NASA Technical Memorandum 109084

ARL Technical Report 476 /

Multilevel Decomposition Approach to_tegrated Aerodynamic/Dynamic / StructuralOptimization of Helicopter Rotor Blades

Joanne L. Walsh and Katherine C. Young

Langley Research Center, Hampton, Virginia

Jocelyn I. Pritchard

U.S. Army Vehicle Structures Directorate, ARL, Langley Research Center,

Hampton, Virginia

Howard M. Adelman

Langley Research Center, Hampton, Virginia

May 1994

Wayne R. Mantay

U.S. Army Aeroflightdynamics Directorate, A TCOM, Langley Research Center,

Hampton, Virgin(NASA-TM-IO9OSZ_) MULTILEVELDECOMPOSITION APPROACH TOINTEGRATED

AERODYNAM IC/DYNAM IC/S TRUC TURAL

OPTIMIZATION OF HELICOPTER ROTOR

BLADES (NASA. Langley Research

Center) 26 p G3t05

N94-33900

Unclas

0012120

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-0001

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Multilevel Decomposition Approach to Integrated Aerodynamic/Dynamic/StructuralOptimization of Helicopter Rotor Blades

Joanne L. Walsh and Katherine C. YoungNASA Langley Research Center

Jneelyn I. PritchardU. S. Army Vehicle Structures Directorate, ARL

Howard M. Adelman

NASA Langley Research Center

and

Wayne R. MantayU. S. Army Aeroflightdynamics Directorate, ATCOM

Hampton, Virginia

A_t

This paper describes an integrated aerodynam-ic/dynamic/structural (IADS) optimization procedure forhelicopter rotor blades. The procedure combines perfor-mance, dynamics, and structural analyses with a generalpurpose optimizer using multilevel decomposition tech-niques. At the upper level, the blade structure and re-sponse are represented in terms of global quantities(stiffnesses, mass, and average strains). At the lower level,the blade structure and response are represented in terms oflocal quantities (detailed dimensions and stresses).

The upper level objective function is a linear combina-tion of performance arid dynamic measures. Upper leveldesign variables include pretwist, point of taper initiation,taper ratio, toot chord, blade stiffnesses, tuning masses,and tuning mass locations. Upper level constraints consistof limits on power required in hover, forward flight, andmaneuver; airfoil drag: minimum tip chord; trim; bladenatural frequencies; autorotational inertia: blade weight:and average strains.

The lower level sizes the internal blade structure at sev-

eral radial locations along the blade. The lower level op-timization assures that a structure can be sized to providethe stiffnesses required by the tipper level and assures thestructural integrity of the blade. The lower level designvariables are the box beam wall thicknesses and several

lumped areas that are analogous to longitudinal stringers ina wing box cross section. The lower level objective func-tion is a measure of the difference between the upper levelstiffnesses and the stiffnesses computed from the wallthicknesses and lumped areas. Lower level constraints areon the Von Mises stress at the box comers for multiple

load cases generated by several flight conditions, limits onwall thicknesses for thin wall theory, and other dimen-sional considerations.

The IADS procedure provides an optimization tech-nique that is compatible with industrial design practices inwhich the aerodynamic and dynamic design is performedat a global level and the structural design is carried out at adetailed level with considerable dialogue and comwomiseamong the aerodynamic, dynamic, and structural groups.The IADS procedure is demonstrated for several cases.

A

AI

aib

cd

CdalI

CF

CD

eL

c_

c t

DVpEEA

EIxx

fb i

Notation

area(ft 2)

autorotational inertia, _ Wjr 2 (lbm'ft2)j=l

ith lumped area (ft2)

box width (ft)airfoil section drag coefficient

maximum allowable section drag coefficient

largest section drag coefficient at azimuthal angle ¥

centrifugal force 0b)rotor coefficient of drag

rotor coefficient of lift

root chord (ft)

tip chord (ft)

pth upper level design variable

Young's modulus of elasticity (lb/ft2)extensional stiffness (lb)chordwise bending stiffness (lb-ft 2)

flapwise bending stiffness (lb-ft2)

ith bending frequency (per rev)

fk

fku

ft i

F

gci

gmax

gi

G

GJh

Ixx

ITER

J

KS

m iNn

nc

nPNDV

OBJPRr

rj

SNff

SNre f

tkv i

V(o,_)

W

wjYi

Y_Af

E

Ey

PO

kth frequency (per rev)

lower bound on kth frequency (per rev)

upper bound on kth frequency (per rev)

ith torsionalfrequency(pet rev)

lower level objective functionith lower level constraint function

maximum lower level constraint function,

max{gci}ithupperlevelconstraintfunction

torsional modulus of elasticity Ob/ft 2)

torsional stiffness (lb-ft 2)

box height (ft)

chc_dwise moment of inertia (ft 4)

flapwise moment of inertia (ft 4)

number of trim iterations

polar moment of inertia (ft 4)

ith weighting factor in objective function

Kresselmeir-Steinhauser function

factor of safety

ith segment tuning mass (slug/ft)

number of blades

integer

number ofconstraintscomponents inlowerlevel

frequencyatn timesrotationalspeedofblade

number ofupper leveldesignvariablesupperlevelobjectivefunction

main rotor power (hp)blade radius from center of rotation (ft)distance along blade from center of rotation (t_)distance from center of rotation to center ofjth

segment (ft)N per rev vertical rotating hub shear in forward

flight (lbf)reference N per rev rotating vertical hub shear in

forward flight (lbf)kill wall thickness fit)

ith lower level design variable

Von Mises stress (lb/ft 2)

total blade weight (Ibm)total weight ofjth structural segment (Ibm)

location of ith tuning mass

point of taper initiation

increment used in frequency window (per rev)coordination parameteraverage strain

Lagrange multiplierpull down factor

bending stress (lb/ft 2)

shear sUess (lb/ft 2)

azimuth angle, zero over tail (deg)

0tw maximumpretwist(deg)

Subscriptsa available or allowable

ff forward flighth hover

m maneuvermax maximummin minimumref reference

SuperscriptsL lower level

U upper levelnondimensional

Introduction

Over the last decade optimization techniques have beenstudied for application to the rotor blade design process. InRef. 1 Miura presents a survey on the application of nu-merical optimization methods to helicopter design prob-lems including rotor blade design. Most optimization pro-cedures have dealt with a single discipline such as aerody-namics (Refs. 2-4), structures (Ref. 5), or dynamics (Refs.2, 6-9). However, the rotor blade design process is multi-disciplinary involving couplings and interactions betweenseveral disciplines such as aerodynamics, dynamics, struc-tures" and acoustics. These couplings and interactions canbe exploited by the optimization procedure if all the disci-plines axe accounted for simultaneously rather than sequen-tially. For instance, in a review (Ref. 10) on the impact ofstructural optimization on vibration reduction, Friedmann

emphasizes the need to include the multidisciplinary cou-plings between aerodynamics, dynamics, and structureseven when optimizing only for minimum vibration.

Techniques and strategies for merging disciplines to ob-tain integrated rotorcraft optimization procedures are de-veloping. In Refs. 11 and 12, a plan is described for inte-grating the disciplines of aerodynamics, dynamics, struc-tures, and acoustics. As part of that plan, aerodynamicsand dynamics have been incorporated systematically intoperformance (Refs. 3 and 4) and airload/dynamic (Ref. 13)optimization procedures resulting in an integrated aerody-namic/dynamic optimization procedure (Ref. 14). Ref. 15

summarizes recent accomplishments based on that plan.

Other multidisciplinary rotor blade optimization work isdescribed in Refs. 16-19. Refs. 16 and 17 describe the

formulation of a multidisciplinary approach to rotor bladedesign for improved performance and reduced fuselage vi-brations. Ref. 18 describes a staged optimization proce-dure for a rotor for combined aerodynamics, dynamics, andstructures. Ref. 19 describes a multidisciplinary optimiza-tion procedure to design high speed prop rotors.

2

What is lacking in previous multidisciplinary rotorblade optimization procedures is an efficient method to in-

tegrate structures or structural properties. Usually struc-tures or structural properties are included in one of two

ways - either as local design variables (indirectly affectingthe response of the blade) or global design variables(directly affecting the response of the blade). When localdesign variables are used, the detail dimensions of a struc-

tural member at one or more radial locations along theblade are used to generate structural properties. When

global design variables ate used, structural properties arethe design variables. Both type of design variables havelimitations. Using local design variables (e.g., Refs.6,7,18-19), such as wall thicknesses of the structural mem-

ber, can lead to a large number of design variables whichcan be computationally expensive. Also, this choice of de-sign variables is at odds with traditional design practicewhere chord, stiffness, and mass distributions along theblade are determined and then a structure is designedwhich matches these distributions. Using global designvariables (e.g., Refs. 2,9,13,14,16-17), such as stiffness and

mass properties, in optimization also has disadvantages.When flapwise bending stiffness, chordwise bending stiff-ness, torsional stiffness, and extensional stiffness distribu-

tions are used as design variables, they are treated as inde-pendent quantities. In reality, these stiffnesses are not in-dependent, and there is no guarantee that a set of wall

thicknesses can be found which will simultaneously givethese stiffnesses.

This paper presents the methodology for incorporatingaerodynamics, dynamics, and structures in an integratedoptimization procedure using both local and global designvariables. Multilevel decomposition techniques based onRef. 20 are used to add structural design variables andconstraints to an existing aerodynamic/dynamic optimiza-tion procedure (Ref. 14). The product is an integratedaerodynamic/dynamic/structural optimization (1ADS) pro-cedure. The multilevel decomposition formulation used inthis paper was presented first in Ref. 15. Another prelimi-nary study of multilevel decomposition techniques appliedto rotor blade design is described in Ref. 21.

The multilevel decomposition approach has been suc-cessfully applied to multidisciplinary problems (e.g., Refs.22-24). As originally proposed in Ref. 25, the coordinationprocedure consisted of an optimum sensitivity analysis(Ref. 26) and a set of equality constraints which relate thedetailed (local) design variables of one subsystem to theglobal design variables on the level above. However, aspointed out in Ref. 27, these equality constraints have

caused difficulties in implementing multilevel decomposi-tion procedures. The IADS procedure is based on the mul-tilevel decomposition approach of Ref. 20 which elimi-nates the equality constraints in the coordination procedureallowing the use of the less computationally costly opti-mum sensitivity derivative found in Ref. 28. However inthe lADS procedure, the set of lower level constraints is

replaced by an envelope function known as theKresseimeir-Steinhauser function (KS-function, Ref. 29)which further reduces computational cost.

First, the general multilevel decomposition strategywith two levels will be discussed (note: systems with morelevels are discussed in Refs. 20, 22, and 25). Next, the

general strategy will be related to rotor blade design.Then, the lADS development including flowcharts of theupper and lower levels and the optimization procedure willbe explained. Results will be presented for several caseswhich demonstrate the strengths of the IADS procedure.

Multilevel Decomposition Optimization Strategy

With a multilevel decomposition approach (Refs. 20,22, and 25), a large complex optimization problem is bro-ken into a hierarchy of smaller optimization subproblems.This hierarchy can be thought of as levels of increasingdetail. At the upper level, the subproblem is formulated interms of global quantities which describe the overall be-havior of the entire system. On the lower level, the sub-

problems are stated in terms of local quantities and localconstraints which have only a small impact on the entiresystem. Each subproblem uses local design variables toreduce the violation of constraints which are unique to thatsubproblem. The coupling between the upper level sub-problem and the lower level subproblems is preservedthrough a coordination procedure such as that described inRefs. 20 or 25. This coupling represents a dialogue be-tween the levels that upon convergence establishes compat-ibility between the two levels.

Fig. 1 illustrates a generic two-level optimization pro-cedure. Note that the analysis proceeds from the upperlevel to the lower level while the optimization proceedsfrom the lower level to the upper level. First, the upperlevel analysis initializes all the global quantities and re-sponses and then provides information to each lower level

subproblem. Individual lower level optimizations are per-formed which reduce local constraint violations as much as

possible and which provides information to the coordina-tion procedure. Next the upper level optimization occurs.The preceding describes one cycle. The entire process isrepeated for several cycles. Convergence occurs when allthe constraints (both upper level and lower level) are satis-fied and the upper level objective function is minimized.

The rotor blade optimization problem can be decom-posed into one subproblem affecting the global response ofthe blade and three subproblems affecting portions of theblade. Quantities such as power required, blade trim, au-torotational inertia, natural frequencies, total blade weight,and average strain describe the global response of theblade. The entire blade must be analyzed to obtain theseresponse quantities. Quantities such as stresses are detailedresponse quantities since only a portion of the blade must

3

be analyzed to obtain these response quantities. Therefore,a two-level decomposed rotor blade optimization problemcan be defined as shown in Fig. 2. The upper leveloptimizes the blade by changing global quantities such asblade planform, twist, and distributions of mass andstiffness. The upper level chord, mass, and stiffnessdistributions are treated as independent quantities. Thereconciliation between these distributions is done on the

lower level. The lower level consists of several indepen-dent subproblems at stations along the blade radius, which

optimize detailed cross-sectional dimensions to satisfystress constraints and reconcile the upper level independentmass, chord, and stiffness distributions with the lower levelcalculated stiffness distributions. This reconciliation is

improved further by a set of upper level coordination con-straints (see Appendix A). First the upper level analysisand optimization will be described, then the lower levelanalysis and optimization, and then the overall IADSsystem.

Upper Level Analysis and Optimization

The purpose of the upper level analysis is to evaluatethe overall rotor blade design on the basis of performance,dynamic, and global structural measures. (For a descrip-tion of the rotor blade design philosophy see Refs. 3-4, 11-12, and 14-15.) The upper level analysis is similar to theintegrated aerodynamic/dynamic analysis reported in Ref.14 with the addition of extensional stiffness design vari-ables, strain conslraints, and coordination consWaints. As

shown in Fig. 3, the blade is analyzed for three flight con-ditions: hover, forward flight, and maneuver. TheLangley-developed hover analysis program HOVT (a bladeelement momentum analysis based on Ref. 30) is used topredict power required in hover. The comprehensive heli-copter analysis program CAMRAD/JA (Ref. 31) is used topredict rotor performance (e.g., trim, airfoil drag, powerrequired), loads, and frequencies for forward flight andmaneuver. The maneuver flight condition simulates a co-ordinated turn in terms of a increased load on the forward

flight rift requirement.

Upper Level Design Variables - The upper level designvariables are the blade planform, stiffnesses, and tuningmasses (see Fig. 4). The blade planform is defined by the

point of taper initiation Ytr, root chord cr, taper ratio Cr/Ct,

and maximum pretwist 0tw. The blade is rectangular from

the root to Ytr and then tapers linearly to the tip. The

pretwist varies linearly from the center of rotation to thetip. Global design variables include the blade chordwise,

flapwise, torsional, and extensional stiffnesscs (denoted byEIxx, EIz_ GJ, and EA, respectively) at three radial loca-

tions: blade root, point of taper initiation, and blade tip.The stiffnesses are assumed to vary linearly between thesepoints and are treated as independent quantities. The re-maining design variables are three tuning masses (denoted

by m I , m2, m 3) and their locations (denoted by y t, Y2, and

),3 ). The total blade mass consists of the structural mass

(which is assumed constant) plus the sum of the tuningmasses. There is no attempt to reconcile the change inweight with the change in design variables since the pre-sent work is based on extending the procedure of Ref. 14 toinclude slructures. However, this reconciliation is possible(see Ref. 15). It is assumed that the center of gravity andaerodynamic offsets are coincident with the blade elasticaxis. The number of blades, rotor radius, rotational veloc-

ity, airfoils, and airfoil distribution are preselected andfixed.

Upper Level Objective function - The objective functionto be minimized is a combination of performance and dy-namics measures and is formulated as follows

OBJ=kl Ph +k2 Pff +k3 Pm +k4 SNff

Phref Pffref Pmref SNref

(1)

where Ph' Pff' and Pm are the powers required in hover,

forward flight, and maneuver, respectively. N is the num-

ber of blades and SNf f is the N per rev rotating vertical

hub shear in forward flight. The terms kl, k2, k3, and k4

are weighting factors chosen by the user. Phref, Pffref'

Pint d , and SNref are reference values used to normalize

and nondimensionalize the objective function components.The usefulness of this objective function was demonstratedin Ref. 14.

Upper Level Constraints - The rotor blade design pro-cess is defined in terms of aerodynamic performance, dy-namics, and global structural requirements. Satisfactoryaerodynamic performance is defined by the following fourrequirements. First, the power required for any flightcondition must be less than the available power. Second,airfoil section drag along the blade radius on the advancingand retreating side of the rotor disk in both forward flightand maneuver must be less than a maximum allowable

value. Third, the rotor must trim at each flight condition.The rotor is trimmed to a constant lift in forward flight anda (different) constant lift in maneuver which ensures that

the rotor has no loss in lift capability or maneuverabilityeven if solidity decreases from the initial to the final de-sign. Incorporation of a maneuver flight condition is usedin place of a constraint on solidity, since low speed ma-neuver determines rotor solidity (Ref. 32). Fourth, theblade tip chord must be larger than a prescribed minimumvalue. Satisfactory dynamics is defined in terms of limitson vibrational frequencies. The blade is designed so thatthe natural frequencies (both bending and torsional) do notcoincide with integer multiples of the rotor speed. Also,lhe blade must have sufficient autorotational inertia as a

safety measure needed in case of engine failure. In addi-tion to satisfying these design requirements, the blade

4

weight must not exceed some upper limit. Satisfactorystructural requirements are defmed in terms of limits on theaverage axial strains for forward flight and the maneuverflight conditions.

This section of the paper discusses the performance,dynamic, and structural constraints. The coordination con-

straints are discussed later in the paper. The performanceand dynamic constraints are the same as those used in Ref.

14. By convention, the ith constraint gi is satisfied if it is

less than or equal to zero.

Performance Constraints - The performance con-

straints are on power required, trim, airfoil section drag,and blade tip chord. The requirement that the power re-quired be less than the power available is given by

for each flight condition (2)

where Pj is the power required for the ith flight condition

and Pa is the power available.

The requirement on the airfoil section drag translates

into a constraint that each airfoil section distributed along

the rotor blade operate at a section drag coefficient cd less

than a specified allowable value Cdau (see Ref. 14). This

leads to 24 constraints per flight condition since the bladeis analyzed in 15 degree azimuthal increments around therotor disk. At a given azimuthal angle W the constraint isformulated as

c_F

gi= dma_x- 1< 0 _F=I5,30,45.....360 (3)

Cdall

where Cdall is the allowable drag coefficient and c _Fdmax is

the largest drag coefficient at any radial station (note: thedrag coefficients in the reverse flow region occurring onthe retreating side of the rotor disc are ignored). In the pre-

sent work the same value for Cdall is used on the advanc-

ing and retreating side of the rotor disk. This simplifyingassumption could easily be lifted.

The trim requirement is difficult to translate into amathematical constraint. The trim constraints in forward

flight and maneuver are implemented using the method de-veloped in Ref. 3 which expresses the constraint in termsof the number of trim iterations (ITER), the maximum

number of trim iterations allowed (ITERmax), and the pth

nondimensional design variable (DVp). The heuristic trim

constraint is given by

NDV

gi =(ITER-ITERmax +1)( _ DVp)<0p=l

(4)

where NDV is the number of design variables. In devel-opment of this equation in Ref. 3, it was found that theaddition of the summation term improved convergence be-cause it allowed calculation of the change in the trim con-sWaint with respect to change in a single design variable.

The final performance requirement is a constraint usedto ensure that the blade tip chord does not become toosmall

gi = 1- ct < 0 (5)Ctmin

where c t is the tip chord and Ctmin is the minimum tip

chord allowed. This is a practical constraint used to assurevalidity of the airfoil tables and address manufacturingconsiderations.

Dynamic Constraints - The dynamic constraints areon frequencies, total blade weight, and autorotational iner-

tia. The constraint on the kth frequency fk (either a bend-

ing or a torsional frequency) is formulated such that thefrequency is separated from integer multiples of the rotorspeed by an amount Af

and

fkgi = --- 1< 0 (upper bound) (0a)

fku

gi = 1 - fk < 0 (lower bound) (6b)fid

where fku has a value that is Af below n+l per rev and fkl

has a value that is Af above n per rev for the applicable n.

For example, suppose Af is 0.1 per rev and f4 is 5.6 per

rev, then nP would be 5 per rev and (n+l)P would be 6 per

rev. Thus f4u and f41 would be 5.9 per rev and 5.1 per rev,

respectively. Formulating the constraints in this mannerallows the frequencies to change from one optimizationcycle to the next cycle provided the frequencies avoid ap-proaching integer multiples of the rotor speed. This for-mulation is different from the approaches used in Refs. 13and 16-17 where the frequencies are kept within prescribedwindows based on the reference blade frequencies. In thiswork, constraints are placed on frequencies in both forwardflight and maneuver since blade collective pitch and theamount of modal coupling may be different for the two

flight conditions and therefore the frequencies could bedifferent.

The constraint that the blade weight be less than somemaximum value is formulated as follows

Wgi = 1 < 0 (7)

Wmax

where W is the total blade weight and Wma x is the maxi-

mum allowable weight. The total blade weight is the

structural mass distribution (which is constant) plus thesum of the tuning masses.

Finally, the blade must have enough autorotationai iner-tia AI for safe autorotation in case of engine failure. Theconstraint is formulated so that the autorotational inertia of

the blade is greater than some minimum value Almi n

AIgi = 1- _ < 0 (8)

Almin

Structural Constraints - The structural constraints areon the average axial swains. The structural constraintswhich are evaluated at the same radial locations used to

defme the design variables (Fig. 4) are imposed on the av-

erage axial strains Cy as follows

and

gi = _- 1< 0 (9a)Ca

where Ca is the magnitude of the allowable strain and

Ey= EA

where CF is the centrifugal force, EA is the extensionalstiffness, and/f is a safety factor on the loads. The strain

constraints are calculated using loads from both the for-ward flight and the maneuver flight conditions.

NDV 0R I

g=g0+ _1 aD-_i ADVi(12)

The assumption of linearity is valid over a suitably smallchange in the design variable values and will not introduce

a large error into the analysis provided the changes ADV i

are small. Errors which may be introduced by use of theapproximate analysis are controlled by imposing "movelimits" on each design variable during the iteration process.A move limit which is specified as a fractional change ofeach design variable value is imposed as an upper andlower design variable bound. At the present time the movelimits are manually adjusted.

Lower Level Analysis and Optimization

This section of the paper describes the lower level anal-ysis and lower level optimization procedure. The purposeof each lower level optimization is to assess whether a

structure at the given radial location can be sized to pro-vide the stiffnesses required by the upper level optimiza-tion and have the strength to withstand loads calculated bythe upper level analysis. The lower level optimizations canbe done in parallel since they are independent.

For simplicity, since closed-form equations can be de-(9b) rived (see Appendix B), the structural member (Fig. 5) is

assumed to be a thin-walled isotropic box. The box crosssection is symmetric about the horizontal axis with wail

thicknesses ti and lumped areas aj which are analogous tolongitudinal stringers in a wing box cross section. The

(10) outer dimensions b (the box beam width) and h (the boxbeam heigh0 are functions of the upper level design vari-ables since b and h depend on the local chord and the local

airfoil thickness. The values of b and h are determined byplacing a box of maximum area within the airfoil crosssection using the method of Ref. 34.

Upper Level Optimization - The upper level optimiza-tion consists of the general purpose optimization programCONMIN (Ref. 33) and an approximate analysis used to

reduce the number of HOVT and CAMRAD/JA analysesduring the iteration process. The approximate analysis isused to extrapolate the upper level objective function and

upper level constraints with linear Taylor Series expan-sions using derivatives of the objective function and con-straints with respect to the design variables

(11)NDV _"11tl[i_OBJ

OBJ=OBJ 0+ _ _ ADV ii=l _DVi[ 0

Lower Level Design Variables - The design variables are

the three wall thicknesses 01, I2, and t4) and the three

lumped areas (al, a2, and a3). The lumped areas are used

to give the lower level more flexibility in matching theupper level stiffnesses. For the present implementation,the lumped areas are assumed to be square areas.

Lower Level Objective Function - The objective func-tion is a measure of the difference between the stiffnessesrequired on the upper level and those determined from thelower design variables

, 2 . 2F Elzz- (Elzz) Elxx - (Elxx). )] . )] ,,3)l, J l, J l, j

6

where a starred quantity ( )* denotes an upper level design

variable. The lower level cross sectional properties Ixx,

Izz, and J are computed (see Appendix B), E is Young's

modulus of elasticity, and G is the torsional modulus ofelasticity.

Lower Level Constraints - The constraints are enforced

on the extensional stiffness, stresses, and the physical di-mensions of the wall thicknesses and lumped areas. Theextensional stiffness constraint which requires the lowerlevel calculated extensional stiffness EA (see Appendix B)to be equal or greater than the upper level extensional stiff-

ness (EA)* (an upper level design variable) is given by

EA

gci = 1 - _ < 0 (14)

at the given cross section. It is noted that the extensional

stiffnesses appear in a constraint rather than in the objec-tive function (Eq. 13) where the other stiffnesses appear.This is done for the following reason. The role of EA in theupper level is limited to satisfying the strain constraints(Eq. 9). The lower level is responsible only for assuringthat the value of EA is at least as large as the value needed

in the upper level -- close matching of EA to (EA)* is notrequired.

The stress constraint which is evaluated at the comer of

the box cross section shown in Fig. 5 has the followingform

gci = I_ 0 (15)cla

where o is the bending stress, "t is the shear stress, and

V(a,x) is the Von Mises stress measure (see Appendix B).Two stress constraints are used - in one x is based on thevertical wall thickness and in the other x is based on thehorizontal wall thickness.

A set of constraints is imposed on the lower level wallthicknesses to assures that the section remains a thin-walled section and that the expression for J remains valid(see Appendix B). These constraints are

_ tjgci - 0.1----_- 1< 0 j = 2and4 (16)

tlgel = --#--- 1< 0 j = 1 and 3 (17)0.1h

where b and h are the width and height of the box crosssection, respectively.

A set of constraints is imposed on the lumped areas andwall thicknesses which require that the dimensions arephysically possible (i.e. the lumped areas can fit inside ofthe box cross section. These constraints are

gci =-[ b-t42 -t2 -Nf_--laf_l <0 (18)

b- t4 -t 1(19)

gc i =-[h-tl - t3 - 23f_-] < 0 (20)

gci=-[h- tl - t3 - 2-¢/_-] < 0 (21)

gci =-[h- tl - t3-2,¢/_-] < 0 (22)

In addition a set of constraints representing upper andlower bounds on the design variables is used. For the kthdesign variable, the lower bound is given by

gci = Vkl - vk < 0 (23)

and the upper bound by

gci = v k - Vku < 0 (24)

where Vkl and Vku are the lower and upper design vari-

able bound, respectively.

For convenience, the set of lower level constraints de-

fined by Eqs.14-24 is replaced by a single cumulative con-straint, an envelope function known as the KS function(Ref. 29), which approximates the active constraint bound-

ary

+ lln[ _eP(gcj -gmax)]KS=gmax P LJ=I _<0(25)

where gmax is the maximum constraint component from

Eqs. 14-24, nc is the number of lower level constraint com-

ponents and p is defined by the user. Initially p is small

and then increases until a maximum value Pmax is reached.

For large values of p, the value of KS approaches gmax"

The KS function is a single measure of the degree of con-straint satisfaction or violation and is positive (violated) if

at least one of the constraints gci is violated. The KS

function is a single-valued function which is continuousand differentiable. This property becomes important whenimplementing the upper and lower levels as described in

7

the section on the overall Organization of the IADS woce-dure.

Lower Level Optimization Procedure - The flowchartfor each lower level optimization procedure is shown inFig. 6. Loads, local chord, box beam width, box beamheight, and upper level stiffnesses are passed down fromthe upper level analysis. The lower level design variables(Fig. 5) are used to calculate lower level stiffnesses. VonMises stresses are calculated using the loads from the for-ward flight and maneuver analyses. The lower level ob-

jective function (Eq. 13) and cumulative constraint (Eq.25) are evaluated. The lower level optimizations are per-formed using the general purpose optimization programCONMIN. Exact analyses are used to evaluate the objec-tive function, the constraint, and any gradients computedby CONMIN. The optimization process is convergedwhen the objective function is minimized and the cumula-

tive constraint is satisfied. After convergence the processreturns to the upper level.

Coordination Between Upper and Lower Levels

The coordination between upper and lower levels is im-plemented by upper level constraints. These constraints

are imposed to encourage changes in the upper level designvariables which promote consistency between the upperand lower level stiffnesses. Specifically, these constraints(one for each lower level optimization) have the form

g=F U-(l+c)F L <0 (26)

Lwhere F ° is the most recent value of the lower level ob-

jective function (i.e., optimmn value of Eq. 13), F U is an

estimate of the change in F L which would be caused by a

change in the upper level design variable values, and c is aspecified tolerance denoted the coordination parameter (seeAppendix A). The importance of this parameter will bediscussed later.

Eq. 26 is the general form of the coordination constraint

as formulated in Ref. 15. Aa shown in Appendix A, thecoordination constraint can be approximated in terms ofthe lower level total optimum sensitivity derivative which

expresses how the optimum lower level objective functionand lower level active constraint will change with a changein upper level design variable.

Overall Organization of 1ADS Procedure

The conceptual IADS procedure is shown in Fig. 2. Itconsists of an upper level analysis (Fig. 3), three lowerlevel optimizations (Fig. 6), and a coordination task. The

actual IADS procedure is more complicated and requiresan upper level sensitivity analysis and three lower level op-timum sensitivity analyses in addition.

The flowchart for the lADS procedure is shown in Fig.7. First, the upper level analysis is executed for the currentset of design variables providing all of the informationneeded to calculate the upper level objective function andconstraints with the exception of the coordination con-

straints. The upper level analysis also provides the loads,local chord, box beam width, box beam height, and stiff-nesses (to be matched) to the lower level analysis. Eachlower level optimization is performed to obtain a set of

lower level design variables which match the current upperlevel bending and torsional stiffnesses as close as possible.

Next, an upper level sensitivity analysis is performedconsisting of forward finite difference derivatives (or gra-clients) of the upper level analysis. These derivatives of the

upper level objective function and upper level constraintsare required to approximate the upper level objective func-tion and upper level constraints during the upper level op-timization. In addition, the loads and local chords corre-sponding to the changes in the upper design variables aresaved. These quantities are used in the three lower level

optimum sensitivity analyses to approximate the coordina-tion constraint (Eq. 26). Appendix A describes how thecoordination constraint is expressed in terms of the total

optimum sensitivity derivative involving changes in the op-timum lower level objective function with respect tochanges in the upper level design variables and changes inthe active lower level constraint with respect to changes inthe upper level design variables.

Finally, the upper level optimization consisting ofCONMIN and approximate analysis occurs. This describes

one cycle of the IADS procedure. The process is repeatedfor additional cycles until convergence is achieved. A verystrict convergence criterion is used for demonstration pur-poses. The overall procedure is converged when thechange in the upper level objective function is less than

0.5x10 "5 over three consecutive cycles and all the con-

stralnts (both upper and lower level) are satisfied. A stepsize of 0.001 is used to compute the finite differencederivatives.

Demonstration of the 1ADS Procedure

This section of the paper describes the analytical blademodel, the mission definition, the optimization problem,and optimization results used to demonstrate the IADS

procedure. Results are presented for two studies - (1) theeffect of initial design and (2) the effect of the coordinationparameter c.

Analytical Blade Model - The analytical blade model

used to demonstrate the 1ADS procedure represents a wind

8

tunnel model of a rotor blade for a four-bladed helicopter.The blade has radius is 4.68 ft. Three sets of advanced air-

foils are used along the blade - the RC(4)-10 airfoil (Ref.35) from the root to 85 percent radius, the RC(3)-10 (Ref.36) airfoil from 85 to 95 percent radius, and the RC(3)-08(Ref. 36) airfoil from 95 percent radius to the tip. Tables

of experimental two-dimensional airfoil data for these threeairfoil types are used in both HOVT and CAMRAD/JA.The analytical model of the blade uses 19 aerodynamicsegments for HOVT, 50 structural segments and 18 aero-dynamic segments for CAMRAD/JA. HOVT is used topredict the power required in hover using nonuniform in-flow (no wake is included) by trimming to a constant liftC L. CAMRAD/JA is used to predict rotor performance,

loads, and frequencies using uniform inflow with empiricalinflow correction factors for the forward flight and maneu-ver flight conditions. Uniform inflow is used to save oncomputational costs (note: even though approximate anal-ysis is used in the upper level optimization 46CAMRAD/JA analyses are required per optimization cy-cle). In CAMRAD/JA an isolated rotor analysis is usedwhich trims the rotor to constant lift C L and drag C D and

zero flapping angle relative to the shaft using collective,lateral cyclic, and longitudinal cyclic pitch. From themodal analyses in CAMRAD/JA using ten bending modesand five torsional modes, it is found that only the first sixbending frequencies are below 10 per rev and need to be

constrained for a four-bladed rotor. Since fbl corresponds

to a rigid body mode and fb2 is the 1 per rev, the first two

frequencies are not constrained. Constraints are placed on

the first four bending frequencies (fb3, fb4, and fb6

flapping-dominated altci fb5 lead-lag dominated) and the

first two torsional frequencies ( ftl representing the rigid

body torsional mode due to the control system stiffness and

ft2 representing the ftrst elastic torsional mode).

Mission Definition - The flight conditions are a constant

lift of 1-g (331 pounds, CL=0.0081), propulsive force of 32

pounds (CD=-0.000811), and an advance ratio of 0.35 for

the forward flight condition and a constant lift of 401

pounds (CL=0.00985), a propulsive force of 23 pounds

(CD=-0.000596), and an advance ratio of 0.3 for the ma-

neuver flight condition. The maneuver flight condition isfor a load factor of 1.22. These flight conditions and loadfactor are similar to those used in Ref. 37.

Optimization problem - The objective function is a com-bination of the power required in hover, forward flight, andmaneuver and the 4 per rev rotating vertical hub shear in

forward flight. The objective function is chosen to be onedominated by performance with little emphasis on dynam-ics. Of the three powers it is assumed that it is most im-portant to reduce the power required in hover - it will havetwice the weight as the other two powers. Several values

were tried for the weighting factor on the hub shear term.It was found that to obtain the proper balance between per-

formance and dynamics, k4 has to be between one and two

orders of magnitude less than k 1 . Thus, for this case, the

weighting factors are chosen to be k I ---10.0, k2=k3=5.0,

and k4 = 0.5.

OBJ=10 Ph +5 Pff +5 Pin +0.5 S4ff (30)

Phref Pffref Pmref S4ref

where Phref, Pffref, Pmref, and S4ref are 15 hp, 13

hp, 12 hp, and 2 lbf, respectively. The reference values arechosen to be representative of the powers required and hubshear for all the initial blade designs used in this work.

The upper and lower bounds for the design variables aregiven in Table 1. On the upper level twenty-two designvariables and 95 constraints are used. On the lower level

six design variables and one cumulative constraint (the KS-function with 24 components) are used at each of the three

spanwise locations (i.e., the root, the point of taper initia-lion, and the tip).

Parameters and flight conditions are summarized inTable 2. Since the blade is made of aluminum, E has a

value of 15.26x108 lb/fl 2, the allowable strain e a has a

value of 0.05 ft/ft, and the allowable stress Oa is 8.352x106

lb/ft 2. The values for minimum tip chord Ctmin, power

available Pa' minimum autorotational inertia, and maxi-

mum allowable drag coefficient Cdall are 0.083 ft, 20 lap,

23.69 lbm-ft 2 and 0.12, respectively. Frequencies must beat least 0.1 away from a per rev value (Af = 0.1 per rev in

Eq. 6).

Study on the Effect of Initial Designs

The lADS multilevel decomposition optimization pro-cedure is demonstrated for three examples using the threestarting points shown in Fig. 8. Example 1 (Fig. 8a) uses arectangular planform with a pretwist of -9.0 degrees, rootchord of 0.3449 It, and upper level stiffnesses design vari-ables initialized to be consistent with the lower level initial

wail thickness and lumped areas (i.e., matched stiffnesses).Example 2 (Fig. 81)) uses a tapered planform with apretwist of -16.0 degrees, root chord of 0.45 ft, andmatched stiffnesses. The blade is rectangular to 80 percentradius and then tapers linearly to the tip with a 3-to-1 taperratio. Example 3 (Fig. 8c) uses the same planform andpretwist as Example 2 but the upper and lower level stiff-nesses are not matched. All these examples use a value of-0.4 for the coordination parameter E in Eq. 26. The impor-lance of the choice of E is examined in a later section of the

paper.

9

Example 1 Rectangular Planform ("initiallymatched stiffnesses") - The starting point for the opti-mization is the rectangular blade shown in Fig. 8a. The

upper and lower level stiffnesses are matched since the up-per level stiffnesses are started with the stiffnesses deter-mined by the initial lower level design variables. This isan infeasible starting point because the lower level stress

constraints at the root are violated. Results are given inTable 3. The initial and final values for the blade planform,performance measures, and dynamics measures are givenin the upper portion of Table 3. The initial and final values

for the constrained frequencies are given in the middle por-tion of the table. Notice that the final value for the fourth

bending frequency fb4 is in a different frequency range

than the initial value. Final values for the lower level de-

sign variables and the upper level stiffnesses ate given inthe bottom portion of the table. The final design is able toimprove the performance characteristics from the initial

blade and satisfy all the constraints. Compared to the ini-tial values, the final design represents a 2.1, 2.3, 2.3, 47.6,and 3.2 percent reduction in the power required in hover,forward flight, and maneuver; hub shear; and upper levelobjective function, respectively.

The final stiffness distributions for the upper (requiredvalues) and lower levels (attainable values) are shown inFig. 9. The matching of the chordwise bending stiffness

Elxx (Fig. 9a), the flapwise bending stiffness Elzz (Fig.

9b), and the torsional stiffness GJ (Fig. 9c) are extremelygood. As shown in Fig. 9d, the lower level is able to ob-tain an extensional stiffness distribution higher than theminimum requirement set by the upper level.

Convergence histories of the individual terms of the

lower level objective function (Eq. 13) are shown in Fig.10 for the three locations - the root (Fig. 10a), point of ta-per initiation (Fig. 10b), and the tip (Fig. 10c). Each term(denoted stiffness deviation) is a measure of how well the

upper and lower stiffnesses match. Initially, the stiffuessesare matched, but the stress constraints are violated at the

root. Therefore, the lower level design variables mustchange to satisfy these constraints while keeping the upperand lower level stiffuesses matched as close as possible.Notice that the chordwise stiffness at the root. torsional

stiffness at the point of taper initiation, and flapwise stiff-ness at the tip are the last stiffuesses to match. Further, itappears that stiffnesses at the point of taper initiation areparticularly difficult to match. This difficulty may be dueto the fact that the point of taper initiation is a design vari-able while the root and tip positions are fixed.

The reason for the deviations in the stiffness is that the

upper and lower levels are in conflict. One component ofthe upper level objective function is the hub shear which

can be reduced significantly by increasing the blade stiff-nesses. On the upper level if the optimizer did not have to

be concerned with stiffness matching, it would increase the

upper level stiffnesses. Without the lower level to keep thestiffnesses in check, a heavy or nonbuildable blade mightresult.

The information shown in Fig. 10 is collected and usedto determine when an upper level design variable movelimit adjustment is necessary during the overall optimiza-tion process (recall that approximate analysis is used on theupper level and exact analysis is used on the lower level).At the present time no automatic move limit adjustment inthe approximate analysis on the upper level is used.Instead, the IADS procedure is run for eight cycles andthen the stiffness deviations are examined. When the stiff-

hess deviation increases (e.g. Cycle 16), the design variablemove limits are manually reduced and the optimizationprocess continued for another 8 cycles. In practicalapplications, the optimization procedure would terminate

after about 30 cycles. However, for demonstration pur-poses the convergence criterion is set to a very small value.Both the upper and lower levels have the same tight con-vergence criterion on each cycle. Overall convergence ofthe 1ADS procedure might improve if the convergence cri-terion is relaxed initially and then tightened as the opti-mization proceeds.

Example 2 Tapered Planform ("initially matchedstiffnesses") - The starting point for the optimization isthe tapered blade shown in Fig. 8b. Initially, the upper andlower level stiffnesses are matched since the upper levelstiffness are determined by the lower level design vari-ables. However, this is an infeasible starting point since athin wall theory constraint is violated on the lower level.

The initial and final values for the blade planform, perfor-mance measures, and dynamics measures are given inTable 4. The final design is able to improve the perfor-mance characteristics from the initial blade. However, thehub shear increases from the initial value.

Fig. 11 shows the final stiffness distributions for the

upper (required values) and lower levels (attainable values)for the chordwise bending stiffness (Fig. 1 la), flapwisebending stiffness (Fig. llb), and the torsional stiffness

(Fig. llc). As shown in the Fig., the stiffness matching isgood, although not a good as in Example 1. The lowerlevel is able to obtain an extensional stiffness distribution

(Fig. 1 ld) higher than the minimum requirement.

Fig. 12 shows the stiffness deviations versus cycle forthe three matching locations - the root (Fig. 12a), point oftaper initiation (Fig. 12b), and the tip (Fig. 12c). Early inthe optimization process, the flapwise and torsional stiff-

ness are both unmatched. After Cycle 10, the matchingsimprove and after 25 cycles, all three matchings are good.At the tip (Fig. 12c) matching proves to be quite difficult.The torsional stiffness is the last to match. The reason forthis is that the blade initial design is tapered and it is diffi-

cult to place a thin wall section in the space near and at thetip and still match the stiffness required on the upper level.

10

Example 3 Tapered planform ("initially un-matched stiffnesses") . In the previous examples, thestarting points used matched stiffnesses. The purpose ofthis example is to demonstrate how the IADS procedurebehaves when it is started from an inconsistent set of stiff-

nesses (i.e., unmatched stiffnesses). The starting point forthe optimization is shown in Fig. 8c. The initial stiffnessesused in the upper level are much larger than the stiffnessesobtained from the lower level design variables. The initialand fatal values for the blade planfonn, performance mea-sures, and dynamics measures are given in Table 5. Thepower required for all three flight conditions has increasedsubstantially along with the hub shear. The initial and f'malconstrained frequencies are also included. Notice that a

bending frequency fb6 has shifted frequency intervals.

The final upper and lower level stiffnesses are shown inFig. 13. As shown in Fig. 13, the optimization procedureis able to match the upper and lower level stiffnesses suc-cessfully. Fig. 14 shows the stiffness deviations for the

three matching locations - the root (Fig. 14a), the point oftaper initiation (Fig. 14b), and the tip (Fig. 14c). As shownin the figure, after 25 cycles the optimization procedure isable to match all three stiffncsses, but it is at the expense ofupper level performance (see Table 5). From these resultsit appears that while the optimization procedure will workwhen starting from an initial point which has unmatchedstiffnesses, it is better to start with a set of consistent stiff-nesses.

Observations on the Effect of Initial Design Study

The 1ADS procedui_ has been exercised for three start-ing blade planforms - a rectangular planform with matchedstiffnesses, a tapered planform with matched stiffnesses,and a tapered blade with unmatched stiffnesses. In allcases the procedure is able to find converged feasible de-signs. Comparing Examples 1 and 2 (Tables 3 and 4, re-spectively), the reader will find two different final bladedesigns (i.e., design variable values are different) with es-

sentially the same objective function value. Apparently,there are many different combinations of design variableswhich satisfy the matching conditions and more than onelocal minimum. The final solution depends on initial con-

ditions. In Example 3 (Table.5), it appears that the opti-mizer converges to a suboptimal solution when comparedwith Example 2. Both examples started from the sameplanform, but Example 2 starts with matched stiffnessesand Example 3 starts with unmatched stiffnesses. Sincethe initial matching of the stiffnesses is relatively easy, thissuggests that the initial matching should always be en-forced.

Comparing all three examples, the reader will also no-tiee that each initial blade has a different frequency rangefor the bending and torsional frequencies and each final

blade design has a frequency which has shifted a frequency

interval (e.g., fb6 in Example 3). During the approximate

analysis, the optimizer can change the upper level designvariables such that a frequency can shift intervals.However, as the design variable move limits are reduced,this shifting is less likely to occur.

At the present time no automatic move limit adjustmentin the upper level approximate analysis is used. However,the stiffness deviation information (e.g., Fig. 10) can becollected and used to determine when an upper level designvariable move limit adjustment is necessary during theoverall optimization process.

Study on the Effect of the Coordination Parameter E

The purpose of this study is to demonstrate the effect ofE in the coordination constraint (Eq. 26) on the optimiza-tion procedure. Results for three e values (+0.4, -0.2, and-0.4) are presented in Table 6 and Figs. 15-16, and 9,respectively. If E is a large positive value, the levels areessentially independent. The upper level is free to changethe upper level stiffness and chord distributions in any waywhich will reduce the upper level objective function. Theonly requirement is that the overall stiffness matchingshould not degrade by more than the amount E from the

best match found on the last lower level optimization. Forexample if e is 0.4, the stiffness matching can degrade by40 percent and still satisfy the coordination constraints. It

is therefore possible that the procedure could convergewith the upper and lower level stiffnesses being mis-matched by as much as 40 percent. A negative value formeans that the upper level must improve the matchingachieved on the lower level by that amount. This sectionof the paper presents results for several values of e usingthe starting point in Fig. 8a which is also used in Example1.

One choice for E would be zero. This would mean that

the upper level cannot degrade the matching achieved onthe lower level. This value was found to be too restrictive

for the optimization process and the procedure convergedin three cycles with very little change in the upper level de-sign variables. The reason for this can be seen by examin-ing the coordination constraint (Eq. 26). At the start of theupper level optimization, the coordination constraint at

each matching location is active (i.e., g=0) since F U isL

equal to F o . As the upper level optimizer tries to change

the upper level design variables, the coordination con-straints become violated. Therefore, the upper level opti-mizer makes only small changes and the process convergesin three cycles.

As shown in Table 6, when E is 0.4, the optimizationprocess is able to improve the performance and dynamics

measures over the initial blade values and improve thelower level (satisfy the stress constraints). This improve-

ll

ment is achieved at the expense of stiffness matching. Fig.15 shows the final stiffness distributions for the upper andlower levels. The lower level is not able to find a set of

stiffnesses to match those required by the upper level. Thisfinal result is technically a feasible design since all theconstraints are satisfied. Recall the upper and lower stiff-

nesses need only be as close as possible (lower level objec-tive function). The upper level coordination constraints donot require the upper and lower level stiffnesses to matchexactly.

When £ is -0.2, the optimization procedure is able toobtain a design that has some improvement over the initialstarting point (Table 6). The upper level objective function

is reduced slightly, but not as much as when e is positive.As shown in Fig. 16, the upper and lower stiffnesses match

well for the chordwise stiffness (Fig. 16a), the flapwisestiffness (Fig. 16b), and the torsional stiffness (Fig. 16c).The lower level is able to obtain an extensional stiffness

which is slightly larger than that required by the upperlevel.

Of the values used in this work, the best value for £ is

-0.4. With this value of £, the optimization procedure isable to obtain improvement on the upper level and find aset of consistent stiffnesses on the lower level. These re-

suits (Example 1) are included in Table 6 for completeness.The stiffness distributions are shown in Fig. 9.

As shown above, positive values of £ result in upperlevel improvement but poor stiffness matching and nega-tive values of £ result in both upper level improvement(although not quite as g,xxl as when £ is positive) and goodstiffness matching. This suggests that a gradual reductionfrom a positive to a negative value for £ could be benefi-cial. The lADS procedure was run with a value of +0.4 for£ for 8 cycles, +0.2 for 8 cycles, -0.2 for 8 cycles, and fi-

nally -0.4 for 8 cycles. This technique of gradually reduc-ing the value of _ did not work. It is felt that the upperlevel planform area and upper level stiffnesses increased toimprove the upper level objective function when £ waspositive so that by the time £ was negative the stiffnessmatching was achieved at the expense of performance anddynamic improvement on the upper level. This situation isanalogous to Example 3 where the mismatched initial con-

ditions resulted in stiffness m_itching at the expense of up-per level improvement.

Concluding Renmrks

An integrated aerodynamic/dynamic/structural (lADS)optimization procedure for helicopter rotor blades has beendeveloped. The procedure combines performance, dynam-ics, and structural analyses with a general purpose opti-mizer using multilevel decomposition techniques. At theupper level, the blade structure and response are repre-sented in terms of global quantities (stiffnesses, mass, and

average strains). At the lower level, the blade structure and

response are represented in terms of local quantities(detailed dimensions and stresses).

The 1ADS procedure consists of an upper level opti-mization, a lower level optimization, and a coordinationtask. The upper level objective function is a linear combi-

nation of performance and dynamic measures. Upper leveldesign variables include pretwist, point of taper initiation,taper ratio, root chord, blade stiffnesses, tuning masses,and tuning mass locations. Upper level constraints consistof limits on power required in hover, forward flight, andmaneuver; airfoil drag; minimum tip chord; trim; bladenatural frequencies; autorotational inertia; blade weight;and average strains.

The lower level optimization sizes the internal blade

structure to provide the stiffnesses required by the upperlevel and assure the structural integrity of the blade. Thelower level design variables are the box beam wall thick-

nesses and several lumped areas which are analogous tolongitudinal stringers in a wing box cross section. Thelower level objective function is a measure of the differ-ence between the upper level stiffnesses and the stiffnesses

computed from the wall thicknesses and lumped areas.The lower level constraints are on Von Mises stresses, ex-tensional stiffnesses, thin wall theory, and dimensionallimits.

The coordination task consists of a set of upper levelconstraints which link the levels and promote consistencybetween the upper and lower level stiffnesses. A coordi-

nation parameter is included in each constraint. This pa-rameter specifies how much the upper level can degrade ormust improve the overall stiffness matching achieved onthe lower level and may also be interpreted as a measure ofhow closely-coupled the two levels are. It is found that a

proper value for the coordination parameter is crucial to thesuccess of the lADS procedure. If the parameter has a pos-itive value, the procedure will converge but the final stiff-ness matching can be unacceptable. If the parameter hastoo small of a value (approximately zero), the optimizationprocess will terminate without improving the dynamics orperformance measures. A small negative value for the co-ordination parameter encourages the upper level to im-prove dynamics and performance using stiffness valueswhich the lower level can match.

The lADS procedure is demonstrated using a model-size rotor blade for several initial blade planforms andvarying amounts of coupling between the levels. In allcases, the lADS procedure achieves successful results. It

converges to a feasible design regardless of whether or notthe initial design had a set of consistent stiffnesses.However, initializing the upper level stiffnesses with thestiffnesses calculated from the lower level design variables,greatly improves the final design.

12

The lADS procedure exploits the couplings and inter-

actions between the disciplines of aerodynamics, dynamicsand structures. It provides an efficient method to integratestructures and/or structural properties into an optimization

procedure since it guarantees that a structure with a consis-tent set of structural properties can be found. The lADSprocedure provides an optimization technique that is com-

patible with industrial design practice in which the aerody-namic and dynamic design is performed at a global leveland the structural design is carried out at a detailed levelwith considerable dialogue and compromise among thegroups.

Appendix A - Coordination Constraint

In a multilevel decomposition approach, the couplingbetween levels is done through a coordination procedure(e.g. Refs. 20 and 25). In the present work, the coordina-tion procedure based on Ref. 20 is used to reconcile thestiffnesses required on the upper level with the stiffnessesthe lower level can actually obtain. This reconciliation re-sults in one upper level constraint at each matching loca-tion

g =FU -(I+e)F L <0 (A1)

Lwhere F o is the most recent value of the lower level ob-

jective function (i.e., optimum value of Eq. 13), F U is anL

estimate of the change in F o which would be caused by a

change in the upper level design variable values, and E isdenoted the coordination parameter. This coordination pa-rameter specifies how much the upper level can degrade ormust improve the overall stiffness matching achieved onthe lower level and may also be interpreted as a measure ofhow closely-coupled the two levels are. If E has a positivevalue, the two levels are not closely-coupled (i.e., they areessentially independent). The upper level can change theupper level stiffness and chord distributions in any waywhich will improve the upper level objective function aslong as the stiffness matching is not degraded by more thanthe amount e. If E has a negative value, the two levels areclosely-coupled and the upper, level is commanded to im-prove the matching by the amount E.

Eq. A1 is the general form of the coordination con-straint as formulated in Ref. 15. The form of the coordina-

tion constraint used in this work is obtained by approximat-

ing F U in terms of the current optimum lower level objec-L L

tive function F o . If F o is expanded in terms of a first or-

der Taylor series about the lower level optimum, then F U

can be approximated by

Ft'=F°L+Y-' v ADVii=l dDVi 1o

(A2)

where DV i is an upper level design variable and dF_----_Vi_ is1¢O

the total optimum sensitivity derivative (Ref. 28) given by

IdDVi = aDVi| o - _-"_ilo

(A3)

OFL

where _ is the derivative of the optimum lower level,lk

objective function with respect to the upper level design0KS

variables, __3---_i is the derivative of the active lower level

constraint (Eq. 25) with respect to the upper level design

variables, _r is the Lagrange multiplier given by

,,=rr sfr Ks lIk0--?/j vl0

(A4)

0KS

where _ is the derivative of the active lower level con-

straints with respect to the lower level design variables at

the lower level optimum. At a lower level optimum, _T

will be positive. If no lower level constraint is active, kT

is set to zero. Substituting Eq. A2 into Eq. A1, the coordi-nation constraint g is approximated by

NDV dFLI 1g= FL+ E _ ADVi[-tl+e] FL<0i=l dDVi o J

(AS)

or simplifying

r,,,ovdFLI -IE aDV - __0 (A6)

Substituting Eq. A3 into Eq. A6, the coordination con-straint becomes

,oviov ,,7>,which is the form implemented in this work.

13

The derivative of the coordination constraint is obtained

by differentiating Eq. A7 with respect to upper level designvariables

ODVi = aDVi Io(AS)

Appendix B - Lower Level Structural Analysis

The purpose of this appendix is to summarize the ele-mentary equations describing the geometry and sU'ucturalanalysis for the lower level structure. A typical cross sec-tion of the thin-walled isotropic box section is shown inFigure 5. For simplicity the top and bottom wall thick-

nesses, t1 and t3 are equal. The total cross-sectional area,A is the sum of the cross-sectional areas of the box beam

elements A i and the lumped areas _ (described in the maintext)

n ill

A = Z Ai + Zaji=l j=l

(B1)

Using the familiar relations, the centroid of the cross-sec-

tion is calculated from the following equations

n nl

_AiX i + _'.ajxj

Xe = i=l j=lA (B2)

and

n m

_'.AiZ i + _-'.ajzj

Zc = i=l j=t (B3)A

where X i and Z i are coordinates in the chordwise and

flapwise directions respectively that specify the distance of

the centroid of the ith element area A i from the reference X

and Z axes shown in Figure 5. Similarly, xj anti zj are co-ordinates that specify the distance of the centroid of the jth

lumped area aj from the reference axes, n is the number ofelements that the cross section is divided into for ease of

calculations and m is the number of lumped areas.

Next, the area moments of inertia of each element aboutits centroidal X and Z axes are calculated from

Ixk = bk_ k=l ..... n+m (B4)12

Izk -- 12 k=l.....n+m (B5)

where bk is the base of the kth rectangular element and hk

is the height relative to the X axis and Ixz i is equal to zero

for symmetric elements. Using the parallel axis theorem,the moments of inertia of each element are found with re-spect to the centroid of the box beam as follows

Icxk = Ixk + Akd2k

lezk = Izk + Ak c2

O36)

where Icxk, and Icz k are the moments of inertia of the

kth element about the centroid of the box beam,

Ixk and Izk are the moments of inertia of the kth element

about its centroidal axes, dk and c k are the distances from

the centroid of the element to the centroid of the box beamin the X and Z directions respectively. The total momentsof inertia for the box beam are equal to the sum of the ele-ment inertias.

Ixx = _ Icxk

Izz= ]_Iczk(B7)

The polar moment of inertia for the box beam is calculatedusing the method described in Ref. 38.

J = 4A2

§ds/t(B8)

where A c is the enclosedareaof the mean periphery of the

box beam wall, ds is the differentialcircumferencial lengthalong the box beam, and t is the local thickness of the wall.

In order to calculate the lower level objective function,the bending and torsional stiffnesses of the box beam are

necessary. For an isotropic beam the moments of inertia,Ixx and lzz, calculated above are multiplied by Young's

Modulus E to acquire the bending stiffnesses, Elxx and

Elzz, in the chordwise and flapwise directions respectively.

Similarly, the polar moment of inertia is multiplied by thetorsional modulus of elasticity G to acquire the torsionalstiffness of the beam, GJ.

The stresses for the constraints in the lower level opti-mization are evaluated at the comers of the box beam us-

hag Von Mises stress measure which is given by

14

V(O,I:) = aJ¢l '_ + 3_ 2 (B9)

where o is the axial bending stress at the outer fiber of the 8.cross section

13__( l__)Xouter+(Mxx_z +CF_i'_-x) outer T (B10) 9.

and _ is the shear stress due to torsion in the wall of thesection with thickness t

MT 10.

x = 2Ae-----_ (B 11)

where Mzz is the flapwise moment; Mxx is the lag mo-

ment; CF is the centrifugal force; and M T is the torque at

the section. The shear stress due to transverse loads has 11.

been neglected for simplicity. Mzz, Mxx , CF, and M T are

computed in the upper level analysis for forward flight and

maneuver, multiplied by a factor of safety _, and then

passed to the lower level.

References

1. Miura, H.: Applications of Numerical OptimizationMethods to Helicopter Design Problems: A Survey.NASA TM-86010, October 1984.

, Bennett, R.L.: Application of Optimization Methodsto Rotor Design Problems. Vertica, Vol. 7, No. 3,1983, pp. 201-208.

3. Walsh, J. L.; Bingham, G. J.; and Riley, M. F.:

Optimization Methods Applied to the AerodynamicDesign of Helicopter Rotor Blades. Journal of theAmerican Helicopter Society, Vol 32, No. 4, October1987.

4. Walsh, J.L.: Performance Optimization of HelicopterRotor Blades. NASA TM-104054, April 1991.

5. Nixon, M. W.: Preliminary Structural Design ofComposite Main Rotor Blades for Minimum Weight.NASA TP-2730, July 1987.

6. Friedmann, P. P. and Shantakumaran, P.: OptimumDesign of Rotor Blades for Vibration Reduction inForward Flight. Proc. of the 39th Annual Forum ofthe AHS, May 9-11, 1983, St. Louis, Missouri.

. Peters, D. A.; Ko, T,; Korn, A.; and Rossow, M. P.:

Design of Helicopter Rotor Blades for DesiredPlacements of Natural Frequencies. Proceedings of

12.

13.

14.

15.

16.

17.

the 39th Annual Forum of the AHS. May 9-11, 1983,St. Louis, Missouri..

Davis M. W. and Weller, W. H.: Application ofDesign Optimization Techniques to Rotor DynamicsProblems. Journal of the American HelicopterSociety, Vol 33, No. 3, July 1988.

Celi, R. and Friedmann, P. P.: Efficient Structural

Optimization of Rotor Blades with Straight and SweptTips. Proc. of the 13th European Rotorcraft Forum,Axles, France, September 1987. Paper No. 3-1.

Friedmann, P. P.: Impact of Structural Optimizationwith Aeroelastic/Multidisciplinary Constraints onHelicopter Rotor Design. AIAA Paper No. 92-1001,Presented at 1992 Aerospace Design Conference.Irvine, California, February 3-6, 1992.

Adelman, H. M. and Mantay, W. R.: IntegratedMultidisciplinary Optimization of Rotorcraft: A Planof for Development. NASA TM-101617 (AVSCOMTM 89-B-004). May 1989.

Adeiman, H. M. and Mantay, W. R.: IntegratedMultidisciplinary Optimization of Rotorcraft. Journalof Aircraft, Vol. 28, No. 1, January 1991.

Chattopadhyay, A.; Walsh J. L.; and Riley, M. F.:Integrated Aerodynamic Load/Dynamic Optimizationof Helicopter Rotor Blades. Journal of Aircraft, Vol.28, No. 1, January 1991.

Walsh, J. L.; LaMarsh, W. J.; and Adelman, H. M.:

Fully Integrated Aerodynamic/Dynamic Optimizationof Helicopter Rotor Blades. NASA TM-104226,February 1992.

Adelman, H. M.; Walsh, J. L.; and Pritchard, J. I.:

Recent Advances in Integrated MultidisciplinaryOptimization of Rotorcraft. Proceedings of the FourthAIAAIUSAFINASAIOAI Symposium onMultidisciplinary Analysis and Optimization.Cleveland, Ohio, September 21-23, 1992. Also avail-able as NASA TM-107665 (AVSCOM TM 92-B-012)September 1992.

Straub, F. K.; Callahan, C. B.; and Culp, J.D.: RotorDesign Optimization Using a MultidisciplinaryApproach. AIAA Paper No. 91-0477, Presented as the29th Aerospace Sciences Meeting. Rent, Nevada,January 7-10, 1991.

Callahan, C. B. and Straub, F. K.: DesignOptimization of Rotor blades for ImprovedPerformance and Vibrations. Proceedings of the 47thAnnual Forum of the American Helicopter Society.Phoenix, Arizona, May 6-8, 1991.

15

18. He, C. and Peters, D. A.: Optimization of RotorBlades for Combined Structural, Dynamic, andAerodynamic Properties. Structural Optimization, 5,pp 37-44, 1992.

19. Chattopadhyay, A.; and Narayan, J. R.: OptimumDesign of High Speed Prop-rotors Using aMultidisciplinary Approach. Proceedings of the 48th

Annual Forum of the American Helicopter Society.Washington, D.C., June 3-5, 1992.

20. Sobieszczanski-Sobieski, J.: Two Alternative Waysfor Solving the Coordination Problem in Multilevel

Optimization. NASA TM-104036, August 1991.

21. Chattopadhyay, A.; McCarthy, T. R.; and Padaldipti,N.: A Multilevel Decomposition Procedure forEfficient Design Optimization of Helicopter RotorBlades. Presented at the 19th European RotorcraftForum, Cernobbio, Italy, September 14-16, 1993.Paper No. G7.

22. Sobieszczanski-Sobieski, J: James, B. B.; and Dovi, A.

R.: Structural Optimization by Multilevel Decomposi-tion. AIAA J., Vol. 23, No. 11, November 1985, pp.1775-1782.

23. Wrenn G. A. ; and Dovi, A.R.: Multilevel

Decomposition Approach to the Preliminary Sizing ofa Transport Aircraft Wing. AIAA Journal of Aircraft,Voi. 25, No. 7, July 1988, pp 632-638.

24. Zeiler, T. A.; and Gilbert, M. G.: IntegratedControl/Structure Optimization by MultilevelDecomposition. AIAA Paper No. 90-1057. Presentedat AIAA/ASME/ASCE/AHS/ASC 31st Structures,

Structural Dynamics and Materials Conference. LongBeach, California, April 2-4, 1990.

25. Sobieszczanski-Sobieski, J.: A Linear DecompositionMethod for Large Optimization Problems--Blueprintfor Development. NASA TM-83248, February 1982.

26. Sobieszczanski-Sobieski, J.,: Barthelemy, J. F.; Riley,K.M.: Sensitivity of Optimum Solutions to ProblemParameters. AIAA J., Vo[. 21, Sept. 1982, pp. 1291-1299.

27. Thareja, R.; and Haftka, R. T.: Numerical Difficulties

Associated with Using Equality Constraints toAchieve Multi-level Decomposition in StructuralOptimization. AIAA Paper No. 86-0854.AIAAJASME/ASCE/AHS 27th Structures, StructuralDynamics, and Materials Conference. San Antonio,Texas. May 1986.

28. Barthelemy, J. F.; and Sobieszczanski-Sobieski, J.."Optimum Sensitivity Derivatives of Objective

9.

0.

31.

32.

33.

4.

35.

6.

37.

38.

Functions in Nonlinear Programing. AIAA J., Vol.22, June 1983, pp. 913-915.

Kresselmeir, G.; and Steinhauser, G.: SystematicControl Design by Optimizing a Vector PerformanceIndex. Proceedings of IEAC Symposium onComputer Aided Design of Control Systems. Zurich,Switzerland, 1971.

Gessow, A.; and Myers, G. C., Jr.: Aerodynamics of

the Helicopters. Frederick Unger PublishingCompany, New York, 1952.

Johnson, W.: CAMRAD/JA - A ComprehensiveAnalytical Model of Rotorcraft Aerodynamics andDynamics - Johnson Aeronautics Version. Volume I:

Theory Manual and Volume II: User's Manual.Johnson Aeronautics, 1988.

Rosenstein, H.; and Clark, R.: AerodynamicDevelopment of the V-22 Tilt Rotor. Paper No. 14.Twelfth European Rotorcraft Forum. Gramisch-Partenkircben, Germany. Setpember 22-25, 1986. '

Vanderplaats, G. N.: CONMIN - A Forwan Programfor Constrained Function Minimization. User's

Manual. NASA TMX-62282. August 1973.

Walsh, J. L.: Computer-aided Design of LightAircraft to Meet Certain Aerodynamic and StructuralRequirements. Master's Thesis. Old DominionUniversity. August 1973.

Noonan, K. W.: Aerodynamic Characteristics of TwoRotorcraft Airfoils Designed for Application to theInboard Region of a Main Rotor Blade. NASA TP-3009, AVSCOM TR-90-B-005. July 1990.

Bingham, G. J.; and Noonan, K. W.: Two-Dimensional Aerodynamic Characteristics of ThreeRotorcraft Airfoils at Mach Numbers From 0.35 to

0.90. NASA TP-2000, AVRADCOM TR-82_B-2.May 1982.

Wilbur, M. L.: Experimetnal Investigation ofHelicopter Vibration Reduction Using Rotor BladeAeroelastic Tailoring. Proceedings of the 47th AnnualForum of the American Helicopter Society. Phoenix,Arizona, May 6-8, 1991.

Bruhn, E. F.: Analysis and Design of Flight VehicleStructures, Tri-state Offset Co. Cincinnati, Ohio.

16

Table 1. Design variable bounds

Design variables

Lower bound Upper boundTwist (deg) -20.0 -5.0

Point of taper initiation (r/R) 0.26 0.985Taper ratio 0.05 5.0Root chord (ft) 0.05 0.833

Elxx (lbm-ft 2) 50.00 20000(_.0

EIzz (lbm-ft 2) 5.00 1000.0

GJ (lbm-ft 2) 5.00 1000.0

EA 0bm) 1000.00 200(ggg)00.0m i (slug/ft) 0.0 0.50

Yi(r/R) 0.24 0.95

t i (ft) 0.00008 0.01

ai (a2) o.o 0.0ooo4

Table 2. Parameters and fli_ht conditionsParameters

Minimum autorotational inertia, Almi n 23.69 lbm-ft 2

Allowable drag coefficient, Cda n 0.12

Minimum tip chord, Ctmin 0.083 ft

Number of blades, N 4

Number of aerodynamic segmentsHOVT 19CAMRAD/JA 18

Number of structural segments 50Number of design variables

Upper levelLower level

Power available, PaBlade radius, RMaximum blade mass, W

Factor of safety, ifAf

Allowable average strain, ea

ITERma x

Pmax

Allowable stress, o a

Young's modulus, E

Table 3 - Rectangular planfonn

stiffnesses - Example 1

FlightRotational velocity

Hover tip Machnumber

C L

C D

Advance ratio

Hover0.00810

conditions

starting point with matched

Initial Final

Hover power (hp) 14.81 14.50Forward flight power (hp) 13.26 12.96Maneuver power (hp) 12.22 11.94Hub shear (lb) 2.1 1.1

Objective function 20.58 19.91Twist (deg) -9.0 -11 A7Taper initiation 0.7 0.701Taper ratio 1.0 1.66Root chord(ft) 0.3449 0.3770

m I (slug/ft) 0.0 0.0002761

m2 (slug/ft) 0.0 0.003199

m 3 (slug/ft) 0.0 0.002014

Yl (ft) 0.450

Y2 fit) 0.583

Y3 fit) 0.453

Cycles to converge 76

fb3 per rev 2.60 2.68

fb4 per rev 3.77 4.57

fb5 per rcv 4.52 4.88

fb6 per rev 7.22 7.55

ftl per rev 7.30 7.30

ft2 per rev 3.61 3.83

Final Design Root Point of taper TipVariables initiationLower Level

t 1 fit) 0.002366 0.002427 0.0004517

22 t2 (ft) 0.003261 0.009954 0.000376618 (6 perlocation) t4 (ft) 0.003414 0.009954 0.0003766

20 hp a ! (ft 2) 0.00003341 0.00003293 0.00001610

4.68 ft a2 (ft 2) 0.00001615 0.00003084 0.000011923.5 Ibm

2.0 a3 (It 2) 0.00003281 0.00003293 0.00001610

0.1 per rev Upper Level0.05 ft/ft Elxx (lb-ft 2) 2057.0 2974.1 153.7

40 Elzz (lb-ft 2) 122.21 140.03 8.61

300 GJ (lb-ft 2) 127.93 128.53 5.87

8.352x106 b/ft 2 EA tlbl 797370. 1647300. 212230.

15.26x108 Ib/ft 2

639.5 RPM (in Freon

density of 0.006 slug/ft 3)0.628

Forward flight Maneuver0.00810 0.00985

-0.000811 -0.000596

0.35 0.30

17

Table 4 - Tapered planform starting point with matched

stiffnesses - Example 2Initial Final

Hover power (hp) 14.85 14.74Forward Flight power (hp) 13.38 13.02Maneuver power (hp) 11.93 11.84Hub shear (ib) 0.6 0.66Obiecfive function 19.88 19.93

Twist (deg) -16.0 -10.85Point of taper initiation 0.8 0.37Taper ratio 3.0 1.64Root chord (ft) 0.45 0.4932m] (slug/fl) 0.0 0.008961

m 2 (slug/fl) 0.0 0.01354

m 3 (sing/fl) 0.0 0.0246

Yt fit) 0.24

Y2 fit) 0.616

y_ (ft) 0.622

Cycles to conver_e 93

fb3 per I'Cv 2.93 2.86

' fb4 per l'eV 5.64 5.33

fb 5 pet I_V 6.22 6.68

fb6 per rev 10.25 9.16

fq per rev 7.30 7.30

ft2 per rcv 6.45 6.12

Table 6- Effect of _ on multilevel decompostim

ol_nization procedure

Initial Filial F'mal Fins!

+.4 -.2 -.4Hover power 14.81 14.44 14.60 14.50

(hp)Forward flight 13.26 12.77 13.11 12.96

power (hp)Maneuver 12.22 11.75 11.96 11.94

power (hp)

Hub shear (lb) 2.1 0.2072 1.85 1.1Objective 20.58 19.48 20.22 19.91

function

Twist (deg) -9.0 -13.32 -11.12 -IIA7Point of taper 0.7 0.786 0.825 .701

initiation

Taper ratio 1.0 3.16 1.41 1.66Root chord (ft) 0.3449 0.3651 0.3606 0.3770

m I (slug/ft) 0.0 0.02571 0.00135 0.0002761

m 2 (slug/ft) 0.0 0.00211 0.0000995 0.003199

m 3 (slug/ft) 0.0 0.00099 0.0000727 0.002014

Yl (ft) 0.4124 0.3115 0.450

Y2 (ft) 0.4154 0.3950 0.583

Y3 fit) 0.4382 0.4292 0.453

Cycles to 90 152 76

conver_e

Table 5 - Tapered planform starting point with unmatched

stiffnesses - Example 3Initial Final

Hover power (hp) 14.85 16.64Forward flight power (hp) 13.27 17.46Maneuver power (hp) 11.89 14.89Hub shear _lb) 0.186 2.45Objective Function 20.01 24.62

Twist (deg) -16.0 -11.98Point of taper initiation 0.8 .889Taper ratio 3.0 1.3148Root chord (ft) 0.45 0.7364

m] (slug/a) 0.008546m2 (slug/fl) 0.007797

m3 (slug/fi) : 0.009030

Yl (ft) 0.323

Y2 (ft) 0.439Y3 (ft) - 0.393

Cycles to conver_e 92

fb3 per rev 2.87 2.90

fb4 per rev 5.54 5.87

fb5 per rcv 8.62 8.10

h6 per rev 9.65 10.5

ftl per rev 7.30 7.30

ft2 per rev 5.48 5.12

Figure 1. General multilevel optimization procedure withtwo levels.

18

Uppe¢Level

Figure 2. Multilevel strategy for rotor bladeaerodynamic,dynamic and structural optimization

h

tl Z

T

._--- t2

-_ b _--

t4

Figure 5. Lower level design variables

line ofsyrnmelr/_X

Upper level design variables ]

@

CAMRAD/J_. IAnalysl-----_s -Jr-. __ __ Design vm.klbl. I CAMRAD/JA

Forward flightperformance

Forward fligMsidoads

,#,Forward fligM

dynamic nmlX)nse

Forward flightstructural amalysis

An_

.I ob_,v.,..d_ I."1 .,d =o,.tr_.t. l"

ManeuverI)ollorlrlriP,¢°

,/,Maneuver

[ °idoadsIk

Maneuver

dynamic response

Maneuver

structural anaiysh;

+

Figure 3. Upper level analysis flowchart

r_R:0.1S rSRwv r/R=1.0Elxx Elxx Elxx

Elzz El:,:, ElzzCent_ GJ GJ GJ

of EA EA EA

m 2 m$ cty: y: T

r

R

Point of tilp_ Intmrection ytr. r/RRoot chord q.

• Tqper ratio q/ct

_/ pretwiet OIw

Maximum

Figure 4. Upper level design variables

I Loads, stiffness, end local chord Ifrom upper level

i Csiculate.tiffnesse. from [4lower level design variables

.k

i•-"..+ +,,re..,

I,

I Objective functionand constraints [+

i O,.m.., I+

lUpdateddesignvariablesI

no

Figure 6. Lower level flowchart

19

, J, ,

t,i

N m

U

lore

C/

I I J0.21 0.14) 0.7l 1.0

Hedkd IooeUo_ rm

9a) Chordwisc bending stiffness

_ m 180--

IO

...._--- u_ _._,_._Figure 7. Overall _.i_zafion flowchart

_lrlti" o his o.io o.7i 1.o

O_ lUladllll IooaUon, rm

rotsUon 9b) Flapwisc bending stiffness

+ " I 4O0

m -

_ -

-_" _ _ " - _ I.ower .vel(--)Twiai = -l_ Tor_J am -- ----_---- up_r _v**(r,_r,d)otlffYlNml

_J' Ib.nt 1S0 --

8b.) Tapered Id_rormTwist"Example2 lOOr_-- ; -1_-16" o o.2s oJo o._s _.o

I_dlel Iocetlon, r_

k.) Tapered l_m_rorm. Example 3 9c) Torsional stiffness

2_E$ --

Figure 8. Initial designs used to start IADS procedure _ _.*, _,._._._.).... _'--- UFq_r I_ml(r_ukN)

a.OIUI --

ff.at_n_on_l

-"I[_lb IlI, ES -- I1_

1.0U, / /

0.SEI_

J I I l0 0._1$ O.llO 0.7i 1.0

Rodlld _

9d) Extensional stiffness

Figure 9. Final upper and lower level stiffnessdistributions - Example 1

2O

10000 -

0.150

StiffnessDeviation

0.100

Flapwisa

.................................Chordwlse

Torsional

0.050

0 0000_- ...... ,, .... ," 25 50 75Cycle

i i

lOa) Root

0.150 1

Stiffness jDeviation G Flapwise

0.100 i ..................................Chordwise

...........ors.oo.0.050

[ Jll!,',,

°°°°o _ 25-- - _'o.... 7's' 'Cycle

10b) Point of taper initiation

0.150

StiffnessDeviation

O.tO0

A

E.20.0000

10c) Tip

Flapwise

.................................Chordwise

..... :".... Torsional

_-_, =c,_ & .... t ..... . . .

25 50 75Cycle

Figure 10. Convergence history of upper and lower levelstiffness deviations - Example 1.

21

=-!_nm

Elxx, lb-ft s

Flepwlee

bending

idlffnew

EIn,llPIt =

To¢IkwIl

utMne_

GJ, IIb4t I

Lower klvel (alludnlble) "_'%,.

----I,---- Upper level (required) "_.

I I I "_,0 0.25 0,80 0.70 1.00

RIdlel Iooatlon, r/R

lla) Chordwise bending stiffness

4(10

3S0 ......... "_X.

-- ,%

300 _'N

200 "_,

150

1()0 -'"_q'-"" Upper lewll (feqM|rad)

0 0.25 0.50 0.75 1.00

Rmdial Iocmlon, r/R

llb) Flapwise bending stiffness

Extlm_ OE$ ,/"/

ii A_,,l_e$ 1.IE 0 MI

1.0110 _ _ Lower level |maln-_le) \,,_,

0 0.215 0.SO 0.76 1.00

I_Kllel Ic_lon, r/R

1ld) Extensional stiffness

Figure l 1. Final upper and lower level stiffnessdistributions - Example 2

3.0E0

400 _ ............ "1_,.

380 __%_ NN %N

_00 "%" N

210

_ N N

180--

100 -- _ LOWer%_U_per k._| (_l.Ired)

=0- X

I I I

0 0.211 0.1110 0.7I 1.00

Radlai location, r4q

l lc) Torsional stiffness

1.00

0.75Stiffness

Deviation50

0.25

0.000UL

12a) Ro(x

Rap,vise

.................................Chordwise

Torsional

.i .... i .... l-ii I

25 50 75Cw_

1.00

0.75StiffnessDeviation

0.50

0.25

Flapwise

.................................Chordwise

........... Torsional

25 50 75Cycle

12b) Point of taper initiation

0.75Stiffness IDeviation

0.50

0,25

Flapwise

.................................Chordwise

........... Torsional

250 O00 .............. ,• 50 75Cycle

12c) Tip

Figure 12. Convergence history of upper and lower levelstiffness deviations - Example 2.

1OOOO

¢hocdvdee ?m

bendingeUfflMmlilxx, Nl_ t

IO00

noo

L

FIIpwlNbeldl_llellffnem

EIn,lb-_ t

Tlmlbnel

_1. Ib_ t

--g--- Lower kvel (emdn_de)

Upper level (required)

1 I I I0 O.Ii O.iO 0.711 1.00

RIcllll Ioolllon,nlq

13a) Chordwise bending stiffness

310 --

3100 --

210 --

200 --

110 --

100 -- _ Lower level (attmble)---+-- Upper level (required)

gO --

I I I0 0.211 O.llO 0,71 1.00

Radlsl Io01dloII ,r/R

13b) Flapwise bending stiffness

--

=glO --

11_ --

_-- upper k_ (required) \

i,I I I I I0 0.26 0.10 O.?l 1.00

Redlel Iooetion.#R

13c) Torsional stiffness

I

l.m -

Lower level (atlalnOb)

UPI_ level (required)

Actuli-- I.---__1.0Eil

ExtemdonlllIllffneu

EA, Ib

O.SEI

Minimum required

P-_+--_-4-- .... J -0 0.25 0.50 0,7S

Redlll kx:ldlon,r/R

13d) Extensional stiffness

Figure 13. Final upper and lower level stiffnessdistributions - Example 3

1.00

22

1i00 --

0.75StiffnessDeviation

0.50

0.25

0.000

Rapwise

.................................Chordwise,!

!ii ........... Torsional

25 50 75Cycle

14) Root

0.75

StiffnessDeviation

0.50

0.25

0.00_

i

Flapwise

.................................Chordwise

........... Torsional

. J . . . * .... , ....

25 50 75Cycle

14b) Point of taper initiation

1.00[ _-!

o.75t!

g':vi".t,Sos h

0.50_

0"25 _1_

0 00_"

14c) Tip

Flapwise

.................................Chordwise

Torsional

25 50 75Cycle

Figure 14. Convergence history of upper and lower levelstiffness deviations - Example 3.

23

1000

Chormldu

heeding=tlffneu

Ellx,llvft =

SO0

FlaI_N 100betidingmlffne_

Elu'lb'lt= SO t

0

UlN=er level (requked) _

I I I _"0 0,2S O.iO 0.70 1.00

RedU-l Iooetkm,

15a) Chordwise bending stiffness

O._*S 0.50 0.7S 1.00Reclhd _, rRI

15b) Flapwise bending stiffness

310 --

Tomkmel 21@ --NMnemQJ, Ib'ft= 20@ --

F.x_mmkmellffilfll_imF.J_,b

Lov_.r level (mttelneble)

---t---- Upper level (required)

0 0.28 O.SO 0.75 1.00Radial location, r/R

15c) Torsional stiffness

I_E$ --

I.OE$ --

1JIEI --

1.0E0 -- A4mml

O.SEI Idln_quqUmqlmqlqld_i

t L I I0 0.21 OJlO 0.71 1,00

R_llml loNlkm, r/R

15d) Extensional stiffness

Figure 15. Final upper and lower level stiffnessdistributions for ¢=+0.4

11o

lOO

Io

----41----- Lower level (mttmlnebb)

---IP--- Upper level (required)

Iio0

1000

NO0

INO

h

1000 -- _ Lower Iov, I (_)

---'_---- _ _wl (r,quOr,d)I00--

I I I0 0.N 0.10 0.76

IooMkmo r_l

16a) Chord_se bending stiffness

1N-

e_mmo

EOR_s.-_

L

Lom_ level (etlalnoble)

----l.---- UPlPW kvol (requital)

I I0,11 0.10

13mltadIoeatio_ r01q0.71 1.0O

Tondo_md*lltna*oQJ, Ik-R |

16b) Fiapwise bending stiffness

NO

NO

110

100

{tO

O

--4--- _w _,4 (m_h, mbk,)

----i--- up_,_ i,ve; (n_,_d)

J IO.N O.IH)

RaNIk,I Iooelion, r/R

10c) Torsional stiffness

0._

I.IEo -

lllonllonll

F_ ,Ib

2.0E0

1.lEe

1.0FJI

OJEo

Lomr level (et_l,al_e)

--+-- umw Ima (m_m')

AeCul

Minimum m¢Fulrod

I I I I0.1tl 0.10 0.71 1.00

Rldhlllooellon,rill

16(1) Extensional stiffness

Figure 16. Final upper and lower level stiffnessdistributions for c=-0.2

24

REPORT DOCUMENTATION PAGE Fo,m ApprovedOM8 No. O7O4O188

_. _F.NCYUS=OMLVtL_ _ =.RSPOWrC_ S. _;,omr WPsJuNOC_T=sCOVEnEO

May 1994 Technical Memorandum4. TITLE ANOSUiTiTLE

MultilevelDecompositionApproach to Integrated Aerodynamic/Dynamic/ 505-63-36-06Structural Optimization of Helicopter Rotor Blades

o. _uJl14o_s)Joanne L. WaishKatherine C. YoungJocelyn I. Pritchard

Howard M. AdelmanWayne R. Mantay

7. mFc_i_i_= _l"m_ N._ur.(s) _ _J_O_ESS(=S)NASA Langley Research Center, Hampton, VA 23681-0001

U.S. Army Vehicle Structures Directorate, ARL,

Langley Research Center, Hampton, VA 23681-0001

U.S. Army Aeroflightdynamics Directorate, ATCOM,

Langley Research Center, Hampton, VA 23681-0001

9. _%Oq_O_lNO I MONITORINGAGENCYNAME(S) ANO ADOREg_ES)

National Aeronautics and Space Administration, Washington, DC 20546-0001U.S. Army Research Laboratory, Adelphi, MD 20783-1145

U.S. Army Aviation and Troop Command, St. Louis, MO 63120-1798

& FUNDING NUMBERS

8. PERFORMING ORGANIZATIONREPORT NUMBER

10. SPONSORING I MONITORING

AGENCY REPORT NUMBER

NASA TM-109084ARL-TR-476

;11. SU_F,i,.EMENTARY NOTES

Presented at AHS Aeromechanics Specialists Conference, San Francisco, CA, January 19-21, 1994.

Walsh, Young, and Adelman, Langley Research Center, Hampton, VA; Pritchard U.S. Army VSD, ARL, LangleyRes. Ctr., Hampton, VA; Mantay, U.S. Army AFDD, ATCOM, Langley Res. Ctr., Hampton, VA.

12_. Di=_-_dNlIONI AVAILAI_UTY STATEMENT _2b. DISTRIBUTION¢OD£

Unclassified - UnlimitedSubject Category- 05

13. AINrrRACT (Mlxknum 200 womV/

This paper describes an integrated aerodynamic/dynamic/structural(lADS) optimizationprocedure for helicopterrotorblades. The procedure combines performance, dynamics, and structuralanalyses with a general purposeoptimizer using multileveldecompositiontechniques. At the upper level, the structureis defined in terms ofglobal quantities (stiflnesses, mass, and average strains). At the lower level, the structure is defined in terms oflocal quantities (detailed dimensionsof the blade structure and stresses).

The lADS procedureprovides an optimizationtechnique that is compatible with industrial design practices inwhich the aerodynamic and dynamic design is performed at a global level and the structuraldesign is carded outat a detailed level with considerable dialogue and compromise among the aerodynamic, dynamic, and structuralgroups. The lADS procedure is demonstrated for several cases.

14. SUBJECT TERMS

rotorblades, hel_pte_, optimization, multidisc_linary design

17. SF.CU_-v CL,Ai,_m--I_ATIONOF REPORT

Unclassified

NSN 7540-01-280-5500

18. SECURITY C_K_.ATION

OF TI.IS PAGE

Unclassified

19. S_CURITY CLASSIFICATIONOF ABSTRACT

Unclassified

llS. NUMBER OF PAGES

25

20. UMrTATION OF ABSTRACT

Standard F¢_m 294 (Rev. 2.49)Pmecd=ed b/ANS_ Std. Z3e-_8296-102