of the hollow fan blade.

Fig. 4 Flow chart of hot-to-clod analysis.

The hot-to-cold analysis is determined by an iterative procedure, as shown in Fig. 4. For a given hot blade, a steady simulation is performed to calculate the aerodynamic pressures and temperatures under the specified operating condition.Then, the blade deflections due to aerodynamic, centrifugal and thermal loads can be obtained by a structural solver. An estimate of the cold blade shape is determined by subtracted the blade deflections from the hot blade shape. The pressure,thermal, and centrifugal loads are then applied to this estimated cold shape in the structural analysis to obtain a new blade deflections. These new blade deflections are subtracted from the hot blade again to get a new estimate of the cold blade. This procedure is iterated until a converged manufacture blade shape is obtained.

Then,a static structural analysis of the converged manufacture blade shape is performed under the consideration of aerodynamic loads and centrifugal loads in the ANSYS Mechanical software.After that,a pre-stressed modal analysis is carried out to determine the mode shape and vibration frequency. As the structural and fluid meshes are not matched exactly, interpolation is used in the mode shape and aerodynamic loads between these two meshes.

2.3. Flutter simulation method

The aeroelastic eigenvalue method and energy method are the most common approaches for flutter simulation, which have been applied and verified by many researchers.24–28A major advantage of the aeroelastic eigenvalue method is that it only needs one calculation to obtain solutions for all NDs, unlike the energy method which needs to calculate each ND individually. However, this method cannot give the details of flow structures and the aerodynamic damping density distribution at the blade surface. Thus, both the aeroelastic eigenvalue method and energy method are used here to study the flutter behaviors of different hollow fan blades. The former method is used to calculate the aerodynamic damping for all NDs.The latter method is used to study the flutter behavior at the most unstable ND. Detailed formula derivation of these two methods can be found in Ref. 26.

The timestep per vibration period in the flutter simulation also influences the results. Fig. 5 shows the relative error of aerodynamic damping at 0ND versus the timestep. Considering that the hollow structure of the fan blade is not determined yet, the flutter simulation is carried out based on the solid blade, which has the same profile shape as the hollow blade.The relative error is defined as the percent change in the damping value relative to the results of 154 timestep. With the increase of timestep in each vibration period,the aerodynamic damping error gradually decreases and stabilizes.Based on the timestep independence verification, 88 timesteps is chosen to balance the simulation accuracy and the computational expenses.

Fig. 5 Relative error of aerodynamic damping versus timestep.

3. Wide-chord hollow fan blade

A shroudless,wide-chord transonic fan blade is adopted as the model to explore the relationship between the hollow structures and flutter characteristics. It is partially outside the flow channel and connected with the disk through a tenon, as shown in Fig. 6. The part in the flow channel (between the shroud and hub) is decided by the aerodynamic design. The design parameters of the fan blade are shown in Table 1.

3.1. Model parameterization of hollow fan blade

The hollow fan blade can be regarded as solid panels and inner-hollow sections.The design of outer profile shape is similar to the solid blade, which is based on aerodynamic performance. Whereas the design of hollow sections needs to consider the weight reduction, fabrication technology and strength requirements. The key step of the hollow fan blade modelling is to parameterize the hollow structure and extract the geometric feature parameters.

In this paper, taking the hollow fan blade with H-shaped reinforcing ribs as the research model. A cross section of the model with three ribs is illustrated in Fig. 7.The main parameters of the hollow fan blade include the distance between the hollow segment and the blade tip or root (M1, M2), the distance between the cavity in hollow section and the blade leading or trailing edge (T1, T2), wall thickness (W1-W5), fillet radius,rib thickness(B1,B2),rib number,rib position(D1,D2).

In order to describe the structural characteristics of the hollow fan blade as simply and comprehensively as possible,meanwhile considering the calculation amount of the flutter sensitivity analysis,the feature parameters of hollow fan blade are simplified as follows:

1) In the hollow section of given blade span,the wall thickness, rib thickness and position characteristics of the cavity are symmetrically distributed;

2) The rib thickness is equal at different blade span and chord length, i.e. B1= B2;

Fig. 6 Model of wide-chord hollow fan blade.

Table 1 Design parameters of shroudless fan blade.

3) The ribs are uniformly distributed along the cavity, i.e.D1= D2;

4) The wall thickness is equal in the same section.In order to investigate the effect of wall thickness at different blade spans on the natural vibration characteristics and flutter performance, five uniformly distributed hollow sections are selected to describe the wall thickness variation of the entire blade,i.e.W1-W5.The wall thickness of other sections is obtained by cubic spline interpolation according to these five sections.

5) In the section of given blade span, the length from the leading and trailing edges to the boundaries of cavity(T1, T2) are determined by the radius of outermostfillet and wall thickness. The position that can accommodate the fillet radius is defined as the boundaries of the cavity. According to the request of fabrication technology,the radii of outermost fillets near the leading and trailing edges are 0.5 mm, and the radii of inner-fillets are 1 mm.

According to the above simplified modeling principle, the boundaries of cavity (T1, T2) can be decided upon when the wall thickness W and radii of outermost fillets are determinate.Likewise, when the rib thickness and rib number are determined, parameter D can also become a certainty. A set of independent feature parameters describing the hollow fan blade are proposed in Table 2. The parameters M1and M2determine the location and dimension of the hollow segment,and N, B, W determine the hollow configuration of each section in the hollow segment.

3.2. Auto-generation of finite element model

In the study of flutter sensitivity analysis, multiple hollow fan blade models with different feature parameters need to be compared. This means that the FEM of the hollow fan blade requires to be updated and reestablished for many times.The conventional method is to construct the threedimensional model of the hollow fan blade by CAD software according to the feature parameters extracted above,and then generate the FEM based on the geometry.This is a tedious and time-consuming work for the flutter sensitivity study. Therefore, it is necessary to realize the auto -generation of FEM based on the blade profile data from aerodynamic design.

3.2.1. Structural frame of hollow fan blade

According to the structure characteristics,the hollow fan blade is divided into solid, hollow and transition segments along the blade span,as shown in Fig.7.The upper and lower solid segments refer to the solid sections above and below the hollow segments respectively,and the same goes for the transition segments. The method of section division is shown in Fig. 8.Before the establishment of the FEM, the positions of the inner-wall surface,ribs,fillets,and the edge boundaries of cavity need to be determined by feature parameters,and the structural frame of the hollow section is shown in Fig. 9.

The key steps to determine the structural frame of the hollow sections are introduced below.

Fig. 7 Model parameterization of hollow fan blade.

Table 2 Feature parameters of hollow fan blade.

·Outer-wall surface:The outer-wall curves of the blade are obtained from the data points of suction surface and pressure surface based on bicubic spline interpolation,and the direction is from the leading edge to the trailing edge.

· Inner-wall surface: The inner-wall curves are determined by offsetting the outer-wall curves inward by the wall thickness along the normal direction. In addition, the variation of wall thickness along the chord length can also be given to simulate the uneven wall thickness in the process of SPF/DB.

· Cavity boundary: Identification of the cavity boundary can also be said to determine the length of the solid parts from the leading edge, the trailing edge, root and tip to the cavity,respectively.Thereinto,the distance between the cavity boundaries and the blade tip or root is directly determined by parameters M1and M2.The boundaries near the leading and trailing edges are related to the outermost fillet radius and the wall thickness. New curves can be obtained by offsetting the inner-wall curves inward by the fillet radius, and the intersection point of the new curves is the center point of the outermost fillet. Then, the tangent of the fillet is the cavity boundary.In order to ensure a reasonable shape of the cavity,the large differences of T1between different hollow sections have to be avoided. Thus, the maximum value of T1in all hollow sections is taken as the cavity boundary, and the same goes for T2.

· Ribs: After the determination of the cavity boundaries,the centerline of ribs can be obtained according to the rib number and the distribution proportion (evenly distributed in this paper).The lines with half rib thickness from the centerline are the curves of rib.

· Fillets: The fillets include the outermost fillets which has been introduced previously, and the inner fillets which is tangent to both the rib and the inner-wall surface. Here, we discuss the determination of the inner fillets. Firstly, a point is selected on the inner-wall curves, and the angle θ between the rib and the tangent direction of this point can be calculated. Similarly, the distance l from the point to the rib can be determined. Then the fillet radius Rfcan be obtained by Eq. (1). According to the fillet radius required by the fabrication technology, the position of point can be adjusted by dichotomy until the radius requirement is satisfied, and the inner fillet is determined.

3.2.2. Establishment of finite element model

The FEM is established by MATLAB,which can be automatically updated with the change of the feature parameters, and the mesh size can be adjusted according to the structure. The solid and hollow segments have their own mesh topologies respectively, and the transition segment needs to realize the transition from the mesh topologies of solid segment to the hollow segment.

Fig. 8 Section division of hollow fan blade.

Fig. 9 Structural frame of the hollow section.

The quadrilateral mesh of each section is established at first according to the structural frame of the hollow fan blade,and then the hexahedral element is composed by the quadrilateral meshes of adjacent sections. The quadrilateral mesh is composed by corresponding nodes on the feature curves of the cross section. The mesh generation of the fillets is more complex, especially those located in the upper and lower boundaries of the cavity. The fillets in the hollow sections are regarded as 1/4 arc, and the fillets in the cavity boundary are regarded as 1/8 sphere, as shown in the Fig. 10.

The FEM of the cross sections in the hollow,transition and solid segments is shown in Fig. 11, in which the topology of meshes in the same color is similar. The blade is divided into 71 elements along the blade span. The transition segment has three elements, and the number of elements in the hollow segment is adjusted according to the parameters M1and M2. In addition, there are 91 elements in the chord length direction and 8 elements in the thickness direction. Each rib has 6 elements in the chord length direction and 4 elements in the thickness direction. The number of nodes is about 70000, and the number of elements is about 60000 in the FEM of the hollow fan blade.

4. Results and discussion

4.1. Steady simulation

The steady simulations begin from the choke point to the last steady-state point,which is considered to be the near stall condition, as shown in Fig. 3. In this section, the steady flows at the design point, which is the flutter simulation concerned, is analyzed.

Fig.12 shows the Mach number at the 50%,70%and 90%span at the design point.The direction of airflow is from left to right, and the direction of rotation is from down to up. It can be seen that there are obvious passage shock waves at three blade spans, and with the increase of blade span, the position of the shock wave gradually moves towards the leading edge.The interaction between the shock wave and boundary layer induces separation,such as the area on the suction surface near the trailing edge and on the pressure surface near the middle chord.Fig.13 shows the static pressure and surface streamline distributions to illustrate the flow field details.Separations are marked as a red dotted line, including the separation after the shock wave.In addition,there is separation on the suction surface near the blade root, which is accompanied by the intense three-dimensional radial flow.

Fig. 10 FEM of fillets.

4.2. Modal analysis

The modal analysis is performed on the FEM of the ‘‘cold”blade as already noted in Section 2.2.The FEM of‘‘hot”blade is established based on the aerodynamic profile data and autogeneration method in Section 3.2. The density of the blade is 4440 kg/m3,the elastic modulus is 109 GPa,and the Poisson’s ratio is 0.34. Fixed boundary conditions are imposed on the tenon of the blade. The centrifugal force and aerodynamic pressure from the steady simulation is applied.Then the static structural analysis is performed under the consideration of the geometric nonlinear effect. Taking a typical hollow blade parameter as an example, the comparison between the ‘‘hot”(red profile) and the ‘‘cold” (blue profile) blade is shown in Fig.14.It can be seen that under the effect of centrifugal loads and aerodynamic pressure,the pre-twisting of the‘‘hot”blade is weakened compared with the ‘‘cold” blade. The largest difference between these two shapes occurs at the tip,and the difference reduces along the blade span and reaches zero at the hub.

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According to the iterative method given in Fig. 4, the predicted ‘‘cold” blade is updated until the convergence criterion is met. The maximum deformation of the blade is gradually stable with the increase of the iterations, as shown in Fig. 15. It can be seen that when the 15th iteration is carried out, a converged displacement of the blade is obtained, which means the cold blade shape is determined.The static structural analysis and pre-stressed modal analysis of the converged cold blade shape are carried out to determine the mode shape and vibration frequency, as shown in Fig. 16.

4.3. Orthogonal experimental design

4.3.1. Test design

The OED is a valuable optimization method proposed by Genichi Taguchi to reduce the number of tests, especially for the systems with multiple factors and levels.29It is a highly efficient multi-factor experimental design method, and has been successfully applied in numerous research areas.30–33For example,for a four-factor and three-level experimental design,81 (34) full factorial experiments must be carried out if each level of each factor is matched for a test. Whereas the OED method only needs 9 representative tests based on orthogonality, and the results can be obtained equivalent to the full factorial experiments with the least number and cost of tests.

Fig. 11 FEM of hollow fan blade.

Fig. 12 Mach number at the design point.

Fig. 13 Contours of the static pressure and surface streamline.

The OED uses orthogonal tables to arrange experiments reasonably. It is described as Ln（rq), in which L represents the orthogonal table, n is the number of orthogonal experiments,r is the number of levels,and q is the number of factors.The determination of factors and levels is the basis of the orthogonal experimental design. In general, three levels are enough for each factor. If a detailed regression analysis is needed, the number of levels can be increased, but no more than five levels are appropriate.34

Fig. 14 Comparison of the hot and cold blade.

Fig. 15 Variation of maximum deformation with iteration number.

Fig. 16 Typical mode shapes and natural frequencies of the hollow fan blade.

In this study, the OED is used to analyze the effect of feature parameters on the aerodynamic damping.The influencing factors include wall thickness W1- W5, rib thickness B and rib number N,which are feature parameters of the hollow section.Three levels are selected for each factor, and the range of the values are determined according to the typical value in Ref.1, as shown in Table 3. The wall thickness W1- W5are given as the percentage of the maximum blade thickness in the corresponding section. Besides, the minimum value is determined by the manufacturability and static strength requirements.The maximum value is 35% of the maximum blade thickness. In addition, when the OED is performed to study the effect of the above seven parameters, the distance from the hollow segment to the blade tip M1and to the root M2are fixed, and expressed as the ratio of the distance to the blade height.

In the flutter simulations, the natural frequency and mode shape are used as input variables, which will directly affect the flutter sensitivity analysis. In addition, according to the research, flutter mostly occurs in low order modes. Therefore,the effect of each feature parameter on the first three modes,including vibration frequency and mode shape, is analyzed at first to extract the more influential parameters,so as to reduce the calculation amount of the subsequent flutter sensitivity analysis.The variation of mode shape is represented by Modal Assurance Criterion (MAC), as shown in Eq. (2). It is widely used in modal tests to check the coincidence between the theoretical and experimental mode shapes.35If the value is equal to unity, these two mode shapes are consistent.

where φ and ψ are the mode shape vectors of different models,T refers to the transpose of the vector.

In this paper, the mode shape of the solid blade is taken as the reference,and the differences of mode shapes between hollow blades are expressed by the MAC values of the solid blade and various hollow blades.According to the seven factors and three levels in Table 3,orthogonal table L18(37)is established,and the simulation tests are shown in Table 4.

Range analysis is adopted to estimate the influence degree of each factor on the test indices. Range value R can be invoked as a sensitivity indicator to rank the impact factors. A factor with the larger range value is the one that has the greater influence on the corresponding indices. It can be calculated as follows:

where i represents the factor levels, from 1 to 3;j is the factor;Rjis the range value of factor j;Kijis the mean value of the test results of factor j at all level i.

The simulation results in Table 4 are analyzed by the range analysis method, and the range values of seven factors are listed in Table 5. Taking the indice mass as an example, the results of mass for factor W1at all level 1(that is,the mass values of cases with W1= 0.16) are 3.82, 4.14, 4.43, 4.13, 4.16,and 4.24 kg. Thus, K1for factor W1is (3.82 + 4.14 + 4.43+4.13+4.16+4.24)/6=4.15.In this way,the mean value K2and K3can be obtained. Then, the range value R of factor W1is 4.28–4.15 = 0.13. It can be seen that, the impact of the wall thickness W2on mass is the largest, followed by W1and W3. The range values of factors W4, W5, B and N are small and differ slightly, which suggest that the wall thickness nearthe blade root is the primary factor for the weight reduction of hollow fan blade.

Table 3 Seven factors and the corresponding levels (M1 = 0.12, M2 = 0.2).

Table 4 Orthogonal experimental design scheme and simulation results.

Table 5 Range analysis for orthogonal experimental design scheme.

Fig. 17 Various trends in the mean values of the first mode under different factor levels.

Fig. 18 Various trends in the mean values of the second mode under different factor levels.

Fig. 19 Various trends in the mean values of the third mode under different factor levels.

The range values R of each factor on the vibration frequencies and mode shapes of the first three modes are calculated in the same way,and the order of influence degree(from the most to least)is shown in Table 5.To further analyze the impact of factor levels on the performance indices,variation trends of the mean values Kijunder three levels of the first three modes are illustrated in Figs. 17-19.

For the first mode, the seven factors have a negligible impact on the mode shape, and the largest variation of the MAC value is only 4×10-5.The range values of its vibration frequency are at the same level as the values of the other two modes. In contrast, the wall thickness W1and W2have a greater influence on the frequency, and the wall thickness W3and W4have a greater influence on the mode shape. For the second mode, the vibration frequency rapidly decreases with the increase of wall thickness W3.Moreover,the wall thickness W3and rib number N have a significant but opposite effect onthe MAC value, and the maximum variations are 2 × 10-3.The MAC value increases with the increasing wall thickness W3but shows a decreasing trend with the increase of rib number N.For the third mode,the variation of MAC value is hundreds or thousands times larger than the first mode. The maximum variations of the frequency and MAC value are both obtained at rib number N. With the rise of rib number,frequency gradually increases, whereas the MAC value falls.Also, as the rib number increases, the frequency and MAC value initially change slightly and then rapidly.

Table 6 Simulation tests for the first torsion mode (M1 = 0.12, M2 = 0.2).

Fig. 20 Aerodynamic damping versus ND of different blade models.

To summarize,the wall thickness near the blade root has a larger impact on the vibration frequency, while for the mode shape, the wall thickness near the blade tip has a more significant influence. In addition, it should be noted that for the third mode (that is, the first torsional mode), the rib number has the greatest influence on the frequency and mode shape due to the effect of the distribution of ribs on the torsional stiffness. In general, the natural vibration characteristics of the first torsional mode are more sensitive to the feature parameters of the hollow fan blades.

4.4. Overall flutter sensitivity analysis

In the flutter simulations, the vibration amplitude is set to 0.75% of the tip chord length to ensure the unsteady flow can be expressed as linear with no negative cells generated.The research on the relationship between the hollow structure and flutter characteristics focuses on the design point condition, as shown in Fig. 3.

4.4.1. Impact of parameters in hollow section

As evidence from Table 5, the third mode is more sensitive to the feature parameters of the hollow fan blade. Therefore, the flutter sensitivity analysis focuses on the third mode. Among the seven feature parameters describing the hollow section,the rib number N and wall thickness W2, W5have a more significant impact on the third mode, whereas other parameters have less effects and can be ignored in the flutter sensitivity study. Thus, the orthogonal table L9 (34) is used for the three-factor and three-level experimental design,and the simulation tests are shown in Table 6.The three levels of the investigated factors are the same as those in Table 3, and the other factors are equal to the values of level 2 in Table 3.

Fig. 21 Various trends in the mean values of the third mode in the flutter simulations.

The overall flutter behaviors of the solid blade and hollow fan blades in Table 6 are simulated using aeroelastic eigenvalue method. In this method, simulations require a multi-passage computational domain, and only one blade (in the middle of the domain) needs to oscillate. The unsteady pressure and aerodynamic damping induced by the blade oscillation are calculated based on the influence coefficient method.The number of passages significantly influences the accuracy of the aerodynamic damping.In general,seven-passage model are sufficient to obtain the unsteady aerodynamic forces even tip gap is included.36Ref. 26 showed that a nine-passage computational domain has a better agreement with the energy method. To guarantee the reliability of results, nine-passage model is adopted in this paper.

Fig. 22 Aerodynamic damping density distribution of different rib number (ND = 1).

Fig. 23 Total displacement distribution of different rib number.

Fig. 20 shows the changes in the aerodynamic damping with the NDs.It can be seen that the variation trends of aerodynamic damping with nodal diameter are similar for the solid blade and hollow blades. For these ten blade models (solid blade and Case 1-Case 9 hollow fan blades in Fig. 20), the most unstable ND all occur at 1, and the corresponding aerodynamic dampings are all positive.

The number of ND for the aeroelastic mode depends on the blade vibration,geometry and flow structure around the blade.The flow will choose the ND which is the most unstable.37Therefore, the minimum aerodynamic damping under the different feature parameters of the hollow fan blade are focused in this paper. It can be seen that for the first torsional mode, the aeroelastic stability of hollow blade is worse than the solid blade.The largest difference of the minimum aerodynamic damping between solid blade and hollow blades(Case 1–Case 9)is 2.3×10-3,accounting for 27%of the damping of solid blade.

The variation trends of the mean values Kijof frequency,MAC value and minimum aerodynamic damping at three levels are shown in Fig. 21. The changes of the frequency and MAC value with the parameters N, W2and W5are thesame as that in Fig.19.The influence degree of factor N is still the largest, followed by W2for frequency and W5for MAC value.

Table 7 Simulation results for the parameters in hollow height.

Fig. 24 Various trends in the minimum aerodynamic damping under different hollow height.

The influence degree of parameters N and W5on the minimum aerodynamic damping is about the same level,which are both about 7.5×10-4,and RNis slightly higher than RW5.In contrast, W2has little effect on the minimum aerodynamic damping. For this blade profile, the minimum aerodynamic damping decreases with the increase of parameters N and W5. Meanwhile, it should be noted that the variation of minimum aerodynamic damping with W5and N is the same as that of the MAC value.With the increase of N and W5,MAC value and minimum aerodynamic damping both decrease.The larger MAC value indicates that the mode shape of the hollow fan blade is more consistent with the solid blade,and thus the minimum aerodynamic damping is larger,which is consistent with the variation trend in Fig. 20. Therefore, we can calculate the MAC value of hollow fan blades with different parameters W5and N through modal analysis, and optimize the hollow section quickly by comparing the MAC value, so as to maximize the aerodynamic damping.

In addition, it can be noticed from Fig. 21(c), with the rise of rib number N, the aerodynamic damping falls slightly at first and then sharply. When N is equal to 3 and 4, there is not a great deal of difference in the aerodynamic damping.This part links the parameters in hollow section with the aerodynamic damping density distribution on the blade surface under the most unstable ND using the energy method. Similarly, using the method of OED analysis, the aerodynamic damping density distribution at three levels of rib number N can be obtained, as shown in Fig. 22.

In general, the aerodynamic damping density distributions are similar for all three conditions. There are large negative damping areas near the leading edge on the pressure surface and positive damping areas near the trailing edge on the suction surface at the 70%-95%span.It is noted that the aerodynamic damping density distributions are almost the same when the rib number is 3 and 4, whereas are different when the rib number is 5. The main differences are the positive damping area near the leading and trailing edge on the suction surface,and the location of the middle chord near the tip on the pressure surface, as shown in the areas enclosed by the red dashed circles in Fig. 22.

Fig. 25 Aerodynamic damping density distribution of different M1 (M2 = 0.2).

Fig. 26 Aerodynamic damping density distribution of different M2 (M1 = 0.12).

To explain this phenomenon,Fig.23 presents the vibration amplitude along the pressure surface in the flutter simulations with the maximum tip displacement of 0.002 mm. It can be seen that when the rib number is 5, the region of 0.0002 contour line is narrower and farther away from the tip, and the other contour lines on both sides are closer to the nodal line.The area near the trailing edge at 95% span (the red dashed box) is a high amplitude region, which is significantly larger in Fig. 23(c) than the other two cases. In addition, it can be noticed from the red dotted line that the location of the contour lines in Fig. 23(c) is lower than the others, which affects the distribution of the aerodynamic damping density.

4.4.2. Impact of parameters in hollow height

The distances from the hollow segment to the blade tip and root are important factors that affect the location and dimension of the hollow segment.In this section,the influence of the parameters M1and M2on the aerodynamic damping is studied by variable-controlling method,as listed in Table 7.The other seven feature parameters, including wall thickness W1-W5, rib thickness B and rib number N are equal to the Case 1 in Table 4.

Fig. 24 shows the variation trend of the minimum aerodynamic damping with the parameters M1and M2. The red line represents that M1is 0.12 and M2changes as 0.1, 0.2 and 0.3.The black line stands for the cases that M2is 0.2, M1is 0.08,0.12 and 0.16 respectively. The results show that the aerodynamic damping increases with the increase of M1and M2.This indicates that reducing the height of the hollow segment is helpful to improve the aerodynamic damping. In Table 7,the largest difference of the minimum aerodynamic damping between the hollow blades is about 2.2 × 10-4for the parameter M1,and 3.7×10-4for the parameter M2.Compared with the impact of parameters in hollow section on the minimum aerodynamic damping, the variation of aerodynamic damping caused by the height of the hollow segment is smaller.

The aerodynamic damping density distribution at three levels of M1and M2are shown in Figs. 25-26. The black lines indicate the location of the upper and lower boundaries of the hollow segment. It can be noticed that the variation of M1mainly changes the aerodynamic damping density in the region of middle chord on the suction surface and the trailing edge at 90%span.The change of M2mainly causes the significant difference of the aerodynamic damping density near the location of lower boundary,including the area below 20%span on the pressure surface and 20%-30%span on the suction surface,as shown in the areas enclosed by the red dashed circles.

5. Conclusions

The relationship between flutter characteristics and feature parameters of H-shaped hollow fan blade is investigated in this paper. Before studying the flutter behaviors of the shroudless,wide-chord transonic fan blade,the model parameterization of the hollow blade is analyzed. Ten feature parameters are proposed to describe the structural characteristics of the hollow fan blade. Based on these parameters, the auto-generation method of FEM is presented.

After the establishment of the FEM, the first three modes are analyzed by the OED method, and the feature parameters of the hollow fan blade which have remarkable influence on the frequency and mode shape are identified.Three more influential parameters and the first torsional mode are concerned in the flutter sensitivity analysis. The flutter behaviors at design point of different hollow fan blades are investigated using the aeroelastic eigenvalue method and energy method.The former method is used to calculate the aerodynamic damping for all NDs through one calculation with nine-passages in the computational domain. The latter method is used to study the relationship between the structure parameters and the aerodynamic damping density distribution at the most unstable ND. The findings are summarized as follows:

(1) In general, the natural vibration characteristics of the third mode (that is, the first torsional mode) are more sensitive to the feature parameters of the hollow fan blades. For this vibration mode, the maximum variations of frequency and mode shape are both obtained at rib number N.The wall thickness near the blade root has a larger impact on the vibration frequency,while for the mode shape,the wall thickness near the blade tip has a more significant influence.

(2) The general trends of the aerodynamic damping curve for the solid and hollow fan blades are similar, and the most unstable ND all occur at 1. Compared with the solid blade, the minimum aerodynamic damping of the hollow fan blade decreases,which indicates that the hollow structure makes the aeroelastic stability worse. The largest difference of the minimum aerodynamic damping between solid blade and hollow blades is 2.3 × 10-3,accounting for 27% of the damping of solid blade.Therefore, the feature parameters of hollow fan blade have little influence on the variation trends of the aerodynamic damping with NDs, but have a noticeable impact on the aerodynamic damping value.

(3) For the parameters in hollow section, the rib number N has the greatest influence on the minimum aerodynamic damping, followed by the wall thickness W5, and the wall thickness W2has little effect.For this blade profile,the minimum aerodynamic damping decreases with the increase of parameters N and W5. The largest variation of aerodynamic damping caused by rib number N and wall thickness W5are both about 7.5 × 10-4. Meanwhile, parameters W5and N have the same influence on the minimum aerodynamic damping and MAC value. Therefore, we can quickly optimize the hollow section by comparing the MAC values of the hollow fan blades with different W5and N, so as to maximize the aerodynamic damping.

(4) For the parameters in the height of the hollow segment,the aerodynamic damping increases with the increase of M1and M2. This indicates that reducing the height of the hollow segment is helpful to improve the aerodynamic damping. However, compared with the impact of parameters in hollow section,the variation of aerodynamic damping caused by the height of the hollow segment is small, and the largest difference is about 2.2×10-4for parameter M1and 3.7×10-4for parameter M2.

(5) With the rise of rib number N, the aerodynamic damping falls slightly at first and then sharply. When N is equal to 3 and 4, there is not a great deal of difference in the aerodynamic damping. However, when N is 5, the aerodynamic damping is about 10.2% less than that when N is equal to 3.It can be seen from the mode shape that when N is 5,the low amplitude region which is near the nodal line becomes narrower and farther away from the tip, resulting in the change of aerodynamic damping density distribution.

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.

Acknowledgements

This study was co-supported by the National Science and Technology Major Project, China (No. 2017-Ⅳ-0002-0039)and the National Natural Science Foundation of China (No.51475022).