An integrated, multiobjective optimization procedure is developed for the design of high speed proprotors with the coupling of aerodynamic, dynamic, aeroelastic, and structural criteria. The objectives are to maximize propulsive efficiency in high speed cruise and rotor figure of merit in hover. Constraints are imposed on rotor blade aeroelastic stability in cruise and on total blade weight. Two different multiobjective formulation procedures, the Min summation of beta and the K-S function approaches are used to formulate the two-objective optimization problems

Chattopadhyay, Aditi

Mccarthy, Thomas R.

English

NASA Technical Reports Server

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DESIGN OF HIGH SPEED PROPROTORS USING
MULTIOBJECTIVE OPTIMIZATION TECHNIQUES
by
Thomas R. McCarthy
and
Aditi Chattopadhyay
Department of Aerospace and Mechanical Engineering
Arizona State University
Tempe, Arizona 85287 - 6106
Proposed Abstract for
1993 AIAA/AHS/ASEE Aerospace Design Conference
February 15-18, 1993
Irvine, California
https://ntrs.nasa.gov/search.jsp?R=19930008876 2020-03-17T09:14:52+00:00Z
Design of High Speed Proprotors Using Multiobjective
Optimization Techniques
by
Thomas R. McCarthyt
and
Aditi Chattopadhyaytt
Department of Aerospace and Mechanical Engineering
Arizona State University
Tempe, Arizona
Introduction
Over the last few years, there has been a revival of interest in VTOL aircraft capable
of operating in fixed wing as well as rotary wing mode. High speed rotorcraft designs,
such as the tilting rotor configuration, pose an entirely new problem in the rotary wing
field. The design goals for this class of aircraft include low downwash velocity in hover,
good low speed maneuverability and cruise speeds of 350 - 500 knots 1. Several new
concepts 2-5 have recently been proposed to meet these design goals. Extensive research
performed in this field have led to the XV-15 research aircraft and ultimately to the
production of the V-22 Osprey tilting rotor for the US Navy.
The combined requirements of efficient high speed performance of a fixed wing
aircraft and good helicopter-like hover characteristics complicates the design process of
tilting high speed proprotor aircraft. It is necessary to maintain good aerodynamic
efficiency in high speed axial flight without degrading hover efficiency. This often leads
to conflicting design requirements. For example, improved efficiency in high speed
"tGraduate Research Assistant, Member AIAA. AHS
tt Associate Professor, Member AIAA, AHS, ASME
2cruisedemandshighdragdivergenceMachnumberswhicharenormally associatedwith
thin airfoils. This however, reducesthe hover figure of merit by reducing CT/a.
Therefore, to maintain the required thrust ceiling in hover, the rotor solidity has to
increase. Also as the forward speedincreases,helical tip Mach number limitations
causedby whirl flutter, requirea reductionin therotor rotationalvelocity. Introducing
blade sweep can alleviate this problem by reducing the effective chordwise Mach
number,which allowsfor higherspeeds,without reducingtherotorRPM. Thereforethe
proper design of proprotor bladescapable of achieving the design objectives must
considerthe right combinationof airfoil thicknessandbladesweepin addition to other
aerodynamicvariablessuchasplanformandtwist.
Severalstudieshavebeenperformed6-9to studydesigntradeoffs betweenthe two
flight modes. For example, Johnson et al.6 performed a detailed study on the
performance,maneuverability and stability of high speedtilting proprotor aircraft,
including the XV-15 andV-22. Liu andMcVeigh7 recently studiedthe useof highly
sweptrotorbladesfor highspeedtilt rotoruse. However,formaloptimizationtechniques
werenot usedin thesestudies. Recentlyaneffort was initiated by Chattopadhyayand
Narayan8,9to developformal multidisciplinaryoptimizationproceduresfor thedesignof
civil high speedtilting proprotor bIades. The propulsiveefficiency in axial flight was
maximizedwith constraintson thefigure of merit in hover,aeroelasticstability in cruise
andotheraerodynamicandstructuraldesigncriteria. The purposeof thepresentpaperis
to formulatetheoptimum designproblemof high speedproprotorsusingmultiobjective
optimizationtechniqueswith the integrationof thenecessarydisciplines.
Problem Definition
A integrated,multiobjective optimizationprocedureis developedfor thedesignof
high speedproprotors with the coupling of aerodynamic,dynamic, aeroelastic and
¢ •
structural criteria. The objectives are to maximize propulsive efficiency in high speed
cruise and rotor figure of merit in hover. Constraints are imposed on rotor blade
aeroelastic stability in cruise and on total blade weight. Two different multiobjective
formulation procedures, the Min ZI3 and the K-S function approaches are used to
formulate the two-objective optimization problem. :_ _ r
Aerodynamic Model
The rotor studied is a wind tunnel model an existing advanced technology proprotor,
which is a three bladed rotor with a rigid hub. Cubic variations are assumed for the chord
(c) and twist (0) distributions to model the blade aerodynamics,
c@) = Co + Cl_- 0.75) + c2@- 0.75) 2 + c3(.g' - 0.75) 3 (1)
0(.9) = Oo + 01(.9 - 0.75) + 02(.9- 0.75) 2 + 03(5' - 0.75) 3 (2)
where _ denotes the nondimensional blade radius. Note that Co represents the chord and
0o the twist at the 75 percent blade radius, respectively. A quadratic lifting line is used
and is defined as follows.
x = f(y) =Ely + 82y 2 (3)
where el, £2 are constants that determine the curvature, and are defined such that
I Ei I < _i (4)
where _i (i = 1, 2) are prescribed bound for the curvature parameters. These bounds
allow for either forward or backward in-plane curvature. When _1 and e2 are equal to
zero the lifting line will be a straight line. The blade sweep, based upon this lifting line
distribution, assumes the following form
4180tan-l(dx" ]
= -7-
= 180tan-l(el +2¢2Y ) (5)
where A(9) is the sweep distribution, in degrees, defined to be positive aft of the straight
lifting line.
Structural Model
The structural model used for the problem consists of a simple two-celled box beam
as shown in Fig. I. The beam is considered to be the principal load carrying member of
the proprotor and the stiffness contributions from the honeycomb and the nonstructural or
tuning weights are placed at the blade tip and are distributed along the planform. The
total blade weight comprises the weight of the box beam, the skin, the honeycomb and
the nonstructural weights. The wall thicknesses of the box beam are assumed to vary in
proportion to the chord distribution.
ts.._ _. _ nonsTctural weight
honeycomb
c(y)
Figure 1 Double-celled box beam configuration
Optimization
Objective Functions: The multiobjective optimization procedure is used to
simultaneously maximize the rotor propulsive efficiency in high speed cruise and the
hover figure of merit.
Design Variables: Both aerodynamic and structural design variables are used. The
aerodynamic design variables include chord, twist and sweep distribution coefficients
(Eqns. 1, 2 and 5). The structural design variables comprise roots values of the wall
thicknesses of the two-cell box beam and the magnitudes of the nonstructural weights, at
the tip, distributed spanwise.
Constraints: To avoid the possible occurrences of air and/or ground resonance associated
with a soft inplane rotor, it is important to maintain the value of the lowest natural
frequency in hover, fl, above I/rev. Therefore the following constraint is imposed.
(i) fl > 1/rev (6)
It is important to impose aeroelastic stability constraints to prevent any degradation of the
rotor stability in high speed cruise. This is all the more important when the blade mass
and stiffness are altered during optimization. The stability constraints are expressed as
follows.
(ii) C_k <- -'Ok k = 1, 2 ..... K (7)
Where K represents the total number of modes considered, and c_k is the real part of the
stability root. The quantity ',)k denotes the minimum allowable blade damping and is
defined to be a small positive number. To avoid incorporation of weight penalties, after
optimization, the total blade weight is constrained as follows.
(iv) w _<wu (8)
6Multiobjective Optimization
A typical optimization problem involving multiple objective functions can be
mathematically posed as follows.
Minimize Fk(_bn)
Subject to
constraints)
k = 1, 2 ..... NOBJ
n= 1,2 ..... NDV
(objEctive functions)
gj(¢n) -<0 j = I, 2 ..... NCON (inequality
enL < qbn < ¢nu (side constraints)
where NOBJ denotes the number of objective functions, NDV is the number of design
variables and NCON is the total number of constraints. The subscripts L and U denote
lower and upper bounds, respectively, on the design variable qbn. A description of the
multicriteria design objective formulation follows.
This study examines three multiobjective function formulation techniques that are
less judgmental than the Pareto based weighting factors and are therefore more suited to
large scale, highly nonlinear optimization problems that are associated with rotary wing
design. The two multiobjective function techniques used are the Minimum Sum Beta
(Min Z_3) and the Kreisselmeier-Steinhauser (K-S) function approaches. A description of
these methods follows.
Minimum Sum Beta (Min Z]3): This method was first used by Weller at al. 10 to
formulate a two objective function rotor vibration problem. Using these technique,
pseudo-design variables that represent tolerances of the individual objective functions
from prescribed tolerances are introduced. The objective function, F:I(_,), is then defined
7asa linearcombinationof thesetolerancesof eachobjective function to their specified
targetvaluesasfollows
NOBJ
FI(_) = 213 k (9)
k=l
where 13k are pseudo design variables with properties such that the original objective
functions Fk remain within a 13ktolerance of some prescribed values. This requirement
introduces new constraints of the following form.
Fk- IZ__k_< 13k k = 1, 2 ..... NOBJ (10)
The quantities Fk are the prescribed target values of the individual objective functions Fk.
Using the above formulation, as the objective function, and correspondingly the values of
13k, are reduced to zero, the values of the individual objective functions Fk are driven to
their prescribed values, Fk. The design variables for the Min 2;13 formulation comprise
the original set of design variables and the pseudo design variables, 13k.
Kreisselmeier-$teinhauser (K-S) function: This technique was first utilized by Sobieski
et al.ll at the NASA Langley Research Center. The first step in formulating the objective
function in this approach involves transformation of the original objective functions into
reduced objective functions 12. These reduced objective functions take the form
* Fk(q_)
Fk(_) = Fk ° - 1.0-gmax -< 0 k=l ..... NOBJ (11)
where Fko represents the value of Fk calculated at the beginning of each iteration. The
quantity gmax is the value of the largest constraint corresponding to the design variable
vector _ and is held to be constant for each iteration. These reduced objective functions
8areanalogousto thepreviousconstraints, and therefore a new constraint vector g2m(_)
(m = 1, 2, .... M) is introduced, where again M = NCON + NOBJ. The new objective
function to be minimized is then defined, using the K-S function as follows:
M
F2((I)) = fmax +_ln 2e p(gm((l))-fmax)
m=|
(12)
where fmax is the largest constraint corresponding to the new constraint vector, g2m((I)),
and in general is not equal to gmax. The multiplier p can be considered analogous to a
draw-down factor where p controls the distance from the surface of the K-S objective
function to the surface of the maximum constraint function. When 19is large the K-S
function will closely follow the surface of the largest constraint function. When 19is
small, the K-S function will include contributions from all violated constraints. The
design variable vector • is identical to that used in the Min 2_ approach.
Analysis
Dynamic. Aerodynamic and Aeroel_a,stic Analyses The aerodynamic, dynamic and
aeroelastic analysis of the high speed proprotor is performed using the code
CAMRAD/JA 13. The code has the capability of analyzing both helicopter and tilting
rotor aircraft. Wind tunnel trim options are used as the reference blade is a wind tunnel
blade model. In cruise, the blade is trimmed to specific rotor lift and drag coefficients
using the rotor collective and cyclic pitch angles. A prescribed wake model, as
implemented in CAMRAD/JA, is used to model the aerodynamics in hover and the rotor
is trimmed to a specific value of the coefficient of power. In axial flight, the components
of the induced velocity are negligible compared to the high forward speed of the rotor.
Therefore, uniform inflow conditions are used to model the aerodynamics in this case.
9The aeroelasticstability analysis in cruise is analyzed with a constant coefficient
approximation.
Structural Analysis: The detailed structural analysis of the rotor blade is performed
based upon the two-celled trapezoidal box beam using an inhouse code that was recently
developed specifically for this application.
Optimization Implementation
The optimization is performed by using the program CONMIN 14. The program uses
the method of feasible directions to solve nonlinear constrained optimization problems.
To reduce the computational effort, an approximate analysis technique is used to compute
the objective function and constraint values during iterations within the optimizer. For
this problem the two-point exponential hybrid approximation technique 15 is used. This
technique takes its name from the fact that the exponent used in the expansion is based
upon gradient information from the previous design point. The technique is formulated
as follows.
NDV
 xr[f0n1po
_(4)) = F((I)o) _'dL['_b°n ) -1.0] Pn 0@n (13)
rl=l
where
10
looJL ;[
I,L )J
Pn = (D¢ln _ + 1.0 (14)
The quantity O1 refers to the design variable vector from the previous iteration and the
quantity O0 denotes the current design vector. The exponent Pn can be considered as a
"goodness of fit" parameter, which explicitly determines the trade-offs between
traditional and reciprocal Taylor series based expansions (also known as a hybrid
approximation technique). Details of this method can be found in Ref. 15.
Results and Discussions
A wind tunnel model of an existing high speed proprotor is used as a baseline design.
The optimization for this problem is performed with a cruise velocity of 400 knots and a
rotational velocity of if2 = 375 RPM (tip speed of 491 ft/s) in axial flight. The operating
condition is 20,000 feet above sea level. In hover, a rotational velocity of f_ = 570 RPM
(tip speed of 746 ft/s) is used at sea-level conditions. The high forward flight speed of
400 knots represents the target cruise value for high-speed rotorcraft. The tip speed is
reduced in the airplane mode so that the helical tip Mach number stays below Mdd (the
drag divergence Mach number). The rotor RPM in cruise (375) is selected after
performing a parametric study on the effect of forward speed and rotor RPM on
propulsive efficiency. A value of CT/CY= 0.08 is used to trim the blade in forward flight,
and a value of Cp/cy = 0.0131 is used to trim the blade in hover. The blade radius is 12.5
11
feet, and the blade is discretized into 10 segments(NSEG = 10). For the Min _13
approach24 designvariablesareused(including thepseudo-designvariables),andin the
K-S functionapproach22designvariablesareused.
Someresults,obtainedto date,aresummarizedin Table 1 andFigs2 and 3. Table 1
presentsa summaryof preliminaryoptimizationresults. From Table 1 andFig. 2 it can
be seenthatit is possibleto obtainsubstantialincreasesin boththehoverfigure of merit,
(21.7- 28.8percent),and theaxialpropulsiveefficiency, flax (24.6- 41.3percent)using
both multiobjectiveformulation techniques.It is of interestto note that themeanchord
(andcorrespondinglythebladesolidity) is increasedby 71percentand40percentin the
K-S functionandMin E_approaches,respectivelyfrom thebaselinevalue. Two possible
explanationsexistfor this largeincreasesin therotor solidity. First, in orderto satisfythe
frequencyconstraint, the root chord is significantly increasedto make the stiffnesses
larger,which in turn increasesthesolidity (seeFig. 3). Secondly,sincethehover figure
of merit is being maximized,a is being increasedto increasethe thrust margin of the
rotor in hover.
Based on the previous experience,the above problem will be formulated with
constraints on rotor solidity. The final paper will presentresults of the integrated
aerodynamic/dynamic/aeroelasticoptimization problemof high speedproprotors with
additionaldesignconstraints.Designtrade-offstudieswill alsobeperformedby varying
the flight conditions and the resultsof correspondingoptimum bladedesignswill be
presented.
Table 1 Summaryof PreliminaryOptimizationResults
12
Reference Bounds Optimum
blade lower upper Min E_ K-S
Objective
Functions
FOM 0.662 0.853 0.936
0.647 0.787 0.805
flax
Constraints
W (lb) 194 194 167 173
fl (per rev) 0.812 1.00 1.01 1.34
or1 0.096 - -0.001 -0.040 -1.529
a2 0.096 - -0.001 -0.040 -1.529
-0.697 - -0.001 -0.732 -0.169
or3
-0.697 - -0.001 -0.732 -0.169
-2.431 - -0.001 -2.443 -2.5020c5
Or6 -0.170 - -0.001 -0.265 -0.073
131 0.150 0.001 0.200 0.006
132 0.150 0.001 0.200 0.010
Mean chord
Ce (ft) 1.48 2.07 2.52
Solidity
0.113 0.158 0.193
Trim
CT/CY 0.110 0.117 0.116
13
1.0
0.8
0.6
0.4
0.2
0.0
41.3%
[] Reference
[] M_n r.13
[] K-S
28.8%
FOM (hover)
21.7%
1"1(axial)
Figure 2 Comparison of individual objective functions
24.6%
14
0.3
0.2
O
..m
¢II
e-
,m
0.1
,m
Z
I
0.2
Reference
"'*-- Min _;13
' ! ', !
0.4 0.6
Nondimensional radius, _,
Figure 3 Chord distribution
! ' 7
0.8 1.0
15
Acknowledgements
This work was accomplished under a grant from NASA Ames Research Center, Grant
number NAG2-771, Technical Monitor, Mr. John Madden.
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16


Design of high speed proprotors using multiobjective optimization techniques

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