multilevel decomposition procedure for efficient design

AIAA JOURNALVol. 33, No. 2, February 1995

Multilevel Decomposition Procedure for Efficient DesignOptimization of Helicopter Rotor Blades

Aditi Chattopadhyay,* Thomas R. McCarthy,1 and Narayanan Pagaldipti1^Arizona State University, Tempe, Arizona 85287

This paper addresses a multilevel decomposition procedure, for efficient design optimization of helicopter blades,with the coupling of aerodynamics, blade dynamics, aeroelasticity, and structures. The multidisciplinary optimiza-tion problem is decomposed into three levels. The rotor is optimized for improved aerodynamic performance atthe first level. At the second level, the objective is to improve the dynamic and aeroelastic characteristics of therotor. A structural optimization is performed at the third level. Interdisciplinary coupling is established throughthe use of optimal sensitivity derivatives. The Kreisselmeier-Steinhauser function approach is used to formulatethe optimization problem when multiple design objectives are involved. A nonlinear programming technique andan approximate analysis procedure are used for optimization. Results obtained show significant improvementsin the rotor aerodynamic, dynamic, and structural characteristics, when compared with a reference or baselinerotor.

NomenclatureAI = autorotational inertia, lb-ft2

CP = power coefficientCT = thrust coefficientc = chord, ftc0-c3 = chord distribution parameters, ftEIXX = lagging stiffness, lb-ft2

EIZZ - flapping stiffness, lb-ft2

Fk = objective functionsFkQ = values of Fk at the beginning of an iteration/ = K-S function constraint vectorfr = 3/rev radial shear, Ibfx = 3/rev in-plane shear, Ibfz = 4/rev vertical shear, Ibg = constraint functionsK = total number of constraints and

objective functionsmc - 3/rev torsional moment, lb-ftmx = 3/rev flapping moment, lb-ftmz = 4/rev lagging moment, lb-ftR - blade radius, ftT - thrust, IbW = total blade weight, Ibi% = nonstructural weight at y'th node, Ibxty,z = reference axesy - nondimensional radial locationot j - real part of the y'th stability rootek = &th objective function tolerance9 = blade twist, deg0!-03 = blade twist parameters, degfji = advance ratiop = K-S function multipliera - area-weighted solidityor - blade root centrifugal stress, lb/ft2

Vj = minimum allowable damping of y'thstability root

0 = design variable vector£2 = rotor angular velocity, rpm

Received Feb. 3,1994; revision received Sept. 21,1994; accepted for pub-lication Sept. 30, 1994. Copyright © 1994 by the authors. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.

* Associate Professor, Department of Mechanical and Aerospace Engi-neering. Senior Member AIAA.

t Graduate Research Assistant, Department of Mechanical and AerospaceEngineering. Student Member AIAA.

Introduction

IN recent years, optimization techniques have received wide atten-tion in helicopter rotor blade design for addressing various design

issues involving dynamics, aerodynamics, structures, and aeroelas-ticity. Considerable research on aeroelastic optimization of metalrotor blades has been performed during the last decade as indicatedby Friedmann.1 Chattopadhyay and Walsh2 addressed vibration re-duction and minimum weight design of rotor blades with stressconstraints. Recently, Ganguli and Chopra3 developed an aeroe-lastic optimization and sensitivity analysis procedure for compositehingeless rotors using an analytical approach. Most of these researchefforts were based on a single discipline.

Rotary wing aircraft design is truly multidisciplinary in nature,and therefore an integration of the necessary disciplines is es-sential for an optimization procedure to be meaningful. Celi andFriedmann4 addressed the coupling of dynamic and aeroelastic cri-teria, using quasisteady airloads, for blades with straight and swepttips. An integrated aerodynamic and dynamic optimization proce-dure was presented by Chattopadhyay et al.5 The integration ofaerodynamic loads and dynamics was achieved by coupling thenecessary comprehensive analysis procedures inside a closed-loopoptimization technique. He and Peters6 performed a combined struc-tural, dynamic, and aerodynamic optimization of rotor blades usinga simple box beam model to represent the structural componentin the blade. Chattopadhyay and McCarthy7'8 developed multi-criteria optimization procedures for the design of helicopter rotorblades. The objectives were to reduce blade vibration with the cou-pling of blade dynamics, aerodynamics, aeroelasticity and struc-tures. Very recently, multidisciplinary optimization efforts have alsobeen initiated to investigate the design of high-speed proprotors byChattopadhyay and Narayan9 and Chattopadhyay et al.10

Since the validity of the designs obtained using optimization tech-niques depends strongly upon the accuracy of the analysis proce-dures used, it is essential to integrate sufficiently comprehensiveanalysis tools within the closed-loop optimization procedure. Suchprocedures are computationally intensive and, hence, can be pro-hibitive within an optimization environment. Furthermore, the prob-lem becomes highly coupled and is accompanied by a large numberof design variables. Therefore, an "all-at-once" optimization pro-cedure in which all of the disciplines are coupled inside a singleloop and optimization is performed based on criteria involving ev-ery discipline can be inefficient and time consuming. Therefore,decomposition techniques are often used to simplify such complexoptimization problems into a number of subproblems. Multileveldecomposition techniques have been applied to problems basedon a single discipline11"15 in structural applications. In a multi-disciplinary design problem, the number of levels in a multilevel

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224 CHATTOPADHYAY, MCCARTHY, AND PAGALDIPTI: DESIGN OPTIMIZATION OF ROTOR BLADES

decomposition procedure depends upon the number of disciplinesinvolved. Individual optimization is performed at each level usinganalysis procedures pertaining to that level. Optimal sensitivity pa-rameters are exchanged between the levels to provide the necessarycoupling between the levels. An optimal design is obtained wheneach individual level is converged and overall convergence isachieved. Therefore, the speed of obtaining a fully converged resultdepends upon the strength of coupling between the various levels.Adelman et al.16 developed a two-level procedure for performing in-tegrated aerodynamic, dynamic, and structural optimization of rotorblades, based on the multilevel optimization strategy described inRef. 14. In the upper level, the blade was optimized for a combi-nation of aerodynamic and dynamic performance with constraintson the aerodynamic, dynamic, and structural behavior. In the lowerlevel, the internal blade structure was designed to provide sufficientstiffnesses and section masses needed at the upper level. However,several important dynamic design criteria were not included in theformulation. Also, aeroelastic stability, which is an important crite-rion in forward flight, was not considered. Finally, a linear combi-nation was used to combine the multiple objective functions. Thisprocedure is judgmental because weight factors are used to com-bine the individual objective functions, which does not guaranteeimprovements of the individual objective functions in accordancewith the weights imposed on them. In this paper, a three-level pro-cedure is developed to further address the complex helicopter ro-tor blade design problem and to introduce additional disciplinarycouplings. Design criteria pertaining to aerodynamics, dynamics,aeroelastic stability, and structures are included. Interdisciplinaryconstraints are imposed on the individual optimizations performedat each level. A multiobjective optimization technique, which is ef-fective in combining several objective functions and constraints intoa composite envelope function, is used where necessary.

Multilevel DecompositionThe multidisciplinary design problem is formulated using a three-

level decomposition procedure illustrated here. Each level is amultiobjective optimization problem characterized by a vector of ob-jective functions, constraints, and design variables. The formulationis outlined next, where NOB J = number of objective functions, NC= number of constraints, and NDV = number of design variables.

Level 1Minimize

subject to

ENDV1

i' - £2J

/ = !,. . . , NOBJ1

k = 1 , . . . , NC1

7 = 1 , . . . , NOBJ2

7 = 1, . . . , NOBJ3

i = 1, NDV1

NDV

; = !,..., NDV3

where F1, F2, and F3 are the objective function vectors at levels1, 2, and 3, respectively; g1, g2, and g3 are the corresponding con-straint vectors; and 01, 02, and </>3 are the corresponding designvariable vectors. The quantities e2 and e3 represent tolerances onthe changes in the y'th objective functions corresponding to levels 2

and 3, respectively. Superscripts L and U refer to lower and upperbounds, respectively, and superscript * represents optimum values.The quantities 8F?*/80/ and 8F?*/30/ are the optimal sensitivityderivatives of the objective functions used at levels 2 and 3, re-spectively, with respect to the design variables at level 1. Finally,802*/80/ and80?/80/ are the optimal sensitivity derivatives of thedesign variables vectors at levels 2 and 3, respectively, with respectto the design variables at level 1. The optimal sensitivity derivativescan be calculated using the technique described in Ref. 17.

Level 2Minimize

subject to

Fj(</>l*,<l>2)

NDV2

7 = 1 , . . . , NOBJ2

• = !, . . . , NC2

7 = 1, . . . , NOBJ3

i = 1 , . . . , NDV2

r ;' o i T j i—V A02 < 0f 7 = 1 , . . . , NDV3o j.2 ^' — ̂ ./ ^ ' '

where 01* is the optimum design variable vector from level 1. Thisvector is kept fixed during optimization at level 2. The optimal sen-sitivity derivatives of the level 3 objective functions and design vari-ables with respect to level 2 design variables are denoted 8F3*/802

and 803*/802, respectively.

Level 3Minimize

subject to

7 = 1 , . . . , NOBJ3

* = !,..., NC3

7 = 1, . . . , NDV3

where the optimum vectors from upper levels, 01* and 02*, areheld constant. The optimization procedure cycles through all threelevels before global convergence is achieved. A "cycle" is definedas one complete sweep through all three levels of optimization.Optimization at an individual level also requires several "iterations"before local convergence is achieved. Cycling between the threelevels is necessary to account for the coupling between the objectivefunctions, constraints, and design variables pertaining to the variouslevels.

Problem FormulationThe decomposition of the helicopter rotor blade optimization

problem is described here. A smart decomposition can decouplethe problem efficiently and help achieve faster convergence. Thiscan be accomplished through a knowledge of the design processand synthesis as well as the magnitudes of the optimal sensitivityderivatives. In the conventional design process of an aircraft, theplanform variables are prescribed by the aerodynamicist. The de-sired stiffnesses of the wing are prescribed by the dynamicist and thestructural engineer designs the wing structure with sufficient stiff-nesses. Based on the previous concept, coupled with a knowledgeof the optimal sensitivity parameters, the rotor design problem isdecomposed into three levels.

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CHATTOPADHYAY, MCCARTHY, AND PAGALDIPTI: DESIGN OPTIMIZATION OF ROTOR BLADES 225

Level 1The rotor is optimized for improved aerodynamic performance

at level 1 of the procedure. The total power coefficient CP is theobjective function. Level 1 design variables include spanwise varia-tions of chord and twist distributions. A constraint is imposed on therotor thrust coefficient CT to insure the same lifting capability of thereference rotor. In order not to deviate too much from the baselineflight condition and also to ensure efficiency in hover a lower boundis imposed on the rotor solidity a . The constraints are formulatedas follows:

nonstructural weight

Fig. 1 Two-celled isotropic box beam.

where the subscript "ref" stands for the reference, or baseline, ro-tor, and the subscript "min" refers to the minimum allowable bladesolidity. As required by the multilevel decomposition procedure,constraints are also imposed on the changes in the optimum valuesof second and third level objective functions and design variables.In this paper, the 4/rev vertical shear and the 3/rev in-plane shear,which are the second level objective functions as described in thenext section, are constrained to be within 20% of their optimumvalues. The total blade weight, which is the third level objectivefunction as described later, is constrained to be within 25% of itsoptimum value.

Level 2At level 2, the objective is to improve the dynamic and aeroelas-

tic characteristics of the rotor and control the vibratory and staticstresses on the blade. For a four-bladed rotor in forward flight, themost critical vibratory hub loads occur at the fundamental bladepassing frequency of 4/rev. This also includes contributions fromthe 3/rev and 5/rev loads as well.18 However, the magnitudes of the5/rev loads are small compared with the 3/rev loads and are there-fore ignored. The six critical vibratory loads are the 4/rev verticalshear fZ9 the 4/rev lagging moment mz, the 3/rev in-plane shear /A,the 3/rev radial shear /r, the 3/rev flapping moment mx, and 3/revtorsional moment mc. In this problem, the 4/rev vertical shear fzand the 3/rev in-plane shear fx are included as objective functionssince their values were found to be more significant. A constraint isimposed on the 3/rev radial force fr. The moments are not includeddirectly as constraints, but their magnitudes are checked during eachdesign iteration. A lower bound is imposed on the autorotational in-ertia A I to insure that the rotor has sufficient inertia to autorotatein the event of engine failure. Constraints are also imposed on thereal part of the stability roots to maintain aeroelastic stability. Theseconstraints assume the following form:

fr < frU

AI > AIL

where the subscripts U and L represent upper and lower bounds,respectively. The quantity o^ represents the real part of the stabilityroot, K denotes the total number of modes, and v denotes a minimumallowable blade damping, which is a positive number.

The design variables used in level 2 are discrete values of theflapping and lagging stiffnesses, EIZZ. and EIXXi, respectively, ateach segment (/ = 1 , . . . , NSEG and NSEG is the total numberof blade segments). Discrete values of the nonstructural weights(Fig. 1) located at the blade tip, w0i,(i = 1, . . . , NSEG) are alsoused as design variables. To reduce the amount of design variablesat this level to reduce the computational effort, the values of thetorsional rigidity GJ are estimated at this level based on changes inthe flapping and lagging stiffnesses. The exact values for GJ are,however, calculated in the third level.

Level 3At level 3, it is of interest to design the spar such that optimum

structural stiffnesses from level 2 match the actual stiffnesses of the

spar. The appropriate objective function is the total blade weight thatis to be minimized. The five independent wall thicknesses (Fig. 1),at each section tjf (j = 1 , . . . , 5), are used as design variables.

To insure that the actual flapping and lagging stiffnesses are closeto the optimum stiffnesses, as determined in level 2, these stiffnessesare constrained to be within a small tolerance of the optimum stiff-nesses. An upper bound constraint amax is imposed on the bladecentrifugal stress, at the root a,.. The constraints are as follows:

\ E I x x - E I X X o f t \ < e ( E I X X o f , )

\EI,z-EIZZopi\<s(EIZZopt)

where £7*.̂ , and EIZZopt are the optimum stiffnesses obtained fromlevel 2, s is the allowable tolerance on the deviations, FS is thefactor of safety, and <7max is the maximum allowable blade stress. Itshould be noted here that at this level all of the structural propertiesare calculated exactly based on the two-celled box beam. Theseexact values that are determined based on thin wall theory are thenused to update all of the structural properties required.

Aerodynamic ModelThe rotor blade aerodynamic model is described in this section.

The blade chord and twist distributions are modeled as nonlinearfunctions of the nondimensionalized radius y. The chord c(y) andtwist distributions 9(y) are defined to have the following spanwisevariations:

c(y) =

0(y) =

(y - 0.75) + c2(y - 0.75)2 + c,(y - 0.75)3 (1)

y- 0.75) + 02(y- 0.75)2 + 03(y - 0.75)3 (2)

These distributions are chosen to closely model the properties of anexisting blade. 19 The nonlinear distributions also allow the optimizersufficient flexibility in the design space, since these parameters areused as design variables.

Structural ModelThe load-carrying structural member in the rotor is modeled as

a two-cell isotropic box beam (Fig. 1) with five independent wallthicknesses that are assumed to vary spanwise. Spanwise nonstruc-tural tuning masses are located at 2.5% chord location. The beamis symmetrical about the x axis and is assumed to be the princi-pal load-carrying member in the rotor. Outer dimensions of the boxbeam at a blade section are based on constant percentages of thechord at that particular section. Titanium is used as the blade mate-rial. Exact structural analyses are performed only in levels 1 and 3since discrete values of the flapping and lagging stiffnesses are usedas design variables at level 2.

Since the flapping and lagging stiffnesses are used as design vari-ables in level 2, the remaining structural properties are estimatedbased on these values. For example, the polar moment of inertia Ie,the torsional rigidity G J , the shear center-center of gravity offset jc/ ,and the blade mass m are estimated based upon the current valuesof the flapping and lagging stiffnesses and the values of the respec-tive stiffnesses from the last real analysis. All other physical bladeproperties are assumed to remain constant. From these values it is

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then possible to estimate the total blade weight W, the autorotationalinertia A/, and the centrifugal stress at the root ar.

Multiobjective Function FormulationBecause the optimization problem in level 2 involves more than

one design objective, the Kreisselmeier-Steinhauser (K-S) func-tion approach20 is used. The first step in formulating the objectivefunction in this approach involves transformation of the original ob-jective functions into reduced objective functions.8 These reducedobjective functions assume the following form:

-1.0- <0 = !,..., NOBJ (3)

The quantity gmax is the value of the largest constraint correspond-ing to the design variable vector $ and is held constant duringeach iteration. A new constraint vector /m(<f>) (m = 1, 2 , . . . , Mwhere M = NC + NOBJ) is introduced that includes the originalconstraints and the constraints introduced by the reduced objectivefunctions [Eq. (3)]. The design variable vector remains unchanged.The new objective function to be minimized is then defined usingthe K-S function as follows:

- /m -log. (4)

where /max is the largest constraint corresponding to the new con-straint vector fm(<&) and in general is not equal to gmax. The opti-mization problem is thus reduced to an unconstrained minimizationof the K-S function. Further details of this approach are found inRefs. 8 and 20.

Analysis and OptimizationThe program CAMRAD21 is used for the blade dynamic, aero-

dynamic, and aeroelastic stability analyses. The code uses liftingline theory, with corrections for yawed flow, to calculate the sectionloading based on the two-dimensional aerodynamic characteristicsof the airfoil. At each cycle of the optimization procedure the blade istrimmed within CAMRAD so that the intermediate designs that arefeasible designs represent trimmed configurations. A wind-tunneltrim option is used since the reference rotor represents an existingwind-tunnel model rotor. The rotor lift and drag, each normalizedwith respect to solidity and the flapping angle, are trimmed usingthe collective pitch, the cyclic pitch, and the shaft angle. An uni-form inflow model is used in the analysis to reduce computationaleffort.

The optimized rotor is trimmed to the same value of the thrust co-efficient CT as the reference rotor using a variable trim procedure.8The blade response is calculated, in CAMRAD, using rotating free-vibration modes equivalent to a Galerkin analysis. Ten bendingmodes that include seven flap-dominated modes and three lead-lag dominated modes and three torsion modes are calculated. Mainblade responses of up to 8/rev are included. Therefore, eight harmon-ics of the rotor revolution are retained in the airloads calculations.The vibratory shear forces and moments are calculated based uponthe airloads information obtained from the aerodynamic analysis.The aeroelastic stability analysis is performed using Floquet theoryfor periodic state equations. Four bending degrees of freedom andone torsional degree of freedom are used for the stability analysis.

The blade section properties are based on thin wall theory andthe structural analysis of the rotor blade is performed using an in-house code.22 The code models a simple two-cell homogeneous boxbeam with one rectangular cell and one trapezoidal cell (Fig. 1). Itis assumed that the flatwise, chordwise, and torsional stiffnesses ofthe blade are provided solely by the box beam.

The optimization algorithm, as implemented in the codeCONMIN,23 is a nonlinear programming technique based upon themethod of feasible directions. A two-point exponential approxima-tion technique24 is used within the optimizer since exact evaluationof the objective functions and constraints during each iteration iscomputationally prohibitive. This technique is superior to the linearTaylor series approximation since it contains gradient informationfrom the current and previous cycles.

ResultsThe optimization procedure is applied to a reference blade of

radius R = 4.685 ft and rotational velocity Q = 639.5 rpm thatis operating in forward flight with an advance ratio ^ — 0.3. Therotor is an existing advanced four-bladed articulated rotor that rep-resents a wind-tunnel model of the Growth Black Hawk rotor.5 Theblade is discretized into 10 segments (NSEG = 10). A summaryof the operating conditions is presented in Table 1. The physicalconstraints imposed during optimization include the blade soliditya, the autorotational inertia A/, the centrifugal stress at the bladeroot crr, the minimum allowable blade damping u, and the 3/revin-plane vibratory shear force fr. A value of a^n = 0.100 is usedas the lower bound on the solidity, and the upper bound used onthe autorotational inertia is 19.8 lb-ft2. A minimum allowable bladedamping of v = 0.01 is used for all modes to ensure that the opti-mum blade retains damping. The reference value of fr ~ 1.11 Ib isused as the upper bound on the 3/rev in-plane shear force. The rootcentrifugal stress is constrained to be below 12.5 x 106 lb/ft2, whichis calculated using the allowable stress for titanium and a factor ofsafety FS — 2. These values are based on the reference rotor.

A total of seven aerodynamic design variables are used in level1 to define the chord and twist distributions. In level 2, 30 designvariables are used to define the discrete spanwise variations of theflapping and lagging stiffnesses and the nonstructural weights. Inlevel 3, the 5 individual wall thicknesses of the two-celled isotropicbox beam lead to a total of 50 design variables along the span.Therefore, the entire multilevel decomposition optimization proce-dure consists of a total of 87 design variables. Global convergence isachieved in only 3 cycles, each cycle requiring 5 iterations in level1, 15 iterations in level 2, and 4 iterations in level 3.

The optimum results are presented in Tables 2 and 3 and Figs. 2-8.It is seen from Table 2 and Fig. 2 that there are significant reductionsin the individual objective functions from all levels. The objectivefunction in level 1, the coefficient of total power Cp, is reduced by19.8%. The 4/rev vertical shear fz, one of the objective functionsin level 2, is reduced by 45.8%. The second objective function inlevel 2, the 3/rev in-plane shear fx, remains equal to the referencevalue. The large reduction in fz and no reduction in fx are possi-bly because the K-S function envelope follows the fz surface moreclosely than fx due to the value of the K-S factor p selected for thisproblem. It must also be noted that the procedure does not guaran-

Table 1 Summary of operating conditions

Blade radius R, ftRotor thrust 7, IbRotational velocity £2, rpmAdvance ratio ju,

4.685277639.50.3

Table 2 Summary of optimum results

Bounds

Lower Upper Reference Optimum

Objective functionsLevel 1

CPLevel 2

0.000470 0.000377

0.2041.41

0.1111.41

Level 3HMb

ConstraintsLevel 1

<T

Level 2frA/, lb-ft2

Level 3ar,x!06lb/ft2

Trim, CT/(j

5.95

0.100 —— 0.116

—— 1.11 1.1119.8 —— 39.4

—— 12.5 1.510.0592

5.73

0.104

1.0033.9

1.510.0659

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CHATTOPADHYAY, McCARTHY, AND PAGALDIPTI: DESIGN OPTIMIZATION OF ROTOR BLADES 227

Table 3 Box beam wall thicknessesBox beam wall thicknesses, in.

NSEG

Referenceh*2(3Uts

1

0.06250.06120.05700.06020.04230.0488

2

0.06250.06130.04950.05510.05850.0422

3

0.06250.06210.06110.06310.05300.0608

4

0.06250.06010.04310.04690.08250.0290

5

0.06250.06210.06990.06580.05110.0728

6

0.06250.06480.11050.08200.06930.1032

7

0.06250.06330.06680.06420.09400.0783

8

0.06250.06310.06700.06570.07660.0711

9

0.06250.06170.05300.05830.05680.0444

10

0.06250.06360.04530.06360.07490.0225

w

10-

6-

2-

0-

-2-

Reference

Optimum

Fig. 2 Summary of optimum results.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Nondimensional radius, y/R

Fig. 4 Blade twist distributions.

0.6 -,

0.5-

0.4-

g12 0.3-o6

0.2-

0.1-

0-

Reference

Optimum


Fig. 3 Blade chord distributions.

tee a global optimum; therefore the solution obtained represents alocal minimum. The total blade weight W, which is the objectivefunction in level 3, is reduced by 3.7%. As summarized in Table 2,the constraints for all three levels are well satisfied. The constraintsimposed upon the autorotational inertia and the centrifugal stressare always well satisfied, whereas the solidity becomes nearly ac-tive for the optimum blade. The 3/rev radial shear force is reducedby 9.5% during the optimization procedure.

The chord and twist distributions, which are defined by level 1design variables, are presented in Figs. 3 and 4. From Fig. 3 it is seenthat optimum blade has a nearly linear tapered planform, despite thefact that the assumed chord distribution is cubic. As a consequence,the blade solidity is reduced from reference to optimum blade. Thisalso implies that the optimum rotor is operating at a slightly higherCT/CT value while maintaining the same rotor thrust as the referencerotor (Table 2). The optimum configuration being tapered representsa more aerodynamically efficient design with a more evenly dis-tributed load distribution. The twist distribution is shown in Fig. 4where significant changes are observed. The reference rotor has alinear twist distribution with nearly 16 deg of total twist, from rootto tip. The optimum rotor has a highly nonlinear distribution, andthe total twist is significantly reduced (approximately 6 deg, from

-0.5

Fig. 5a Lift coefficient distributions for reference rotors.

0.5

-0.5

Fig. 5b Lift coefficient distributions for optimum rotors.

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30000-,

cg\ 25000-

o :g 20000-

| 15000-

SD 10000-.s2 5000-

0-

Reference

Optimum


Fig. 6 Lagging stiffness distributions.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Nondimensional radius, y/R

Fig. 9 First elastic flap dominated mode.

1600̂

1400-

1200-

1000-

800-

600-

400-

200-

0

Reference

Optimum

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Nondimensional radius, y/R

Fig. 7 Flapping stiffness distributions.

0

Fig. 10 Second elastic flap dominated mode.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Nondimensional radius, y/R

0.7-,

0.6-

lo.5-

•80.4-

|0.3-

|0.2-

0.1-

0.0-

Reference

Optimum


Fig. 8 Nonstructural weight distributions.

root to tip). As seen in Figs. 5a and 5b, the overall reduction in theblade twist has the effect of reducing the region of negative anglesof attack in the optimum rotor map compared with the referencerotor, thereby improving aerodynamic efficiency. It is of interest tonote that the advancing blade encounters negative angles at the tipin the reference rotor (Fig. 5a). The performance is significantly im-proved by optimization, and this region of negative angles of attackis completely alleviated in the optimum rotor (Fig. 5b).

The level 2 design variables are shown in Figs. 6-8. Both thelagging stiffness EIXX and the flapping stiffness EIZZ have simi-lar trends. Because of the optimal sensitivity derivatives, there isa very strong dependence of the stiffnesses on the chord distribu-tion determined in level 1. This is because as the chord changes,the dimensions of the box beam also change. Figure 6 shows thelagging stiffness distribution where the similarities between the stiff-ness and chord distributions are observed. Similar trends are alsoobserved in the flapping stiffness distribution (Fig. 7). The nonstruc-tural weight wo distributions are presented in Fig. 8. When comparedwith the reference rotor, there is an increase in the magnitudes of

these weights from root to 75% span followed by a subsequentdecrease. This trend represents a compromise between the opti-mizer's effort in altering the dynamic response while reducing theblade weight. The nonlinearities in the stiffnesses and nonstructuralweight distributions are due to the optimizer's effort in altering themode shapes (make them more orthogonal to the forcing function)to reduce vibration. The elastic mode shapes of the optimum andreference rotors for the first two flap and lead-lag dominated andthe first torsion mode are shown in Figs. 9-13. The general trend ineach of these distributions is to reduce the amplitude of the modesat the inboard and midspan locations of the blade. This has theeffect of reducing the vibration over a significant portion of theblade, which reduces the overall vibration at the root. The trendis particularly evident in the first elastic torsion mode (Fig. 13).It must be noted that the principal nature of the modes remainsunaltered during optimization. Although the autorotational inertiais used as a constraint, this constraint is always well satisfied andtherefore does not influence the nonstructural weight distributionsignificantly.

The design variables of level 3 are presented in Table 3, whichshows the discrete spanwise variations of all five independent boxbeam wall thicknesses of the reference and optimum blades. Thenonlinearities in the distributions are a result of the objective in level3, which is to obtain a minimum weight structure that satisfies thestiffness distributions as predicted by level 2. Such a beam would bevery impractical to build using isotropic materials, but the stiffnessdistributions can be easily achieved through composite tailoring.

Figure 14 shows the real and imaginary parts of the characteris-tic exponents, which are level 2 constraints, for the optimum andreference rotors. It is seen that aeroelastic stability is maintainedafter optimization and that the blade has higher damping than theminimum prescribed value (v = 0.01).

In summary, the multilevel procedure developed is efficient informulating and addressing multidisciplinary design optimizationof rotary wing aircraft. The decomposition helps in understand-ing the coupling between the various disciplinary criteria and thedesign variable linkage. The design objectives and constraints are

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CHATTOPADHYAY, MCCARTHY, AND PAGALDIPTI: DESIGN OPTIMIZATION OF ROTOR BLADES 229

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Nondimensional radius, y/R

Fig. 11 First elastic lead-lag dominated mode.


Fig. 12 Second elastic lead-lag dominated mode.


Fig. 13 First elastic torsion mode.

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i i i

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Fig. 14 Characteristic exponents for p, = 0.30.

all satisfied in the example shown. The results obtained must beviewed within the context of the modeling assumptions used in theanalysis. However, the procedure provides realistic design trendsin rotary blade design and important tradeoffs associated withthe coupling of aerodynamics, dynamics, aeroelastic stability, andstructures.

Concluding RemarksA multilevel, multidisciplinary decomposition procedure has

been developed for the optimal design of helicopter rotor blades.Blade aerodynamic, dynamic, aeroelastic, and structural design re-quirements have been coupled within a three-level decompositionoptimization procedure. In level 1, the coefficient of total power isreduced using aerodynamic design variables. Level 2 comprises twoobjectives functions to be reduced, the 4/rev vertical shear and the3/rev in-plane shear. The two objective function problem is formu-lated using the Kreisselmeier-Steinhauser function approach. Thedesign variables used are discrete span wise distributions of the flap-ping and lagging stiffnesses and nonstructural weights. At level 3,the objective is to obtain a minimum weight spar design to satisfythe stiffnesses predicted by level 2. The design variables are discretespan wise distributions of the five independent box beam wall thick-nesses. A nonlinear programming approach is used in conjunctionwith a hybrid approximation technique. The procedure yields sig-nificant design improvements. The following specific observationsare made within the context of the modeling assumptions used.

1) The multilevel decomposition optimization procedure yieldedsignificant reductions in the objective functions in all three levels.The use of the decomposition technique allowed for more identifi-able interpretations of the results, and its inclusion within the op-timization procedure made the overall optimization problem moreefficient.

2) The optimization procedure yielded a more efficient rotor withimproved lift distributions.

3) The coefficient of total power was significantly reduced due tothe more optimal chord and twist distributions obtained in level 1.

4) Blade vibration was improved significantly through stiffnessand mass tailoring. The mode shapes were also altered in an effortto reduce vibration. The flapping and lagging stiffness distributionsfollowed the chord distribution due to the use of optimal sensitivityderivatives, which linked the two levels.

5) The optimum rotor configuration remained aeroelastically sta-ble throughout the optimization procedure even though stiffness andmass distributions were significantly altered.

6) The procedure yielded a significantly lighter rotor with im-proved overall performance.

AcknowledgmentsThis research was supported by NASA Ames Research Center

under Grant NAG 2-771, with technical monitors John F. Madden IIIand Sesi Kottapalli, and Grant NCA 2-778, with technical monitorThomas A. Edwards.

References^riedmann, P. P., "Helicopter Vibration Optimization Using Structural

Optimization with Aeroelastic/Multidisciplinary Constraints," Journal ofAircraft, Vol. 28, No. 1, 1991, pp. 8-21.

2Chattopadhyay, A., and Walsh, J. L., "Minimum Weight Design of Rotor-craft Blades with Multiple Frequency and Stress Constraints," AIAA Journal,Vol. 28, No. 3, 1990, pp. 565-567.

3Ganguli, R., and Chopra, L, "Aeroelastic Optimization of a HelicopterRotor with Composite Tailoring," American Helicopter Society 49th AnnualForum, St. Louis, MO, 1993.

4Celi, R., and Friedmann, P. P., "Efficient Structural Optimization of RotorBlades with Straight and Swept Tips," Proceedings of the 13th EuropeanRotorcraft Forum (Aries, France), 1987.

5Chattopadhyay, A., Walsh, J. L., and Riley, M. F, "Integrated Aerody-namic/Dynamic Optimization of Helicopter Blades," Journal of Aircraft,Vol. II, No. 1, 1991, pp. 58-65 (special issue on Multidisciplinary Opti-mization of Aeronautical Systems).

6He, C., and Peters, D. A., "Optimization of Rotor Blades for CombinesStructural, Dynamic and Aerodynamic Properties," Proceedings of the ThirdAirforce/NASA Symposium on Recent Advances in Multidisciplinary Analy-sis and Optimization (San Francisco, CA), 1990.

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7Chattopadhyay, A., and McCarthy, T. R., "Recent Efforts at MulticriteriaDesign Optimization of Helicopter Rotor Blades," NATO/DFG ASI, Bercht-esgaden, Germany, Sept. 1991, see also Stuctural Optimization (submittedfor publication).

8Chattopadhyay, A., and McCarthy, T. R., "Multidisciplinary Optimiza-tion of Helicopter Rotor Blades Including Design Variable Sensitivity," En-gineering Composites, Vol. 3, No. 1, 1993.

9Chattopadhyay, A., and Narayan, J., "Optimum Design of High SpeedProp-Rotors Using a Multidisciplinary Approach," Engineering Optimiza-tion, Vol. 22, 1993, pp. 1-17.

10Chattopadhyay, A., McCarthy, T. R., and Madden, J. E, "An OptimizationProcedure for the Design of Prop-Rotors in High Speed Cruise Including theCoupling of Performance, Aeroelastic Stability and Structures," Mathemat-ical and Computer Modelling, Vol. 19, No. 3/4, 1994, pp. 75-88 (specialissue on Rotorcraft).

11 Schmit, L. A., and Merhinfar, M., "Multilevel Optimum Design of Struc-tures with Fiber-Composite Stiffened-Panel Components," AIAA Journal,Vol. 20, No. 1, 1982, pp. 138-147.

12Schmit, L. A., and Chang, K. J., "A Multilevel Method for StructuralSynthesis," AIAA/ASME/AHS/ASCE 25th Structures, Structural Dynam-ics, and Materials Conference, Palm Springs, CA, 1984.

13Kirsch, U., "An Improved Multilevel Structural Synthesis Method," Jour-nal of Structural Mechanics, Vol. 13, No. 2, 1985, pp. 123-144.

14Sobieszczanski-Sobieski, J., James, B. B., and Riley, M. F., "StructuralOptimization by Generalized, Multilevel Optimization," AIAA/ASME/-AHS/ASCE 26th Structures, Structural Dynamics, and Materials Confer-ence, Orlando, FL, 1985.

15Barthelemy, J. F. M., and Riley, M. F, "Improved Multilevel Optimization

Approach for the Design of Complex Engineering Systems," AIAA Journal,Vol. 26, No. 3, 1988, pp. 353-360.

16Adelman, H. M., Walsh, J. L., and Pritchard, J. I., "Recent Ad-vances in Integrated Multidisciplinary Optimization of Rotorcraft,"AIAA/USAF/NASA/OAI 4th Symposium on Multidisciplinary Analysisand Optimization, AIAA Paper 92-4777-CP, Cleveland, OH, 1992.

17Sobieszczanski-Sobieski, J., and Barthelemy, J. F. M., "Sensitivity ofOptimum Solutions to Problem Parameters," AIAA Journal, Vol. 20, No. 9,1982,pp. 1291-1299.18Gessow, A., and Myers, G. C., Aerodynamics of the Helicopter, College

Park Press, 1985.19Chattopadhyay, A., and Chiu, Y. D., "An Enhanced Integrated Aero-

dynamic/Dynamic Approach to Optimum Rotor Blade Design," StructuralOptimization, Vol. 4, 1992, pp. 75-84.20Kreisselmeier, A., and Steinhauser, R., "Systematic Control Design by

Optimizing a Vector Performance Index," Proceedings of the IFAC Sympo-sium on Computer Aided Design of Control Systems (Zurich, Switzerland),1979,pp. 113-117.21 Johnson, W., "A Comprehensive Analytical Model of Rotorcraft Aerody-

namics and Dynamics," Pt. II, Users' Manual, NASA TM 81183, 1980.22McCarthy, T. R., "An Integrated Multidisciplinary Optimization Ap-

proach for Rotary Wing Design," Master's Thesis, Arizona State Univ.,Tempe, AZ, Dec. 1992.23 Vanderplaats, G. N., "CONMIN—A FORTRAN Program for Constrained

Function Minimization," Users' Manual, NASA TMX 62282, 1973.24Fadel, G. M., Riley, M. F, and Barthelemy, J. M., "Two Point Exponential

Approximation Method for Structural Optimization," Structural Optimiza-tion, Vol. 2, pp. 117-224.

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multilevel decomposition procedure for efficient design

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