Optimum rotor interdisciplinary design with a finite state aeroelastic system

Muthl. Comput. Modelling Vol. 18, No. 314, pp. 37-51, 1993 Printed in Great Britain. All rights reserved

0895-7177/93 $6.00 + 0.00 Copyright@ 1993 Pergamon Press Ltd

Optimum Rot or Interdisciplinary Design with a Fi .nite State Aeroelastic System*

C.-J. HE Acoustics and Dynamics Department, Lockheed Engineering and Sciences Company

Hampton, VA 23666, U.S.A.

D. A. PETERS Department of Mechanical Engineering, Washington University in St. Louis

One Brookings Drive, St. Louis, MO 63130, U.S.A.

Abstract-This paper presents an analytical formulation of an optimum rotor interdisciplinary design. A finite-state aeroelsstic rotor model, coupling simultaneously a generalized dynamic wake model with blade finite elements, is applied to perform the optimum rotor blade design for improved aerodynamic performance and vehicle vibration. A feasible direction nonlinear optimizer provides the optimization algorithm. The uniqueness of the present approach is the systematic rotor aeroelastic model, which offers an efficient analytical tool, and retains necessary aerodynamic and blade dynamic building blocks for a sufficient rotor dynamic response analysis. The formulation is well-suited for an efficient design sensitivity computation without resorting to finite differencing, and thus provides a practical design tool.

NOMENCLATURE

size of leading edge weight

blade chord

airfoil section lift curve slope

rotor power coefficient

size of lumped mass

nondimensional blade root cutout, dimensionless on R

harmonic rotor hub forces in Z, y, and .z direction, respectively

nondimensional blade sectional lift, dimensionless on pR2R3

harmonic rotor hub moments in I, y, and z direction, respectively

total number of inflow harmonics

rotor power required

number of rotor blades

ith blade generalized coordinate, equation (14)

rotor radius

blade radial coordinate, nondimensionalized by R

box beam web thickness

number of radial mode functions of the mth harmonic inflow

box beam flange thickness

nondimensional time, nondimensionalized by R

tangential airflow at blade section

blade nodal variable vector

‘th .I design variable

air density

modal inflow states

steady state of ~7

advance ratio

inflow forcing functions

blade normal mode matrix

inflow radial modal function

rotor azimuth

blade pitch

blade ith mode natural frequency, nondimensionalized by R

*The major portion of this research was performed while the authors were at the Georgia Institute of Technology. The optimization work in this paper was funded under NASA Grants NAG-l-710 and NAG-1-1373, Howard Adelman, Technical Officer. The inflow model was developed under NASA Grants NAG-2-462 and NAG-2-728, Robert Ormiston, Technical Officer.

37

38 C.-J. HE AND D. A. PETERS

R rotor rotational speed Ai induced inflow at rotor disk

x total inflow ( 1’ derivative with respect to f

1. INTRODUCTION

The design of rotor blades involves a close coupling among aerodynamics, dynamics, and the

blade structural details. First, each of these disciplines presents a challenge. Second, they in-

teract strongly. Finally, there is a large number of design variables, which further intricates the

problem. On the other hand, this complication offers a potential opportunity for the applica-

tion of numerical design optimization techniques. Many attempts have been made toward the

development of a design tool based on computer programming. There is work emphasizing a

proper design of blade dynamic characteristics for lower vibrations, which is always a large con-

cern in helicopter design [l-5]. There is also optimization of rotor aerodynamic performance [6].

In [7-lo], rotor aeroelastic design optimization is performed in the optimum design of rotor blades

for vibration reduction or combined performance and vibration improvement. The work of Young

and Tarzanin [lo] provides experimental validation of an optimized design for a low vibration ro-

tor and shows substantial reduction in the 4/rev vertical hub load and in the 4/rev hub moments

based on wind tunnel tests.

This research has greatly increased our understanding of helicopter rotor optimum design. But

the credibility of results is diminished either by model simplification (such as no unsteady wake

in the modeling) or by the prohibitive computational costs and inaccuracy due to system noise or

system nonlinearity of finite difference derivatives with comprehensive helicopter analysis. These

problems remain a challenge to the interdisciplinary design optimization of helicopters even with

today’s most powerful computers.

The present work is a further effort continued from our previous interdisciplinary design opti-

mization [ll]. We have removed the limitation of using specified loads in the design and devel-

oped a finite-state aeroelastic rotor model in a formulation that can perform design sensitivity

computation through a direct solution of a system of rotor dynamic equations. In particular,

a coupled simultaneous system of ordinary differential equations combining finite-state aerody-

namics, blade dynamics, and blade controls is established and solved. This results in rotor hub

loads (both steady and vibratory) and their sensitivity derivatives in one simultaneous solution.

By taking advantage of vectorization of the code (available with most supercomputers), one can

handle a large number of design variables without an inordinate increase in computation time.

2. FORMULATION OF ROTOR AEROELASTIC EQUATIONS

The rotor blade is assumed to be an elastic beam undergoing flapwise bending, chordwise

bending, and torsion. Such important blade design parameters as chord variation, blade pretwist,

and chordwise offsets of the blade center of gravity and of the aerodynamic center from the

elastic axis have been included in the analysis. A finite element approach is applied with twelve-

degree-of-freedom beam elements [12]. Aerodynamic loads are computed based on a finite-state

aerodynamic theory. In this theory, the blade sectional aerodynamic lift, drag, and moment

are developed based on an unsteady thin airfoil model. However, the unsteady induced how is

computed from a three-dimensional generalized dynamic wake theory [13]. Thus, the resultant

airloads are both 3-D and unsteady.

The dynamic wake theory is developed from an incompressible potential flow solution [13]. In

this solution, the unsteady induced flow is related to the blade aerodynamic lift by a system of

first-order ordinary differential equations * + [ L" 3-l CP n r rmc n

Optimum Rotor Interdisciplinary Design 39

where [MC] and [MS] are the same except that [MC] contains elements for the fundamental zeroth harmonic:

with

and

, form=0,1,2 ,..., N,

. .

[ MS ] =

[’ .I

$H,” , form=1,2 ,..., N,

. .

H,” = cn +cm; l;$” 1 m); ‘)!!, n m . . n m . .

(3)

(4)

(5)

[ L"]_l=2[ i”]_‘[ V”], (6) [ L"]-l=2[ is]-‘[ V”]. (7)

The [L”] and [La] are inflow influence coefficient matrices which have been obtained in closed-

form [13]. The [V”] and [V”] are the mass-flow parameter matrices. They are also the same except that [V”] contains mass-flow parameters for the azimuthally uniform inflow state. Therefore,

, for m = 0, 1,2, . . . , IV, (8)

[ Vs ]= ’ VT

1 .I

, form=1,2 ,..., N, (9) . .

with

v-p = &GYP, (10) Vm = P2 + X(X + ai:)

n CL2 +x2 ’ for m # 0 and n # 1. (11)

The CY,” and /3,” are the modal inflow states. They are, in fact, the induced flow expansion coefficients:

Xi(T,lhQ = 2 2s,,, +m- 1

c $,“(r)[aT(Q cos(m?l) + KY8 sin(m$,)ll (12) m=O n=m+l,m+3,...

where the $F(T) are the radial inflow shape functions [13]. The rz’s are the generalized inflow forcing functions, and are obtained by spanwise integration of weighted blade aerodynamic lift, and then summation over all the blades at each instant of time [13]. The detailed formulation of


the inflow forcing functions utilized in this application has been presented in [14], based on

unsteady thin airfoil theory.

The blade dynamic equations of motion are in terms of second-order differential equations

an

Pfl~x~** + M{xl* + F{X) = {F), 03)

where F are external nodal forces which depend upon both blade control pitch, blade motions,

and the induced flow.

The blade dynamic equations are converted into the modal domain with generalized coordinates

based on normal modes from an eigenvalue solution, [@I:

They are further combined with rotor finite-state aerodynamics and with rotor auto-pilot

control to form the final rotor aeroelastic system in the following form:

PI

with

PI=

P =

[II 0

0

0

0

r “,

*

+ [El

[Y] 0 0 0 0

0 [M”] 0

0 0 Pw

I -[I 0 ,L& 0 0 O I 0 0 0

0 0 0 [L$ 0 0 0 0 (kP-1 : 0 0 0 0 -[II 0

(15)

(16)

(17)

where [T], [ICI are time constant and gain matrices for the auto-pilot [15]. In this study, rotor

thrust and rotor tip-path-plane orientation are utilized as trim objectives, i.e.,

At each time step, the auto-pilot works to eliminate the difference, ACT, AD,, and A&, between

current thrust coefficient and tip-path-plane orientation and the desired values by adjusting the

rotor trim variables (0). The rotor trim variables consist of both collective pitch, 00, and cyclic

pitch, 8, and 0,. The rotor steady trim state is achieved when {AT} approaches zero within a

specified error together with a periodic rotor response.

The above finite-state aeroelastic model has been validated by correlation with experimental

data [13,14]. It has proved to be applicable for design.


3. DESIGN SENSITIVITY DERIVATIVE COMPUTATION

The computation of design sensitivity derivatives is the most time-consuming part of the design

optimization. This is especially true when the design model is comprehensive and the number of the related design variables is large. The rotor aeroelastic equations established in the previous section offer a direct analytical formulation for computing these derivatives.

The finite-state aerodynamics removes the implicit dependence of aerodynamics on the design

variables. Therefore, a direct computation of design sensitivity derivatives becomes feasible

without relying on finite differences. Since the variation of induced inflow influence coefficients due to small design perturbation

is negligible, a direct differentiation of the inflow dynamic equations (1) and (2) results in the

sensitivity equations of the induced flow in the following form:

The derivatives of the inflow forcing functions on the right-hand side of the equations are related

to the derivative of blade sectional lift:

(23)

The derivative of blade sectional lift with a design variable depends on changes in blade chord, twist, blade response, and induced inflow with that variable. This can be obtained as

8Ek 1 -= -cluuT axj 2 {L

(WbO - &O) + ;(wbl - Ail) 1 T& + CT& (WbO - &O) + f(w,, - &I) [ II , 3 3 (24)

where wbc and ‘u&r are the first two Glauert expansion coefficients of the relative air how due to the blade pitch settings and blade motions, and Ai0 and Air are those for the induced how, Ai [14].

The sensitivity equations of the blade response can be obtained from a direct differentiation of equation (14)

a =- dxj (25)


Finally, the trim equations are perturbed with design change:

ITI{ g}**+{ g}*=rkl{ e}. Incorporation of these trim variable perturbations into the design sensitivity derivative calculation

accounts for the effect of trim variable change due to design perturbation. It would be formidable to consider such influence in the finite difference approach due to an implicit dependency of rotor

trim on design variables. The above equations, (19), (20), (25), and (26), are combined to form a system of sensitivity

equations of rotor aeroelasticity:

where

, (27)

(23)

The system of design sensitivity equations can now be solved together with rotor aeroelastic

equations (15) at a trimmed state by a fourth-order Runge-Kutta integration scheme. A trim

state of the helicopter rotor is first achieved with the auto-pilot trim, equation (15). Then,

the rotor design sensitivity equations (27) are combined with the trimmed rotor equations and

solved simultaneously. Once the sensitivity derivatives of blade dynamic response and induced

flow are obtained, the design sensitivities of blade airloads and inertial loads are straightforward algebraic operations. Finally, the design sensitivity of rotor hub loads can be obtained by spanwise

integration of blade sectional load sensitivity and summation over all the blades. The convergence of the sensitivity derivatives is determined by the periodicity condition in

steady flight. For all the cases tested, the transient response of the sensitivity equations takes five to eight cycles to subside, and a steady periodic solution is obtained. Although the number of equations is very large, computation time is minimized by the vectorization of the code. In fact, about 159 seconds of CPU time on a CRAY-2S is needed for a typical comprehensive solution of

both trimming the aeroelastic rotor and computing the design sensitivity with 63 design variables.

As compared with the CPU time for the finite-difference technique, based on function evaluation time, the current method results in about a 90% reduction in computer time. This CPU time saving is proportional to the number of design variables, as shown in Figure 1. For more design variables, more benefits with current method in computer time will result.

4. OPTIMIZATION IMPLEMENTATION

The optimum blade design is posed as a constrained minimization problem. The multiobjective

programming is to minimize the power required for a specified lifting capacity and vibratory rotor hub loads in a nonrotating frame. The weighting method is applied to reduce this multiobjective minimization to the single objective problem:

F(x) = kzFz + kyFy + k,F, + k,,M, + kmyMy + k,,M, + k,P, (29)

where F,, F,, F,, M,, My, M, are the vibratory rotor hub loads, P is the rotor power required, and k,, k,, etc. are the weighting coefficients which reflect the relative importance of the objectives. These weighting coefficients are first determined based on dimensional considerations.

Optimum Rotor Interdisciplinary Design

loo%7

43

0 10 20 30 40 50 60

Number of Design Variables Figure 1. Variation of CPU time saving with the number of design variables

h t

Figure 2. Blade cross-sectional geometry.

Thus, k, = k, = k, = 1, k,, = k,, = km, = R-l, k, = (OR)-‘. Naturally, they can also vary

within their dimensions to emphasize certain specific objectives.

The first constraint imposed on the optimization is the blade mass inertia for autorotation in

case of engine power failure. This also prevents the tip chord from becoming too small since

the blade weight is related to the size of the blade chord. For an appropriate maneuver flight

and stall margin, the rotor solidity is constrainted to be fixed while blade chord distribution is

optimized. For aeroelastic stability, the center of mass is constrainted to be forward of the elastic

axis of the blade at all cross sections. This excludes any classical bending-torsion flutter with

articulated or teetering rotors. The blade natural frequencies up to the seventh mode are kept

away from coincidence with rotor speed harmonics. Finally, the rotor is trimmed to a specified

lifting capacity and tip-path-plane orientation at a given flight condition.

The design variables include both aerodynamic parameters and blade structural details. The

aerodynamic design variables are the spanwise variation of blade chord and twist. The blade

is modeled as a box beam for the cross-sectional properties, Figure 2. The primitive structural

design variables are the flange and web thickness of the box beam, which is placed within the

blade sectional geometry defined by the local chord. There are also internal lumped mass and

leading edge weights for additional freedom of weight distribution. Thus, we have five design

parameters at each cross section, along with given material properties that define the blade mass

and structural properties. Naturally, they are constrainted such that the pieces must fit within

each other (e.g., d < b - s1 - ~2). There are side constraints imposed on all these design variables

to assure a meaningful design.

The CONMIN optimization code has been used as an optimizer [16]. It is a first-order program-

ming method that uses a feasible search direction obtained from a compromise between gradients

of objective function and imposed constraints. Thus, in each iteration, CONMIN requires as


0.08

h 0.06 .z * .I

.* E 0.04

S

fil 0.02

‘$

9’ 0

-0.02

Figure 3. Design sensitivity w.r.t. taper, p = 0.35.

inputs the objective and constraint functions, and their gradients with respect to the design variables. As presented in the previous section, an analytical formulation has been established for the design gradient computation which is computationally efficient. For this efficient design

gradient computation, there can be a reduction of more than 80% in CPU time as compared to

the finite difference approach in the overall design since the computation of sensitivity derivatives

consumes more than 90% of the total design CPU time.

5. RESULTS AND DISCUSSION

The rotor has four blades and each baseline blade has a rectangular planform with an initial -8”

linear twist. It resembles a typical McDonnell Douglas AH-64A blade in geometry, but without

tip sweep. The box beam material is taken to be 6061-T6 Aluminum alloy.

5.1. Design Sensitivity

The design sensitivity refers here to the rate of change of rotor power required or vibratory

hub loads (normalized by the trimmed value) with respect to a design variable. Also, the design

sensitivity results presented in the following are associated with the baseline rotor blade at a

specified flight condition. For a nonlinear problem, the design sensitivity can vary for a different

rotor blade configuration or at a different flight state. Figure 3 shows the effect of blade taper on the rotor power consumption and 4/rev harmonic

hub loads at p = 0.35. The taper reduces the rotor power consumption, but raises the hub

vibratory loads. The effect of blade negative twist on the design is to lower both the rotor power

required and the vibratory hub pitch and roll moments, but to increase the oscillatory shears and yaw moment, Figure 4. The conflicting effects of blade taper and twist present a problem in searching for an optimum design with both improved performance and vibration without applying

a numerical design optimization technique. The effect of rotor trim variations due to a design change has almost exclusively been ignored

in existing rotor design optimization practice. No quantitative justification has really been made

for it. The simultaneous system of blade structural dynamics, aerodynamics, and blade control presented in this paper offers a unique opportunity to investigate this unanswered question. Figure 5 shows the effect of trim variation on blade taper sensitivity. The difference in the taper

sensitivity with or without trim effect is minor. There is, however, a remarkable influence on

blade twist sensitivity, Figure 6. Fortunately, the changes seen in blade twist sensitivity bear the

same trend (the same sign), with or without trim considered. Figure 6 has only shown the design sensitivity of power required and the three major vibratory hub loads. There is a similar effect of the trim on less important vibratory hub in-plane shears and yaw moment. Therefore, for the

Optimum F&or Interdisciplinary Design

q Powa

Fx

E FY

I I 4’ I I

1

Figure 4. Design sensitivity w.r.t. twist, 1-1 = 0.35.

0.1

0.075

0.05

0.025

.0.025 ’ I I I I

P FZ Mx MY

8 No Trim Effect

N With Trim Effect

Figure 5. Effect of trim variation on taper sensitivity, p = 0.35.

-5’ , 1 1 I I

P Fz Mx MY

No Trim Effect

With Trim Effect

45

Figure 6. Effect of trim variation on twist sensitivity, p = 0.35.

HCH 18:3/4-E


4.0

-5 2.0

g

8 c

Q 0" 0.0

-2.0

2.0

-1.a

-~.-. . , , . . . , ,

- Fx ‘\ ‘&.

__--- w ---_ Fr

_._._._._ MX

-- w . . . . Mz

0.25 0.50 r 0.75

Figure 7. Effect of box beam flange thickness, p = 0.35.

- Fx _---- Fy

---- Fz

_._.e.-- m

---*--- MY . . . . Mr

0 0.50 0.75 1.00 r

Figure 8. Effect of box beam web thickness (sl), IL = 0.35.

2.0

Optimum Rotor Interdisciplinary Design

I' * mm I'. m s I' * *.

47

-FX _____

-1.0 - w

--_-_-_-R w--_._ Mx -1.._1 My . . . . Mz

-2.o- ""'*".'. *. .".'. 0.00 0.25 o&o 0.76 1.00

r Figure 9. Effect of box beam web thickness (sz), ~1 = 0.35.

sake of computational efficiency, it may be reasonable to neglect the trim variation from a design

perturbation in the calculation of design sensitivity with little or no effect on the design search

direction.

Next, the effect of structural variables on vibratory rotor loads is investigated. Figure 7 shows

the impact of the box beam flange thickness (t) on vibratory (4/rev) hub loads at p = 0.35. This

box beam thickness is directly related to blade flapwise bending stiffness, and thus affects blade

flapwise bending. It also affects the torsional mode. It is most effective to suppress the harmonic

(4/rev) normal shear and pitch or roll moments acted at the hub by reducing the box beam

flange thickness at around the 35% radial station. For outboard blade stations, the opposite (an

increase in the flange thickness) is beneficial for vibration suppression.

Figure 8 shows how the box beam web thickness on the leading side (sr) affects the vibratory

loads. Figure 9 is for the web thickness on the trailing side (~2). Both sr and s2 are the main

contributors to the blade edgewise bending stiffness. They also change the offset of sectional

chordwise center of gravity from the elastic axis. It is interesting to note that sr and s2 have

significantly different influences on rotor vibration. Increase in sr around midspan would lower

vibratory F,, Adz, Mv, and M,, but increase Fy and F,, while s2 does the opposite (except

for Fy). This is due to the fact that changes in sr and sz move the chordwise c.g. in opposite

directions. Also, the change in sr is more influential beyond the 75% radial station, while the

greater effect of s2 occurs inboard of T = 0.75.

The size of the leading edge weight (u) is mainly utilized to move the chordwise c.g. ahead of

the elastic axis. An increase in leading edge weight would move the blade sectional chordwise c.g.

toward the leading edge. Its impact on rotor vibration is shown in Figure 10. The placement of

the leading edge weight near the midspan would reduce the 4/rev normal shear (F,) at the rotor

hub, but increase all the other of the vibratory load components. However, changing the leading

edge weight at the outboard blade tip region would provide an opportunity to reduce all of the

harmonic rotor hub loads.

The lumped mass inside the box beam is available for spanwise mass distribution tuning. Its

effect on rotor vibration is shown in Figure 11. This mass placement has a greater impact on

vibratory roll and pitch moments than on the others.

48 C.-J. HE AND D. .‘k. PETERS

1.0 -

.c”._.__ ._._. -.-.-. _._.-.--._.~_*_

_--_*_ “..

0.0 - - -----__ __-“_<<_b :.,’ *\.

tl;N.+ *\ ‘, .’ -

‘N.’

- Fx

-1.0 - ----- FY --_--Fz

Mx

-"---*- MY . . . . Mz

-2.0 0.00

I I a. .I... * 1 * ' *. 0.25 0.50 0.75 1.00

r

Figure 10. Effect of leading edge weight, @ = 0.35

2.0 1 ’ ‘B ‘I’. “I””

. t

1.0 -

i ._**- _M.--

_._._.d-'-'-'..~. _A.

r% .' \

'\

-1.0 - ----- FY ---_ Fr

-a-*-s-e- Mx

---- MY . . . * Mz

-2.0," " " " " " " I"' * 0.00 0.25 0.50 0.75 1.00

r

Figure 11. Effect of lumped mass, p = 0.35.

5.2. Integrated Performance and Vibration Optimization

The optimum rotor blade is designed to minimize both the power required and the vibratory

hub loads. For rotor aerodynamic performance, the power required by the rotor in both hover

(p = 0.0) and high forward flight speed (p = 0.35) was chosen as design objectives (there is

no effect of fuselage parasite drag in this isolated rotor design). Hover flight is a critical design

point for the rotor performance. In the process of optimization, it is more efficient to have a

multistep approach to accomplish the multiobjective design goal. First, the power required in


q Baseline

q Optimum

Figure 12. Power required for baseline and optimum design.

hover is minimized using the chord and twist as design variables with autorotational inertia as a

constraint. This process results in a satisfactory power reduction for a specified lifting capacity. It

produces about 2.4% improvement in rotor performance as compared to the baseline rotor blade

which already has a good performance design, Figure 12. This performance improvement results from a tapered blade and a further negative blade twist. The taper is attained with a constraint

on rotor solidity. This prevents any unacceptable design that gains performance benefits from reducing the rotor solidity. Compared to the original rectangular blade, the final improved blade

planform for this design has a taper ratio of 1.6 (chord at T = 0.30 to chord at blade tip). This taper is mainly from the aerodynamic performance design. The original blade has an initial -8’

twist. The final design increases the negative twist to -8.8”. Again, this twist variation is mainly

from the performance optimization. Before a comprehensive integrated performance and vibration minimization is performed, the

vibratory hub loads are reduced from pure structural design (box beam design). This takes into

consideration the increase of rotor vibratory hub loads from the performance design, mainly the

increase of in-plane shears (20% more than original design). This additional step is important for the suppression of all components of rotor hub vibratory loads in the final design.

Finally, the parallel optimization is performed to minimize both power and vibratory hub

loads in forward flight (CL = 0.35) with autorotational inertia, sectional chordwise c.g., and natural

frequencies as constraints. In this step, both aerodynamic and structural design parameters enter the optimization; hence, the design is of interdisciplinary nature. The final design results show

that an improvement is achieved in both performance and hub vibratory loads, Figures 12 and 13. The vibratory hub loads reduction occurs for normal shear and all three vibratory moments for

the optimum blade design. The reduction of 4/rev pitch and roll moments is significant. There

are 68% and 52% decreases in amplitude for vibratory pitch and roll moments, respectively.

Although there is no significant change in the in-plane shears, the normal shear, roll and pitch moments are the major contributors to the vehicle vibration.

Figure 13 also shows the effects of the blade natural frequency constraints on the final design.

For example, the torsional mode (4.08/rev) of the baseline blade has two possibilities in its frequency constraint window setting. If the natural frequency of this mode is set such that its frequency moves into the lower window (between S/rev and 4/rev), indicated as Design 1 in

the figure, it becomes less coupled with the third flapwise bending mode (4.8/rev). However, it becomes closer to the second flapwise bending mode (2.6/rev). This results in a satisfactory reduction in both vibratory hub shears and moments. On the other hand, if the upper window

(4/rev to 5/rev) is set for this same mode (a stiffer blade), denoted as Design 2, a stronger coupling with the third flapwise bending mode occurs which gives less vibratory loads reduction.

50 C.-J. HE AND D. A. PETEFLS

4

q Baseline

3-

2-

q Design1

q Design2

Fx FY FZ Mx MY MZ Figure 13. Hub vibratory loads for baseline and optimum design, p = 0.35.

6. SUMMARY AND CONCLUSIONS

Fully integrated aeroelastic optimization has been performed for an articulated rotor. The

objective function includes vibratory loads and power required. The ability to obtain meaningful

results in a reasonable computing time is due to an efficient, finite-state rotor wake model. This

model has allowed the analytic computation of design variable derivatives with greater than 80%

saving in CPU time for the optimization.

The fully integrated performance and vibration reduction design has been fulfilled by a multi-

step approach which is both efficient and effective. The blade natural frequency constraint

influences the final design in the vibration suppression. The trim variation due to a perturbation

in design change has an effect on the design sensitivity derivatives, but does not change the trend.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

REFERENCES

R.B. Taylor, Helicopter vibration reduction by rotor blade modal shaping, Proceedings of the 5’ph Annual National Forum of the American Helicopter Society, pp. 90-101, Anaheim, CA, (1982). P.P. Friedmann and P. Shanthakumaran, Optimum design of rotor blades for vibration reduction, Journal of the American Helicopter Society 29 (4), 70-80 (October 1984). D.A. Peters, M.P. Rossow, A. Korn and T. Ko, Design of helicopter rotor blades for optimum dynamic characteristics, Computers Math. Applic. 12 (I), 85-109 (1986). J. Pritchard and H. Adelman, Optimal placement of tuning masses for vibration reduction in helicopter rotor blades, AIAA Journal 28 (2), 309-315 (February 1990). W.H. Weller and M.W. David, Experimental verification of helicopter blade design optimization for minimum

vibration, Proceedings of the 44th A nnual National Forum of the American Helicopter Society, pp. 263-280,

Washington, DC, (June 1988). J.L. Walsh, G.J. Bingham and M.F. Riley, Optimization methods applied to the aerodynamic design of helicopter rotor blades, Journal of the American Helicopter Society 32 (4), 39-44 (October 1988). J.W. Lim and I. Chopra, Aeroelastic optimization of a helicopter rotor, Journal of the American Helicopter Society 34 (l), 52-62 (January 1989). A. Chattopadhyay and Y.D. Chiu, An enhanced integrated aerodynamic load/dynamic approach to optimum rotor blade design, Proceedings of the 4dh Annual Forum of the American Helicopter Society, pp. 459-468, Washington, DC, (June 1990). C.B. Callahan and F.K. Straub, Design optimization of rotor blades for improved performance and vibration, Proceedings of the 47th Annual Forum of the American Helicopter Society, pp. 869-882, Phoenix, AZ, (May 1991). D.K. Young and F.J. Tarzanin, Structural optimization and Mach scale test validation of a low vibration rotor, Proceedings of the 47th AHS Annual Forum, pp. 955-968, Phoenix, AZ, (May 6-8, 1991).


11. C.-J. He and D.A. Peters, Optimization of rotor blades for combined structural, performance, and aeroelastic characteristics, Strzlct~ral Optimization 5 (l-2), 37-44 (December 1992).

12. T.W.H. Ko, Design of helicopter rotor blades for optimum dynamic characteristics, Doctor of Science Thesis, Department of Mechanical Engineering, Washington University, St. Louis, MO, (August 1984).

13. D.A. Peters and C.-J. He, Correlation of measured induced velocities with a finite state wake model, Journal of the American Helicopter Society 37 (3), 55-70 (July 1991).

14. C.-J. He and D.A. Peters, An aeroelastic analysis with a generalized dynamic wake, Proceedings of the American Helicopter Society International Specialists’ Meeting on Rotorcraft Basic Research, Paper No. 19, Atlanta, GA, (March 1991).

15. D.A. Peters, M. Chouchane and M. Fulton, Helicopter trim with flap-lag-torsion and stall by an optimized controller, Journal of Guidance, Control, and Dynamics 13 (5), 824-834 (Sept-Ott 1990).

16. G.N. Vanderpleats, CONMIN-A FORTRAN program for constrained function minimization, User’s Man- ual, NASA TMX-62282, (1973).

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Optimum rotor interdisciplinary design with a finite state aeroelastic system