helicopter vibration - an overview

8/13/2019 Helicopter Vibration - an overview

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8 J. AIKCRAFT VOL.28, NO. 1

Helicopter Vibration Reduction Using Structural Optimizationwith Aeroelastic/Multidisciplinary Constraints-A Survey

Peretz P. Friedmann*University of California at LosAngeles, LosAngeles, California90024

This paper presents a survey of the state-of-the-art in the field of structural optimization when applied to

vibration reduction of helicopters in forward flight with aeroelastic and multidisciplinary constraints. Itemphasizes the application of the modern approach where the optimization is formulated as a mathematical

programming problem, the objective function consists of the vibration levels at the hub, and behavior con -

straints are imposed on the blade frequencies and aeroelastic stability margins, as well as on a number of

additional ingredients that can have a significant effect on the overall performance and flight mechanics of the

helicopter. It is shown that the integrated multidisciplinary optimization of rotorcraft offers the potential for

substantial improvements, which can be achieved by careful preliminary design and analysis without requiring

additional hardware such as rotor vibration absorbers or isolation systems.

m

mns

Nomenclature

= cross-sectional dimension, Fig. 2= semichord= thrust coefficient= weight coefficient=W/nQ2R4= vector of design variables= root offset ofblade= lag and flap stiffness, respectively= flap and lag stiffness, respectively, at

= objective function, Eq. (12)= generalized forcing function for ith

blade mode= four per rev harmonic component of

shear in hub plane (x andy direction)and vertical z direction), respectively

blade root

= blade torsional stiffness= cross-sectional dimension, Fig. 2= cross-sectional dimension at blade root

and at tip, respectively= flapping inertia of blade= generalized inertia for ith blade mode= mass polar moment of inertia of rotor= objective function

= initial value of mass polar moment ofinertia of rotor

= weight factors equal to zero or one

= length of elastic part of blade= length defining station, where outboard

blade segments start, Fig. 3= mass per unit length ofblade= nonstructural mass per unit length of

= reference mass per unit length= maximum peak-to-peak value of

oscillatory hub rolling moment= four per rev harmonic component of

pitching, rolling, and yawing moment,respectively

blade

Received Jan. 20, 1990, revision received May 11, 1990, acceptedforpublication May 11, 1990. Copyright 1990by the AmericanInstitute of Aeronautics and Astronautics, Inc. All rights reserved.

*Professor and Chairman, Mechanical, Aerospace, and NuclearEngineering Department. Associate Fellow AIAA.

= maximum value of oscillatorypeak-to-peak vertical hub shear

= blade radius= cross-sectional dimension, Fig. 2= thickness of front and rear vertical wall

of structural box= vibration index for i th mode= weight of helicopter= offset from elastic axis to cross-sectional

aerodynamic center= offset f rom elastic axis to cross-sectional

center of mass= offset from counterweight t o elastic axis= chordwise offset of nonstructural mass

from elastic axis= chordwise offset of c.g. from elastic

axis, baseline configuration= real part of characteristic exponent,

Eq. (14)= precone= viscous modal damping in ith mode= real part of eigenvalue in hover= sweep angle= taper ratio ofblade= advance ratio= blade solidity= allowable normal stress= allowable shear stress= mode shape for ith mode, Eq. (12)= rotor angular velocity= forcing frequency in vibration index,

Eq. (I@, or natural frequency= natural frequency of ith mode= imaginary part of eigenvalue= upper and lower bounds o n natural

frequency, respectively

I. Introduction and Historical PerspectiveIBRATION levels encountered in helicopters have beenV one of the most important criteria for distinguishing

between successful and less successful designs. During the lasttwo decades, at least two major helicopter production con-

tracts have been awarded on the basis of criteria associatedwith vibration levels. During the same period, criteria forvibration levels at typical locations in the fuselage, such as the

pilot seat, have become much more stringent. For example, 15years ago vibration levels in the neighborhood of 0.10 g were


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JANUARY 1991 HELICOPTER VIBRATION REDUCTION 9

considered acceptable, in modern helicopters 0.05 g levels aredesirable, and the requirements for the next generation ofhelicopters will probably be in the 0.02-0.05 g range. It isevident from two excellent reviews dealing with a wide arrayof vibration problems and their control in rotorcraft, written

by Reichert' and Loewy2, that such problems have been fre-quently treated, by passive or active means, after they have

manifested themselves in the flight test phase.Since the principal source of vibrations is the main rotor, ithas been recognized a long time ago that reducing vibrationsat their source is very cost effective. Attempts were made inthe mid-1950s to reduce vibration levels in the rotor by achordwise and spanwise placement of concentrated masses,denoted tuning masses, and to reduce vibration levels in rotor

blades. References 3-6 are representative of such early at-tempts. Much later, as the art of rotor blade design becamemore sophisticated, aerodynamic means such as blade twistand use of swept tips on rotor blades were identified in anexcellent paper by Blackwel17 as a practical means for con-trolling vibrations at their source, Le., the rotor.

Helicopter vibrations are generated by a combination ofaerodynamic, dynamic, and structural sources that form the

basis of any aeroelastic response calculation. However, itshould be emphasized that for a helicopter these three groupsrepresent only a restricted set of disciplines when one seeks the

best possible design.Among all flying vehicles the helicopter is inherently inter-

disciplinary since the flexibility of the blades couples with theaerodynamics and dynamics so as to influence vehicle perfor-mance, flight mechanics, and handling qualities as well asacoustics. Furthermore, in the next generations of helicopters,the disciplines cited above will also couple with the controlsystem, which can and probably will be used for stabilityaugmentation, vibration reduction, and suppression ofpoten-tial aeromechanical instabilities. Finally, it should be notedthat the vibration problem is further complicated by the cou-pling between the rotor and fuselage, which determines the

precise manner in which the vibratory loads from the rotorexcite the fuselage.

The optimum design of a helicopter for minimum vibrationsin forward flight clearly represents a complex structural opti-mization problem with multidisciplinary constraints. Thus,the simultaneous consideration of the disciplines mentioned isessential for the successful design of helicopters. Ideally, the

best designs will be obtained only if vibration levels are consid-ered early in the design process before the numerous designvariables are fixed due to the conflicting considerations intro-duced by the interdisciplinary nature of the problem. Thus,automated structural design techniques have to be appliedearly in the preliminary design process when the largest num-

ber of design variables can be determined based on the simul-

taneous application of multidisciplinary constraints. This isthe only approach that can guarantee substantial benefits andproduce a vibration-free, optimally designed helicopter.

The field of structural optimization, or structural synthesis,has become a practical tool in recent years as a result of threedecades of extensive research and development.8-'0 This is dueto significant advances in computers coupled with a large bodyof research, which has led to more efficient methods using acombination of mathematical programming techniques foroptimization, approximation concepts'' for improved effi-ciency, and the finite element method for structural modeling.Structural optimization has found considerable use in theaerospace industry; however, until recently most of the appli-cations have been for fixed-wing aircraft. 12~13Application offormal structural optimization to helicopter problems started

in the early 1980s, and it generated considerable interest inindustry, academia, and research organizations. A detailedreview of the application of numerical optimization to heli-copter design problems, including structural optimization, can

be found in a paper written by Miura.14 The various tech-niques available for vibration reduction in rotorcraft using

structural optimization were explored and reviewed in consid-erable detail in Ref. 15.

It is interesting to note that while the helicopter communitystarted using structural optimization techniques at least 10years after such studies emerged in the fixed-wing area, thecommunity has been quick to recognize the remarkable pay-offs available, and presently the use of structural optimiza-

tion, with interdisciplinary constraints, for these two types ofvehicles is almost on an equal footing. The primary reason forthe rapid acceptance of structural optimization in the rotary-wing field is due toboth the inherently interdisciplinary natureof the problem and the fact that the high potential payoff forvibration reduction of helicopters in forward flight, usingstructural optimization, has a central role in rotorcraft and isless important in fixed-wing aircraft.

The main objective of this paper is to present the currentstate-of-the-art in the field of structural optimization whenapplied to vibration reduction of helicopters in forward flightwith aeroelastic and multidisciplinary constraints. For conve-nience, the research described here is separated into twogroups. The first group consists of research carried out atuniversities; the second group presents the research performed

in industry and government organizations. This body of re-search enables one to draw useful conclusions on the impor-tant ingredients that need to be incorporated in current andfuture research. Furthermore, it also enables one to makeuseful recommendations on the future developments that aredesirable for the design of the next generation of vibration-free rotorcraft.

11. Research Performed at UniversitiesMost of the research done at universities during the last nine

years has been aimed at the design of blade configurationswhich would have low vibration characteristics in forwardflight. A description of this research in chronological order is

presented next.

The first documented application of optimum structuraldesign to the vibration reduction problem of a helicopter inforward flight while simultaneously enforcing aeroelastic con-straints was presented in Refs. 16-18. A concise description ofthis research is provided below.

The helicopter vibration reduction problem is expressed as ageneral class of structural synthesis problems consisting of

preassigned blade properties and helicopter performanceparameters, design variables, objective functions, behaviorconstraints, and side constraints. This optimum design prob-lem, which is solved using mathematical programming meth-ods, is stated in the following mathematical form.

Find the vector of design variables D such that

and J(D)-min (3)where gq D)is the qth constraint function in terms of thedesign variablesD; i is the ith design variable, superscriptsLand U denote lower and upper bounds, respectively; and J D)is the objective function in terms of the design variables.

The system considered in these studiesI6-'* was a four-bladed hingeless rotor, attached t o a rigid fuselage. The fuse-lage degrees of freedom were not included, and the compli-cated mechanism of vibration transfer to a flexible fuselagewas also excluded. This idealized configuration is shown in

Fig. 1.Quantities defining the helicopter blade configurationsuch as b , el, and xA were treated as preassigned parame-ters. Helicopter performance parameters that define the heli-copter flight condition in trimmed flight are the advance ratiop and the weight coefficient Cw.These were assumed to bespecified parameters describing the configuration.


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10 P. P. FRIEDMANN J. AIRCRAFT

FUSELAGE

Fig. 1 Helicopter rotor fuselage model.

=0

t

LEADING EDGETUNING WEIGHT

h

YO

Fig. 2 Typical blade cross section.

Fig. 3 Spanwise and chordwise location of nonstructural mass (the figure shows the leading-edge location; when x =0, mass is on elastic axis;E. A. and A. C. denote the elastic axis and aerodynamic center, respectively.

The design variables are associated with a typical crosssection of the rotor blade shown in Fig. 2. For each crosssection, the design variables are the breadth bs, the heighth, ,and the thicknesses t b and th of the thin-walled, rectangular

box section representing the structural member at each span-wise station. A total of seven spanwise stations are used for afinite element analysislgof the blade which produces the freevibration mode shapes and frequencies of the rotating blade.Elastic properties of the blade in bending and torsion as well

as the structural mass properties are expressed in terms ofthese design variables. The nonstructural mass of the blade isassumed to consist of two parts. The first part is the nonstruc-tural skin and honeycomb case surrounding the structural cellshown in Fig. 2, which provides the appropriate aerodynamicshape and is assumed to be a fixed percentage of the critical

blade mass. The second contribution to nonstructural mass isrepresented by m, in Fig. 2, which is a counterweight used asa tuning device for controlling blade frequency placement andthe section center of gravity location. The nonstructuralmassesm, at three outboard stations of the blade, see Fig. 3,corresponding to the two outboard finite elements, are alsoused as design variables, and the offsets x from the elasticaxis are given parameters.

Two types of behavior constraints are used. The frequency

constraints are expressed in terms of nondimensional rotatingfrequencies u2 of the blade in the flap, lag, and torsionaldegrees of freedom. These uncoupled rotating frequencies areobtained from a Galerkin-type finite element ana1y~is.I~Thefundamental frequencies in flap, lag, and torsion are requiredto fall between preassigned upper and lower bounds. The


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JANUARY 1991

VIBRA TION ANALYSIS GENERATE FREQUENCYIGFEM

-

6 ELEMENTS1 CONSTRAINTS1

1

GENERATE AEROELASTICCONSTRAINTS I N

HOVER

AEROELA STIC ANAL YSISIN HOVER BY GLOBAL

2MODE SLN.)GALERKIN METHOD -I

AEROELA STIC A NALYSIS GENERATE OBJECTIVE

IN FORWARD FLIGHT BY FUNCTIONMETHOD I2 MODE SLN.)GLOBAL GALERKIN - HUB SHEARS AND/OR -HUB MOMENTS)

HELICOPTER VIBRATION REDUCTION

GENERATE APPROXIMATEPROBLEM

CONSTRAINTS

OBJECTIVE FUNCTI ON

- -

frequency placement constraint is expressed mathematically inthe form:

2g D ) = 7 -1502g D)=l- 7 5 0"L

(4)

Equations (4) and ( 5 ) are written for each of the threefundamental frequencies of the blade in flap, lag, and torsion,

providing a total of six behavior constraints. The higher fre-quencies are constrained to avoid four per rev (4/rev) reso-nances in the four-bladed hingeless rotor considered in thisstudy.

The second type of frequency constraints are placed onaeroelastic stability margins. The aeroelastic constraints repre-sent the requirement that stability margins in hover remainvirtually unchanged during the optimization process. For soft-in-plane blade configurations, (almost all hingeless rotors in

production belong to this category), the effect of forwardflight is usually stabilizing,20therefore, this assumption isquite reasonable. The aeroelastic stability constraints are ex-

pressed mathematically in the form

For hover, the aeroelastic analysis, based on six modes,20produces eigenvalues that occur in complex conjugate pairs

7)The blade is stable when {k


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12 P. P. FRIEDMANN J. AIRCRAFT

-

0.006

4) The mathematical programming problem representedby Eqs. (1-3) is replaced by an approximate problem wherethe constraints g, D) and the objective function J D ) areexpressed by Taylor series approximations. The approximate

problem is solved by the NEWSUMT optimizer to obtain animproved design.

5) The entire optimization process is repeated with theimproved design as starting point until the sequence of vectors

D converges to a solution D* where all inequality constraintsare satisfied and J D*) is at least a local minimum.

It should be noted that due to the excessive computer timeinvolved, the optimization procedure described above was notexercised in a completely automated manner. It was used in aninteractive manner, with manual interruptionsso as to reducecomputing costs associated with generating the substantialamount of numerical results presented in Refs. 16-18. Theseresults have indicated that by applying modern structural opti-mization to the design of soft-in-plane hingeless rotors, vibra-tory hub shears in forward flight can be reduced by 15-40%.The reduction is achieved by relatively small modifications ofthe original design, which yielded optimal frequency place-ment in fundamental flap, lag, and torsion modes. These

percentages represent conservative estimates since the originaldesign was a soft-in-plane blade resembling an MBB 105 hin-geless blade, which is known to be a good design. The best

-- NONLINEAR LINEAR

0.06

w

-I>Y 0.04a

a

W

i,

0.02

w

>2= -0.004;52ksiR -0.002

0.0._0.0 0.1 0.2 0.3 0.4r

Fig. 6 In-plane hub shears, peak-to-peak values, nondimensional-ized Pvll/Q2Zb)vs p comparison on initial and final designs aftertwo stages of optimization, (Do-initial design, DII-design after twostages of optimization).

INBOARD SEGMENT

\ / R O T O R H U BPITCH CHANGE BEARING\

Fig. 7 Swept tip hingeless rotor blade model.

results were obtained when using nonstructural mass, as addi-tional design variables, in the outboard segment of the bladeand placed on the elastic axis. An interesting byproduct ofoptimization were final blade configurations, which were9-20 lighter than the initial uniform blade. This result wasobtained without using blade weight as the objective functionin the optimization process. Finally, it should be noted thatuse of the linear oscillatory hub shears at an advance ratio of

p = 0.30 represents as acceptable the objective function for theoptimum design of soft-in-plane hingeless rotor blades.

From the numerous results presented in Refs. 16-18, twofigures are worthwhile to include here because they have some

basic features that were also noted in other studies reviewedhere. The nonlinear vertical hub shears over the advance ratiorange of O


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The behavior constraints used were similar to the con-straints used in the earlier work. For frequency constraints,Eqs. (4) and (5) were used again. Two types of aeroelasticstability constraints were used. Again, it was assumed that forsoft-in-plane blade configuration, maintaining aeroelastic sta-

bility margins for hover would be adequate. The more strin-gent aeroelastic stability constraint used was

5 0,D)= 1-- k = 1,...,6k0 9 5 l k B (9)which expresses the requirement that the loss of stability in agiven mode should not exceed 5% of the baseline value SjrB. Amore relaxed aeroelastic stability constraint

g D)= ti 5 0, k = 1,2,...,6 (10)which was also used, represents the requirement that the blade

be stable in hover. This more relaxed stability constraint wasfound to be ineffective in the numerical studies.

In addition to the behavior constraints on frequency place-ment and aeroelastic stability margins, a new constraint on

autorotation was also introduced. The autorotation constraintis expressed mathematically as

J

0.9Jog D)= 1 . . . 5 0This constraint, Eq. (ll) , expresses the requirement that themass polar moment of inertia of the rotor J retain at least 90%of its initial value loduring optimization.

The objective function used in this study was the peak-to-peak value of the 4/rev vertical hub shears, corresponding tothe first part of Eq. (8); for the swept tip case, it was foundthat normalizing it by the thrust coefficient, Le., Pzlmax/Cw,isuseful since it compensates for the deficiencies associated withthe trim procedure.

The optimization problem described so far resembles theprevious formulation.16-'sTherefore, it is important t o men-tion the main differences between this and the previousstudy.16-18From the structural optimization point of view, themain differences are associated with the construction of theapproximate problem and the solution of the optimization

problem. In this study it was recognized that the most expen-sive function to evaluate is the objective function. An effectivemethod for building an approximation to the objective func-tion is to use a Taylor Series approximation in terms of thedesign variables"

where F D)is the objective function, Do is the current design,and VF Do)and [H Do)]are respectively the gradient and theHessian matrix at the current design. The Hessian matrix is thematrix of the second partial derivatives of the objection func-tion with respect to the design variables. The vector 6D isdefined as

In Refs. 16-18, finite differences were used for the construc-tion of the derivative information in Eq. (12). This approachled to excessive computing cqsts. In this study an alternativeapproximation technique, introduced by Vanderplaat~,2~,*'was used. This alternative technique is based on the idea ofapproximating the gradient the Hessian in Eq. (12),by usingdesign information available at the time. The details of theimplementation of this technique for the hub shear minimiza-tion problem can be found in Refs. 24 and 25.

The approximate constrained optimization problem wassolved using the feasible direction code CONMIN.28 Theaeroelastic analysis from which the vibratory hub loads and

aeroelastic constraints were obtained is described in Refs. 29

and 30. The trim procedure used was similar to that used in theprevious study.16-18With the exceptions noted here, the basicorganization of the optimization process was similar to thatdescribed in Fig. 4.

The optimization results ~ b t a i n e d ~ ~ ~ ~indicated vibrationreductions between 20-50 . The additional vibration reduc-tion obtained from using the sweep angle as an additionaldesign variable was on the order of 10 . These results can beconsidered to be conservative, since tuning masses were notused as design variables in the optimization. The method usedfor constructing the approximate problem (objective function)was found to be quite efficient. However, it was felt thatbetter approximations could be obtained by using analyticalgradients.

The most recent and to some comprehensive study of vibra-tion reduction in helicopter rotor blades with aeroelastic con-straints has been carried out by Lim and C h ~ p r a . ~ ' . ~ ~Whereasthis study resembles the previous studies cited, i t also containsa number of new and substantial contributions. The optimumdesign problem is represented again by Eqs. (1-3). The config-uration considered is a four-bladed soft-in-plane hingeless

rotor attached to a fuselage of infinite mass, and the preas-signed parameters resemble those used in the previous stud-ies.16-18,24,25The configuration of the airfoil, including a tun-ing mass and the representation of the blade by finiteelements, is illustrated in Fig. 9.

The design variables a t each blade cross section are mm oE I , EI,, and GJ. The blade is represented by five finiteelements, leading to a total of 30 design variables. The linkage

between the stiffness properties and the actual cross-sectionaldimensions is not made in this study.

11

Fig. 8 Single cell and double cell blade cross sections.

1wnsn

X

Fig.9representation of blade finite elements.

Configuration of the airfoil including a tuning mass and


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14 P. P. FR

The behavior constraints consist of the requirement that thefundamental flap, lag, and torsional mode of the blade bestable in forward flight. Mathematically, this constraint isexpressed as

g D) ak 5 0, k = 1,2,3 14)where Yk is the realpart of the characteristic exponent associ-ated with three fundamental modes. Although this constraintappears to be more accurate than the hover constraint used in

previous s t ~ d i e s , ~ - ~ , ~ ~ , ~ ~for the case of a soft-in-plane hinge-less blade, it may actually be a somewhat weak constraint.First, because the soft-in-plane blades are most unstable inhover,20imposing such a constraint in forward flight withoutenforcing a similar constraint in hover can be dangerous.

Next, requiring that the blade be stable is not sufficient; ablade should have a predetermined aeroelastic stability marginthat should not change substantially during the optimization

process.Furthermore, it should be noted that frequency placement

constraints and autorotation constraints, imposed in Refs. 24and 25, are not used in this study. Indeed, the authors impose

the side constraints on blade and mass stiffness properties,which the authors imply will produce favorable frequencyplacement.

Two possible alternatives are considered for the objectivefunction. One is the sum of the 4/rev harmonics of the hubloads in the hub nonrotating frame:

+K ~ ~ M ; ;+K ~ ~ M ~ ; (15)where the hub forces and moments are nondimensionalized bymoQ2R2and moQZR, respectively. A second objective functionis defined as the sum of the hub force resultant and momentresultant in the hub-fixed nonrotating frame:

J D)=K d ( F L ) 2+ (F ,)2 + (F )2

Both objective functions are evaluated for an advance ratio ofp = 0.30, as in Refs. 16-18, 24, and 25. The optimization

problem is solved using the CONMIN code.28Each optimiza-tion iteration involves updating the search direction from thesensitivity analysis and determining the optimum move pa-rameter by polynomial approximation in the one-dimensionalsearch.

An important contribution made in Refs. 31 and 32 consists

of the formulation of a direct analytical approach for thecalculation of the derivatives of the hub loads and bladestability with respect to the design variables. A detailed de-scription of the sensitivity analysis of the vibratory hub loads,using a finite element formulation in space and time, is de-scribed in Refs. 32 and 33. A similar sensitivity analysis for theaeroelastic stability of the blade as represented by the real partof the characteristic exponent was presented in Refs. 32 and34. These sensitivity derivatives are obtained at a fraction ofthe computational cost associated with the more conventionalfinite difference method used in Refs. 16-18. Although the

benefits of using sensitivity analyses in structural optimizationare well k n o ~ n : ~ ? ~ ~this is the first time that such an analysishas been developed for complicated problems such as hubloads and blade stability in forward flight. Another contribu-

tion in the analysis used for this optimization study was incor-poration of a t rim procedure which is fully coupled with theaeroelastic response a n a l y s i ~ . ~ ~ . ~ ~This represents a substantialimprovement on the trim procedures used in the previouss t ~ d i e s , ~ ~ - ~ ~ , ~ ~ ~ ~ ~where the propulsive trim procedure was notfully coupled with the aeroelastic response analysis.

IEDMANN J. AIRCRAFT

The optimization studies c o n d ~ c t e d ~ , ~ ~showed that use ofEq. (16) with KF =K M = 1 is preferable to using Eq. (15) withK F ~= 1 and KFX=KFY = K M X =KMY=K M z = 0. Next, awide range of combinations of cross-sectional design variableswas explored. It was found that a combination of cross-sec-tional design variables consisting of mns,yns,EZy, EZz, and GJrepresents the most effective combination, since it takes ad-vantage of both distribution on nonstructural masses and theirlocation as well as modifications of blade stiffnesses. Theresults obtained after eight iterations, with 25 design variables,are shown in Fig. 10. It is evident from Fig. 10 that a 77%reduction of the objective function was achieved, which repre-sents the largest reduction obtained in this study. For thevarious other cases considered, reductions in the objectivefunction in the range of 20-50 were observed.

The most important conclusions obtained in these studieswere 1) analytical sensitivity derivatives for hub loads andaeroelastic constraints produce large reductions in computertimes and should be used in future studies, and 2) the use ofcombined objective functions such as Eq. (16) is preferable tominimizing vertical hub shear alone.

In addition to the fairly comprehensive studies described

above, there were a number of less ambitious studies con-ducted at a number of universities, where the main objectivewas to explore some fundamental aspects of the structuraloptimizaticn problem. A representative study in this category,Ref. 36, was aimed at application of structural optimization torotor blade frequency placement. This study is based on the

premise that separation of blade flap, lead-lag, and torsionfrequencies from the aerodynamic forcing frequencies, whichoccur at integer multiples of the rotor revolutions per minutewill also reduce vibration in forward flight. The optimumstructural design problem is one in which the mass and stiff-ness distributions are selected by an optimization process,such that the uncoupled flap, lag, and torsional rotating fre-quencies are placed in certain predetermined windows,which are separated from the integer resonances. Aeroelastic

stability constraints were not included in Ref. 26, and directminimization of hub loads resulting from the aeroelastic re-sponse of the blade was not incorporated in this study.

Again the optimization problem is cast in the form repre-sented by Eqs. (1-3). The cross section of the blade model wassimilar to that shown in Fig. 2, except that the nonstructuralmass at each cross section consisted of a tuning mass mnsrshown in Fig. 2, together with a second lumped mass MLinside the structural box (this mass is not shown in Fig. 2). Thedesign variables were the dimensions b,, h and the thicknessesas t b and t h of the thin-walled rectangular box section, repre-senting the structural member, at each spanwise station. Thevibration analysis, from which mode shapes and frequenciesare obtained, is based on a tapered finite element model, using

10 spanwise stations.36The behavior constraints were frequency windows on thefirst and second flapwise bending mode frequencies, expressedas

These constraints are recast in the form of Eqs. (4) and ( 5 ) .Constraints were also imposed on rotor inertia for autorota-tion, in a form similar to Eq. (11). A third constraint on themaximum stresses due to centrifugal loads were also imposed.Side constraints were placed on the design variables to preventthem from reaching impractical values. A constraint was also

placed on the physical dimensions of the structural cell so thatit would fit within the airfoil.

Two objective functions were used: 1) minimize discrepan-cies between desired and actual frequencies, and 2) minimizeweight. The optimization problem was solved using CON-MIN.28The actual helicopter rotor blade optimization studies

- _ -


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were somewhat limited in scope. First, an articulated rotorblade was optimized, and a weight reduction of 26% wasobtained. Next, a teetering rotor blade was optimized, withoutany significant reduction in blade weight.

Peters and Cheng37have continued the research on the bladefrequency placement problem.36A blade resembling the AH-64A blade geometry was investigated, and the principal em-

phasis was on satisfying stress constraints, assuming that theharmonic blade laads are obtained from a flight loads survey.The results obtained indicated that stress constraints based onfatigue life considerations can be added to the frequency

placement problem. However, the authors concluded that thefeasibility of a completely integrated (Le., including additionaldisciplines such as aerodynamic performance) and totally au-tomated design process is still in the distant future.

A very interesting study, which has a truly multidisciplinaryflavor, was recently completed by CeL3* In this study thetorsional stiffness of a hingeless rotor blade and its cross-sec-tional offsets are designedso as to stablize the phugoid oscilla-tion of the helicopter by increasing the stablizing effect of therotor. The optimization problem is again cast in the forms ofEqs. (1-3). The objective function is the real part of a complex

eigenvalue which characterizes the phugoid mode, at p = 0.30.The design variables are blade torsional stiffness GJ and bladecross-sectional offsets xA and x I . The behavior constraints are1) aeroelastic stability constraints in forward flight, same asEq. (14); 2) blade root load constraints, in the rotating system,expressing the requirement that the maximum peak-to-peakflap bending moment and root torsional moment should notincrease by more than 10% from their baseline values; and 3)a cyclic pitch response constraint representing a handling qual-ity type of constraint. The objective function and the behaviorconstraints are approximated by Eqs. (12) and (13), using theapproach described b e f ~ r e . ~ ~ , ~ ~The optimization problem issolved using CONMIN. The results obtained indicated thatapproach is efficient, and phugoid modes can be stabilized.Although this study was not aimed at vibration reduction, it

nevertheless illustrates how handling quality type of con-straints can be incorporated in vibration reduction studies.

111. Research Carried Out in Industry and

Government Research Organizations

The research described in this section is divided into anumber ofparts, each associated with a particular organiza-tion where the research was carried out. An attempt was madeto describe this research in a chronological order.

A. Research at United Technologies Research Center

United Technologies Research Center has traditionally hada longstanding interest in rotor blade design with dynamic

constraints. In 1971, B i e l a ~ a , ~ ~in a pioneering study, consid-ered the minimum weight design problem of a helicopter rotorblade in hover. Behavior constraints were imposed on bend-ing-torsion (flap-pitch) aeroelastic stability as well as on thefrequency placement. The five design variables included thewidth and depth of the spar at the blade tip, chordwise posi-tion of the spar center, chordwise dimension of the counter-weight (tuning mass, Fig. 2), and its chordwise position (at30 span and 70% span). A linear aeroelastic analysis wasused and analytical expressions for the flutter eigenvalue sensi-tivity were obtained. The optimization algorithm was a La-grange multiplier method. The weight reduction achieved fluc-tuated between 3.5-9 of the initial weight of the blade. Theimportant contribution of this study was the recognition of theimportance of structural synthesis in blade design, almost a

decade before the topic became popular, and also the first useof a sensitivity analysis for this class of problems. However,this study did not address the vibration reduction problemdirectly.

Subsequently, Taylora considered the vibration reductionproblem in rotor blades in forward flight using modal shap-

OPTIMUM HUB LOADS

.5

0 .4-XvTI0.3

-3 .2n

a

3Id

n

F: F F M MFig. 10 Optimum 4/rev hub forces and moments for p = 0.30, CT/u = 0.07; Nb =4 (Ref. 31); hub forces and moments are nondimen-sionalized with resect to m@R* and moflZR3,respectively.ing. This approach is based on the idea of reducing vibration

levels by modifying the mass distribution and, to a lesserextent, the stiffness distribution of the blade, in order toreduce a so-called model shaping parameter (MSP). TheMSP was based on the flapwise modes, and it was not afunction of the mode natural frequency or the forcing fre-quency. However, MSP could be used to reduce the sensitivityof the blade modes to selected harmonics of the air load andthus can be interpreted as an ad hoc type of optimality crite-rion. Strictly speaking, Taylors study is not an optimumdesign approach; however, MSP can be easily incorporated ina blade structural optimization study by using it as an objec-tive or constraint function.

Building upon this foundation, some of the most importantcontributions to blade design optimization have been mademore recently by Davis and Weller.41-43In the firstfour different problems representative of a hierarchy of opti-mization problems, from relatively simple to complex, wereconsidered. The analytical studies dealt with four specific

problems, namely: 1) maximization of bearingless rotor struc-tural damping; 2) blade natural frequency placement; 3) mini-mization of hub modal shears; and 4) minimization of modalvibration indices. Only the last three topics, which are withinthe scope of this survey will be discussed.

The three problems are cast in the standard mathematicalprogramming form represented by Eqs. (1-3). The optimiza-tion problem is solved using the automated design synthesis(ADS) package,44and various options of the program wereexercised to determine which optimization algorithm providedthe best performance. The coupled free vibration modal anal-

ysis of the rotating blades was based on a Holzer-Myklestadmethod.41The most important aspects of these three structuraloptimization problems are briefly reviewed below.

Blade Natural Frequency Placement

The design variables used were EI,, E I , , and blade mass, at11 spanwise sections resulting in 33 design variables. For anarticulated rotor, the first three elastic flapwise modes and thesecond chordwise mode were placed. A constraint of f 5%was placed on total blade weight and rotor inertia changes. Amulticomponent objective function

J D)= J k l ~ l -0 + k 2 ~ 2- ~+ k3 03- ) ~ k 4 ~ 4- ) ~(17)

was used to produce frequency placement. Other alternativesto Eq. (17) were also explored; however, overall this objectivefunction appeared to be the most effective.

Minimization ofHubModal Shears

The design variables and constraints were similar to theprevious problem. Modal hub shears for three critical modes


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J. AIRCRAFT18 P. P. FRIEDMANN

B. Research at Bell Helicopter Company

The application of formal mathematical programming tohelicopter design problems was initiated at Bell in the early1980sby Bennett.45Among the various examples ~ o n s i d e r e d , ~ ~a simple problem representing minimization of vertical hubshears due to blade flapping was treated using formal opti-mization. The design variables were the mass and stiffnessproperties of the blade at 15 radial stations. The objectivefunction consisted of a mode shape parameter, somewhatsimilar to the modal shaping parameter developed by Taylor 40A Myklestad program was used to calculate the coupled natu-ral frequencies and mode shapes of the blade. Equality con-straints, requiring that blade weight and flapping inertia re-main fixed during the optimization process, were imposed.The optimization problem was solved using the OPT program.As a result of the optimization, vertical shear was reduced by60%.

Additional research on this topic was described by Yen.46He showed that significant reduction in oscillatory hub loadscould be achieved by structural optimization of rotor blades.Yen emphasized the important role of the interaction betweenrotor structural properties and unsteady aerodynamic loads

for minimizing hub loads. The analysis used simplified, linearrotor dynamic equations. A discussion of limited correlationstudies among theory, wind-tunnel, and flight tests was alsopresented.

C. Research at McDonnell Douglas Helicopter CompanyThe application of numerical optimization methods to heli-

copter design problems at the McDonnell Douglas HelicopterCompany (MDHC) is described in a recent paper by Banerjeeand Shan t hak~m ar an . ~~Structural optimization has been ap-

plied to three separate problems: 1) aeroelastic tailoring ofrotor blades; 2) structural optimization of the fuselage; and 3)optimal design of advanced composite rotating flex beam^.^^ Abrief description of these topics is provided next.

The studies on aeroelastic tailoring of rotor blades47werebased on combining a Rotor/Airframe Comprehensive Aeroe-lastic Program (RACAP) with an optimizer such as ADS orCONMIN. The RACAP program is based on a coupled flap-lag-torsional model of the blades using a transfer matrix ap-p r o a ~ h . ~ ' . ~ ~The rotor combined via a six degree of freedomcomplex hub impedance model to a NASTRAN finite elementmodel of the fuselage. An iterative trim, air loads, and aeroe-lastic approach is used to obtain a response solution. Theoptimization problem is formulated using Eqs. (1-3). Thedesign variables are EZy EZz GJ, and the massm along theblade. Side constraints imposed on blade properties are de-scribed in Ref. 47. The behavior constraints considered wereupper and lower bounds on blade moment of inertia to satisfyautorotation requirements. The objective function minimized

was the blade response. Frequency constraints were used onlyin a selective manner. The constraints and the objective func-tions were approximated using second-order Taylor series ex-pansion, Eq. (12), and the gradient information was generatedby finite differences. Surprisingly low, 5-10 , reductions invibration levels were obtained.

The structural optimization of the fuselage was based on adetailed NASTRAN model, with considerable design variablelinkage.47The weight was minimized while requiring accelera-tion levels below desired values at specific locations in thefuselage. The lowest fuselage modes were also constrained to

be within specified frequency bounds. In addition, the dy-namic displacements at selected locations and the elementdynamic stresses were constrained to be below specified upperbound values.

A detailed study of optimal design of an advanced/compos-ite flexbeam for bearingless rotor applications is presented inRef. 48. The flexbeam configuration optimized is shown inFig. 15. The optimization is accomplished by combining afinite element analysis model for a multiple load pathcantilevered beam under centrifugal loads called HUBFLEX

with the ADS optimizer. The design variables are b and tshown in Fig. 15, for each element of the flexbeam. Theconstraints are stress constraints

~yyr/ llow I: 1 o; ~ y r / ~ a I l o w 1where uy is the normal stress and T ~ .is the shear stress. Theobjective function was to minimize the peak value of the

combined stresses along the flexbeam for conditions of limit,maximum fatigue, and endurance loads. An alternative objec-tive was to maximize the first in-plane damping represented bydamper motion per degree lag while constraining stresseswithin allowable limits. Side constraints were also imposed oncross-sectional dimensions and effective virtual flap and laghinge offsets. The principal accomplishment of this researchwas to reduce the time required to design an optimal configu-ration from six months to one week, as a result of the auto-mated design procedure.

D. Other Industrial Research

A few additional studies are briefly mentioned here for thesake of completeness. These represent preliminary attempts toapply structural optimization to problems that fall within thescope of this survey. Reference 51 represents an early attemptto design the flexbeam of a bearingless rotor so as to minimizecombinations of bending and axial stresses for a given oscilla-tory load. In Ref. 52, Perley describes the use of regressionanalysis as a design optimization tool; among the variousapplications, the vibration reduction problem is also brieflydiscussed.

E. Research at NASA Langley Research Center

A very substantial research activity in the field of structuraloptimization of rotor blades has been nurtured under theauspices of the Interdisciplinary Research Office and theAerostructures Directorate USAARTA-AVSCOM at NASALangley Research Center. It should be emphasized that this

activity is quite broad and extends beyond the topics discussedin this paper. The most important facets of this researchactivity, which are aimed at vibration reduction in helicopters,can be found in Refs. 53-55.

In Ref. 53 the authors address the problem of finding theoptimum combination of tuning masses on a rotor blade so asto minimize blade root vertical shear, while avoiding excessivemass penalty. The design variables in this study are the tuningmasses and their spanwise locations along the elastic axis of anidealized beam, which is assumed to represent the blade. Theblade is represented by a simple finite element model of arotating beam and only flapping motion is considered. Fre-quency constraints similar to Eqs. (4) and (5 ) are imposed onthe first two flapwise modes. Three different objective func-

tions are developed and tested. The optimization problem wassolved using CONMIN. The first objective function minimizesthe MSP associated with the two modes, multiplied by thetotal sum of the tuning masses. The second objective functionreduces the shears, multiplied by the total sum of the tuningmasses. The third minimizes the shear as a function of time,multipled by the sum of the tuning masses. Sensitivity deriva-tives are obtained analytically for the performance parametersand constraints to facilitate optimization. The first objectivefunction was found to be ineffective when dealing with Torethan one mode. The other two objective functions proved tobe effective for the multiple mode and multiple load case andproduced excellent shear reduction with a low mass penalty.This study represents a useful fundamental contribution tooptimizing rotating beams under prescribed dynamic loading

conditions; however, it is yet to be applied to represent apractical helicopter vibration problem.

Reference 54 describes the minimum weight design of heli-copter rotor blades with constraints on flap and lag frequen-cies; autorotation and stresses due to centrifugal loads wereconsidered. This study represents essentially an improvement


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on Ref. 35 and is based on a similar blade model. The typicalcross section is similar to Fig. 2, except that the tuning mass isreplaced by a lumped mass inside the structural box, on theline of shear centers (elastic axis), and the thickness of thevertical wall of the structural box th (see Fig. 2) can have avalue thl in the front and th2 at the rear (or trailing edge). Thestructural box can also have linear taper, and for this case, theheight at the root h, and the taper ratio h = h,/h,, are addi-tional design variables. The objective function is a combina-tion of structural weight and the nonstructural weight. Theautorotation constraint is similar to Eq. (1 l), the frequencyconstraints are similar to Eqs. (4) and (5). Frequency con-straints are placed on five frequencies associated with first twoelastic flap and the first three elastic lag modes. The rotor isassumed to consist of four articulated blades. The blade isassumed to be torsionally rigid; the torsional stiffness is repre-sented by a root spring, simulating control system stiffness.The stress constraints consist simply of the requirement thatthe stress due to centrifugal forces not exceed a given value inany segment of the structural box representing the blade. The

blade was divided into 10 segments, and the mode shapes andfrequencies were calculated using the free vibration module ofthe CAMRAD code.56First-order Taylor series are used togenerate approximations to the objective function and theconstraints. A sensitivity analysis is used to obtain analyticalderivatives of the objective function, the autorotational, andthe stress constraints. A central finite difference scheme isused to generate the derivatives of the frequency constraints.The CONMIN program is used to solve the optimizationproblem.

Optimization results for both a rectangular rotor and ta-pered rotor blade were obtained. For each blade three separatecases were considered: 1) all five frequency constraints, au-torotation and stress constraints are imposed; 2) all five fre-quency constraints and autorotation constraints are used; and3) frequency constraints on two flap modes only and autorota-tional constraints are used. The weight savings for cases 1through 3 were 2.67, 3.04, and 8.50%, respectively, for theuniform blade. Somewhat higher numbers were obtained forthe tapered blade. These results indicate that as additionalconstraints are imposed, the weight savings become fairlymodest.

A much more ambitious extension of this research is pre-sented in Ref. 55, which describes a study where integratedaerodynamic/dynamic optimization was performed so that

blade weight and 4/rev vertical hub shear in forward flightwere minimized simultaneously for a four-bladed articulatedrotor. The rotor blade model, shown in Fig. 16,represents amodified Black Hawk blade developed for wind-tunnel test-ing. The design variables are EZxr, EZ ,,, h c , k , and wJ atseven spanwise stations. Three different objective functionsare used: 1) blade weight consisting of the sum of structuraland nonstructural weight; 2) 4/rev vertical shear F, (at anadvance ratio p = 0.30); and 3) weight and 4/rev vertical shearare minimized simultaneously using a global criteria ap-proach.ss The behavior constraints imposed are similar tothose in the previous except that only constraints onthe first four flexible frequencies are introduced. The analysisis based on the CAMRAD code,s6 which is used to trim the

Ch=C t

SectionA-A

X

Fig. 16 Simplified rotor blade model with linear t a ~ e r . 5 ~

Preassignedparameters variables

currentdeslgnvarlabksJ

-*hytructuralanalysis anatysls

Fig. 17 Basic organization of the optimization p r o c e d ~ r e . ~ ~rotor, calculate the 4/rev shears, and compute the modeshapes and frequencies. The approximation for the objectivefunction and constraints is similar to the previous Thegradient information for hub shears is obtained by finite dif-ferences. The basic organization of the optimization proce-dure is shown in Fig. 17.The optimization problem is solved

using the CONMIN code.Three sample optimization test cases were used for thismodel rotor. For the first case, the weight alone is used as anobjective function. This leads to a 7.5% reduction in bladeweight and reduction of 40.6% in 4/rev vertical shear. In thesecond case, the 4/rev hub shear alone is minimized, whichleads to a 85.6% reduction in hub shear, accompanied by an8.5% reduction in blade weight. For the third case, the objec-tive function represents a simultaneous minimization ofbladeweight and hub shear using the global criteria approach. Forthis case a 77.6% reduction in hub shears and a 10.6% reduc-tion in blade weight is obtained. This paper contains numer-ous results together with a careful interpretation of the results.It is also shown that the power requirement of the rotor isr e i x e d as a byproduct of optimization.

IV. Concluding Remarks

Based on the detailed survey presented in this paper, onecan assess the state-of-the-art in the field of vibration reduc-tion in helicopters using structural optimization with aeroelas-tic and multidisciplinary constraints. One can also try to pre-dict some of the future developments in this field.

Although activity in this field started at least 10years aftersimilar endeavors were initiated in the fixed-wing area, thelevel of activity devoted to rotary-wing vehicles has advancedrapidly, and the level of sophistication evident in recent stud-ies is comparable to the fixed-wing field. In one particulararea, Le., experiment verification of optimization designbased on wind-tunnel testing of dynamically scaled modelrotor^,^^,^^ the rotor-wing research has surpassed the workdone on fixed-wing vehicles, since comparable tests have notbeen reported in the literature. This r e s e a r ~ h ~ ~ , ~ ~has also clear-ly demonstrated that frequency placement alone is not aneffective method for reducing vibration levels in forwardfight.

It is evident that useful vibration reduction studies in for-ward flight need to include the hub shears and, for hingelessrotors, even the hub overturning moments in the objectivefunction. Behavior constraints need to be imposed on fre-quency placement, rotor inertia for autorotation, stresses dueto centrifugal effects, and aeroelastic stability margins. Stud-ies that have included most of these constraints and are repre-sentative of the current state-of-the-art include Refs. 16-18,24, 25, 31, 32, 41, 43, and 55. The range of 4/rev hub shearreduction, achieved in four-bladed rotors, evident from these

studies varies between 20-75qo. The higher end of the range(75%) can be achieved only by using nonstructural tuningmasses in addition to modifications in the mass and stiffnessdistribution of the blade. Thus, use of such tuning masses


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20 P. P. FR

plays an important role in this class of problems. Further-more, in a number of studies, weight reductions on the orderof 7-17 were obtained as a result of optimization, and it isinteresting to note that such weight reductions were obtainedeven without blade weight as the objective function.

In most studies where 4/rev hub shears were used in theobjective function, the hub shears were calculated at a partic-ular value of the advance ratio, usually p=0.30. This ap-proach, first used in Ref. 16, was adopted in other stud-ies,,ZS,31,32,47,ssand it produces fairly uniform hub shearreduction in the whole range of advance ratios for straightblades.

Two approaches for minimizing vibration levels haveemerged. The first approach is based on actually calculatingthe 4/rev hub shears from an aeroelastic response analy-sis,16-18~24~2s~47~ssincluding a trim calculation. This approach,which produces uniform reduction in hub shear throughoutthe whole range of advance ratios, is analytically more compli-cated and computationally much more expensive than thesecond approach. It also significantly increases the complexityof the structural optimization process. The second alternativeis based on using modal shaping parameters, modal shear, or

vibration i n d i ~ e s . ~ ~ - ~ ~ , ~ ~ , ~ ~Among these the vibration indexused in Refs. 41-43 appears to be promising. This secondapproach reduces the complexity of the calculations, since anaeroelastic response calculation is circumvented. However, itis important to note that the vibration indices are single-valuedand cannot reflect a variation in air loading or response sensi-tivity due to changes in operation conditions. This is perhapsthe reason that the actual measured vibration reductions inRefs. 42 and 43 exceed the predicted values, which were usu-ally found to be conservative. It is possible that calculation ofthe vibration indices was based on an assumed air load distri-bution corresponding to a fairly high advance ratio; therefore,the more impressive vibration reductions, shown in Figs. 11,13, and 14, were evident mainly at high advance ratios,0.30


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helicopter vibration - an overview

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