This paper describes a joint activity involving NASA and Army researchers at the NASA Langley Research Center to develop optimization procedures aimed at improving the rotor blade design process by integrating appropriate disciplines and accounting for all of the important interactions among the disciplines. The disciplines involved include rotor aerodynamics, rotor dynamics, rotor structures, airframe dynamics, and acoustics. The work is focused on combining these five key disciplines in an optimization procedure capable of designing a rotor system to satisfy multidisciplinary design requirements. Fundamental to the plan is a three-phased approach. In phase 1, the disciplines of blade dynamics, blade aerodynamics, and blade structure will be closely coupled, while acoustics and airframe dynamics will be decoupled and be accounted for as effective constraints on the design for the first three disciplines. In phase 2, acoustics is to be integrated with the first three disciplines. Finally, in phase 3, airframe dynamics will be fully integrated with the other four disciplines. This paper deals with details of the phase 1 approach and includes details of the optimization formulation, design variables, constraints, and objective function, as well as details of discipline interactions, analysis methods, and methods for validating the procedure

Adelman, Howard M.

Mantay, Wayne R.

English

NASA Technical Reports Server

This paper is primarily concerned with the phase 1 activity; namely, the rigor- 
ous mathematical optimization of a helicopter rotor system to minimize a combination 
of horsepower required at various flight conditions and hub shear transmitted from 
the rotor to the fuselage. The design will satisfy a set of design requirements 
(constraints) including those on blade frequencies, autorotational inertia, 
aerodynamic performance, and blade structural constraints. Additionally, the design 
is required to satisfy constraints imposed by response of the fuselage and also those 
constraints related to acoustics requirements. * 
N 9  0 5s lm!:;,;.- ,! ) 3 
w - - 9, -05 GENERAL APPROACH AND SCOPE 
2 2 3  013 
Howard M. Adelman and Wayne R. Mantay 5, {
The general approach for the activity is illustrated in figure 1. In phase 1 
the blade aerodynamic analysis, blade dynamics, and blade structural analysis are 
coupled and driven by the optimizer. 
geometry as well as the internal structure (spar, leading and trailing edge, ballast, 
etc.) takes place inside the box in figure 1. The influences of the airframe dynam- 
ics and acoustics are accounted for in terms of design requirements (constraints) on 
the blade design. These requirements are described in the next section of the 
paper. For a check on the efficacy of representing the acoustics requirements 
indirectly, the "final" design will be input to an acoustics analysis. The acoustics 
analysis calculates the acoustic constraints and derivatives of these constraints 
with respect to the design variables. 
well the design was able to satisfy the actual acoustics design requirements. 
The phase 2 procedure, wherein acoustics is fully integrated with the blade 
The optimization of the blade aerodynamic 
This information will be used to determine how 
aerodynamics, blade dynamics, and blade structural analysis, is also illustrated in 
figure 1. The design produced in phase 2 (when converged) will satisfy acoustics 
goa l s .  Airframe dynamics in phase 2, as in phase 1, is accounted for by effective 
constraints on the blade dynamics, aerodynamics, and structural behavior. Finally, 
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in phase 3 airframe dynamics is integrated and the result is a fully integrated 
optimization strategy. 
This section of the paper consists of details of the integrated rotorcraft 
optimization problem. Included are descriptions of the following: the objective 
function (the quantity to be minimized for obtaining an optimum design); the design 
variables (dimensions and other parameters of the design); constraints (a set of 
behavioral or characteristic limitations required to assure acceptable and safe 
performance); and definitions of the interactions among the disciplines. 
Ob j ec t ive Function 
The objective function will consist of a combination of the main rotor horse- 
power at five flight conditions plus a measure of vibratory shear transmitted from 
the rotor to the hub. Although several multiple objective function techniques are 
available (ref. 11) one leading candidate is a linear combination whereby 
where F is the objective function 
kl through k6 are weighting factors 
HP1 through HP5 are required horsepower at various flight conditions 
S is the vertical hub shear 
A candidate set of flight conditions would be: 
Flight condition 
1 
2 
3 
4 
5 
Description 
Hover 
Cruise 
High speed 
Maneuver 
C 1 irnb 
Velocity (kts) 
0 
140 
200 
120 
1000 fpm 
W O C )  
Load factor 
1.0 
1.0 
1.0 
3 . 5  
4 
Blade Model and Design Variables 
Figure 2 is a depiction of the rotor blade model to be used in the phase 1 opti- 
Also shown in figure 2 are the design variables which are defined mization activity. 
in table 1. 
linearly tapered from root to tip. 
location (referred to as the point of taper initiation) and may be linearly tapered 
thereafter to the tip. Design variables which characterize the overall shape of the 
blade include the blade radius, point of taper initiation, taper ratios for chord and 
depth, the root chord, the blade depth at the root, the flap hinge offset, and the 
blade maximum twist. Tuning masses located along the blade span are characterized 
by the mass values and locations. Design variables which characterize the spar box 
beam cross section include the wall thicknesses at each spanwise segment and the ply 
thickness at O o  and +45 . 
blades, the rotor angular speed, and the distribution of airfoils. 
The blade model may be tapered in both chord and depth. The depth is 
The chord is constant from the root to a spanwise 
0 Additional design variables include the number of rotor 
Constraints 
As previously described, the phase 1 activity is based on integrating the blade 
aerodynamic, dynamic, and structural analyses within the optimization procedure. The 
acoustics and airframe dynamics analyses are decoupled from the first three disci- 
plines and their influences are expressed in terms of constraints. Accordingly, the 
total set of constraints is made up of two subsets. The first subset consists of 
constraints which are evaluated directly from the first three disciplinary analyses 
and are a direct measure of the degree of acceptability of the aerodynamic, dynamic, 
and structural behavior. The second subset represents indirect measures of the sat- 
isfaction of constraints on the acoustics behavior and the requirement of avoiding 
excessive vibratory excitation of the airframe by the rotor. 
The constraints are summarized in table 2.  The first two constraints are for 
aerodynamic performance and require that for all flight conditions, main rotor 
5 
horsepower not exceed available horsepower and that airfoil section stall not occur 
at any azimuthal location. The next nine constraints address blade dynamics. The 
first requires that the blade natural frequencies be bounded to avoid approac'ning any 
multiples of rotor speed. 
and inplane loads, transmitted hub shear, hub pitching, and rolling moments. The 
next three dynamic constraints are an upper limit on blade response amplitude, a 
lower limit on blade autorotational inertia, and finally, the aeroelastic stability 
requirement. The structural constraints consist of upper limits on box beam 
stresses, blade static deflection, and blade twist deformation. The acoustic con- 
straints are expressed as an upper bound on the tip Mach number and an upper llound on 
the blade thickness to limit thickness noise; and an upper bound on the gradient of 
the lift distribution to limit blade vortex interaction (BVI) and loading noise. The 
effective airframe constraints are expressed first as a separation of the fundamental 
blade inplane natural frequency in the fixed system from the fundamental pitching and 
rolling frequency of the fuselage. 
quency to avoid the proximity to any fuselage frequency. 
upper limit on the blade mass which will avoid any designs which satisfy the (:on- 
straints at the expense of large mass increases. 
The next five impose upper limits on the blade vertical 
Second is a bounding of the blade passage fre- 
The final constraint: is an 
Interdisciplinary Coupling 
Phase 1 of the effort will utilize several design variables which have 
historically been significant drivers of disciplinary phenomena. In addition, other 
variables are being included to provide other unexplored design opportunities. 
Table 3 shows an attempt to quantify the interactions among the disciplines th.rough 
the design variables. 
based solely on acoustics, performance, or dynamics. This variable also influences 
blade structural integrity and fixed system response to transmitted loads. 
provides the strong interdisciplinary coupling for tip speed shown in table 3 .  
For example, rotor tip speed has driven past rotor designs 
This 
There 
6 
are variables, such as blade twist, which can strongly influence some disciplines, 
such as aerodynamics, while not perturbing others (e.g., structures) and other 
variables such as a hinge offset which, heretofore, have not greatly influenced 
conventional rotor design. 
A significant part of the current effort will explore not only the obvious 
strong design variable couplings, but will also address those variables which may 
provide design synergism for multidisciplinary design goals. 
design key for missions which have not been accomplished with today's rotorcraft. 
This may provide a 
Organization of System 
The overall organization of the system to optimize a blade design for aerodynam- 
ics, dynamics, and structural requirements is shown schematically in figure 3 .  In 
order to perform the aerodynamic, dynamic, and structural analyses indicated in the 
blocks in figure 3 ,  it is first necessary to transform or "pre-process" the design 
variables into quantities needed in the various analyses. For example, the dynamic 
and structural analyses both need stiffnesses E1 and GJ, and laminate properties. 
The aerodynamic analysis needs lift and drag coefficients for the airfoils used. 
above information is obtained by the design variable pre-processors which act as 
translators of the global design variables into local variables needed in the analy- 
ses. The output of each analysis block, in general, serves two purposes. First, 
response-type output may be transmitted to another analysis block (e.g., airloads 
from aerodynamics to dynamics); second, information entering into the objective func- 
tion or constraints is supplied to the objective function and constraints block 
(e.g., stress constraints from the structural analysis). A key part of the procedure 
is the sensitivity analysis. This block corresponds to the calculation of deriva- 
tives of the constraints and objective function with respect to the design variables. 
The derivatives quantify the effects of each design variable on the design and, 
The 
7 
thereby, identify the most important design changes to make enroute to the optimum 
design. 
The sensitivity data are passed to the optimizer along with the current values 
of the design variables, constraints, and objective function. The optimizer uses the 
information to generate a new set of design variables, and the entire procedure is 
repeated until a converged design is obtained. 
converged when all constraints are satisfied and the objective function has reached a 
value which has not changed for a specified number of cycles. 
For our purposes, a design is 
Optimization Algorithm 
The basic optimization algorithm to be used in this work is a combination of the 
general-purpose optimization program CONMIN (ref. 12) and piecewise linear approxi- 
mate analyses for computing the objective function and constraints. Since the opti- 
mization process requires many evaluations of the objective function and const:raints 
before an optimum design is obtained, the process can be very expensive if complete 
analyses are made for each function evaluation. However, as Miura (ref. 3 )  points 
out, the optimization process primarily uses analysis results to move in the clirec- 
tion of the optimum design; therefore, a complete analysis needs to be made only 
occasionally during the design process and always at the end to check the final 
design. Thus, various approximation techniques can be used during the optimization 
to reduce costs. In the present work, the objective function and constraints will be 
approximated using piecewise linear analyses that consist of linear Taylor series 
expansions. 
CONM1N.- CONMIN is a general-purpose optimization program that performs con- 
strained minimization using a usable-feasible directions search algorithm. In the 
search for new design variable values, CONMIN requires derivatives of the objective 
function and constraints. The user has the option of either letting CONMIN determine 
the derivatives by finite differences or supplying such derivatives to CONMIN. The 
8 
second option will be used in this work. 
ever possible - for example for vibration frequencies, mode shapes, and modal shear. 
Analytical derivatives will be used when- 
Eventual incorporation of the Global Sensitivity Equation (GSE) approach is planned. 
As described in reference 13, the GSE approach is potentially very effective for 
integrated problems such as a helicopter rotor. Finite difference schemes will be 
used for derivative calculations where analytical approaches are unavailable. 
Piecewise linear approximation.- In the approximate analysis method, deriva- 
tives of the objective function and constraint functions with respect to the design 
variables are used for linear extrapolation of these functions. 
linearity is valid over suitably small changes in the design variable values and wi.11 
not introduce a large error into the analysis provided the changes remain small. 
The assumption of 
Specifically, the objective function Fo, the constraints go, and their respec- 
tive derivatives are calculated for the design variables 
analysis. 
(ref. 14). The first-order Taylor series approximations for the new objective func- 
tion and the constraint values are as follows: 
V,,k using an accurate 
For example the aerodynamic performance constraints are supplied by CAMRAD 
and 
where NDV is the number of design variables, F is the extrapolated value of the 
objective function, g is the extrapolated value of the constraint, and Vk is the 
updated design variable value determined by CONMIN. 
Errors introduced by the piecewise linear approach are controlled by imposing 
"move limits" on each design variable. Move limits are specified as fractional 
9 
changes in each design variable value. Additional information and examples of the 
Joanne L. Walsh and Kevin W. Noonan b ,I 
This section of the paper deals with the aerodynamic performance aspects of 
rotor blade design. Design considerations, aerodynamic constraints and design vari- 
ables are described. 
Design Considerations 
An important aspect of aerodynamic design of a helicopter rotor blade is ;:he 
selection of the airfoils which could be applied over various regions of the b:Lade 
radius. 
section drag divergence Mach number on the advancing side of the rotor disc, avoid 
The choice of airfoils is controlled by the need to avoid exceeding the 
exceeding the maximum section lift coefficients on the retreating side of the :totor 
disc, and avoid high oscillatory pitching moments on either side of the rotor disc. 
Since airfoils with high maximum lift coefficients are advantageous in high speed 
forward flight and pull-up maneuvers, high lift sections are generally used from the 
rotor blade root out to the radial station where the advancing side drag divergence 
Mach number precludes the use of the section. From that station outward, other air- 
foil sections which have higher drag rise Mach numbers are used. 
Once the airfoils and an initial airfoil distribution are selected, the induced 
and p r o f i l e  power components become functions of twist, taper ratio, point of taper 
initiation, and blade root chord (ref. 16). For the hover condition, the majority of 
the power is induced power and the remainder is profile power. 
which minimize both induced and profile power are desirable. 
Rotor blade designs 
The induced power is a 
function of blade radius, chord, and section lift coefficient. The profile power is 
a function of blade radius, chord, and section drag coefficient. The induced and 
profile power can be reduced (provided the aerodynamics of all retreating blade 
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