cfd murugan paper

Computational Fluid Dynamics JOURNAL vol.12 no.1 April 2003 (pp.–)

Influence of Inflow Models on Helicopter Aeroelastic Optimization

Senthil Murugan ∗ Ranjan Ganguli †

Abstract

This paper presents a multi-objective optimization study for helicopter vibration reduction inforward flight using an evolutionary algorithm. The objective is to reduce the vibratory forcesacting on the rotor hub which are the main sources of helicopter vibration. A nonlinear aeroelasticmodel of the helicopter is used and the numerical results are obtained for a 4-bladed hingeless rotorusing finite elements in space and time. The optimization is performed using two different multi-objective formulations: 1) an Euclidean norm method and 2) a min-max method. A real-codedgenetic algorithm capable of finding the global minima is used for aeroelastic optimization. Themain rotor blade elastic stiffnesses and mass are taken as design variables. The min-max methodshows better reduction in the objective functions than the Euclidean norm based multi-objectiveformulation. The influence of linear inflow model and free wake aerodynamic models on aeroelasticoptimization is then studied. The aeroelastic optimization results with free wake model show aquite different optimal design when compared with the linear inflow model based optimization. Avibration reduction of 8 − 15 percent is achieved with blade elastic stiffness and mass as designvariables. The real-coded genetic algorithm is computationally efficient and is recommended foraeroelastic optimization with advanced aerodynamic models.

Key Words: Helicopter vibration, Free wake models, Real-coded genetic algorithm,Min-max method.

1 INTRODUCTION

In contrast to the fixed wing aircraft, flow-field aroundthe helicopter main rotor is highly influenced by thewake effects. The tip vortex shed from a helicopterrotor blade causes a rapid change in the local flow con-dition of the successive blades. These blade vortex in-teractions lead to a high level of vibration and aerody-namic noise for the vehicle. Aeroelastic optimizationstudies have been carried out to minimize these vibra-tions [1, 2]. The main elements needed for a rotorcraftaeroelastic optimization process are: a comprehensiverotor analysis code and a robust optimizer. The ro-torcraft aeroelastic analysis code used in the optimiza-tion should represent the blade aeroelasticity and ro-tor aerodynamics accurately to predict the blade re-sponse and performance parameters. Compared tothe structural modeling of the rotor blades in compre-

Received on.

† Graduate Student, Indian Institute of Science, India‡ Associate Professor, Indian Institute of Science, India

c©Murugan & GanguliComputational Fluid Dynamics JOUR-NAL 2002

hensive rotor analysis code, mathematical modelingof the rotor flow-field which plays a key role in thecalculation of vibratory air loads have not been yetmatured. This is due to the fact that the rotor bladepass through various flow-fields such as transonic flowin the advancing side, stalled and reversed flow in theretreating side and swept flow in the fore and aft re-gion of the rotor disk. The aerodynamic models de-veloped in the last two decades ranges from the liftingline vortices with a deformed helix to the full Navier-Stokes solutions [3]. The computational time and stor-age data for an aeroelastic simulation increases enor-mously when the full Navier-Stokes solutions are used.For example, the Euler/Navier-Stokes simulation foraerodynamic analysis often consume weeks of simula-tion time to map the entire flight regime [4]. There-fore, a full CFD analysis for aeroelastic optimizationin forward flight is difficult. A tightly coupled nonlin-ear aeroelastic simulation with the CFD for aerody-namics and CSD (computational structural dynamics)for structural dynamics is still in the research stage[5]. However, hybrid methodologies developed in therecent years show a promising feature to couple theCFD analysis with a comprehensive aeroelastic anal-ysis code [4]. In hybrid methodologies, an enormousreduction in the computational time is achieved by

2 Murugan and Ganguli

using Navier-Stokes analysis only in the unsteady vis-cous flow regions and potential flow analysis in theother regions. Therefore, with the increasing com-puter power, it will be quite possible to couple thehybrid CFD analysis with comprehensive aeroelasticanalysis code for aeroelastic optimization in the nearfuture.

Other than the modeling issues in aeroelastic anal-ysis, a robust optimizer is essential for a successfulaeroelastic optimization. The rotorcraft optimizationproblems are highly nonlinear and nonconvex in na-ture. Most of the previous aeroelastic optimizationstudies have used gradient based methods for design-ing a rotor blade for low vibration [1]. The gradientbased rotorcraft aeroelastic optimization studies en-counter the problems such as selection of initial design,local minima and difficulties and errors in the calcu-lation of gradients. However, gradient based methodsare preferred than the evolutionary algorithms in theprevious studies due to the computationally expensivenature of the helicopter aeroelastic analysis [1]. Thetime requirement is due to the fact that the rotor gov-erning equations are function of both radial and tem-poral coordinates and need to be solved together withthe nonlinear trim equations for accurate load predic-tions. At least, four loops are involved in rotorcraftaeroelastic optimization with the optimization loop asthe outermost [6]. For a set of design variables fromthe optimizer, the trim equations are solved in the in-ner second loop and for each set of trim values, aero-dynamic load distribution over the blades for a fullrevolution of the rotor is calculated as the third loop.The wake analysis form the innermost fourth loop asfinding the wake geometry and induced velocities foreach airload distribution. Because of these multipleloops, evolutionary algorithms are generally avoided inaeroelastic optimization to reduce computation time.Ribera and Celi [6] derived sensitivity derivatives forthe free wake models to reduce the computation timeof an aeroelastic optimization.

In recent years, the power of computers has in-creased dramatically. Computer codes which used totake hours to run ten years ago now run in minutes.Therefore, it appears possible to use stochastic opti-mization methods such as genetic algorithms for rotor-craft optimization [7, 8]. Lee and Hajela used the par-allel genetic algorithm for rotor blade design [9]. Theuse of high fidelity aerodynamic models such as freewake models is restricted in the rotorcraft optimiza-tion studies because of the large number of functionalevaluations. Real-coded genetic algorithms (RCGA)developed in the recent years show a reduction in thecomputation time when compared to the conventionalbinary coded genetic algorithms [10, 11].

One more interesting issue in the rotorcraft aeroe-lastic optimization is the multi-objective nature of theproblems. Therefore, proper multi-objective formula-tion is needed rather than the simple scalar additionof objective functions. A weighted linear combinationof vibratory forces or norm of the forces is used asthe objective function in most of the previous vibra-tion reduction optimization studies [1]. The main dis-advantage of this type of multi-objective formulationis finding the proper weights for each sub-objectivefunction. Therefore, alternative multi-objective opti-mization methods have to be investigated to combinethese objectives and constraints. A survey of multi-objective optimization methods is given in Reference[12].

This paper focus on the development of a robust op-timization technique and multi-objective formulationsfor rotorcraft optimization studies. Rotorcraft opti-mization is carried out for vibration reduction withtwo multi-objective formulations. Then, the influenceof linear and free wake inflow models on aeroelasticoptimization is studied. Real-coded genetic algorithmis used as the optimization tool to design a low vibra-tion rotor blade.

2 ROTOR AEROELASTICITY

In this work, a comprehensive aeroelastic analysiscode, based on finite element method is used to eval-uate the helicopter vibration. The rotorcraft struc-ture is modeled as a nonlinear representation of elas-tic rotor blades coupled to a rigid fuselage. The bladeis modeled as a slender elastic beam undergoing flapbending, lag bending, elastic twist, and axial deflec-tion. The effect of moderate deflections is included byretaining second order non-linear terms. The bladeis discretized into beam finite elements, each with fif-teen degrees of freedom. The finite element equationsare reduced in size by using normal mode transforma-tion. This results in the non-linear ordinary differen-tial equation with periodic coefficients as given below.

Mp + Cp + Kp = F(ψ,p, p) (1)

Here M, C, K and F represents the finite elementmass matrix, structural stiffness matrix, damping ma-trix and finite element force vector, respectively. Also,p(ψ) is the modal coordinate and ψ = Ωt is the az-imuth angle. Here, Ω is the rotor rotational speed.These equations are then solved using finite elementin time in combination with the Newton-Raphsonmethod. The solutions to the equations are then usedto calculate rotor blade loads using the force summa-tion method, where aerodynamic forces are added tothe inertial forces. The blade loads are integrated overthe blade length and transformed to the fixed frame

Influence of Inflow models on Helicopter Aeroelastic Optimization 3

to get hub loads. The steady hub loads are used to ob-tain the forces acting on the rotor and combined withfuselage and tail rotor forces to obtain the helicopterrotor trim equations.

F(Θ) = 0 (2)

The above nonlinear trim equations are also solvedusing the Newton-Raphson method. The helicopterblade response equations and rotor trim equations in(1) and (2) are solved simultaneously to obtain theblade steady response and hub loads. This coupledtrim procedure is important for capturing the aeroe-lastic interaction between the aerodynamic forces andthe blade deformations. Further details of the analysisare available from Ref. [13].

3 AERODYNAMICS

The aerodynamic modeling used in this study con-sists of two parts : a local blade element model anda global wake model. The blade element model con-sists of a linear attached flow model and a separatedflow model. In the attached flow model, the unsteadylift, drag and pitching moment consists of componentsfrom circulatory and non circulatory loads. The effectof near shed wake is considered in the circulatory part.The trailing edge flow separation is modeled using theKirchoff model which accounts for the nonlinear liftvariations [14].

The wake behind the rotor disk determines the in-duced inflow distribution over the disk and plays a keyrole in the calculation of blade response and loads. Atlow speed flight condition, the wake stays close to therotor disk and has a dominating influence on blade air-loads. The comprehensive analysis code used in thiswork has three types of inflow models: 1) linear inflowmodel, 2) prescribed wake model, and 3) free wakemodel. These models are briefly discussed below.

The simplest rotor wake models assume a uniformor linear inflow distribution over the rotor disk. Thesesimple inflow models may capture the global effects ofthe rotor wake and are usually satisfactory for highspeed flight condition where large portions of the nearwake are carried away by a high incoming velocity.The linear inflow model used in this study is based onthe Drees model in which the rotor induced inflow iscalculated by [14]

λ = µ tan αs + λi

λi = CT /2√λ2+µ2

(1 + kx x cosψ + ky x sinψ)

kx = 43

[(1− 1.8µ2)

√1 + (λ

µ )2 − λµ

]

ky =−2µ

where µ, αs and CT are the advance ratio, rotorshaft tilt and rotor thrust coefficient. The above in-flow model is useful for performance predictions butgreatly unpredict vibratory loads. The linear wakemodel becomes less accurate at low speed and hover-ing flight conditions when the inflow distribution be-comes highly nonuniform over the rotor disk.

The prescribed wake and free wake models consid-ers the spatial and temporal variations of circulationaround the rotor blade to predict the induced inflows.The wake is divided into three regions: 1) Near wake,2) rolling up wake and 3) far wake. In the wake anal-ysis, the induced inflow depends on the blade motion,free stream velocity and vorticity strength and its ge-ometry. The free wake model used in this study isbriefly given below.

The main part of the wake analysis is to find thegeometry of the wake. The free wake model itera-tively solves for wake geometry accounting for bothfree stream and self induced velocity fields. The wakeanalysis is said to be converged when the geometryhas attained the steady state solution. The distortionof wake geometry is considered only for blade tip vor-tex where as the rigid or prescribed wake is consideredfor blade inboard vorticity. The geometry of the tipvortex behind the reference blade is given by

r(Ψ, φ) =rb (Ψ− φ) + φµ + D (Ψ, φ)(3)where rb is the position vector of the blade tip, µ is

the free stream velocity vector, φ is the wake age andD is the distortion due to wake self induced velocity.Once the geometry of the wake is solved, the influencecoefficients are calculated using the Biot-Savart law.The induced velocity due to the vortex line segmentis given as

4v =−14π

∫Γ× r

r3dσ (4)

where r is the vector from the vortex line segmentdσ to the point P. For the vortex sheet element, theinduced velocity is expressed as

4v =−14π

∮r × ω

r3dA (5)


In the final stage of wake analysis, the induced ve-locity at the discrete points on the rotor disk λi(ri, Ψj)is calculated as the product of bound circulation andinfluence coefficients and summing the contributionsfrom all the vortex filaments. Further details of therotor wake model are given in Ref [13]. Use of thefree wake model in the aeroelastic analysis helps inpredicting the magnitude of the helicopter vibrationmore accurately [15].

4 REAL-CODED GENETIC ALGORITHM

Helicopter vibration reduction problem consideredin this study is optimized using a real coded geneticalgorithm. GAs are stochastic optimization tech-niques based on the Darwin’s theory of survival ofthe fittest [8]. GA is a search algorithm based onthe mechanism of natural selection that transforms apopulation (a set of solutions) into a new population(i.e., next generation) using genetic operators such ascrossover, mutation and reproduction [7]. A survivalof the fittest strategy is adopted to identify the beststrings and subsequently genetic operators are usedto create a new population for the next generation.The GAs differ from the conventional optimizationmethods and search methods in the following way [8]:

1. Genetic algorithms work with a coding of the pa-rameter set, not the parameters themselves.

2. Genetic algorithms search from a population of solu-tion points instead of from a single solution point.

3. Genetic algorithms use objective function informa-tion, not derivatives or other auxiliary knowledge intheir search for the optimal or best solution.

4. Genetic algorithms use probabilistic transition rulesi.e., randomized operators and not deterministicrules for information exchange among the strings.

More details about how genetic algorithms workfor a given problem can be found in the literature [7].

A good representation scheme for the solution isvery important in obtaining the best solution for agiven problem using GA. The most commonly usedsolution representation is the binary vector (0 and 1).Recently in [16], it is shown that a string can consistof binary digits, integers and floating point numbers.It is also shown in [16] that a natural representationof strings is more efficient and produces better results.The advantage of the real-coded GA are

1. The binary coding of the design variables in thealgorithm discretizes the design space as a set ofdesign points whereas most of the design variable

values are continuous in the design space. This pre-assigned constraint for the design space may resultin a local optimum.

2. A major disadvantage of the binary representationis the length of the chromosome. The length ofthe chromosome increases exponentially with the re-quired decimal accuracy which in turn reduces theefficiency of the GA.

3. The mapping between the real space and the binaryspace creates a problem for the crossover operatorused in GAs. In the discrete variables case, theGA operators may produce an invalid offspring andtherefore repair algorithms are used to avoid theoffspring’s falling outside the discrete regions.

The quality of solution in RCGA depends on the typeof genetic operators used in the problem. A hybrid ge-netic algorithm employs different crossover and muta-tion operators at different stages of the genetic processand can provide an effective solution to many practicalproblems. A detailed study on effects of hybrid oper-ators for real-coded genetic algorithm is presented in[17].

The genetic operators used in real-coded genetic al-gorithm of the present study is discussed in the fol-lowing paragraphs. Let X be the possible solution

X = [x1 x2]

where x1 and x2 are real numbers. The parents in thepopulation are randomly initialized and also satisfythe geometric constraints. The main genetic operatorsfor this problem are defined such that the solutiongenerated always satisfies the geometric constraints.Next, we describe the genetic operators and the fitnessfunction.

4.0.1 Genetic operators:

The crossover and mutation operators used in thisstudy are described below. Let X and Y be the twoparents selected for the crossover operation.

X = [x1 x2]Y = [y1 y2] (6)

where xi and yi are real numbers.

a) Averaging Crossover: This crossover operator se-lects a scalar value α in the range of 0 ≤ α ≤ 1 andoffspring’s are produced by averaging as follows

X ′ = X + α(Y −X)Y ′ = Y + α(X − Y ) (7)

where X is less than Y . This operator alwaysgenerates the offspring that satisfies the geometric


constraints. The new offspring will have values inbetween their parents.

b) Heuristic Crossover: This crossover operator se-lects a random value α in the range of −1 ≤ α ≤ 1and offspring’s produced by this operator are

X ′ = X + ηcαδ

Y ′ = Y + ηcαδ (8)

where δ is half of the difference between the upperand lower bounds of the individual design variablesand ηc is the crossover rate.

The mutation operators are used to introduce diver-sity in the population.The mutation operators used inthis study are explained below. Let X be the parentselected for mutation operation.

X = x1, x2

a) Heuristic mutation : In this paper, a single mu-tation point is randomly selected. The offspring pro-duced in this operation is described below.

X ′ = X + ηmαδ (9)

where ηm is the mutation rate.In this work, hybrid operators are used in the real-

coded genetic algorithm. In hybrid real-coded geneticalgorithm, the same parents are used to generate moreoffspring’s using different crossover and mutation op-erators. The offspring with the better fitness valuewill replace the parents in the next generation.

5 OPTIMIZATION

The general unconstrained multi-objective optimiza-tion problem is of the form

Minimize : F (x) = [F1(x),F2(x), ...,Fk(x)]

withxL ≤ x ≤ xU (10)

The superscripts L and U refer to the lower and upperbounds on vector of design variables x, respectively.The subscript k refers to the number of objective func-tions. An Nb-bladed helicopter rotor transmits NbΩforces and moments to the fuselage as the principalsource of vibration. Here, Ω refer to the rotationalspeed of the rotor. For vibration reduction of a four-bladed rotor, the objective is to reduce the 4Ω forcesand the 4Ω moments transmitted to the rotor hubfrom the rotating rotor system. The 4Ω forces are

the longitudinal (Fx), lateral (Fy) and vertical (Fz)forces. The 4Ω moments are the rolling (Mx), pitch-ing (My) and yawing (Mz) moments. The 4Ω forcesare normalized by the rotor steady thrust, and the 4Ωmoments are normalized by the rotor steady yawingmoment. Therefore, for vibration reduction problem,the six objective functions can be given by

Minimize : J(x) = [Fx, Fy, Fz,Mx,My,Mz] (11)

The rotor blade structural stiffnesses, i.e, flap bending(EIy), lag bending (EIy) and torsional stiffness (GJ),and blade mass per unit length (m) are consideredas design variables. Move limits are placed on thesedesign variables to ensure that the design does notbecome impractical.

EI lowy ≤ EIy ≤ EIhigh

y

EI lowZ ≤ EIZ ≤ EIhigh

z

GJ low ≤ GJ ≤ GJhigh (12)

mlow ≤ m ≤ mhigh (13)

The above optimization problem is solved using areal-coded genetic algorithm as discussed below.

6 RESULTS

A four bladed soft-inplane hingeless rotor similar tothe BO-105 rotor is considered for the numericalstudy. The baseline blade properties considered inthis study are given in Table (1). The initial values

Table 1: Baseline Hingeless Blade Properties

Number of blades 4Radius, R (m) 4.94Advance ratio, µ 0.3Rotational speed, Ωrad/s 40.10Lock number 5.2mo (kg/m) 6.46solidity 0.07CT /σ 0.07c/R 0.055

of design variables are non-dimensionalized and aregiven below

Xbaseline = [EIy/(m0Ω2R4), EIz/(m0Ω2R4),GJ/(m0Ω2R4), m/m0]

Xbaseline = [0.0108, 0.0268, 0.00615, 1.0]

(14)


Initially the optimization is performed with twodifferent multi-objective formulations: 1) Euclideannorm method with linear inflow model, case I and 2)Min-max method with linear inflow model, case II.The optimization results of both the methods are com-pared. Then, the influence of free wake model on theaeroelastic optimization is studied in the case III. Thethree cases are discussed in the following section.

6.1 Case I: Euclidean norm method withlinear inflow model

One way of solving the multi-objective problem is byminimizing the difference between the potential opti-mal point and a utopia point (also called as an idealpoint) [12]. The Euclidean distance N(X) between thecurrent objective function values and utopia points isgiven by

N(X) = |F (X)− F o| (15)

where F o is the vector of utopia points for the vec-tor F (X). For the vibration reduction problem con-sidered, the Euclidean distance N(X) for vibratoryforces and vibratory moments can be written as

N1 =√

(Fx − F ox )2 + (Fy − F o

y )2 + (Fz − F oz )2

N2 =√

(Mx −Mox)2 + (My −Mo

y ) + (Mz −Moz )2

where the design vector X is given in the Eqn. (14)The norm of the forces and moments are added with

weights W1 and W2 to form a single objective functionas given below

J = W1N1 + W2N2 (16)

Generally, the utopia points for forces and momentsare considered as zero in the vibration reduction opti-mization studies since zero vibration is the ideal goal.And the weights of objective functions W1 and W2 aregiven a value of one which is equivalent to giving equalweights for both forces and moments. Therefore, theobjective function can be written as [1]

J =√

F 2x + F 2

y + F 2z +

√M2

x + M2y + M2

z (17)

Now, the optimization is performed with the abovesingle objective function, J . The upper and lower lim-its on the blade stiffnesses are taken as thirty percentabove and below the baseline values given in the Eqn.(14) and twenty percent above and below the base-line value for mass per unit length. The real-codedGA is run with the above objective function with lim-its on the design variables. A vibration reduction of18% is achieved by the optimization for the objectivefunction J . The percentage reduction in each of the

six vibratory forces or sub-objective functions givenin Eqn. (11) is shown in Fig. 1. The vertical force(Fz) acting on the rotor hub which is considered asthe dominant force in causing vibration is reduced byabout 20 %. The remaining loads show a vibration re-duction of about 20% except for the yawing momentMz which shows a reduction of about 60%. The opti-mal design values are given as

Xoptimal = [0.0122, 0.0189, 0.00689, 1.09] (18)

The optimal design shows an increase in the blademass, flap and torsional stiffness whereas lag bendingis reduced when compared to the baseline values givenin Eqn. (14).

The main disadvantage of this method of scalarizingthe multi-objectives is that of the loss of individual-ity of objective functions. The yawing moment Mz

show a higher reduction when compared to the otherfive vibratory forces as shown in the Fig. 1. This isbecause the net effect of forces is considered by thesingle objective function given in equation (17). Theincrease in one objective function can be hidden bythe decrease in other objective functions. Therefore,alternative methods are needed to perform the multi-objective optimization such that individual forces orobjective functions are optimized as per the designerpreferences.

6.2 Case II: Min-max method with linearinflow model

In this case, min-max approach is used for scalarizingthe multi-objectives [12]. The basic min-max formu-lation can be posed as

Minimize: F (x) = max [F1(x) F1(x) ... Fk(x)]

where the maximum value of objective functionsFi(x) is taken for functional or fitness evaluation inGA. The multi-objective function for vibration reduc-tion can then be written asMinimize:J1(x) = max [Fx(x)

F bx

Fy(x)F b

y

Fz(x)F b

z

Mx(x)Mb

x

My(x)Mb

y

Mz(x)Mb

z]

where the superscript b refer to the baseline valuesof objective functions. This objective function tries tominimize the objective function with maximum valueat the current design value. By this approach, theGA tries to minimize all the objective functions si-multaneously. A reduction of 15−30% is achieved forloads other than Mz and 50% reduction is achievedfor Mz with this min-max method. The percentage ofreduction in each of the forces are given in Fig. 2. Inthe min-max method, each of the objective functionsshow an almost equal amount of reduction. In the Eu-clidean norm method, the objective function Mz show


a reduction of 60% whereas other objective functionsFx, Fy, Fz,Mx,My show a reduction of 20%. The op-timal design values of min-max method are given as

Xoptimal = [0.0120, 0.0206, 0.005965, 1.15] (19)

The optimal design shows an increase in flap stiff-ness and blade mass whereas lag bending and tor-sional stiffness are reduced from their baseline values.However, the optimization results are more evenly dis-tributed.

6.3 Case III: Min-max method with freewake model

In the previous optimization cases I and II, a linearinflow model is used to perform airload calculations.The rotorcraft aeroelastic optimization is often per-formed with the linear inflow aerodynamic model withan implicit assumption that the same optimal designor at least the same direction in the design space willresult even with the high fidelity aerodynamic model.To check this assumption, the objective functions areevaluated at the optimal design values of the case IIwith the free wake model in the aerodynamic analysisand the results are shown in Fig. 3. In contrast to theassumption, the objective functions Fz and Mz showan increase from their baseline values when the freewake model is used whereas the same optimal designvalues show a reduction in all objective functions withthe linear inflow model as shown in Fig. 2. Therefore,it is necessary to use the proper high fidelity aerody-namic models in the aeroelastic optimization.

In case III, a free wake model is used for predictingthe vibratory loads instead of linear inflow model as inthe case II. The GA is run with the same parameters asin case II. The min-max method is used to formulatethe multi-objective function as in case II. It is foundthat the vibratory forces are more sensitive to designvariables while using the free wake model. The GAneeded more runs to find an optimal design when freewake models are used. The percentage of reductionin rotor hub forces and moments is given in Fig. 4.The hub shear, pitching and rolling moments show areduction of about 8−15%. The optimal values of thedesign variable are given as

Xoptimal = [0.0124, 0.0306, 0.00633, 1.16]

The optimal blade stiffness and mass per unit lengthof this case show an increase from their baseline designvalues. Now, the six vibratory forces are evaluatedwith the linear inflow model for the current optimaldesign and the percentage of reduction is shown inFig. 5. All the six vibratory forces show a higherpercentage of reduction with the linear inflow model

when compared to the free wake model results shownin Fig. 4. Using the free wake model therefore gives arobust design and is much more realistic.

The optimal design values for cases I, II and IIIare shown in Fig. 6. The stiffness values EIy and mshow a similar trend whereas the EIz and GJ differswith the multi-objective formulation and aerodynamicmodels in aeroelastic optimization. In particular, us-ing the free wake model (case III) results in a com-pletely different blade design in terms of the lag stiff-ness which is not accurately predicted by the linearinflow models.

The flap, lag and torsion response of the helicopterrotor blade for optimal results of the case II and caseIII are shown in Fig. 7 to Fig. 9. The blade responsefor the optimal design values with linear inflow modeland free wake model show a similar change from itsbaseline value. However, while the inflow models donot cause much change in the flap response, they sig-nificantly affect the lag and torsion response.

The pitching and roll moments and hub shear areparticularly important for helicopter vibration. Theseloads for the optimal and baseline design values ofcase II and III are shown in Fig. 10 to 12. It is clearfrom the figures that these loads are underpredictedwith linear inflow model. Therefore, aeroelastic opti-mization results overestimate the vibration reductionif the low fidelity aerodynamic models are used andcan lead to the actual designs not being able to de-liver the vibration reduction levels predicted by theanalysis.

7 CONCLUSIONS

The following conclusions are drawn from this opti-mization study:

1) The min-max method for multi-objective formu-lation is found to be an effective and practical way ofcombining the multi-objectives of the helicopter vibra-tion reduction problem. In contrast to the Euclideannorm method, the min-max method reduces all the sixhub loads in an equitable manner.

2) The aerodynamic inflow models play a key rolein the prediction of vibratory loads. The optimizationresults obtained using free wake analysis are quite dif-ferent from that using linear inflow model. Vibrationreduction of 15 - 30 percent obtained using linear in-flow model and min-max optimization reduces to 8 -15 percent when free wake model is used.

3) Real-coded genetic algorithm is efficient for solv-ing aeroelastic optimization problems with free wakemodeling. Therefore, real-coded genetic algorithm canbe also a candidate to the hybrid CFD methods basedrotorcraft design optimization.


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[17] F. Herrera, M. Lozano, A. M. Snchez, A taxon-omy for the crossover operator for real-coded ge-netic algorithms: An experimental study, Interna-tional Journal of Intelligent Systems, 18, 309-338,2003.

0

10

20

30

40

50

60

Vib

ratio

n re

duct

ion

(%)

Fx F

y F

z M

x M

y M

z

Fig. 1: Euclidean norm method with linear inflowmodel, case I.

0

10

20

30

40

50

60

Fx F

y F

z M

x M

y M

z

Vib

ratio

n re

duct

ion

(%)

Fig. 2: Min-max method with linear inflow model,case II.

−50

−40

−30

−20

−10

0

10

20

30

40

50

60

Fx F

y F

z M

x M

y M

z

Vib

ratio

n re

duct

ion

(%)

Fig. 3: Case II optimal design with free wake model.

0

10

20

30

40

50

60

Vib

ratio

n re

duct

ion

(%)

Fx F

y F

z M

x M

y M

z

Fig. 4: Min-max method with free wake model, caseIII.

1

0

10

20

30

40

50

60

Fx F

y F

z M

x M

y M

z

Vib

ratio

n re

duct

ion

(%)

Fig. 5: Case III optimal design with linear inflowmodel.

−30

−20

−10

0

10

20

30

40

50

% c

hang

e fr

om b

asel

ine

valu

e

Case ICase IICase III

EIy EI

z GJ m

Fig. 6: Optimal design values for cases I, II and III.

0 50 100 150 200 250 300 3500.02

0.03

0.04

0.05

0.06

0.07

0.08

Azimuth angle (deg)

Fla

p re

spon

se (

non−

dim

ensi

onal

)

Baseline ( Linear inflow )Optimal ( Linear inflow )Baseline ( Free wake )Optimal ( Free wake )

Fig. 7: Blade tip flap response.

0 50 100 150 200 250 300 350−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

Azimuth angle (deg)

Lag

resp

onse

(no

n−di

men

sion

al)

Baseline ( linear inflow)Optimal ( linear inflow)Baseline ( Free wake )Optimal ( Free wake )

Fig. 8: Blade tip lag response.

0 50 100 150 200 250 300 350

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Azimuth angle (deg)

Tor

sion

rep

onse

( d

eg )


Fig. 9: Blade tip torsional response.

0 50 100 150 200 250 300 350−2

−1.5

−1

−0.5

0

0.5

1x 10

−3

Azimuth angle (deg)

Pitc

hing

mom

ent (

non

−di

men

sion

al ) Baseline (Linear inflow)

Optimal (Linear inflow)Baseline ( Free wake)Optimal ( Free wake)

Fig. 10: Reduction in pitching moment My.

2

0 50 100 150 200 250 300 350

0

5

10

15

20x 10

−4

Azimuth angle (deg)

Rol

ling

mom

ent,

Mx (

non

−di

men

sion

al) Baseline ( Linear inflow)

Optimal ( Linear inflow) Baseline (Free wake) Optimal (Free wake )

Fig. 11: Reduction in rolling moment Mz.

0 50 100 150 200 250 300 350

0.075

0.08

0.085

0.09

0.095

0.1

0.105

Azimuth angle (deg)

Ver

tical

hub

she

ar, F

z ( n

on−

dim

ensi

onal

)


Fig. 12: Reduction in vertical hub shear Fz.

3

cfd murugan paper

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