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Produced by the NASA Center for Aerospace Information (CASI)
September 15, 1982

IFS 10,9&

qsq ^`

FIRST SEMI-ANNUAL STATUS REPORT

on

' NASA-Langley Grant No. NAG-1-250

for

Period Covering 18 January 1982 through 16 July 1982.

"DESIGN OF HELICOPTER ROTOR BLADES FOR

OPTIMUM DYNAMIC CHARACTERISTICS"

by

David A. Peters, Professor and Chairman

and

Timothy Ko, Research Assistant

Department of Mechanical Engineering

Washington University, Campus Box 1185

St. Louis, MO 63130

Alfred E. Korn, Professor

and

Mark P. Rossow, Professor

Department of Engineering and Technology 

Southern Illinois University

hdw ,rdsville, IL 626126

(NASA-CB- 169352) DESIGN Ji BELICOPTE6 fiCTOE N82-33374

BLADES FOB OPTIMUM DYNAMIC LHIRACTERISTICS

Semiannual Status Report, 18 Jan. - 16 Jul.

1982 (Washington Univ.) s3 p HC A03/bF A01 Unclas

CSCL 01C 63/0535,320
CONTENTS

Wage

1. Introduction. . • • + • , • • • , • , , , , • . • • • • , , ,, , + ^.

1, 1 Rotor Blade Design , • • , • , 1

1.2 Overview of Optimal Structural Design, • , • . , . + 4

2. Background, , . . . . . . . . , , • , . . . • • . . , . . • + • 6

2.1 Formulation of 'Problem • • . . , , . . o+ . + • , , ,

2.2 Definition of 'tasks. . • . . . , , 

2.3 Finite Element Madel . 9

3. Statement of Work . . • • . . , . • 13

3.1 Phase 1 - The Inverse Problem . . , . • 13

3.2 Phase 2 Forced Response. . . • , 14

3.3 Phase 2 - Improynments to Model. . • 15

3.4 Phase 4 - Applications to Existing Designs . , 15

4. Illustrative Examples of Structural Optimization. • , 17

4. 1 Cantilever team* • . 4 • . . . . . . . . . . , . . 17

4.2 Cantilever with Given Weight . . . . . . . , , • 17

4.3 Cantilever with Tip Mass . . . . . . . . . . • . , . , 17

5. Preliminary Calculations for Rotors . . . . . . . . . . . . . 21

5.1 Wind Turbine Blade . . . . . . . . • , • • 0 0 21

5.2 Other Considerations . . . . . . . . .6 . . . . . . . 21

5.3 Helicopter Blade . . . . . . . . . .. . . • . . . . 0 23

6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7. References. . . . . . . . . . . . . . , . . . . . . . . . . 29

-. notation. . . . . . . . . . . . . . . . . . a 0 . . . . . 6 31
1, INTRODUCTION

1.1 Rotor Blade Design

The design of helicopter rotor blades involves not only considerations 

strength, survivability, fatigue, and cost, but also requires that blade

natural frequencies be sigm1f cantly separated from the fundamental aerodyn

forcing frequencies (e.g. Rol. 1). A proper placement of blade frequencies

is a difficult task for several reasons. First, there are many ,forcing

frequencies (at all integer-multiples of the rotor RPM) which occur at rath

closely-spaced intervals. For example, 5/rev and 6/rev are less than 20%

apart. Second, the rotor RPM may vary over a significant range throught the

flight envelope, thus reducing even further the area of acceptable natural

frequencies. Third, the natural modes of the rotor blade are often coupled

because of pitch angle, blade twist, offset between the mass center and

elastic axis, and Large aerodynamic damping. These couplings complicate the

calculation of natural frequencies. In fact, the dependence on pitch angle

makes frequencies a .Function of loading condition, since loading affects

collective pitch. Fourth, the centLtfugal stiffness often dominates the

lower modes, making it difficult to alter frequencies by simple changes in

stiffness.

In the early stages of the development of the helicopter, it was belie

that helicopter vibrations could be reduced (and even eliminated) by the co

choice of structural coupling and mass stiffness distributions. However, it

is easy to imagine how difficult it is to find just the proper parameters su

that the desired natural frequencies can be obtained. The difficulties in

placement of natural frequencies have led, in many cases, to preliminary

designs which ignore frequency placement. Then, after the structure is
-2-

"finalized" (either on paper or in a prototype blade), the frequencies are

calculated (or measured) and final adjustments made. Reference (2) describes

the development of.the XII-17 helicopter in wlAtch a 300-1b Height was added t

each blade in order to change the spanwise and chordwise mass distribution a

thereby move the first flapwise frequency away from 3/rev. However, these

types of alterations are detrimental to blade weight, aircraft development

time " and blade cost. In addition ' corrections usually are not satin

factory, and the helicopter is often left with a noticeable vibration problem

The state-of-the-art in helicopter technology is now to the point,

however, that it should be possible to correctly place rotor frequencies

during preliminary design stages. 'There are several reasons for this. First

helicopter rotor blades for both main rotors and tail rotors are now being

fabricated from composite materials (Refs. 3 and 4). This implies that the

designer can choose, with limited restrictions, the exact El distribution

desired. Furthermore, the lightness of composite blades for the main rotor

usually necessitates the addition of weight to give sufficient autorotational

blade inertia. Thus, there is a considerable amount of flexibility as to how

this weight may be distributed. Second, the methods of structural optimizati

and parameter identification are now refined to the point where they can be

efficiently applied to the blade structure. Some elementary techniques have

already been used for the design of rotor fuselages (Ref. 5). It follows tha

the time is right for the use of structural optimization in helicopter blade

design. Some work on this is already under development, and, although not

published, some companies are already experimenting with the optimum way to

add weight to an existing blade in order to improve vibrations.
-3-

The purpose of the work discussed here is to invest ip the possibili

s

(as well as the limitations) of tailoring blade mass and stiffness distribu

to p:.qo an optimum blade design in terms of weight, inertia, and dynamic

characteristics.

The major objectives of the work are:

1) To determine to what extent changes in mass or stiffness distributi

can be used to place rotor frequencies at desired locations.

m

2) To establish theoretical limits to the amount of frequency shift.

3) To formulate realistic constraints on blade properties based on weig

mass moment of inertia, size, strength, and stability.

4) To determine to what extent the hub loads can be minimized by prope

choice of EI distribution,

5) To determine if the design for minimum hub loads can be approximat;e

by a design for a given set of natural frequencies.

6) To determine to what extent aerodynamic couplings might affect the

optimum blade design.

7) To determine the relative effectiveness of mass and stiffness distri

on the optimization procedure.

8) To determine to whet extent an existing blade could be optimized wit

minimal changes in blade structure.

9) To develop several "optimum profiles" for rotor blades operating und

various standard conditions.

The work 1s.td focus on configurations that are simple enough to yiold

clear, funiamental insights into the structural mechanisms but which are

sufficiently complex to result in a realistic result for an optimum rotor bl
-4-

1.2 Overview of Optimal Structural Design

Most approaches to optimal structural design may be classified into three

categories. (For recent review articles see Refs. 6 and 7.) One sitch category

is "variational methods." 'These generally rely on techniques from the mathematical theory of the calculus of variations, and, when applicable, often

provide useful physical insight into the nature of an optimal design.

Unfortunately, only relatively simple problems can be solved by this approach,

since the mathematics becomes intractable when complex engineering structures

are considered.

A second category of structural optimi.zatlon techniques consists of the

application of mathematical programmin% methods together with the discreti,-

zation of the structure by finite element techniques. This approach to

optimization was founded in 1960 (Ref. 8) with the hope that more complex

structures could be analyzed than were possible when using the analytical

techniques of the calculus of variations. However, in the late 6098t

became apparent that mathematical programming methods had limitations of their

own, namely, unacceptably long computation times occurring when the number of

design variables become large (over 20-100, depending on the type of structure)

Fortunately, several improvements developed over the last few years appear to

have significantly extended the capability of the mathematical programming

approach, and, as a result, it is this approach we intend to draw upon for

solution techniques in the proposed research. In a later section of this

proposal,, after our design problem has been formulated, we will discuss these

recent improvements in the approach.

A third co.egory of structural optimization approache is the "optimality

criterion" approach in which an equation expressing some necessary condition of

optimality is used as the basis for constructing an iterative (successive
I

-Kre-design} procedure. Originally developed because of dissatisfaction with

mathematical programming tachnques, the optimality criterion approach init

relied on intuitive optimality criteria such as constant straws-ratio and

uniform strain-energy density conditions. More recently, optimality criteri

(and associated re-design equations) hove been derived from the Kuhn-Tucker

conditions (see, e.g., Ref. 9) for a constrained minimization problem.

The optimality criterion approach seems especially well-suited to probl

with a large number of design variables. Since our proposed design problems

will have a moderate number of variables and since deriving efficient to-des

equations for our problems is not immediately straightforward, we initially

prefer the mathematical programming approach over the optimality-criterion

approach.

A structural optimization computer program, called CONMIN, is available

from NASA. It is this program that is used in our present work. °CONMIN

is based on the mathematical nonlinear programming method of :feasible directi

Nevertheless, if COWIN proves to be too expensive computationally (or 

any other way unsuccessful for our work) we may turn co other approaches.
-b2. BACKGROUND

2.1 Formulation of Problem

Because numerically-based optimization is best carried out with discrete

variables, tloe finite element technique stands as the most logical choice for

the blade model. A recent research project (Ref. 10) has resulted in a fini

element computer program that is ideally suited to the work proposed here. T

program allows for tapered, twisted finite elements in a rotating environment

The existing code can calculate natural frequencies, (with or without aerodynamic terms) and forced response.

Another important aspect of the rotor blade optimization problem is the

selection of the optimality criteria and constraints to be imposed. Our d

dnsign problem has certain features which ate unusual compared to typical

problems occurring in the structural optimization literature. There are

basically three catAgories of k;r teria. In the first class, one would

minimize weight given eonskiaints on the natural frequencies (i.e. frequency

"w ndowo 11 ). In this case, a constraint on rotary inertia is also implied sin

a rotor must have sufficient inertia to autorotate. The advantage of this

approach is that it is directly related to the physical realities of design.

The disadvantage, however, is that the first guess will probably not be feasi

(that is will not have frequencies that fall in the "windows"), This can be 

stumbling block to convergence. A second type of criteria is one in which th

objective is to minimize the discrepancies between desired frequencies and

actual frequencies. The constraint then becomes a window on autorotational

inertia. Although this avoids unfeasible solutions, it does not directly

minimize weight (although weight is limited by the autorotational constraint)

An objective function could be constructed that combined blass mass and frequ
-7-

placement, but the relative weightings of the two componenti is not obvious.

The third category of constraint is to minimise vibrations directly without

r"gard to frequency placement. Although this appears on the surface to be

the perfect solution, there are problems. First, calculation of vibrations

is an order-of-magnitude more difficult than the calculation of frequencies.

Second, past efforts at this have resulted in strange designs, incompatible

with standard helicopter practice. Third, there is still the problem of the

weight-vibration trade-off. In this work, we intend to concentrate on the

first two categories with some attention to the third.

Another type of constraint involved in the problem is the limitation on

structural properties. The blade planform, airfoil, and twist arc chosen

by the aerodynamicist on the basis of performance. The structural engineer

must choose his design to fit in the aerodynamic envelope given, , There are

five structural parameters to be chosen: 1) flapping stiffness, 2) inplane

stiffness, 3) torsional stiffness, 4) mass, and S) torsional moment of

inertia. In practice, these cannot be chosen completely independently.

Figure 1 shows the envelope of a typical blade section.. All stiffness is

assumed to reside in a box-beam of dimension b x h with thicknesses a and t.

This beam is placed as far forward as possible (to keep the elastic axis

near the 1/4 chord). Mass properties are due to the box-beam, skin, honeycomb,

and two lumped masses. The lumped mass in the tip is typical of rotor blades

and is used to keep the mass center forward of the aerodynamic center. A

second lumped mass is included to allow independent choice of mass and massmoment. The constraints of this construction are clear and are listed on the

f igure .

In addition, there are minimum constraints on b, h, s, t to hold centifuga

loads and4 to remain within... manufacturable limits. For example a simple ninimum p p 
ORIGINAL PAQE IS

OF POOR QUALITY ME

dSo

..

yr

c

0. •-

V .c

rr

b

w ^'

C O V

udHN

N

R N N

o

vi VI VI v1 v1 as

VI VI VI VI VI

00000 .

r.. NQC.

^

2Hd

C.1

OD

ZQHZU7

_OHW

V / GQ

coVRZW

CD

-7b-

EJ

a

a i=
-aconstraint on area could come from the centrifugal ccoost raint (not considerin

bending stress). Thus, if 

an 

is the maximum stress and if f is a safety fact

then

n

(om) ? f f E (Mj)11 ril /A1

imj+l

n

ft lSiSZ2i I

A > ^'n (l)m

Of course, when we enter the vi-hrutory-response phase of the work (discussed

later) bending stresses will be included.

Similarly, our early work (wnich neglects flutter boundaries) can nevertheless include flutter criteria in a simplified manner. First, we can

choose frequency placement such that no coalescence occurs between flap-lag,

flap-torsion, or lag-torsion. Second, we can constrain the five parameters

in Figure 1 such that t1te mass center is always forward of the 1/4-chord, a

common design practice to prevent torsion-flutter In rotor blades.

For speoific examples, a smaller space of design variables may be used.

For example, in the general case one could force the elastic axis to the

quarter chord by choosing b - C/2 - 2h; or one could choose to allow no lumpe

mass in the blade interior (d . C). This would reduce the number of design

variables from 5 to 4 or 3. If one considers flapping deflection only, then

all variables except t and d may be fixed with no loss of generality. (This

is equivalent to simply using El and mass as variables.) Similarly, for

inplane one can consider only s and d; and for torsion, one can consider only

b and a. Thus, for each of the three uncoupled cases one has three possibilit

1) vary stiffness only, 2) vary inertia only, 3) vary stiffness and inertia.
-9--

2.2 Definition of Tasks

The work, as proposed hers, consists of four primary phases;

1) solution of the inverse problem - $Ivan a sat of frequencies find the mass

and stiffness distribution; 2) solution for optimum forced response - given

a loading condition, minimize blade stresses and hub reactions, subject to

weight restrictions; 3) improvements to include flap-lag-torsion coupling;

and G) application to existing rotor blades - given an existing blade,

improve its dynamic characteristics by realistic structural modification.

These phases are described in detail below. In each phase, work can be

performed with nondimensional equations. * Nondimensional quantities to be

considered in optimization are P (mass distribution), n (flapping stiffness),

nz (iuplane stiffness), n x (torsional stiffness), and a (e. g. - e a. offset).

'The nondimensional parameters to be included as 'relatively fixed are

Y (Lock number), 9 (twist), k (stiffness of control system), d (radius of

,,,yration about e.g.) and ^ (aspect ratio).

Aerodynamics assumptions and other constraints are described in the

individual tasks indicated below.

2.3 Finite-Element MoOel

Although tapered, twined elements are within our capabilities, we

introduce, here a simpler case which is also of value. The stiffnesses Gx,

EIZZ , El yy are assumed to be constant along the length of the element. The

mass of the element is assumed to be evenly distributed on the two nodes.

Let the deflection of an element in the y and z directions at a distance

x be denoted as w(x)-and"v(x), fur-which the displacement models are assumed

to be polynomials of third degree. The expressions are given as

*Dimensional cases will also be considered, however, to keep insight.
-lp ORIGINAL PAQV I

OF POOR QUALITY

w(x) * R (2x3 32x2 + Z3

) + V6 (31X2

 - 2x3)

R^(x3 - 2 x2 + X X) - Z(x - ltx2)

w(x) . 3(2x3 - Ux2 + z3) + R7(3Rx2 _ 2x3)

2(x3 - 2Rx2 + lZ2

x) - 2 (x3 - Rx2)

where vi , v30 vd and vg represent the banding degrees of freedom in the

zx plane and u 2 , u7

, u4 and ug represent the bending degrees of freedom

in the yx plane

The strain energy due to bending deformation ran be expressed as

t El 2 2 $wav2 2 El 2 2 ^

v ! (—= ) (A-4 + Eryz ---

f 8x2 + 2

zz v

o

2

) Idx

ax ax ( x

(ii) The potential energy in tension from the centrifugal force field,

which is equivalent to the negative of kinetic energy due to radial

displacement, is given by

TK a ^o' [(aX) 2 + (

Sv}2ldx

where T, tension force, is assumed to be constant along each element.

(iii) The kinetic energy due to inplane displacement is given by

2

Tic 1/2 10 mv2Q2dx

which is equivalent to a u . -T
ORIGINAL pA%,, t

-ll- OF Poop QLIAL17Y

uj

u

Ao

IN ^9

Meanwhiieo the pretwiat angle ^(x), and the torsional deformation 8(x)

are assumed to be polynomials of first degree, and can be expressed as

O(x) - 01 (l - P + OAP

0 (x) 0 v5 (1 -) + vi0 (^,)

where 01 , 02 represent the pretwiat angle at node 1 and 2 0 and v 5 v10

represent the elastic torsionai degree of freedom at each end.

( {v) The torsional energy, due to elastic deformations and centrifugal

terms, can be expressed as

U . f

0

(1 1,.^z2 T (O' + 8') 2 +-1

ka 2 T ( $ 1 + 0 1 ) 2 + 2 GJ612 )dx

where ka2 . ffz 2' dydz . IyY

ka2 n ffy 2 dydz . Izz

(v) The "torsion-rotation" energy under the effect of rotation is given b

`I'^ a - f &12 (km2 - km2) (^ + 0)2dx

where km2, km2 are mass moment of inertia which can be expressed as

km2 tl pz2dydz ' 

pIyy

^'

M2 Ir py 2 dydz - Alxz
ORIQ1t4A►i. PAM IS

OF POOCH QOALI-ty

-12-

Total diplacement energy now con be used to form the stiffness matrix from

U 2 uT(KIU

where u is the vector of nodal displacements, in the order as u 1 , u6t u3,

ug, u20 U70U40u90 us, u10 ; (K) is the elemental stiffness matrix of order 

(vi) The mass matrix will be obtained by the kinetic energy of an eleme

which is given by

T fo {2 (- * 2 ( t)2 + 2 1z2 (8t x)2

2

+ 2 lyy ( atax)2 + 2 (kml + km2)9'2}dx

Written in matrix form, the kinetic energy can be expressed as

T^1uT (m)u

where (m) is the mass matrix.
ORIGINAL PACE 1S

OF POOR QUALITY.

3. STATEMENT OF WORK

3.1 Phase 1 - The Inverse Problem

The first phase of the research effort will involve solution of an inverse

problem - that is, find the system structure given the eigenvalues. The

solution to the inverse problem is difficult from a mathematical point of view,

The inverse problem that will be treated in this phase will be done in

successive steps. First, a nonrotating cantilever beam will be used to study

bending frequencies. Flapping will be considered itJ the first beginning.

Then, in-plane and torsion will be added. No twist or offset will be included

so that the two bending frequencies and the torsion frequency will be uncoupled

A conventional optimal design formulation of one of the problems might be

(1) minimize total weight

n

W a E }► i4ri

i-1

With respect to design variables U and n, subject to constraints

E r2 Ari -1

SZi-a<wi <01+e

Pi > umin

nmin ? n > nmax

2 xiKxi

and w

i = T

ximxi

where K and m are structural stiffness and mass matrices with the rigid

degrees of freedom removed, x is the eigenvector corresponding to the natural

frequency, and w is the prescribed iatural frequency.
ORIGINAL RAGE Z" 

-14- OF POOR QUALITY

The solution of the above optimization procedure is effected by the

optimization program, CONMIN. The efficiency of that program is greatly

improved when the derivative of the constraints (with respect to the paramete

p

is known. Thus, we must consider the derivative of frequency, with respect t

the parameters:

2

X

T

[aK ) x - xT t am x

a(wi) a^i i aui

xT m xi

3.2 Phase 2 Forced Response

In phase two, we will solve for blade structures that minimize forced

response. In this step a very simplified model will be used in that we will

simply apply a given forcing function. We will not take into account that

the blade motions themselves may affect this loading, except that aerodynamic

damping will be included. We will assume a very simple lift distribution, su

as u = r2 J1 - r 2 , that oscillates with integer-multiple frequencies

10

in^ Lift E wnu(r)e

n-1

We will then formulate the problem as:

4) Minimize ( e root shear + boot momentl); that iss minimize

n

E

l (w

iui - vlu i ) max Ari (1 + eri)

is

given constraints omax < CT critical'

n

I uiria°i 1^ ui > umin

inl

The forcing functions will be weighted probably as w i (.3) i or some simila

decreasing weightings. The a will include tension stress plus the oscillato
OF

OF POOR QUALf"` y

-l5- 

stress. One of the major purposes of this will be to decide if minimizing h

loads ir, equivalent to spacing frequencies half -way between integers. If th

Is so, then future optimizations would get by without a complicated loads 

calculation. There is some reason to suspect that this is so. Conceptually

the hub loads could be expressed as

w

L E it

i,j 1-w2I/12

Thus, minimizing loads is similar to maximizing (1-w2)(1-w2/4)...

(1-w2 /n2 ). The simple factor (1-w2)(1-w2/4) has a maximum at w - 1.58 and

the factor (1-w2/4)(1-w2/9) has a maximum at w - 2.56. Thus,at (or slightly

above) these points may well be optimum.

3.3 Phase 3 - Improvements to Model

In phase three, two parallel efforts are planned. In the first one, th

blade structural model will be expanded so as to {cclude twist and c.g. offs

These will couple flap, inplane, and torsion. Therefore, uncoupled analyses

will no longer be adequate. We already have computer programs that do this

(Ref. 10) but the coupling will mean a more difficult convergence task for

the optimization programs. In this phase, c.g. offset will be included as a

variable parameter. This implies that "no flutter" must be an added

constraint (Ref. 13).

A parallel effort in Phase 3 will be the improvement of the aerodynamic

model to include aerodynamic coupling between flap, lag, and torsion. The

extent of this work will be partially determined by the results of Phase 2.

Phase 4 - Application to Existing Designs

Finally, these two parallel improvements (structural coupling and aerod

coupling) will be combined in a general program. Phase 4 will be to apply th
t

-16-

program to existing rotor-blade designs to determine what structural

modifications would be recommended. The, fact that our programs already agree

well with existing results constitutes a strong starting point for structural

improvements. 
-17-

4. ILLUSTRATIVE EXAMPLES OF STRUCTURAL OPTIMIZATION

4.1 Cantilever Beams with Given Frequency

Some simple examples will be examined and discussed before the utilizat

of the program CONMIN. In each case the results will be compared with those

obtained by previous researchers, if it is avmilable.

The first limiting example is the problem of determining the optimal de

of an elastic cantilever beam, such that with a specific natural frequency, 

weight of the structure attains the Minimum value.

We start with a uniform beam, modeled by ten elements, with a given

lengtn of 10 inches, E = 1.0 lb-in 2 , El - 10 lb-in, density - 0.042 lb/in3,

and a specified first lowest natural frequency - 0.6489 rad /sec. We obtain

the final stiffness profile shown in Figure 2. Figure 2 is the present

result with ten elements.

4.2 Cantilever Beam with Given Weight

A related problem has also been treated by Niels Olhoff 1111. He seeks

the design of a cantilever beam that yields a maximum value of a particular

higher natural frequency wn (i.e., of specified order, n) with the volume

and length of the beam specified. His work is the dual problem of the examp

shown in Figure 2. Optimization with respect to the fundamental frequency

under the constraint of volume is similar to the one of minimizing weight

(or volume) under the constraint of specified natural frequency.

Figure 3 gives the profile of the optimal cantilever for n - 1 by Olhof

One can see that the shapes in Figures 2 and 3 are very similar.

4.3 Cantilever with Tip Mass

Another example problem is to minimize the weight of a cantilever carry

a mass at the tip, subject to the constraint that the fundamental natural
1,

' 

s18-

ORIGINAL PAGE 19

OF POOR QUALITY

r

2 4 6 8 10

Figure 2. Area Moment of Inertia for Optimum Beam in

Present Work.

.15

.10

.05

Figure 3. Area Moment of Inertia for Optimum Beam

from Reference 11.
-19-

frequency must be greater than or equal to a specified value. The problem w

originally formulated by Turner (12).

Kahn and Willmart (14) used an optimality criterion method to solve

Turner's problem. In this example, four finite elements are used, with the

areas of each as the design variables, as illustrated in Figure 4. The

specified natural frequency is 17.752 rad/sec. The other initial data are

Modulus of elasticity 10.3 x 106 psi

Mass density . 2.5 x 10 4 lb-s2/in4

Radius of gyration (A1) - 2.0

Radius of gyration (A2 ) . 1.5

Radius of gyration (A3) . 1

Radius of gyration (Ad w 0.5

Concentrated mast+ . 1 1b-s2/in

Length of each element - 60 in

where I A (radius of gyration).2
-20-

The results of the optimization are shown in Table 1 The feasiblo

starting design is described by AI a ZOO, A2 0 150 0 A3 - 60 and A4 0 35.

Table 1

Comparison of Cantilever Beam with Concentrated Mass .^

Ref.(12] Ref. [131 This Paper

Iteration - 25 10

AI 136,81 136.63 134.60

A2 118.73 118.7 116.58

A3 83.591 83.586 82.734

A4 34.427 34.608 34.898

Weight 2243.0 2242.9 2214.41

It can be seen that excellent results have been obtained using the pres

CaNMIN optimization program.
-21-

5. PRELIMINARY CALCULATION FOR ROTORS

5.1 Wind Turbine Blade

The next example is the optimization of a wind turbine rotor blade at

30 rpm. A tent-element model, is used. Only the flapping is considered. The

area moment of inertia, 1 0 and the lumped weight of each element are taken

as the design variables. Young's Modulus, R * 0.2 x 10 $ lb-in, and

density » 0.4334 Win 3 are assumed to be constant. Slade radius, R - 750 inch

Table 2 shows the profile of moment of inertia and the distribution of

added weight; 46or the initial, and final configurations.

The .final profile of the area moment of inertia along the blade is

similar as the one in the previous example. The Lumped mass is concentrated

at the tip of the blade as might be expected given a minimization of weight

with a fixed moment of inertia. That is, the moment of inertia remains at

the minimum value, as expected. We also note that most of the lum;ed mass

is removed so that only mass necessary to the stiffness elements (or necessary

for the autorotation constraint) is maintained,

5.2 Other Considerations

An important aspect of the optimization problem is the existence (or lack

of it ) of a feasible solution .

. A "feasible solution" is defined as any set o

design variables that satisfy the constr:«nts (whether or not that particular

solution is an optimum). It is possible that, if the problem is poorly

formulated, that no feasible solution exists. What is more often the case,

however, is that there are Feasible solutions rtt that the optimization scheme

may not be able to find them. Thus, it is advantageous to have a feasible

initial guess so that one is assured that at least a local optimum is possible.

For example, Table 2 illustrates that the first guess is .feasible

(w1 > 2.82 per/rev). Here we found that CONMIN was able to move from.this
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-23-

first guess through the space of feasible solutions. In other cages, however

when the first guess is not feasible we have found that CONNIN is not able to

reach a solution. In such cases, one must add or remove some weight (or add

or remove some EI) from the first guess to move into the feasible space. Fr

there, optimization is obtained.

For example, Table 3 present s data for the same wind turbine as in `fabl

but the constraint on the first natural frequency has been lowered to removo

it from the dangerous 3/rev range. This implies that the first guess in Tabl

is no longer feasible. In order to overcome this, a lumped mass is added to

station 9 (225.4 vs 49.50). This lowers w 1 below 2.621 rev but oleo lowers

w2 to 0,25/rev. This could be alleviated in one of two ways: 1) move the

mass to the node of the second node, or 2) simply widen the w 2 window. We ha

done the latter. It is interesting that the added weight is ultimately

rearrangou to other places and other weight removed such that the new design

is no heavier than the optimum in 'fable 2. Furthermore, w 2 is rai,.ed to 8.5

so that the "widened window" had no effect on the solution.

5.3 Helicopter blade

The design and analysis of a helicopter blade is discussed. Similar to

Section 3, only flapping is considered and a tcn-e'4 4t^kent model is used. Den

is constant along the blade and equal. to 0.18 x 10

-3

slugs/in3 . Young's

Modulus is equal ko 0.49 x 107 lb-in 2 at the root equal to 0.585 x 10 lb-in2

eleewhere. Blade radius is usual. to 193 inches. Results are given in Table

4 and 5.

In Table 4, w1 is in the desired range but w 2 is Goo small. Furthermore

the autorotational inertia is larger than necessary. In this case, the CONMI

program is able to remove mass and stiffness in such a way to raise w 2 and

lower wl . The minimum I sat as a constraint (0. 4) is reached at c j y point
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-26-

except the root. The root remained high to keep w l > 1.05. The new blade is

one-third the original mass. In Table 5, a stiffer initial guess is used and

the frequency w is restrained from significant decrease. Furthermore, w 2 i

near its maximum value. In this case, the program GONMIN would like to

decrease EI and m, but any removal of material could lower w beyond its lowe

bound of 1.24/rev . To counter this, the optimization scheme adds EI near th

root (to maintain w > 1.24/rev). Furthermore, the lumped mass necessary

to maintain aottorotational constraint is moved slightly inboard to have less

effect on w1 (keep it high) but more effect on w2 (keep it low).
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-27-

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6. smm

Thus far we have completed the following objectives:

1) Define the optimization problem and obtain physically reasonable

constraints.

2) Make CONMIN operational on our own computers and successfully apply

it to beam problems with known solutions.

3) Apply optimization procedures to typical rotor configurations

for flapping.

What remains in the first year is

1) Try many more flapping optimization runs to determine the effects

of mass versus El variations, to study the limits on range of frequen

placement, to study the limit on number of frequencies placed, to

determine to what extent a feasible first guess is necessary.

2) Determine the feasibility of using frequency-placement as an objectiv

function rather than as a constraint

3) Try inplane and torsion optimization.

After this, we intend to follow the statement of work as planned.
-29-

7. REFERENCES

1. Niebank, C. and Girvin, W., "Sikorsky S-76 Analysis, Design and Develop

For Successful Dynamic Characteristics", proceedings of the 34th Annual

Natural Forum of the American Helicopter Society, May 1975, pp 78-23-1

through 78-23-17.

2. Hirsh, Harold, Dutton, Robert E., and Rasumoff, Abnar, "Effect of Spanw

and Chordwise Mass Distribution on Rotor Blade Cyclic Stresses," Journa

of the American Helicopter Society, Vol. 1, No. 2, April 1956.

3. Ellis, C.W. at al., "Design, Development, and Testing of the Booing Ver

Army XUH-61A," Proceedings of the 32nd Annual National 'Forum of the Ame

Helicopter Society, Preprint 1010 0 May 1976.

4. Fenanghty, Ronald R. and Noehren, William L., "Composite Bearingless

Tailor Rotor for UTTAS," Journal of the American Helicopter Society, Vol

No. 3, July 1977, pp 19-26.

5. Hanson, H.W. and Calapodas, M.J., "Evaluation of the Practical. Aspects 

Vibration Reduction Using Structural, Optimization Techniques," Proceedi

of the 35th Annual National Forum of the American Helicopter Society,

May 1979 0 pp 79-21-1 through 79-21-12.

6. Niordson, F.I. and Pedersen, P., "A Review of Optimal Structural Design

Proceedings of the 13th International, Congress of Theoretical and Appli

Mechanics, Springer-Verlag, Moscow (1973).

7. Venkayya, V.B., "Structural. Optimization: A.Review and Some Recommendat

Int. J. for Num. Meth. Engrg., Vol. 13, 1978, pp 203-228.

8. Schmit, L.A., "Structural Design by Systematic Synthesis," Proc. 2nd

Nat. Conf. Electronic Computation, AJCE, 105-132 (1960).

9. Fox, R.L., Optimization Methods for Engineering Design, Addison-Wesley

Reading, PA 1971.

10. Ko, Timothy, "Use of Tapered, Twisted Finite Elements," M.S. Thesis,

Washington University, 1979.

11. Niels Olhoff, "Optimization of Vibrating Beam with Respect to Higher Or

Natural Frequencies," J. Struct. Mach. 4(1), 87-122 (1976).

12. M. J. Turner, "Design of Minimum Mass Structures with Specified Natural

Frequencies.. AIAA J. 5(3), 406-412 (1967).

13. M.R. Khan and K.D. Wllmert, "An Efficient Optimality Criterion Method

for Natural Frequency Constrained Structures," Computers S Structurca,

Vol. 14, No. 9-6, pp 501-507, 1981.
-3Q8, NOTATION

a slope of lift curve, rad`1

a size of tip mass, m

b width of box beam, m

c blade chord, m

d width of lumped mess, m

e weighting function

f safety factor

ti height of box beam, m

1 nondimensional mass/unit length, m R/m

p CJ/m2R3

PY EIy/m2R3

uZ EIz/m2R3

Y lock number, p3cR4/1

K control system stiffness, N-m/rad

K K/m2R2

8(r) blade twist, rad

e nondimensional c.g. elastic-axis offset (offset/R)

d radius of gyration`41/1-R2 x

inverse aspect ratio c/r

P density of air

e small parameter

qi S1t

Q rotating speed

c7m maximum stress

CT mass of skin kg/m

fi,,p density of lumped mass, box beam, honeycomb kg/m3
-31-!

I necessary inertia for autorotaton, kg-m2

Ixtorsional mass moment of inertia (about c.g.)

per unit length, kg-m

El flapping stiffness, N-m2

91 inplane stiffness, N-m2

GJ torsional stiffnese, N-m2

m mass/unit length, kg/m

M reference mass, I/R2 , kg

r x/g

ArI length of element, m

R blade radius, m

g ot thicknesses of box beam, m

u(r) lift distribution, N/m

vI displacements, m

wl weighting functions

x length along blade, m