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Produced by the NASA Center for Aerospace Information (CASI)
W t;'

t

FOURTH SEMI-ANNUAL STATUS REPORT

O?'1

NASA RESEARCH GRANT NO. NAG-1-250 

for

Period Covering 16 July 1983 -through 17 January 1984

"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 Korn, Professor

and

Mark P. Rossow, Professor

Department of Civil Engineering

Southern Illinois University

Edwardsville, IL 62026

January 17, 1985

(NASA-CR- 175380) DESIGN CF HELICOPTER ECTO.R N84- 17171

BLADES FOR OP`IIMUM DYNAMIC CHARACTERISTICS

Semiannual Status Report, 16 Jul. 1983 - 17

Jan. 1984 {Washington Univ .) 29 p Unc.las

HC A03/MF A01 CsCL 01C G3/05 18246
TABLE OF CONTENTS

 page

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . U . . .. . . . . . . . . . . . . . . . . . . 1

2. Response of Starting Design and Optimal Designs for Varying

Forcing Frequencies ..............................2

3. Blade Response to Harmonics of Rotor Speed ...................7

4. other Studies Conducted During the Reporting Period ..... ....10

5. Future Work .................................................12

6. APPENDIX - Derivation of Mass and Stiffness Matrices

as Functions of Natural Frequencies... ........ o.14

7. References ..................................................16

8. Figures .....................................................17
!

t

a

1. INTRODUCTION

This report is the fourth semi-annual report of a research

project concerned with the optimal design of helicopter rotor

blades. The goals of the project and the approach adopted have

been described in previous reports, and these descriptions will

not be repeated here. The present report will be confined

primarily to describing several studies comparing the forced

responses of an initial (i.e., non-optimized) blade to those of a

final (optimized) blade,

At the 39th Annual Forum of the American Helicopter society

last spring, we presented a paper describing our work on optimal

rotor blade design. In the ensuing discussion, several questions

were raised about the manner in which the optimal design problem

had been formulated. One question was whether or not the Forced

response of the blade can be adequately controlled, as we 

assumed, by our approach of "frequency plact=xtn:nt", that is, of

restricting the natural frequencies of the blade to lie within

narrow intervals ;located away from certain integer multiples of

the rotor speed. A second question was whether or not aerodynamic

damping substantially reduces the resonant peaks, in which case

concern about avoiding resonances through proper selection of 

frequency windows would be unnecessary. similar, questions had

been mentioned in the original proposal for this project, when it

was stated that the sensitivity of the optimal design to the

choice of frequency window would be studied, and if it was 

that frequency placement did not in fact reduce vibrations, then

some other objective such as minimizing root shear would be

1

V
explored. Because of the timely coincidence of the questions

raised at the Foruin and the intent expressed in our original

proposal, we decided the time had arrived to investigate the

appropriateness of "frequency placement'.

The investigation of this question was carried out through

two, somewhat overlapping, problems. First, the forced

responses of an initial (i.e., non-optimized) design were

compared to those of a final (optimized) design as the frequency

of the forcing function was changed; cases with and without

aerodynamic damping were considered. Next, the responses of

initial and final designs were evaluated as one natural frequency

was varied (the ethers were held fixed), and a forcing function

containing harmonics of the rotor speed was applied. Again ` cases 

with and without aerodynamic damping were considered. The general

finding from these studies is that frequency placement is a

viable means of reducing vibration, although it is by no means

the only method and should be used in conjunction with others.

In the penultimata section of the report, several topics are

briefly described in which studies-have peen ini{-iated during the 

reporting period, but not yet completed. The final section of the

report contains a sketch of plans for future work.

2. RESPONSE OF STARTING AND OPTIMAL DESIGNS FOR VARYING FORCING

FREQUENCIES

In this section of the report, the response of both the

initial and final (optimal) designs to an external forcing

function is studied as the frequency of the forcing function is

varied. Blades both with and without aerodynamic damping are

2
M I

considered. To formulate these problems, consider the forced

behavior of a rotor-blade.

The equation of motion for the finite-element representation

of a rotor blade subjected to an external excitation may be

written in matrix form as

(M] {X(t) } + (C] {X(t) ) + [K] {X(t) } = {F (t) }

where

[M] = mass matrix,

{X(t)} = column vector of nodal displacements,

[C] = damping matrix,

(K] = stiffness matrix, and

{F(t)} = forcing function column vector.

where

The forcing function may in turn be expressed as

{F(t)} = {Vo } eiWtI

W = forcing frequency, and

Vo

forcing amplitude.

After some calculation, it can be shown that the amplitude

of the response -- written as {X}, independent of t -- can be

given as

{x} _ [K + iWC - W2M] - 1{V0}

r

:•; In the present section, the flapping response is considered.

r

The in-plane response is inferred from the results without

3

_ter -•,-+^• ^ re.^... ^^.^ . y' .. ^. s . —
damping, since there is little aerodynamic damping in the inplane

direction.

Fig. i shows a plot of the forcing amplitude v o (Ref.l)

used in the study. Given the forcing amplitude, we can calculate

the response of each node of the finite-element representation of

the blade as the value of the forcing frequency, W, is varied.

The tip (finite-element node farthest from the hub) response is

of special interest. Before the results obtained from this study

are presented, it is useful to examine the frequency placement

results which are described in the Third Semi-Annual Report

(pp.21-26). The results for the frequencies (in units of

cycles/rev) are, for flapping mode only,

MODE INITIAL DESIGN FINAL DESIGN

lst 1.18 1.118 

2nd 3.22 3.09

3rd 5.89 5.67

Blade dimensions are given in Table 8, p. 26, of the Third SemiAnnual Report.

The frequencies in the above table correspond to the

symmetric modes of a teetering rotor. Thus, only even harmonics

of the rotor speed have been considered as forcing frequencies. 

As a result, the optimized blade (Final Design) finds the third

mode moved away from the critical 6.0/rev (front 5.89 to 5.67).

Similarly, the movement of the second mode to 3.09/rev removes it

from 2.0 and 4.0/rev. In the comparison study to follow, however, 

we will apply the entire spectrum of frequencies to this blade

(not just even harmonics). Thus, the "Final Design" can no longer

. 4 
be considered optimum. A comparison of the two blades, however,

does indicate the strong effect of resonance because each case

has a distinct resonance (6 and 3/rev)

We shall now consider the results of the present study. Fig.

2 shows the tip responses of both the initial design and final

(i.e., optimized) design as functions of the forcing frequency.

Aerodynamic damping has been neglected (Alternatively, the

results can be interpreted as giving the inplane response.). It

can be seen that near 1.18 cycles/rev, the responses of the two

designs are very similar. However, the responses corresponding

to the second and third modes differ significantly. For example,

in the second mode the peak of 3.22 cycles/rev (initial design)

moves to 3.09 cycles/rev (final design). similarly, the peak of

the third mode moves from 5.89 cycles/rev to 5.67 cycles/rev,

which is especially important since it is highly desirable to

keep the frequency away from the integer frequency of 6

cycles/rev. 'We conclude from these results that the frequency

placement approach does have a significant effect on the forced

tip response when damping is not considered.

Next, the effect of aerodynamic damping is considered,, that

is, the results to be presented correspond to flapping.

Mathematical details of the damping formulation are available in

the thesis by Ko [Ref.2]. The effect of aerodynamic damping on

zedueing the resonant peaks of the tip response of the initial

blade is shown in Fig. 3. Fig. 4 shows the damped responses of

both the initial and optimized blades so that the effect of

frequency placement can be studied. It is interesting to observe

5

i
here that when damping is included, no apparent advantage is

gained by optimizing the blade, at least in terms of reducing the

tip response, except in the range of 3-4/rev, in which a thirty-

five percent reduction occurs. However, we must also examine the

effect of optimization when the response is measured by the

average shear force existing in the blade.

Consequently, the shearing force in the blade is considered

next. As a measure of the average shear in the rotor, we consider

the sum of the squares of the shear force (abbreviated SSS),

SSS = Y1 2 + Y2 2 + ... + Y102

In this equation, Yi represents the shear force at node i in

the (ten-element) finite-element model. Note that the root shear 

is necessarily included as one of the terms on the right-hand

side of the equation, so that a large value of root shear will

cause SSS to also be large,. 

Fig. 5 shows the variation of SSS with respect to the forcing

,frequency for the initial design with and without aerodynamic

damping. Fig. 6 shows the same quantities for the final

(optimized) design. Fig. 7 compares the quantity SSS corresponding

to initial and final designs when aerodynamics is considered.

Inspection of these figures shows that, in contrast to behavior 

of the tip response, the shear response is significantly af

by optimizing the blade, even when aerodynamic damping is

included. The 3/rev loads are increased by fifty percent due to 

the movement of w 2

from 3.22 to 3.09/rev. Similarly, the 6/rev

loads are reduced by seventy percent due to the movement of w3

6
from 5.89 to 5.67/rev. Thus, even with damping, frequency

placement is a powerful driver of loads. it follows that

frequency placement can be justifiably considered an important

part of blade optimization.

3. BLADE RESPONSE TO HARMONICS OF ROTOR SPEED

In the study just described, the response of the blade to

changes in the forcing frequency was considered. Now we consider

a different approach. In effect, we examine how the blade

responds to a forcing function "during the optimization

procedure" -- in the sense that during optimization, the

optimization algorithm varies the natural frequency of the blade

(to force it to satisfy the frequency constraints). In obtaining

the results to be presented next, we simulated the optimization

procedure by varying the natural frequency. Thus we can observe

what happens to the forced response during frequency placement. 

The formulation of the approach is as follows. Through

appropriate transformations (described in the Appendix), the

system mass matrix can be written as

[M] = [U] -T [U] -1

and the system stiffness matrix as

[K] = [M] [U] diag [ (wi t ) ] [U] T [M]

in which w i are the natural frequencies of the system, [U] is a

matrix whose columns are eigenvectors and the notation "ding"

indicates a diagonal matrix (all off-diagonal terms vanish). From

7
examination of these expressions, it can be seen that the

stiffness and mass matrices can be considered functions of the

natural frequencies. Thus it becomes poss^ble to fix all

frequencies but one, and then study the response of the system as

that one frequency is varied. in particular, the response to the

following forcing function will be studied:

(F(t)) = {vl}eiWt + (v2)ei2Wt + + {Vn}einWt

where

W = the .rotor speed, and

{Vn } = (1/n){vo},

and {v o } was defined previously in Fig. 1. Since the arguments of

the exponentials are integer multiples of W, resonance will occur

at harmonics of the rotor speed. The particular forcing function

given above is known from empirical observation to provide an

approximate, but physically realistic representation of the

radial anti harmonic variations of the amplitude of the load on a

real blade. As in Section 2, the blade response will be defined

through the tip displacement and the sum of the squares of the

shears, except that, here, the n = 1 term has been omitted from

the expressions for calculating tip displacement and shears

because this term represents a tip-path plane tilt that is

controlled by the pilot for trimming purposes. It is not part of

the true vibratory loads we are considering.

Results for the problem just formulated are shown in Fig. 8,

where the sum of the squares of the shears is plotted as a

function of w 2 , the second natural frequency, with the other

8
natural frequencies being fixed. This figure corresponds to the

initial blade design (Blade dimensions are given in Table O, p.

26, of the Third Semi-Annual Report.) Fig. 9 shows the same

quantity for the case where the third natural frequency is

varied. It is interesting to note that the response curve for the

damped case in Fig. 9 lacks resonant peaks -- apparently the

damped response is so completely dominated by the resonance of

the second natural frequency, whic ► i is fixed near 3/rev, that the

(damped) resonant peaks for the third frequency are negligible by

comparison. it is worth mentioning, in passing, that the value

of the damped response corresponding to w l = 1.18, w2 = 3.22, and

w 3 = 5.89 can be read from Fig. 8 as well as Fig. 9 and can be

seen to be the same (approximately 6.0 lbs). This observation

provides a re-assuring check that the figures represent actual

behavior and not a programming error.

For the final (optimal) design, the analogous quantities are

plotted in Figs. 10 and 11. Again, no resonant peaks are present

in the damped response when the third natural frequency is

varied. Comparison of magnitudes of ordinates in Figs. 8 and 10

(no damping) shows that the overall shear measure is reduced in

the final design in the regions away from resonance. Also, the

choice of scale on the vertical axis in Fig. 10 highlights the

effect of frequency placement. Note that by inspecting Figs. 8-

11, a designer may select the design frequency which minimizes

the average shear as measured by the SSS.

One of the most interesting results of Fig. 10 is

information about the width of valleys and peaks, since

9
this gives design information. First, let us examine the nodamping curve (inplane response). Here, the minimum points are

nearly at the centers of the regions (2.55/rev) and (3.55/rev).

The frequency windows to maintain no more than thirty percent

increase in loads are 2.40 - 2.70/rev and 3.40 - 3.70/rev (plus

or minus 0.15/rev) -- a fairly narrow window. For the damped

curves (flapping response), minima are also near the one-half

points, but the window for thirty-percent increases are much

wider -- 2.20 - 2.90/rev and 3.20 to 3.Q0/rev (plus or minus

0.30/rev). Stated another way, inplane frequencies should be no

closer than a 0.4/rev from integers, but flapping frequencies may

be as close as 0.2 from an ,integer. It should be emphasized that

these observations apply to this particular example and may not

be generalized for other frequency constraints. In future work,

we will apply similar reasoning to optimized articulated rotors

for whicb the frequency spectrum is more meaningful.

Another conclusion to be drawn from the above results is

that the undamped response curve has very flat-bottomed "valleys"

when one of the fixed frequencies is near an integer value (cf.

Figs. 10 and 11) .

4. OTHER STUDIES CONDUCTED DURING THE REPORTING PERIOD

As an extension of work reported previously, data were

obtained for six rotor-blades produced by several different

helicopter companies, and considerable effort was ex -)ended in

attempting to choose box-beam dimensions and other stiffness and

ma y ~ parameters in our finite-element model in order to match the

natural frequencies of these given blades. The motivation for

10

F
4

this work is, once a given blade can be properly represented by

the finite-element model, to improve it through the application

of optimization techniques. The task of matching frequencies

necessarily proceeds by a certain amount of trial-and-error and

is, as a result, time-consuming and tedious. The results of this

frequency-matching effort will be described in the final report.

Another study conducted during the reporting period has been

concerned with the application of an alternative optimization

technique -- the optiA,,,. -ity-criteria method. This method, a

generalization of the traditional stress-ratio approach to

improving a structural design, has received considerable

attention among structural optimizers in the last five to ten

years [Refs.3-5), and thus it appears appropriate to make at

least a preliminary investigation of its applicability to rotorblade dos> ..igvi * At this writing, however, our efforts to implemetit

the opt^-T,ality-criteria method have not been successful; since

there exist a number of different ways of implementing the

method, one should not conclude that it cannot be made to work

for rotor-blade design. Only the particular implementation we

have chosen appears in doubt. At present, we have no plans for

continuing work on this method, since it threatens to divert

effort from more promising topics.

Yet another topic of study during the reporting period was

the effect on natural frequency calculations when secondary

structural items such as shear deformation, restraint of warping

during twist, and filler stiffness are ignored. A simplified

elliptic blade profile, approximating a true helicopter blade,

11

..^ ...._.. _ .. ^, _.. a ,..► 
i

was studied to obtain rough estimates of modelling errors caused

by neglect of the secondary items. It was concluded that the

mathematical model we have been using should be quite accurate

(perhaps only a one percent change at most due to the

consideration of secondary items.). However, it is imperative

that accurate filler properties, dimensions, and locations be

known in order to represent the mass distribution with a

reasonable degree of precision.

A final topic of study, during the reporting period has been

the implementation of a faster subroutine for eigenvalue

calculation. Computer code for the subspace iteration algorithm

has been obtained and integrated into our blade analysis program.

Difficulties involving missing eigenvalues have surfaced,

however, and further development is required.

V

5. FUTURE WORK

Yn addition to continuing work in the area described above

(matching data for some actual rotor blades to finite-element

models, and then optimizing the blades), we also plan to work in

another area; that of investigating the validity of the

dimensional constraints we have chosen. In previous reporting

periods, we have described many examples in which constraints on

the thicknesses of the walls of the box beam have been present.

It can be concluded that such constraints can be handled in an

optimization procedure with little difficulty. Another, less

clearly defined, type of dimensional constraint is that of

constructibility; for example, if an actual blade is to be

constructed, abrupt thickness changes from one finite element to

12
13

1

the next present manufacturing difficulties. In the remaining

year of the research project, one area of investigation will be

to formulate, through consultation with helicopter manufacturers,

realistic constraints for constructihility.

a
14

6. APPENDIX - Derivation of Mass and stiffness Matrices

as Functions of Natural Frequencies

Define

[K*l = [ M l

-1/2 [K] [ ,1] -1/2

and construct a square matrix [U * ] by using the eigenvectors of

[K

*

 l as columns. If the eigenvectors are normalized to the

identity matrix, that is, if

[U

* l T C U * ] = [I] ,

it then follows that

[ U * ] T [ K * ] [ U * l = diag [ ( w i 2 ) ] ,

where wi 2 are the eigenvalues of [K*].

Next, let

[U] = [M]-1/2[U*],

from which it follows that

[U] T = [ U

*]T[M]-1/2,

[Ul -1 = [ U * ] T[M] 1/2,

[U] T [K] [U] = diag [ (wi 2 ) l ,

and

[U]T[M] [U] [I] .

Finally, then, the stiffness and mass matrices can be

written as functions of the eigenvalues, wi2;
[ M] _ [U]-T[U]-1

(M] 1 /2 (M] 1/2,

and

(K] = (U] `Tdiag( (wi 2 ) ] [U]-1

= [M] l/2 [U * ]diag [ (wil) ] [U * ] T [M] 1/2

= [M] [U] diag [ (wi 2 ) ] (U] T [M) .

Note that the eigenvectors, [U], and eigenvalues, wi2,

appearing on the right-hand side wera originally calculated from

the stiffness and mass matrices, [K] and [M]. If we consider only

relatively small changes in the frequencies, w i , then the

eigenvectors should relatively unchanged. Thus the last two

equations for [M] and (K] with [U] held fixed can be considered

as expressing the mass and stiffness matrices as explicit

functions of the natural frequencies.

15
0

i

7. References

1. Pitt, D., "Rotor .Dynamic Inflow Derivatives and Time

Constants From Various Inflow Models". D.S. Thesis, Washington

University, 1980.

2. Ko, T., "Use of Tapered, Twisted Finite Elements", M.S.

Thesis, Washington University, 1979.

3. Venkayya, V.B., "Structural Optimizatrlon: A Review and

Some recommendations", Int. J. for Num. Meth. Engrg., Vol. 13,

1978 0 pp. 203-228.

4. Khot, N.S., Berke, L., and Venkayya, V.B., "Comparison of

Optimality Criteria Algorithms for Minimum Weight Design of

Structures", AIAA Journal, Vol. 17, No. 2, 1978, pp. 182-190.

5. Khan, M.R., Wi 1 lmert, K.D., and Thornton, W.A., "An

Optimality Criterion Method for Large-Scale Structures", AIAA

Journal, 'Vol. 17, No. 7, 1979, p ,p. 753-761.

16
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