Research ArticleNew Method for Analyzing the Flutter Stability of HingelessBlades with Advanced Geometric Configurations in Hovering

Si-wen Wang ,1 Jing-long Han ,1 Quan-long Chen,2 Hai-wei Yun,1 and Xiao-mao Chen1

1State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China2Chongqing Enstrom General Aviation Institute of Technology, Chongqing 401135, China

Correspondence should be addressed to Si-wen Wang; nmghappy@126.com

Received 6 October 2019; Accepted 20 December 2019; Published 17 February 2020

Academic Editor: Joseph Morlier

Copyright © 2020 Si-wen Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new method used to analyze the aeroelastic stability of a helicopter hingeless blade in hovering has been developed, whichis especially suitable for a blade with advanced geometric configuration. This method uses a modified doublet-lattice method(MDLM) and a 3-D finite element (FE) model for building the aeroelastic equation of a blade in hovering. Thereafter, theflutter solution of the equation is calculated by the V-g method, assuming blade motions to be small perturbations aboutthe steady equilibrium deflection. The MDLM, which is suitable to calculate the unsteady aerodynamic force of nonplanarrotor blade in hovering, is developed from the doublet-lattice method (DLM). The structural analysis tool is thecommercial software ANSYS. The comparisons of the obtained results against those in the literatures show the capabilitiesof the MDLM and the method of structural analysis. The flutter stabilities of swept tip blades with different aspect ratiosare analyzed using the new method developed in this work and the usual method on the basis of the unsteady striptheory and beam model. It shows that considerable differences appear in the flutter rotational velocities with the decreaseof the aspect ratio. The flutter rotational velocities obtained by the present method are evidently lower than those obtainedby the usual method.

1. Introduction

Hovering is the most important flight state of a helicopter.The flutter stability of hovering is directly related to flightsafety. Numerous methods on stability characteristics of hin-geless rotors in hovering have been developed to obtain anaccurate flutter stability boundary. The aeroelastic equationsof their methods are usually obtained by aerodynamicmodels, which are formed by quasisteady [1] or unsteadystrip theories [2], and the structural dynamic models, whichare formed by moderate [3] and large deflection beam [4]theories or other refined beam theories [5–7], consideringthe geometrical nonlinearity. Generally, for the fluttersolution of the aeroelastic equations, the nonlinear steadystate is obtained first and then the flutter solution, assum-

ing that the flutter motion is a small perturbation aboutthe equilibrium position [1–8]. The blades of aforemen-tioned researches are made of isotropic [3] or composite[8] material with high aspect ratios and simple planformshapes. With the development of computational fluiddynamics (CFD), the method of rotorcraft aeroelastic tightcoupling of computational structural dynamics (CSD) andCFD has been used for rotor blade aeroelastic stability anal-ysis recently [9, 10]. Although the CFD/CSD coupledmethod can get better predictions of rotor stability, thecomputational cost of this method is very high [11], andCSD models are mostly built with beams mentioned above.

The majority of manufacturers choose blades withadvanced geometric configurations, such as the ERATO[12] blade with reduced Blade Vortex Interaction (BVI) noise

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 1891765, 16 pageshttps://doi.org/10.1155/2020/1891765

signature to increase the flight speed and have good aero-acoustic behavior. The advanced blade with a complicatedtip and a small aspect ratio deviates largely from the clas-sical rectangular one. The modal analysis using the beammodel of blades may be unsuitable due to the deviationof a planform shape. Truong [13] studied the dynamiccharacteristics of the ERATO using 3-D finite elementand 1-D beam analyses in MSC software. Natural fre-quencies are calculated at various rotor rotational speeds.The predictions of the 3-D finite element model that cor-relate with the experimental results are more precise thanthose of the (1‐D + 2‐D) model, within a 5% relativeerror. Yeo [14] et al. are motivated by the above researchto study the difference between 1-D beam and 3-D finiteelement analyses for advanced geometry blades, whichhave tip sweep, tip taper, and planform variations nearthe root and physics behind them. Differences occurwhen coupling generated from geometry (tip sweep) ormaterial (composite) emerges, especially for high-frequency modes. Thereafter, Kee and Shin [15] use aneighteen-node solid-shell finite element to model theblade structures and give the differences of natural fre-quencies between the one-dimensional beam and three-dimensional solid models. As the blade aspect ratiodecreases, considerable differences appear in the bendingand torsion modes. Evidently, the accuracy of modal cal-culation directly determines the reliability of flutter analy-sis. Here comes the conclusion that the commonly usedaeroelastic model constructed by strip theory and the 1-D beam model is unsuitable for the flutter stability analy-sis of blades with advanced geometric configurations inhovering. It is also easy to conclude that the CFD/CSDtight coupling approach using 3-D solid structure modelsinstead of beam models will increase computing costs inev-itably when there is already a large amount of computation.A new high-efficiency approach of advanced blade flutteranalysis is sorely necessary. So far, no relevant literature hasbeen reported on that.

A new method of rotor flutter analysis, using a 3-Dfinite element model and a modified doublet-latticemethod (MDLM) to establish the aeroelastic equation, isdeveloped to solve the abovementioned problems. ANSYSis applied to the numerical analysis of blade structure,and the blades are discretized by the 3-D solid element.In aerodynamic calculation, compared with the aerody-namic models mentioned above, one of the major advan-tages of the DLM [16] is that motions of parts of thewing section can be considered, which makes the DLMmore suitable for being coupled with the 3-D finite ele-ment model to establish the aeroelastic model than thestrip theory. The other one is that the DLM is muchmore effective than the CFD method. However, theDLM cannot be used for rotor flutter analysis directlybecause the inflows change over the location of the bladeand the pretwist cannot be ignored. The MDLM, whichovercomes the above difficulties to calculate the unsteadyaerodynamic of rotor blades in hovering, is developedfrom the DLM. The method of solving the aeroelasticequations is basically the same as the abovementioned

one. The nonlinear steady state is obtained first to linear-ize the flutter equations, based on the small perturbationassumption. Thereafter, the linearized flutter equationsare solved through the V-g method [2]. The structuralanalysis method and MDLM of rotor blades are verifiedby literature examples. Thereafter, the new and usualmethods of rotor flutter analysis are used for solving theflutter rotational velocities of advanced blades in hovering,which have different ratio aspects. The comparison ofthese two methods is conducted systematically to under-stand the differences.

2. Analysis

2.1. Modified Doublet Lattice Method (MDLM)

2.1.1. Correction of the Nonplanar Hypothesis. The DLMassumes the lifting surfaces as a planar plate without pre-twist angles; that is, in the aerodynamic coordinate frameof the sending panel, the chordwise component of the nor-mal vector of the receiving panel is equal to zero [17].Pretwist angles are widely used in helicopter rotor blades,which have a great influence on aerodynamic forces.Hence, the DLM’s assumption is unsuitable for rotorblades. On the basis of the DLM, the MDLM is developedfor calculating unsteady aerodynamic loads on nonplanarlifting surfaces in subsonic flows.

Normal wash velocity [18] of receiving point ðx, y, zÞinduced by the surface motion can be written as follows:

�w = 18π

ðA

U∞Δ�pK �x, �y, �z, ω,M∞ð Þqd

dA ð1Þ

where �x = x − bξ , �y = y − bζ , �z = z − bη , Δ�p, qd , U∞, andM∞are the pressure loading, velocity pressure number, free-

flow velocity, and Mach number at sending point ðbξ , bζ , bηÞ,respectively. ω is the circular frequency of vibration ofthe blade. The kernel function [18] can be expressed asfollows:

K �x, �y, �z, ω,M∞ð Þ = limnc→0

∂∂nc

e− iω�x/Vð Þð�x−∞

∂∂nd

1Re iω λ−M∞Rð Þð Þ/M∞

� �dλ

� �,

ð2Þ

where

λ = x − bξ , r = ffiffiffiffiffiffiffiffiffiffiffiffiffi�y2 + �z2

p, R =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 + �β

2r2q

,

�β2 = 1 −M2

∞, �ω = ω

a∞�β2 , i =

ffiffiffiffiffiffi−1

p:

8>><>>: ð3Þ

Partial differentiations ∂/∂nc and ∂/∂nd are taking thederivative of the receiving point and the sending point. ∂

2 International Journal of Aerospace Engineering

/∂nc and ∂/∂nd can be expressed by the following formu-lations:

∂∂nc

= cos nc, xð Þ ∂∂x

+ cos nc, yð Þ ∂∂y

+ cos nc, zð Þ ∂∂z

,

∂∂nd

= cos nd , bξ� � ∂

∂bξ + cos nd , bζ� � ∂

∂bζ + cos nd , bηð Þ ∂∂bη :

8>>><>>>:

ð4Þ

In the case of small perturbations, the following rela-tions are obtained:

cos nd , bξ� �< <cos nd , bζ� �

, cos nd , bηð Þ: ð5Þ

cos ðnc, xÞ cannot be ignored due to blade elastictorsion and pretwist. Formulation (4) can be rewritten asfollows:

∂∂nc

= cos nc, xð Þ ∂∂x

+ ∂∂nc0

,

∂∂nd

= cos nd , bζ� � ∂

∂bζ + cos nd , bηð Þ ∂∂bη ,

∂∂nc0

= cos nc, yð Þ ∂∂y

+ cos nc, zð Þ ∂∂z

:

8>>>>>>>><>>>>>>>>:

ð6Þ

The kernel function K can be also written as follows:

K �x, �y, �z, ω,M∞ð Þ = cos nc, xð Þ −iωV

e−iω�x/V∂∂nd

�I + e−iω�x/V∂∂x

∂∂nd

�I

+ e−iω�x/V limnc→0

∂∂nc0

∂∂nd

�I,

ð7Þ

where

�I =ð�x−∞

1Rei�ω λ−M∞Rð Þ/M∞

� �dλ: ð8Þ

Because

∂

∂bζ1Re i�ω λ−M∞Rð Þð Þ/M∞

� �= −

∂∂y

1Re i�ω λ−M∞Rð Þð Þ/M∞

� �,

∂∂bη 1

Re i�ω λ−M∞Rð Þð Þ/M∞

� �= −

∂∂z

1Re i�ω λ−M∞Rð Þð Þ/M∞

� �,

8>>><>>>:

ð9Þ

and using ψs for representing the anhedral of thereceiving point and ψd for representing the anhedralof the sending point, we can write certain equationsas follows:

cos nc, yð Þ = −sin ψs, cos nc, zð Þ = cos ψs,

cos nd , bζ� �= −sin ψσ, cos nd , bηð Þ = cos ψσ:

8<: ð10Þ

And then

According to the following equations:

∂�I∂z

= ∂�I∂r

∂r∂z

= �zr∂�I∂r

, ∂�I∂y

= ∂�I∂r

∂r∂y

= �yr∂�I∂r

,

∂2�I∂z2

= 1r∂�I∂r

+ �z2

r∂∂r

1r∂�I∂r

, ∂

2�I∂y2

= 1r∂�I∂r

+ �y2

r∂∂r

1r∂�I∂r

,

∂2�I∂yz

= �y�zr

∂∂r

1r∂�I∂r

,

8>>>>>>>>><>>>>>>>>>:

ð12Þ

and substituting Equations (11) and (12) into Equation (7),the kernel function can be rewritten as follows:

K �x, �y, �z, ω,M∞ð Þ = cos θe−iω�x/V −sin ψσ�y + cos ψσ�zð Þ

� −iωV

K1r2

+ ∂K1∂x

1r2

+ e−iω�x/VK1T1r2

+ K2T2r4

,

ð13Þwhere

∂∂nc0

∂∂nd

= − sin ψs sin ψσ

∂2

∂y2+ cos ψs cos ψσ

∂2

∂z2− sin ψs cos ψσ + cos ψs sin ψσð Þ ∂2

∂y∂z

" #,

∂∂nd

= − −sin ψσ

∂∂y

+ cos ψσ

∂∂z

:

8>>>><>>>>:

ð11Þ

3International Journal of Aerospace Engineering

The computational procedure of K1 and K2 are given byLandahl [18]. ∂K1/∂xof Equation (13) can be calculated bytaking the partial derivative of K1 with respect to x. Theresults are given as follows:

where

The constants an and c0 of Equation (16) are given byKalman et al. [19].

If u1 ≥ 0, then I1, I2, and ∂I1/∂x will be written as follows:

K1 = r∂I∂r

, K2 = r3∂∂r

1r∂I∂r

,

T1 = cos ψs − ψσð Þ, T2 = �z cos ψs − �y sin ψsð Þ �z cos ψσ − �y sin ψσð Þ,

I = −ð�x−∞

1Re i�ω λ−M∞Rð Þð Þ/M∞dλ:

8>>>>>><>>>>>>:

ð14Þ

K1 = I1 +M∞re−ik1u1

R 1 + u21� �1/2 ,

K2 = −3I2 −ik1M

2∞r2e−ik1u1

R2 1 + u21� �1/2 −

M∞re−ik1u1

R 1 + u21� �3/2 1 + u21

� � �β2r2

R2 + 2 + M∞ru1R

" #,

∂K1∂x

= ∂I1∂x

− ik1∂u1∂x

M∞re−ik1u1

R 1 + u21� �1/2 + −

∂R∂x

M∞r

R2 1 + u21� �1/2 −

∂u1∂x

M∞ru1R 1 + u21� �3/2

!e−ik1u1 ,

8>>>>>>>>>>><>>>>>>>>>>>:

ð15Þ

u1 = −�x −M∞R

�β2r

, R =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x2 + �β

2r2q

, ∂u1∂x

= −1�β2r+ M∞�x

�β2rR

,

I0 = 〠11

n=1an

nc0 − ik1n2c20 + k21

e−nc0u1 , ∂I0∂x

= 〠11

n=1− nc0an

∂u1∂x

nc0 − ik1n2c20 + k21

e−nc0u1 ,

J0 = 〠11

n=1

ane−nc0u1

n2c20 + k21� �2 n2c20 − k21 + nc0u1 n2c20 + k21

� �− ik1 2nc0 + u1 n2c20 + k21

� � �� �:

8>>>>>>>>>><>>>>>>>>>>:

ð16Þ

I1 = 1 − u11 + u21� �1/2 − ik1I0

" #( )e−ik1u1 , ∂I1

∂x= −ik1

∂u1∂x

I1 +u21

1 + u21� �3/2 −

11 + u21� �1/2

" #∂u1∂x

− ik1∂I0∂x

( )e−ik1u1 ,

I2 =13 2 + ik1u1ð Þ 1 − u1

1 + u21� �1/2

" #−

u11 + u21� �3/2 − ik1I0 + k21 J0

( )e−ik1u1 ,

8>>>>><>>>>>:

ð17Þ

4 International Journal of Aerospace Engineering

And if u1 < 0, then I1, I2, and ∂I1/∂x can be rewrittenas follows:

I1 = 2 Re I1 0, k1ð Þ½ � − Re I1 −u1, k1ð Þ½ � + i Im I1 −u1, k1ð Þ½ �,I2 = 2 Re I2 0, k1ð Þ½ � − Re I2 −u1, k1ð Þ½ � + i Im I2 −u1, k1ð Þ½ �,∂I1∂x

= 2 Re ∂I1∂x

0, k1ð Þ� �

− Re ∂I1∂x

−u1, k1ð Þ� �

+ i Im ∂I1∂x

−u1, k1ð Þ� �

:

8>>>><>>>>:

ð18Þ

2.1.2. Application of the MDLM on Blade. In the MDLM,the lifting surface of a blade is divided into trapezoidal panels,which have different freestream velocities. The freestreamvelocity and the reduction frequency are assumed to be con-stant within the same panel to satisfy the basic assumptionand condition of the MDLM based on the DLM. Each panelcontains a line of constant strength point doublets along thequarter-chord line and a control point located at the 3/4chord point at midspan. Substituting the coordination ofthe control point ðxi, yi, ziÞ and the doublet point ðxj, yj, zjÞinto Equations (1) and (13), the normal wash velocity �wij ofthe control point can be given as follows:

�wij =U j

8πqdjΔ�pj∬Sj

KijdS: ð19Þ

In Equation (19), Uj is the freestream velocity of the jthpanel, Sj is the area of the jth panel, and Δ�pj is the pressuredifference per unit area of the jth panel. The dynamic pres-sure is given by qdj = 0:5ρ∞U2

j , where ρ∞ is the air density.Let the pressure difference on the unit length of 1/4 chordof the jth panel be Δ��p. To make the distribution of Δ��p on1/4 chord equivalent to that of Δ�pj on the jth panel, theirrelationship is as follows:

Δ��pjl j = Δ�pjSj, ð20Þ

where l j is the length of the 1/4 chord of the jth panel. Here,Sj = ðl j cos χjÞΔxj, where χj is the sweep angle of the doubletline and Δxj is the average chord. The pressure differenceacross surface Δ�pj can be written as Δ�pj = ð1/2Þρ∞U2

j�cpj,where �cpj is the coefficient of the pressure difference.Substituting Sj and �pj into Equation (20), ��p can be rewrit-ten as follows:

Δ��pj =12 ρ∞U2

jΔxj cos χjΔ�cp jð Þ: ð21Þ

Transform the area integral of Equation (19) into a lineintegral, and then the normal wash velocity is obtained asfollows:

�wij =U j

8πΔxj cos χj

ðl j

Kijdl

!Δ�cp jð Þ =DijΔ�cp jð Þ, ð22Þ

where

Dij =U j

8πΔxj cos χj

ðl j

Kijdl: ð23Þ

The downwash factor Dij can be obtained by the steadydownwash factor D0ij and incremental oscillatory down-wash factors D1ij and D2ij, where

Dij =D0ij +D1ij +D2ij: ð24Þ

The steady downwash factor D0ij is derived fromhorseshoe vortex considerations. The difference from thetraditional method given by Hedman [20] is that the infi-nite free vortex is replaced by the wake shed from theblade. The induced velocities of the wake on the panelsare calculated by the classical blade element momentumtheory and considered to be steady and uniform alongthe blade length [21]. For brevity, the derivation processesof incremental oscillatory downwash factors D1ij and D2ijare not presented, which can be referred to the literature[20]. The numerical form of Equation (1) in matrix nota-tion becomes

w½ � = D½ � Δcp �

, ð25Þ

The differential pressure coefficient vector ½Δcp� can be

obtained by ½Δcp� = ½D�−1½w�. For the jth panel, the pres-sure difference located at the 1/4 chord point at the mid-span of the panel can be obtained byFaj = qdjΔSjΔcpj.Thereafter, the aerodynamic load vector is obtained inthe form of a matrix by the following:

Fa½ � = qd½ � ΔS½ � D½ �−1 w½ �, ð26Þ

where

qd½ � =qd1 ⋯ 0⋮ ⋱ ⋮

0 ⋯ qdn

0BB@

1CCA,

ΔS½ � =ΔS1 ⋯ 0⋮ ⋱ ⋮

0 ⋯ ΔSn

0BB@

1CCA:

ð27Þ

2.2. Structure Modeling. Considering the geometric nonlin-earity, the nonlinear forced undamped vibration equationof the blade can be simply obtained by

Mss Ω, qsð Þ €qs½ � +Kss Ω, qsð Þ qs½ � = Fs½ �, ð28Þ

where ½qs� is the displacement vector and ½Fs� is the struc-tural load vector. Mss and Kss are the structural mass

5International Journal of Aerospace Engineering

matrix and stiffness matrix, respectively, and the functionsof rotational speed Ω and qs. Equation (28) is transformedinto the modal space by writing

ps½ � =ΦTs qs½ �, ð29Þ

where Φs is the modal matrix and ½ps� is the vector of eightgeneralized coordinates in the modal space. SubstitutingEquation (29) into Equation (28), the normal equation canbe rewritten as follows:

�Ms Ω, psð Þ €ps½ � + �Ks Ω, psð Þ ps½ � = �Fs

�, ð30Þ

where �Ms =ΦTs MssΦs, �Ks =ΦT

s KssΦs and ½�Fs� =ΦTs ½Fs�.

2.3. Aeroelastic Modeling. Given the difference of meshingbetween the aerodynamic and the structure models, an inter-connecting approach must be established between them foraeroelastic analysis, and it can be made using a spline matrixGas [22]. The transformation from the structural grid nodemotions ½qs� into the aerodynamic grid node motions ½w�can be given by the following:

w½ � =Gas qs½ �: ð31Þ

The transformation formula of the aerodynamic ½Fa� andthe equivalent value acting on a structural grid ½Fs� can beobtained by the following:

Fs½ � =GTas Fa½ �: ð32Þ

Substituting Equations (31) and (32) into Equation (26),we can obtain the following:

Fs½ � =Q qs½ �, ð33Þ

whereQ is a function of k and rotor rotational velocityΩ and

Q k,Ωð Þ =GTas qd½ � ΔS½ � D½ �−1Gas: ð34Þ

Combining Equations (28), (33), and (34), the aeroelasticequation is obtained by

Mss Ω, qsð Þ €qs½ � +Kss Ω, qsð Þ qs½ � =Q Ω, qsð Þ ~qs½ � ð35Þ

Transforming Eq. (35) to the modal space, the equationcan be rewritten as follows:

�Ms Ω, psð Þ €ps½ � + �Ks Ω, psð Þ ps½ � = �Q k,Ωð Þ ps½ �, ð36Þ

where �Qðk,ΩÞ =ΦTs Qðk,ΩÞΦs.

2.4. Nonlinear Steady-State Solution. The conclusion of theliterature [3] shows that the static deformation of the bladewill change the structural stiffness, due to the considerationof geometrical nonlinearity, and the aerodynamic force dis-tribution, which will affect the blade flutter stability. Hence,the first step in solving Equation (35) is to get the steady trim

solution in hovering. Dropping the acceleration term, we canwrite Equation (35) as follows:

Kss0 Ω, qs0ð Þ qs0½ � = Fs0½ �, ð37Þ

where stiffness matrix Kss0 is a function of the steady displace-

ment ½qs0� and Ω and structural load vector ½Fs0� is given bythe following:

Fs0½ � =GTas qd½ � ΔS½ � D0½ �−1 w0½ �: ð38Þ

The velocity vector ½w0� of Equation (38) is a vector con-sisting of the normal downwash velocity, which is the normalcomponent of a combined velocity of freestream and inducedinflow of the wake, at each panel. For solving the nonlinearaeroelastic equation (37), an iterative algorithm is establishedin the Isight platform integrating the steady aerodynamic cal-culation part of self-made MDLM program and the ANSYSsoftware. The process of the algorithm is as follows:

(1) The initial displacements of the structural grid nodesare set by ½qs00� = 0. The displacements of the struc-tural grid nodes of the upper wing surface are takento represent the overall displacement of the blade.The V-g method is used to calculate the blade flutterrotational velocity ΩF and flutter frequency

(2) In the ith iteration, where i = 1, 2, 3,⋯, n, the dis-placements of the structural grid nodes are ½qs0i−1�.The location of structural and aerodynamic gridnodes is refreshed. The normal downwash velocityvector ½w0� and the steady aerodynamic influencecoefficient matrix ½D0�−1 are recalculated due to thechange of the normal direction and the coordinatesof sending and receiving points at each panel

(3) The interpolation matrix Gas is obtained using theThin Plate Spline (TPS) interpolation method [23].The steady structural load ½Fs0� is obtained solvingEquation (38)

(4) The structural grid node displacement vector ½qs0i� iscalculated using the static structural analysis ofANSYS software when the structural load is appliedto the upper wing surface

(5) If j½qs0i� − ½qs0i−1�j < ε, then, the computation con-verges, and if it does not, then, Steps (2)–(4) arerepeated

After the calculation converges, the structure displace-ment ½qs0�, the location of structural and aerodynamic gridnodes, and the structural load ½Fs0� in a static trim state areobtained. Given the large deflection and the geometric non-linearity, the natural modes of vibrationΦs0, the mass matrixMss

0 ðΩ, qs0Þ, and the stiffness matrix Kss0 ðΩ, qs0Þ about the

equilibrium position are obtained by the prestress modalanalysis of ANSYS.

6 International Journal of Aerospace Engineering

2.5. Flutter Solution. The flutter motion ½qs� is assumed to bea small perturbation ½~qs� about the equilibrium position ½qs0�and can be written as follows:

qs½ � = qs0½ � + ~qs½ �: ð39Þ

Substituting Equation (39) into Equation (35), we canlinearize the flutter equation, based on the assumption ofsmall perturbation, by the following:

Mss0 Ω, qs0ð Þ €~qs

h i+Kss

0 Ω, qs0ð Þ ~qs½ � + qs0½ �ð Þ =Q Ω, qs0ð Þ ~qs½ � + Fs0½ �,ð40Þ

Simplified by Equation (37), Equation (40) can be rewrit-ten as follows:

Mss0 Ω, qs0ð Þ €~qs

h i+Kss

0 Ω, qs0ð Þ ~qs½ �ð Þ =Q Ω, qs0ð Þ ~qs½ �: ð41Þ

Transforming Equation (41) into the modal space, theequation can be rewritten as follows:

�Ms0 Ω, ps0ð Þ €~psh i

+ �Ks0 Ω, ps0ð Þ ~ps½ � = �Q k,Ωð Þ ~ps½ �, ð42Þ

where �Ms0 =ΦTs0Mss

0 Φs0, �Ks0 =ΦTs0Kss

0 Φs0, �Qðk,ΩÞ =ΦTs0Q

ðk,ΩÞΦs0, ½~ps0� =ΦTs0 ½~qs0�, and ½~ps� =ΦT

s0 ½~qs�.Based on the assumption of harmonic motion

(½~ps� = ½~pss�eiωt) transforming Equation (42) into a fre-quency domain, the aeroelastic equation can be rewrittenas follows:

−ω2 �Ms0 Ω, ps0ð Þ + �Ks0 Ω, ps0ð Þ − �Q k,Ωð Þ �~ps½ � = 0: ð43Þ

Equation (43) is solved in the frequency domain bythe V-g method to obtain the flutter rotational velocityof the rotor blade in hovering. A specific process canrefer to Ref. 24.

Given the effect of centripetal force, the mass and stiff-ness matrixes of the blade are the functions of rotationalvelocity. To match the current rotational velocity to theflutter rotational velocity, we can use an iterative algo-rithm as follows:

(1) In the ith iteration, where i = 1, 2, 3,⋯, n, for therotational velocity Ωi and the collective pitch θ0,the static steady-state solution is carried out, andthe specific steps can be referred to theabovementioned

(2) The V-g method is used to calculate the blade flutterrotational velocity ΩF and flutter frequency f F

(3) If jΩi −ΩF j < ε, the computation converges, and ifit does not, Ωi+1 =ΩF and Steps (1)–(2) arerepeated

After the calculation converges, the flutter rotationalvelocity ΩF and flutter frequency f F is obtained for the cur-rent blade pitch.

3. Presentation of Results

3.1. MDLM Validation. In this work, the steady-state defor-mation and flutter stability of a hovering rectangular hinge-less blade are analyzed. The aeroelastic model is establishedby combining the moderate deflection beam theory withthe MDLM.

The blade is made of isotropic material. Nondimen-sional natural frequencies in the flap, lead-lag, and torsiondirections are ωw = 1:15, ωv = 1:5, and ωϕ = 3:0 or 5.0,respectively. Table 1 shows other blade nondimensionalparameters [3].

Figure 1 exhibits the steady tip deflection wtip, flap andlead-lag bending deflection vtip, and elastic twist angle θtipwith respect to collective pitch θ0 and the results of Siva-neri and Chopra [3]. The agreement between the tworesults is good.

Figure 2 shows the flutter stability boundary respect withβp and the results of Sivaneri and Chopra [3]. The fluttersolution is calculated by the V-g method. A general agree-ment emerges between the two results.

3.2. Rotational Modal Analysis. In the present study, thenatural frequencies of the swept tip blades were investi-gated in ANSYS, and the results were compared withthose of the experiment [25] to assess the availability ofANSYS for analyzing the dynamic characteristics of rotat-ing hingeless isotropic rotor blades. The blade parametersare the same as those of the beams tested by Epps andChandra [25]. The radius of the rotor is 1,016mm, andthe hub length is 63.5mm. In addition, the length of theswept tip segment is 152.5mm, the width is 25.4mm,and the thickness is 1.6mm. The tip sweep angles Λ are15 and 45 degrees. The isotropic material of the bladesis aluminum. The model is 3-D FE model of 2908 nodesand of 360 elements, and the meshing is shown inFigure 3. Figures 4 and 5 show the natural frequencieswith respect to the rotating speeds. The results show thatthe present predictions have a good correlation with theresults of the experiment.

3.3. Flutter Solution. The flutter rotational velocities ofhingeless rotor blades that have swept tips and differentaspect ratios are obtained for different collective pitchesby the present method mentioned above and the usual

Table 1: Values of parameters of uniform blade for numericalresults.

km1/R = 0 c/R = π/40 a = 6km2/R = 0:025 σ = 0:1 βp = 0:05kA/km = 1:5 γ = 5 CD0 = 0:0095

7International Journal of Aerospace Engineering

method constructed by unsteady strip aerodynamic theoryand large deflection beam theory [4]. The airfoil sectionof the blade is naca8h12 and is shown in Figure 6. InFigure 6, the elastic axis position is ea, and the distanceto the leading edge of the ea is XE. Table 2 shows theparameters of the section. There exists a destabilizing offset,XA, between the aerodynamic center and the elastic axis.

The isotropic material is aluminum. Figure 7 showsthe blade platform, which is based on the blade tested byHarder and Desmarais [22] and includes a hub. Theradius of the rotor blade is R = 4:252m. Two hub lengths(hi = 0:367m and 1.5645m) are considered in investigatingthe effect of aspect ratios (AR = 18:07 and 12.5). The

length of the swept tip segment is n = 0:6378m. The tipsweep angle Λ is 45 degrees.

Table 3 shows natural frequencies of the blade(AR = 18:07) at the rotational velocity 400 rpm, as the totalnumber of elements is varied from 4880 to 6710. The resultsshow, for the case considered, that 6100 elements are suffi-cient for reasonably good convergence.

Figures 8–9 show the flutter rotational velocities, andFigures 10-11 show the V-g diagrams, with respect tovarious aspect ratios, where Ω = 500 rpm. The results showthat the flutter rotational velocities obtained by the pres-ent method are evidently lower than those obtained bythe usual method. The differences in flutter rotational

0.14

PresentRef

0.12

0.10

0.08

0.06

0.04

Wtip

0.02

0.00

–0.02

–0.04

–0.060 5 10 15

𝜃0 (degree)20 25 30

(a)

00.00

5 10 15 20 25 30

–0.02

–0.04

–0.06

–0.08

–0.10

PresentRef

V tip

𝜃0 (degree)

(b)

0.00 5 10 15 20 25

–0.2

–0.4

–0.6

�et

a tip

(deg

ree)

–0.6

–1.0PresentRef

𝜃0 (degree)

(c)

Figure 1: Comparison of steady tip deflection by the MDLM and the quasisteady strip theory (ωw = 1:15, ωv = 1:5, and ωϕ = 5:0).

8 International Journal of Aerospace Engineering

14

12

10 Stable

Unstable8

6

41 2 3 4 5 6 7

𝜃0 (

degr

ee)

𝛽p (degree)

PresentRef

Figure 2: Effect of precone on the stability boundaries (ωw = 1:15, ωv = 1:5, and ωϕ == 3:0).

0.000

0.075

0.150

0.225

0.300 (m)

Y

XZZ

Figure 3: Meshing of the blade model.

800700600500400300200Rotation speed (rpm)

100–100

140

120

t1

f5

f4

f3

f2

f1

100

80

60

Nat

ural

freq

uenc

y (H

z)

40

20

00

PresentRef

Figure 4: Natural frequencies of the blade (Λ = 45°).

9International Journal of Aerospace Engineering

velocities between the two methods increase with thedecrease of the aspect ratio. When AR = 18:07, the resultsof the two methods are close, and the differences betweenthe two methods are less than 10%. When AR = 12:5, thedifferences between them are 10% to 20%.

The cause of the differences may mainly consist of thefollowing two parts:

(1) The flutter of both conditions occurs between the 2nd

mode and the 3rd mode. The modal analysis of thesetwo modes directly affects the flutter result.Figures 12–13 show the frequencies of these modesabout steady deflected positions with respect to therotating speeds (theta0 = 6°). Figures 14-15 show theshapes of these modes of the blade (AR = 12:5). The2nd mode is the flap and torsion coupled mode, andthe 3rd mode is the lead-lag and torsion coupledmode. The differences in modal analysis betweenthe 1-D beam model and the 3-D finite elementmodel, increasing with the decrease of the aspectratio, because of strong coupling, generated fromgeometry, between flap, lag and torsion [12], maycause differences of flutter results

(2) In the modal analysis, the blade sections of the 3-Dfinite element model have deformations, which arenot included in the strip theory aerodynamic load-ings. But the motions of parts of sections can be

800700600500400300200Rotation speed (rpm)

100–100 0

t1

f5

f4

f3

f2

f1

Nat

ural

freq

uenc

y (H

z)

140

160

120

100

80

60

40

20

0

PresentRef

Figure 5: Natural frequencies of the blade (Λ = 15°).

z

y

XE

×ea

Figure 6: Blade section.

Table 2: Values of parameters of blade section.

A = 0:001229m2 Iz = 0:441 × 10−5 m4 XA = 0:006704mIy = 0:766 × 10−7 m4 J = 0:244 × 10−6 m4 c = 0:215mXE = 0:060454m

x

R-n

n

Λ

hi

y

Figure 7: Blade planform with tip sweep (Λ = 45°).

Table 3: Natural frequency of hingeless rotor blade (AR = 18:07).

Number ofelements

Mode 1(Hz)

Mode 2(Hz)

Mode 3(Hz)

Mode 4(Hz)

4880 7.57 11.837 15.508 26.472

5490 7.57 11.837 15.506 26.471

6100 7.5697 11.836 15.505 26.469

6710 7.5696 11.836 15.504 26.467

10 International Journal of Aerospace Engineering

Flut

ter r

otat

iona

l vel

ocity

/𝛺

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.02 4 6 8 10

PresentCho &lee

𝜃0 (degree)

Figure 8: Flutter rotational velocities of the blade (AR = 18:07).

0.8

0.6

0.4

0.2

0.00.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

–0.2

g

Rotational velocity (𝛺)

3-D modelBeam mode

Figure 10: V-g diagram of the blade (AR = 18:07).

Flut

ter r

otat

iona

l vel

ocity

(𝛺)

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.02 4 6 8 10

PresentCho &lee

𝜃0 (degree)

Figure 9: Flutter rotational velocities of the blade (AR = 12:5).

11International Journal of Aerospace Engineering

0.8

1.0

0.6

0.4g

0.2

0.00.2 0.4 0.6 0.8 1.0 1.2 1.4

–0.2 Rotational velocity (𝛺)

3-D modelBeam mode

Figure 11: V-g diagram of the blade (AR = 12:5).

20191817161514131211

Nat

ural

freq

uenc

y (H

z)

1098765

400 420 440

3rd mode

2nd mode

Rotation speed (rpm)460 480 500

3-D modelBeam mode

Figure 12: Natural frequencies (AR = 18:07).

Nat

ural

freq

uenc

y (H

z)

3rd mode

2nd mode

Rotation speed (rpm)

30292827262524232221201918171615

400 420 440 460 480 500

3-D modelBeam mode

Figure 13: Natural frequencies (AR = 12:5).

12 International Journal of Aerospace Engineering

considered in the MDLM. Figures 16–17 show thedisplacements of the 3-D finite element nodes atr = 0:75R in the Flap/torsion coupled mode aboutthe steady deflected positions. The results showthat the deformations of sections increase withthe decrease of aspect ratio

4. Conclusion

(1) The results of steady-state deflection and the flutterstability boundaries show that the MDLM is reliablein the steady and unsteady aerodynamic calculations

(2) The dynamic characteristics of rotating bladesobtained by the modal analysis of ANSYS show goodagreement with the experiment

(3) The flutter solutions of the blades with advancedgeometric configurations are obtained by the newmethod, based on the MDLM and ANSYS-basedstructural analysis, and the usual method, basedon strip theory and the beam theory model. Theflutter rotational velocities obtained by the presentmethod are lower than those obtained by the usualmethod. The differences of flutter rotational veloc-ities between the two methods increase with thedecrease of the aspect ratio. The distinctions mayarise mainly from the differences of modal analysisbetween the 1-D beam model and the 3-D finiteelement model and the warping of the blade sec-tions. Hence, the method of this paper is moresuitable for the flutter stability analysis of advancedgeometric blades than the usual method

0.030247 Max0.024437

N: ModalDirectional Deformation 4Type: Directional Deformation (Z Axis)Frequency: 25.209 HzSweeping Phase: 0.°Unit: mGlobal Coordinate System2019/12/21 15:45

0.0186270.012817

0.00700690.0011968

–0.0046133–0.010423

–0.016234–0.022044 Min

0.000

0.250

0.500

0.750

1.000 (m)

Z

Y X

(a) Front view

N: ModalDirectional Deformation 4Type: Directional Deformation (Z Axis)Frequency: 25.209 HzSweeping Phase: 0.°Unit: mGlobal Coordinate System2019/12/21 15:43

0.030247 Max0.0244370.0186270.0128170.00700690.0011968–0.0046133–0.010423–0.016234–0.022044 Min

0.0000.250

0.5000.750

1.000 (m)

Z Y

X

(b) Top view

Figure 14: Lead-lag/torsion coupled mode.

13International Journal of Aerospace Engineering

0.0000.250

0.5000.750

1.000 (m)

0.74052 Max0.624240.507960.391680.27540.159120.042844–0.073435–0.18971–0.30599 Min

Y

Z

X

N: ModalDirectional Deformation 3Type: Directional Deformation(Z Axis)Frequency: 21.204 HzSweeping Phase: 0.°Unit: mGlobal Coordinate System2019/12/21 15:44

Figure 15: Flap/torsion coupled mode.

0.025

Orginal sectionDeformed 3-D section

0.020

0.015

0.010

0.005�ic

knes

s (m

)

0.000

0.00 0.05 0.10Chord (m)

0.15 0.20 0.25

–0.005

Figure 16: Deformed sections of 3-D model (AR = 18:07).

Orginal sectionDeformed 3-D section

0.025

0.020

0.015

0.010

0.005

�ic

knes

s (m

)

0.000

0.00 0.05 0.10Chord (m)

0.15 0.20 0.25

–0.005

–0.010

Figure 17: Deformed section of 3-D model (AR = 12:5).

14 International Journal of Aerospace Engineering

Nomenclature

Δ�p: Pressure loadingqd : Velocity pressure numberω: Circular frequency of vibrationΨs,Ψd : Anhedral of the receiving point and sending pointSj: Area of the jth panel

Δp: Pressure difference on the unit length of 1/4chord of the jth panel

D0ij: Steady downwash factor½cp�: Differential pressure coefficient vectorMss: Structural mass matrix½qs�: Displacement vectorΩ: Rotor blade rotational velocityGas: Spline matrix�Ms: Structural mass matrix in modal spaceKss0 : Structural stiffness matrix at trim position

½Fs0�: Steady structural load vectorωv, ωw, ωϕ: Fundamental coupled rotating lead-lag, flap, and

torsion natural frequencies, respectivelykm: Mass radius of gyration of blade cross section

(km =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikm1

2 + km22p)

σ: Solidity ratioa: Lift curve slopeCD0: Blade section profile drag coefficientea: Elastic axis positionIy, Iz , J : Cross-section moments of inertia in the flap,

lead-lag, and torsion directions, respectivelyU∞: Free-flow velocityM∞: Free-flow Mach numbernc, nd : Normal vector of the receiving point and the

sending pointUj: Freestream velocity of the jth panelΔ�pj: Pressure difference per unit area of the jth panelΔ�cpj: Coefficient of pressure differenceD1ij,D2ij: Incremental oscillatory downwash factors½Fa�: Aerodynamic load vectorKss: Structural stiffness matrix½Fs�: Structural load vectorΦs: Modal matrix½Ps�: Displacement vector in modal space�Ks: Structural stiffness matrix in modal space½qs0�: Blade steady displacement½w0�: Steady normal downwash velocity vectorkm1,km2: Principal mass radii of gyration of blade cross-sectionkA: Polar radius of gyration of blade cross sectionγ: Lock numberβp: Blade precone angleXE: Distance to the leading edge of eaXA: Destabilizing offset between aerodynamic center

and elastic axis.A: Area of the blade section.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was cosupported by the National Natural ScienceFoundation of China (No. 11472133) and a project funded bythe Priority Academic Program Development of JiangsuHigher Education Institutions.

References

[1] P. Friedmann, “Aeroelastic instabilities of hingeless helicop-ter blades,” Journal of Aircraft, vol. 10, no. 10, pp. 623–631,1973.

[2] P. Friedmann and C. Yuan, “Effect of modified aerodynamicstrip theories on rotor blade aeroelastic stability,” AIAA Jour-nal, vol. 15, no. 7, pp. 932–940, 1977.

[3] N. T. Sivaneri and I. Chopra, “Dynamic stability of a rotorblade using finite element analysis,” AIAA Journal, vol. 20,no. 5, pp. 716–723, 1982.

[4] M. H. Cho and I. Lee, “Aeroelastic stability of hingeless rotorblade in hover using large deflection theory,” AIAA Journal,vol. 32, no. 7, pp. 1472–1477, 1994.

[5] D. H. Hodges, “Geometrically exact, intrinsic theory fordynamics of curved and twisted anisotropic beams,” AIAAJournal, vol. 41, no. 6, pp. 1131–1137, 2003.

[6] M. R. Amoozgar, H. Shahverdi, and A. S. Nobari, “Aeroelasticstability of hingeless rotor blades in hover using fully intrinsicequations,” AIAA Journal, vol. 55, no. 7, pp. 2450–2460, 2017.

[7] M. V. Fulton and D. H. Hodges, “Aeroelastic stability of com-posite hingeless rotor blades in hover–Part II: Results,” Math-ematical and Computer Modelling, vol. 18, no. 3-4, pp. 19–35,1993.

[8] P. P. Friedmann, K.-A. Yuan, andM. de Terlizzi, “An aeroelas-tic model for composite rotor blades with straight and swepttips. Part I: Aeroelastic stability in hover,” International Jour-nal of Non-Linear Mechanics, vol. 37, no. 4-5, pp. 967–986,2002.

[9] H. Yeo, M. Potsdam, and R. A. Ormiston, “Application ofCFD/CSD to rotor aeroelastic stability in forward flight,” in66th American Helicopter Society Annual Forum Proceedings,Phoenix, Arizona, May 2010.

[10] J. Han, Q. Chen, and H. Yun, “Aeroelasticity of helicopters,”Acta Aeronautic et Astronautica Sinica, vol. 36, no. 4,pp. 1034–1055, 2015.

[11] W. Zhang, B. Wang, and Z. Ye, “High efficient numericalmethod for limit cycle flutter analysis based on nonlinear aero-dynamic reduced order model reduced order model,” in 51stAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynam-ics, and Materials Conference, Orlando, Florida. New York,April 2010.

[12] B. G. van der Wall, C. Kessler, Y. Delrieux et al., “From aeroa-coustics basic research to a modern low-noise rotor blade,”Journal of the American Helicopter Society, vol. 62, no. 4,pp. 1–16, 2017.

[13] K. V. Truong, Dynamics studies of the ERATO blade, based onfinite element analysis, vol. 2005, 31st European RotorcraftForum, Florence, Italy, 2005.

15International Journal of Aerospace Engineering

[14] H. Yeo, K. V. Truong, and R. A. Ormiston, “Comparison ofone-dimensional and three-dimensional structural dynamicsmodeling of advanced geometry blades,” Journal of Aircraft,vol. 51, no. 1, pp. 226–235, 2014.

[15] Y. J. Kee and S. J. Shin, “Structural dynamic modeling forrotating blades using three dimensional finite elements,” Jour-nal of Mechanical Science and Technology, vol. 29, no. 4,pp. 1607–1618, 2015.

[16] A. B. Guimarães Neto, R. G. A. da Silva, and P. Paglione, “Con-trol-point-placement method for the aerodynamic correctionof the vortex- and the doublet-lattice methods,” Aerospace Sci-ence and Technology, vol. 37, pp. 117–129, 2014.

[17] M. Brendel and E. Sulaeman, “Improved doublet lattice for-mulation for calculating aerodynamic loads on lifting surfacesin unsteady subsonic flow,” in 12th Applied AerodynamicsConference, Colorado Springs, USA. New York, 1994.

[18] M. T. Landahl, “Kernel function for nonplanar oscillatingsurfaces in a subsonic flow,” AIAA Journal, vol. 5, no. 5,pp. 1045-1046, 1967.

[19] T. P. Kalman, W. P. Rodden, and J. P. Giesing, “Application ofthe doublet-lattice method to nonplanar configurations in sub-sonic flow,” Journal of Aircraft, vol. 8, no. 6, pp. 406–413, 1971.

[20] S. G. Hedman, “Vortex lattice method for calculation of quasi-steady state loadings on thin elastic wings,” Tech. Rep. 105,Aeronautical Research Institute of Sweden, Stockholm, Swe-den, 1967.

[21] D. H. Hodges and R. A. Ormiston, “Stability of elastic bendingand torsion of uniform cantilevered rotor blades in hover,” in14th Structures, Structural Dynamics, & Materials Conference,Williamsburg, New York, USA, March 1973.

[22] R. L. Harder and R. N. Desmarais, “Interpolation using surfacesplines,” Journal of Aircraft, vol. 9, no. 2, pp. 189–191, 1972.

[23] R. Franke, “Thin plate splines with tension,” Computer-AidedGeometric Design, vol. 2, no. 1-3, pp. 87–95, 1985.

[24] P. C. Chen, “Damping perturbation method for flutter solu-tion: the g-method,” AIAA Journal, vol. 38, no. 9, pp. 1519–1524, 2000.

[25] J. J. Epps and R. Chandra, “Experimental-theoretical study ofthe effects of tip sweep on the natural frequencies of rotatingcomposite beams,” in 35th AIAA Structures, StructuralDynamics and Materials Conference, Hilton Head, South Car-olina. New York, April 1994.

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