NASA-CR-169940 

19830010445 

A Reproduced Copy 

OF 

Reproduced for NASA 

by the 

NASA Scientific and Technical Information Facility 

FFNo 672 Aug 65 1111111111111 1111 11111 1111111111111111111 1111 

NF02590 
. ~~'" 

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SECOND SEMI-AtnruAL STAT1JS REPORT 

on 

NASA RESEARCH GRANT NO. UAG-1-250 

for 

Period Covering 16 July 1982 through 16 January 

"DESIGN OF HELICOPTER ROTOR BLADES FOR 

OP'I'IMtTM DYlWITC CHARACTERISTICS" 

by 

David A. Peters, Professor and Chairman 

. and 

Timothy Ku. Research Assistant 

Department of ~echanica1 Engineering 

Washington University, Campus Box 1185 

St. Louis, MO 63130 

Alfred E. Korn, Professor 

and 

-

Mnrk P. Rossow, Professor 

Department of Engineering and Technology 

Southern Illinois University 

Edwardsville, IL 62026 

February 7, 1983 

...... 

,. • 

I 

{ 

(NAS~-Cn-16g9~1) DE:'TGN OF IlELICCF'lEP ROTOR 1'183-18716 

ELAD"!:S Fen Cr~HllJ'-I DYtnU'\IC C'IIIHC"'!HIST!CS 

5eDiannuill St,lt\1;, Report, 16.Jul. 1982 - 16 

Jan. 1981 (Washinqton Univ.) '31 p Unclas 

HC ACS/MF 710' C5CL 0 Ie G3/05 08769 

,", 

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TABLE OF CONTENTS 

1. Introduction. 

l.~ Overview of Research Project 

1.2 Overview of Current Report •• 

2. Experiments with the CONMIN Program and Parameters. 

2.1 The Basic Problem: Two Frequency Constraints. 

2.la Analytical Gradients vs. Finite Differences. 

2.lb Various Forms of the Frequency Constraint Function 

2.lc Influence of Initial Design on Convergence and 

Optimal Design • • • 

2.ld Parruneters: ITMAX, ITRM, DELFUN, DABFUN • 

2.1e Parameters: THETA, PHI. 

2.2 Placement of Frequencies (More Severe Constraints) 

2.2a Further Studies of Convergence Parameters. • 

3. Optimal Designs for a Non-Rotating Cantilever • • • 

3.1 Optimization with Three Fre~uency Constraints. 

3.2 Addition of ~umped Weights •••••• 

3.3 Addition of Auto-Rotational Constraint 

4. Optimal Designs for a Rotating Cantilever Berun. 

5. Numerical Aspects of the Problcm. • • • 

5.1 Convergence with Increasing Number of Elements 

5.la Convergence of Frequencv •• 

5.lb Convervence of Optimal Desir-n •• 

5.2 Accuracy of Eigenvalue Calculations. 

5.3 Sensitivity of Frequency to !ir.lall Changes in Thickness 

. . . . 

Page 

1 

1 

4 

5 

7 

9 

10 

12 

13 

14 

15 

15 

17 

17 

18 

21 

23 

25 

25 

25 

26 

29 

31 

'~ .. 

,'" 
6. Concluaions. 

7. Near Future Plans. 

8. References • 

9. Appendices • 

10. Tables . 

11. Figures. 

TABLE OF CONTENTS (Cont.) 

Pago 

33 

35 

36 

37 

40 

70 

.', 

\ 

i 

• I 

I I 

~ 
1. InTRODUCTION 

1.1 Overview of Research Project 

The design of helicopter rotor blades involves not only considerations of 

strength, survivability, fatigue, and cost. but also requires that blade I \ 

natural frequencies be significantly separated from the fundamental ap.rodynamic 

forcing frequencies (e.g. Ref. 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 rather 

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

apart. Second, the rotor RPl-1 mav vary over a signific311t range throughout the 

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

frequencies. TIlird, 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 ot natural frequencies. In fact, the dependence on pitch angle 

makes frequencies a function of loading condition. since loading affects 

collective pitch. Fourth, the centrifugal stiffness often dominates the 

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

stiffness. 

In the early stages of the development of the helicopter. it was believed .-" 

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

choice of structural coupling and mass stiffness dist,,:ibutions. Uowevet'. it 

is easy to imagine how difficult it is to find ,1ust the proper parameters such 

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

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

designs which ignore frequency placeMent. Then, after the structure is 

! 

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-2-

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

calculated (or meaaured) and final adjustments ~~de. Roference (2) describes 

the development of the XlI-17 helicopter in which a 300-1b weight W3S added to 

each blade in order to change the spanwise and chordwiae mass distribution and 

thereby move the first f1apwise frequency away from 3/rev. However. these 

types of alterations are detrimental to blade weight, aircraft development 

time, and blade cost. In addition, corrections usual1v ar~ not satisfactory; 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 prelioinary 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 ~ertain restrictions. the exact EI 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 freedom as to how 

this weight may be rlistributed. Second, the methods of structural optimization 

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 that 

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

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

published, some companies are already e:~rerimenting with the optimum way to 

add weight to an existinR blade in order to improve v:l.brations. 
-3-

" 

'.' 

The purpose of the work discussed here is to inv2stigate the possibilities 

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

to give an optimum blade design in termo of weight. inertia. and dynamic 

characteristics. 

The major objectives of the work are: 

1) To dotermine to \Ihat extent changes in maGS or stiffness diatribution 

can be used to place rotor frequencies at desired locations. 

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

3) To formulate realistic constraints on blade properties based on weight, 

mass moment of inertia, size, strength, and stabil!t~. 

4) To determine to' what extent the hub loads can be minimi:ed by 

proper choice of E1 distribution. 

5) To determine if the design for minimum hub loads can be approximated 

by a design for a given set of natural frequencies. 

6) To determine to what extent aerodynamic ~ouplings might affect the 

op~~mum blade design. 

7) 1~ determino the relative effectiveness of mass and stiffness distribution 

on the optimization procedure. I 8) To determine to what extent an existing blade could be optimized with 

minimal changes in blade structure. 

9) To develop severnl "optimum profiles" for rotor blades opernting under 

various standard conditions. 

The work Is to focus on confiRuratlons that nrJ simple enough to yield 

clear, fundamental insights into the structural mechanisms but which are 

,;ufficiently complex to result in n realistic result for nn optir.lum rotor blade. 

1 

• ________ ._ ....... ""'-...... ___ ............ _ ... ' .......... ~~ .... ""' .... ' ........................... c-. _ ... _Hi ........... __ .... ·_.-.d-.. I;,,'-;-~; • ..:..J .\;L-.....t;;~'-:r-~_ '-
-4-

1.2 Overview of Current Report 

This second s~i-annual report Derves two purposes:' 1) it inforos our 

sponsors and other interested parties what we have accomplished during the 

last six months, and 2) it serves as an archive of data to which we expect 

to refer frequently in coming months, as our research proceeds. As a result \ 

of the latter, archival purpose, the report contains much information which 

is not necessarily new, but which needs to be recorded in an orderly fashion 

so that it may be easily accessed in the future. 

The first section of the report details our experience with the CONMIN 

optimization program applied to the problem of finding the optimal design 

of a vibrating cantilever beam (see Fig. 1). This section gives the results 

of parameter changes, the results obtained with various constraint forumulations, 

and the effect of allowing lumped weights at discrete points. Other aspects 

that are studied include the autorotational constraint and the effect of 

rotation. The principal conclusion of these investigations is summarized 

as follows: CO~IIN works reasonably well for all problems we have considered 

so far. 

The second part of the report discusses some numerical aspects of the 

problem. One important aspect is the effect of the rlmber of finite elements 

on both the frequency calculation and the sequence of optimal designs. Also 

of importance is the effect of errors in the eigenvalue analysis as well as 

the sensitivity of the frequency to small changes in blade dimensions 

(manufacturing tolerances). The conclusion to be drawn from these numerical 

studies is that if reasonable care is taken, nuwerical difficulties arc not 

signifif ant for the problems we have considered thus far. 
··5-

2. EXPERUfENTS WITH THE CONllIN PROGRAM AIm PARAMETERS 

The CONMIN (Ref. 6) program was employed to minimize the weight of the 

cantilever beam, described in the introduction. Before extensive optimization 

problems are solved, it is necessary to experiment with the progr~ parameters 

and determine the best values for the particular class of p~oblems at hand. 

The problem chosen for the numerical experiments involved minimization of 

weight subject to two frequency constraints and to side constraints on the 

thickness. The non-rotating cantilever bcam undergoing flapping vibrations 

was examined. Specifically, the following questions were to be answered: 

1) Do the gradients produced by analytical techniqueo match those 

obtained by finite differences? 

2) Is there any difference between results obtained with constraints on 

the frequency in Hz. and those obtained with constraints on the square 

of the circular natural frequency in (rad/sec)2? Can scaling of the 

constraint function improve convergence? 

3) Under what conditions do the starting values for thickness influence 

the convergence properties of the problem? Can an original, infeasible 

design (i.e., one which violates one or both of the frequency constraints) 

be expected to converge to a feasible, optimal design? 

The determination of the so-called "optimal design" in CONHIN is influp.nced 

by several important parameters. Changes in the parameters can change the 

final answer and can influence the speed of convergence. The most important 

parameters are ITRM, DELFUN, DABFUN. Convergence is defined whenever ITRM 

consecutive iterations are encountered such that the v~lues of DELFUN or DABFUN 

(or both) are less than the stated values input to the prograc. The parameters 

and default values are defined aR follows: 

\ 

.-
-6-

ITRM: Default Value - 3. Number of consecutive iterations to indicate 

convergence by relative or absolute changes (DELFUN or DABFUN 

respectively). 

DELFUli: Default Value - 0.0001. }linimum relative change in the objective 

function to indicate convergence. DELFUN - ABS (1.0 - OBJ(J-l)! 

OBJ(J», where the objective functions for the current, J th, 

st iterate and the previous,J-l ,iterate are tested. 

DABFUN: Default Value - 0.001 times the initial objective function 

value. Minimum absolute change in the objective function to 

indicate convergence. DABFUN - ABS (OBJ(J) - OBJ(J-l». 

Note that a practical criterion for convergence is employed by the program. 

Thus, slight differences in answers can be expected if problems are started 

from different initial points, or if different values of ITRM, DELFUN, and 

DABFUN are employed. 

The parameter CT is used to define whether or not a constraint is active. 

th The exact satisfaction of the J constraint, G(J) - a is numerically unusable. 

Rather, a band, CT in magnitude, on each side of the exact zero is employed. 

The default value and formal definition of this parameter is: 

CT: Default Value - -0.1. th The J constraint, G(J). is considered to 

be active if CT ~ G(J) ~ ABS(CT). The value of CT is sequentially 

reduced in magnitude during the optimization process. 

The parameter THETA is called the "Push Off Factor". and is used by the 

programmed Method of Feasible Directions (Ref. 7) to go from one feasible 

design to another feasible. improved design. The default va~ue is 1.0. 

Larger values of THETA are appropriate for highly non-linear constraint 

functions. Lower values are appropriate as the constraints approach linear 

functions. 

.-

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::·i·1 ~,.,.:~.-' 
-7-

The parameter PHI controls how quickly an infeasible design will be 

mo"rod in the direction of the feasible region. The default value • 5.0. 

Values of PHI above 5 should be employed if a feasible solution cannot be 

obtained. 

Finally. the value of ITMAX defines the maxicum number of interations 

in the optimization process. If a solution cannot be obtained in ITMAX 

iterations. the program is teroinated. The default value is 10. 

Herein. experiments with the parameters ITRM. DELFUN. DABFUN. CT, THETA. 

PHI and ITMAX were performed to enable definition of the numerical values 

that best fit the class of problem at hand. There are numerous other 

parameters within CO~~IN. but in this study the default values for these 

other parameters were considered to be adequete. 

2.1 The Basic Problem: Two Frequency Constraints 

The problem to be considered is the non-rotating cantilever shown 

in Figure 1. Each 24 inch long element has a different thickness of 

flange. Letting tl be the thickness of the flan~e nearest the fixed end. 

element thicknesses are numbered in order. such that the tip end has a 

thickness denoted by t

lO• Corresponding values of moment of inertia of area 

are denoted by 11 th~JuSh 110• A material density and modulus of elasticity 

of 0.1 lbf/cu.in. and 10 x 106 psi are used. 

(in2) (lbf) It can be shown that the area. Ai • weight. Wi • and 

moment of inertia, I (in4) are the following functions of the thickness, i 

t (in). 

i . 

Ai - 0.50 + 7.6 ti 

Wi - 1.20 + 18.24 ti 

25 ti 2 

Ii - 96 + 24 {28S - 228 ti + 60.8 ti } 

(1) 

(2) 

(3) 

- .- * 

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-8- OF'poon QUAUl"l 

The thicknesaes are constrained to lio within the range 0.05 inches 

to 1.25 inches (the value for which the crOGS section would be a solid 

rectnngle). 

A uniform cantilever uith a thickness of 0.10 inches for each flange 

would have the follouing first tuo frequencies: 

fl - 1.98 Hz 

f2 - 12.4 Hz 

To ensure that a feasible starting solution could be obtained with at least 

one chosen thickness. the frequencies are constrained to be within ± 0.2 Hz 

of the frequencies 1.98 and 12.4. Thus. 1.8 ~ fl ~ 2.2 and 12.2 ~ f2 ~ 12.6 Hz. 

The mathematical progr~ng problem becomes: 

10 10 

Min.W - Min r Wi - 12 + 18.24 t ti 

i-I ~l 

S.T. 1.8 Hz ~ fl ~ 2.2 Hz 

12.2 Hz ~ f2 ~ 12.6 Hz 

and. 0.05 ~ ti ~ 1.25 

i M 1.2 •••. 10 (4) 

The above problem has ~!n thickness decision variables. To preserve 

generality for later work with more complex sections. it was decided to 

use Moment-of-Inertia decision variables. The problem then becomes: 

10 

Min.Z - E Q(Ii ) 

i-I 

S.T. Gl - 1.8 - fl ~ 0 

G2 - fl - 2.2 < 0 

G3 - 12.2 - f2 ~ 0 

G 4 - f 2 - 12. 6 ~ 0 

and, 0.83073 ~ Ii ~ 5.20833 

i ... 1,2 ••. 10 (5) 

' , .. 

' . • ,c 

• ~I 

.. '~ 

.-

I 
In the above formulation, the i th thiclr.ness has been written as a· 

non-linear function. Q, of the ith moment of inertia 

(6) 

The problem is for~u1ated in terms of ten moment of inertia decision variables, 

11 through 1

10, four frequency constraint functions, and ~enty side constraints. 

The objectiv~ fur.ction and frequency constraints are non-linear tunctions of 

the decision variables. 

2.1a Analytical Gradients vs Finite Differences 

The CONMIN program has the option to compute gradients of the objective 

f~ction and constraints via Finite Differences. If possible, however, it is 

more efficient for the user to provide analytically-derived gradients. In 

Appendix I, the method used to obtain gradients is given in detail. By using 

both methods on the same problem, it is possible to provide checks on the 

deriv."ltion and programming of the analytical gradient method. 

Tables lA and 1~ present the results from using the two options. An 

original design with constant thickness of 0.05 inches leads to the designs 

in Table lA, whereas a starting thickness of 0.10 inches is the basis for 

Table lB. It is noted that the finite difference method leads to essentially 

the same results as obtained with analytical gradients. 

The four designs all converge to a common design defined bv: 

1) A thickness of element 1 of aobut 0.086 inches - 0.091 inches 

2) The thickness of all other elements are governed by the 

lower bound constraint (0.05 inches) 

3) The second natural frequency is governed by the lewer bound 

constraint (f2 - 12.2 Hz) 

4) The first natural frequency is not actively constrained, and 

ranges from 1.94 to 2.00 Hz. 

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I 

J 

1 

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-10- OF POO~ QJ,.I.:'lY"{ 

The sensitivity of the first (lowest) frequency to s~lll changes in 

thickness will be considered later, along with the accuracy of the single 

precision routine used for eigenvalue extraction. 

It is instructive to make a further check on the solutionb by running 

an optimization problem with Jn1y one decision variable (Ii) and one 

frequency constraiat (f2 ~ 12.2 Hz). All other thicknesses are held 

constant at 0.05 inches. The S'l1.ut:ion is started with ti ., 0.10 i4ch"!s, 

and in 6 iterations converges to the results that follow: 

4 I i (ti );1.256 inches (0.0902 in~hes) 

Ueight: 21. 85 1bf 

These results can be considered to be the values toward which all four 

runs should approac~ Within the limits of numerical accuracy, all four 

cases do converge to the values obtained in the calibration run. 

Hereafter, all analyses are based on the use of analytical gradie~ts. 

2.1b Various Forms of the Frequency Constraint Functj.on 

The constraint ~e1ations, described earlier, have been 

G1 - 1.8 - fl2. 0 

G2 - f1 - ::.2 2. 0 

G

3 os 12.2 - f2 :. 0 

G4 = f2 - 12.6 .. < 0 

where, f

1, f2 are the natural frequencies in Hz. (7) 

The authors of CO~~IN recommend use of constraint functions tpat are 

mutually of the same order of magnitude. and that pcoperty is satisfied 

by the formulations in Eqn. 7. 

I 

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I 

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-J 

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i' ':" l"" ,;;,;.;; c~ " '.-j ... ....a. .. !j.~-~~~ ... d.J:I,. ........ ~I.~;- • .>!.t .. 
....... 

-11-

It is instructive to sec whether the use of eigenvalue conatrail,ts 

alter the convergence of the problem. 

Let ~i • ith eigenvalue 

2 

• wi (the square 

(8) 

of the circular natural frequency) 

2 If ~i is extracted in units of (rad/scc) , the alternate form 

of the constraints becomes 

G; - (1.8(2n»2 - ~l 

* 2 G2 • ~l - (2.2(2n» 

G; • (12.2(2n»2 - ~2 

c: · A2 - (12.6(2n»2 

where, Al , ~2 are the eigenvalues in (rad/sec)2 (7a) 

• The Gi constraints in Eqn. 7a nrc no longer expected to be of the same order 

of magnitude and convergence difficulties l1\ay result. 

To test the convergence properties, the two forms of the constraint 

function arc used on a problem with initial thickness of 0.05 inches. As 

seen in Table 2, the formulation with eigenvalue constraints does converge, 

but the optimal solution is not as good as that obtained by using frequency 

conatraints • 

When the eigenvalue constraints arc used on a problem starting with 

t • 0.10 inches, a very poor optimal solution is obtained. A good optimal 

solution could b~ obtaiucd by scalinR the objective functions hy 10 ;lIld 100. 

11<. 

G '" l • (\/10 

"'''' 11< r.2 • G,,/lO .. 

*'" ... G) • G)/lOO 

"'''' « (~4 • G4/1OO (71)) 

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, • ¢ .'h .t,.,_ v ",W. • 4 WW..,. 'if. 'i, ' " ~> 

-12-

Alternately. G* i constraints could lead to a good optimum if the constraint 

thickness, CT. were changed from -0.1 (the def~ult value) to -800.0. 

When the eigenvalue c~a~traints are used on a problem starting with 

t - 1.25 inches, no feasible design is found. Attempts to move tovard a 

feasible design by changing the parameter PHI from 5 to 50 to 150 are , 

. , 

I 

unGuccessful. , , 

"7 

For all further studies, frequency constraints of the form shown 

in Eqn. 7 are employed. It is to be noted that if troubles are encountered, 

a more efficient form of Equation 7 can be employed to ensure objective 

function values of the same order of magnitude. 

and (fi/fiU) - 1 ~ 0 (9) 

where, f - th in Hz i i frequency, 

fiL - lower bound on ith frequency, in Hz 

fm - upper bound on ith frequency, in Hz 

" 2.lc Influence of Initial Design on Convergence and Optimal Design ~. 

Consider the three following initial designs: 

Case 1 - constant thickness of 0.05 inches 

Case 2 - Constunt thickness of 0.10 inches 

Case 3 - Constnnt thickness of 1.10 inches 

In Case 1. the initial design violates the lower bound on f2' while in 

Case 3. the lower bound on both frequencies is violat~d. Only in Case 2 

does the initial design result in a feasible initial solution. 

The information for the three runs is presented in Table 3. In all 

cases, feasible solutions were readily obtained, and eventually, the optimal 

1 
-13-

solution was obtained. If tho most difficult case (Case 3) had been 

required to meet more rigorous convergence criteria, a few more iterations . .. 

would have resulted in an improved optimal design. .. , Certainly, initial designs which are feasible and close to optimal ~~' 

\ 

.' are ideal. nut it is possible to start with designs which are not feasible, 

and which are far from opti~l. Unfortunately, such conclusions are problem ..... , 

dependent. For more severe frequency constraints, or for added problem 

conacraints, it may be necessary to start with feasible or close-to-feasible .. 

;, 

designs in order to optimize. , 

, 

., 

2.ld Parameters ITItAX, ITRM, DELFUN, DABFUN .~ 

, 

All data collected are based on the following values of the CONMIN , 

convergence control parameters. < ~ 

Il1MX: Maximum number of iterations (default value - 10) 

Values used: 40, 80 

IT~t: Consecutive iterations for convergence (default value - 3) 

Values used: 3, 5, 8 

DELFUN: Relative change parameter (default value • 0.0001) 

Values useJ: 0.0001, 0.00005 

DAllFUN: Absolute change parameter (default value - 0.001 tines the 

initial objective function) 

Values used: 0.011 (default valu:! for t • 1.10 inches) 

0.001 (default value for t .. 0.10 inches) 

O.OGOs (default value for t • 0.05 inches). 

In general, the d~fault v~tlues for ITRM. DELFUN, nnd DAB FUN le.:\d to 

good convergence prop(!rties. In about half of the runs. the objective 

function does not change. or just barely changes. during the last ITRM 

iterations. The other half have conver~ence governed by the DABFUN parameter. 
..... 

-14-

Whon larser thicknessen ara used, higher initial objcctiv~ function reQulta, 

nnd tho dofault valuo for DAEFUN can bo lerger then deoited. 

To teot the ~dequacy of the dofnult valuos, Caso 2 in Tabla J oa~ be 

rerun with tightor controls. The results. iterAtion by lteration. aro shown 

in Tabla 4. Tho tishtar controllod ~un o8fiontially doubles the n,~ber of 

iterations, and halveo DELFUN nnd DAEFm~. Yet, tho fin~l ro,ulta ara 

essentially identical. 

The reccmmended parameters for Buch rllntl are: 

tTl-lAX • 40 

tTRM - J (default value), or 5 

DELFL~ - 0.0001 (default value) 

DABFtlN - default value (0.05 inchen < t ~ 0.25 inches) 

• 0.0025 (t ' 0.25 inches). 

The pnrnmeters stated mny not be upplicable either fllr l:.rRer problom:t. or 

for more severoly conntrained problems. The parameter will bo critically 

examined nt severnl stnges of tho IItudy. 

2.1c Pnrametcrs THETA, PHI 

111~ht1r values of THETA (Tho Push Off Fuctor) nrc roc:ommcnded for hiShly 

non-linear conl-ltraint fllnctiona. The dl'fulIlt vnllle (0 • 1), 10, 100, Hnll 

700 wero tried on d specific initial dcsiRn (t • 0.10 inches 4 t • 1.355 inches ). 

Th", results Ilrc ::hown 1n TIlbl0 5. Clo!lI\'"ly, chllonAing tlw pnrllmctl"lr from the 

dt,Cnlllt vnllltl dot's not improve tho r:1plditv llf tho con\,tlrRt'nc~ or tho qlllllitv 

5 (tIll' dl'fault vllhll!) to 50 lind t5n did "lIt l'n:lblo nn initlallv lnft'ludhlt' 

'-... ~-~- ...... -..... -; 

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OF POOu QUriLilY 

Further designs are hereDiter based on default values for THETA and 

PHI. 

2.2 Placement of Frequencies (More Severe Constraints) 

The constraints defined by Eqn. 7 can be generalized as follows: 

(10) 

th where, the lower and upper bounds on the i frequency, 

in Hz, is given by fiL and f

iU respectively. 

Three cases are examined, as follows: 

Case 4: Cases 1, 2, 3 were constrained by a band of ± 0.2 Hz 

around 2.0 Hz and 12.4 Hz. The band is now narro~1ed to 

± 0.1 Hz. 

Case 5: The first two frequencies are separated, such that a 

± 0.2 Hz band is defined around 1.7 Hz and 13.0 Hz. 

Case 6: The first two frequencies are brought closi!r together, such 

that a ± 0.2 Hz band is defined around 2.3 Hz and 11.8 Hz. 

The results of the optimizations are shown in Table 6. Thus, within reason, 

it is possible to re"proportion initial designs (feasible or infeasible) such 

that frequencies are placed where desired during an optimization of weight. 

2.2a Further Stujies of Convergence Parameters 

The recommended IT1L\X, IT~~, DELFm~, and DABF~ parameters (30, 3, 

0.0001, 0.0025) are used in Case 4. The convergence of the objective 

function to 3 significant figures seems to be incomplete in the 3rd figure. 

., 

'j 

J 

1 

I 

I 

1 

J 

i 

i 

• j 

I 
.~"'--. 

-16-

For Case 5, ITRM is changed from 3 to 5 and a surprisingly substantial 

reduction in objective function is obtained. Five iterations appear to be 

appropriate to maintain objective functions of 3 significant figures. For 

Case 6, another check (not shown in Table 6) was made by tightening the 

parameters to 40, 6, 0.00005, 0.0001. After 40 iterations, the convergence 

criteria had not been satisfied, but probably would be in another few cycles. 

The objective function was ~n1y changed from 0.593 to 0.590, but elements 

2 and 3, originally 0.0703 and 0.0886 inches, were appreciably changed to 

0.0832 and 0.0791 inches respectively. The objective function is quite flat 

near convergence, and small changes in objective function can be accompanied 

by appreciable changes in structural configuration. The accuracy of 

eigenvalue extraction, herein done by a standard library routine in single 

precision, obviously has an effect on the defined final configuration. 

A numerical study of eigenvalue calculations is presented in section 5. 

.' 

j 

I 

I 

1 

! 

1 

1 

j 
-17-

3. OPTIHAL DESIGUS FOR A NON-ROTATING CAtITILEVER 

With the experience gained by studying ~o frequency constraints, it 

is possible to intelligently formulate the more difficult problems that 

follow: 

1) Added Constraints - A third frequency 

2) Addcd Design Variables - Non-structural lumped mass 

3) Added Constraint - The auto-rotation constraint (i.e., minimum 

mass moment of inertia about an axis norms1 to the beam, and 

located at the root). 

3.1 Optimization with Three Frequency Constraints 

In addition to the c~nstraints shown in Eqn. 10, two more must be 

added: 

where, the lower and upper bounds on the 3rd frequency, 

in Hz, is given by f3L and f3U respectively 

Three cases are examined, as reported in Tables 7A and 7B: 

(11) 

Case 7: A.±. 0.2 Hz constraint band is pls.ced around the first three 

frequencies of the initial design. 

Case 8: The spread between the desircd values of fl and f2 is 

narrowed, and the desired value of f3 is decreased by 

1.0 Hz from that defined in Case 7. 

Case 9: The spread between the desired values of fl and f2 is increased, 

and the desired value of f3 is raised by 1.0 Hz from that 

defined in Case 7. 

.. , 

.~ 

l 

t 

1 

j 

I 

1 

I 

i 

1 

l 

.1 

I 

I 

j 
-18-

The more difficult optimizations (Cases 8 and 9) are repeated with different 

starting points, and (in Case 9) with different convergence criteria. Convergence to an estimated 99% accurate value of objective function is 

accocplished for runs using ITMAX • 80, ITRM • 5, DELFUN • 0.0001, and 

DABFUN • 0.0005. Designs initiated with various constant thickness values 

converge to similar optimal designs. For example, Cases 9B and 9C both 

result in minimum thickness for elements 2, 3, 4, 5, 9, and 10. Despite 

identical objective functions, individual thicknesses for elements 6, 7, 

and 8 in Cases 9B and 9C are far from identical. Again it is noted that 

near optimum there can be appreciable changes in structural configuration 

with very minimal effect on the objective function. 

3.2 Addition of Lumped Weights 

'l'he introduction of lumped weights (assumed to have mass, but not to 

contribute to the moment of inertia of area, I, used in defining the stiffness 

matrix) increases the number of decision vA~i3bles in the optimization 

problem. For one such weight at the center of each element, there are now 

twenty decision vari3Lles: 

Il(root element) through 110 (tip element) 

W1 (weight on root clement) through \.r10 (weight on tip element) 

. ~ 

" " 

I 

I 

1-· 

" • I 

'I 

I 

1 

J 
-19- ORIGINAL pr .. m~ [3. 

OF POOR QUALiTV 

The problem is formulated as follows: 

10 10 

Min Z • 182.4 y r Q(Ii ) + r tli 

i-1 i-I 

and. 0.83073 ~ Ii ~ 5.20833 

i-1.2 •••• 10 

o ~ Wi ~ 100 

i .. l.2, ••• 10 (12) 

The newly defined symbols in Eqn. 12 are as follows: 

y - element density, lbm/cu.in. 

Wi • weight of lumped mass at ith element 

center. Ibm 

Very minor modifications are necessary to include the analytical 

gradients of the objective function with respect to the lumped weights 

and of the constraints with respect to the lumped weights: 

dZ 1 aW

i -= (i=-1,2 •••• 10) 

()K -. [0] aWi 

(13) 

" 

! 

i 

I 

i 

I 

I 

I 

I , 

.-

j 

1 

'I 

1 

1 

j 

1 

1 

l 

j 
where. 

-20-

Z is the objective function 

K is the stiffness matrix 

M is the mass matrix 

I is the identity (unity) matrix 

2 8 is the acceleration of gravity. 386.4 inchea/sec 

The two ~trix derivatives in Eqn. 13 are used to define the partial derivative of the eigenvalue with respect to the decision variable. (dAi/dWK). 

The remainder of the operations are outlined in Appendix I. Thus. the 

optimi:ation can still be based on analytical gradients. 

Three cases have been invcstigated. The results are shown in 

Tablcs SA and 8B. 

Case 10: A previous design with t - 0.10 (Case 7) is modified by 

reduction of the density from 0.10 Ibf/cu.in. to 0.05 Ibf/cu.in •• 

and by replacement of lost weight bv 1.512 Ibf lumps at each 

element center. There is more freedom to choose the decision 

variables in Case 10 (i.e •• the non-structural mass can be used 

efficiently to move frequencies). The final result is a 

constant-section beam of minimum thickness with lumped weight~ 

as shown in the Table. The optimal weight is onlv about twothirds that of the optimal weight for Case 7. 

Case 11: Two runs were made with only one difference in initial design. 

In 11A, 1 1hf lumped weights were used at each node, and in 

lIB lumped weights were not used. The final optimal designs 

were almost identical in all respects. Since all of the initial 

frequencies had to he raised, it was most efficient to remove 

all of the lumped "'eights of Case llA. 

.' 

J 

• I 

j 

i 

i 

I 

1 

t 

j 

I 

1 , 
L ',y 

-21-

Case 12: The initial design of case 1lA was used with a compressed 

range of frequency constraints (from 1.8-35 to 2.6-32 Hz). 

The optimal design involved a combination of thicknesses 

greater than the minimum, and luoped weights. The tip 40% of 

the beam was made of minimum thickness elements without 

lumped weights. 

Convergence for all the runs was excellent. Note that since the objective 

function is now weight (with numerical values of 20 to 30 lbs) previous Iv 

used values of DABFUN are not appropriate. Herein, DABFUN was raised to 

0.001. The other parameters were kept the same as before. 

3.3 Addition of Auto-Rotational Constraint 

This constraint is intended to be applied to rotating systems. However, 

the constraint is added here as the next step in developing the larger 

problem to be considered. Denoting the minimum mass moment of inertia about 

a vertical axis through the root of the beam as Ii' and the actual moment 

mn 

of inertia as I , it is required that I > Ii' Rearrangi,1g the information m m - m n 

into a more usable form, the seventh constraint, to be added to Eqn. 12 

becomes: 

G7 - 1 - (I /1 i ) < 0 m m n - (14) 

The gradients of G7 with respect to the decision variables are analytically 

obtained, as shovn in Appendix II. 

Twc optimization runs are made. In the first run, Case 10 is re-run 

with a I i value of 500 lbf inches m n 

2 

sec • This value is deliberately chosen 

to be low, such that the constraint remains inactive throughout the run. 

As exp'ected, the results reoained identical to that in Case 10. The optimal 

2 design has an Im value of 807.7 Ibf inches sec. The next case is reported 

\ 

" 
in Table 9. 

-22-

A value of Ii. 1100 Ibf inches sec

2 is demanded, such that 

mn 

the auto-rotational constraint is active during the optimization. For CODparison, the new Case 13 and the previous Case 10 (uithout auto-rotational 

constraint) are included in Table 9. Although satisfaction of the formal 

convergence cri~eria is not met in 80 iterations, the optimization is close 

to being finishei. (DF.LFUN and DABFUN ~ 0.0004 and ~ 0.005, respectively, 

for the last five iterations). As expected, the tip element is thickened 

and the lumped weight increased, since that is the most efficient way to 

satisfy the auto-rotational constraint. At optimum the auto-rotational 

constraint was, for all practical purposes, one of the active constraints. 

\ 

" 

l 

1 

I 

t 

i 

-j 

I 

I 
-23-

4. OPTIMAL DESIGNS FOR A ROTATUm CANTILEVER BEAM 

If the cantilever beam is rotating in a horizontal plane, centrifugal 

forces are created which stiffen the system and increase the natural 

frequencies in vertical vibration. The first natural frequency has as a 

lower limit equal to the speed of rotation. Thus, for high speed of 

rotation, placement of frequencies must be done w:f.th due consideration of 

the centrifugal effects. These trends are shown in Tabl~ 10. For 300 RPM, 

one bound on the fundamental frequency is 5 Hz. Despite vast ranges in 

thickness and lumped weights, fl ranges only between 5.1 and 5.7 Hz for a 

uniform cantilever. Frequency placement is much less dependent on stiffness 

and mass distribution than for a non-rotating beam. The calculations in 

Table 10 are based on Ref. 8. 

The optimization formulation remains the same as shown in Eqn. 12 

and supplemented by Eqn. 14. The major modification required involves the 

contribution of element tension to the element stiffness matrices. The 

added contribution also cau.ses a modification of the frt::::uency constraint 

gradients. Some details of the derivations are shown in AppendiX III. All 

eigenvalue calculations are based on double precison routines. 

In Table llA, results for three optimizations are shown. The single 

difference in input is the speed of rotation. 

Case 14: This run is a repeat of the single precision run of Case 10. 

The non-rotating beam solutions are almost identical. The 

major difference is a slight re-crrangcoent of the lumped 

weights. 

I 

.-1 

i 

.~ I 

J 

.1 

j 
-24-

Case 15: The speed of rotation is the low value of 30 RPM. There is 

no difficulty in placing the frequencies in the same range 

as required for the non-rotating beam. The optimization 

results in a slight increase in weight ov~r that for the 

previous case. 

Case 16: With a speed of 100 RPM, the p~oblem docs not fully converge ,-

in 80 iterations. The objective function appears to be 

accurate to two (rather than the requested thr~e) decimal 

places. The tip element alld tip weight are now relatively 

large. The extra mass is needed to lower the frequency and 1 

counteract the effect o{ the high speed. I 

With speeds of 300 RPM, the requested frequency placement 

must be modified. Also. the convergence criteria is relaxed. 

as shown in Table lIB. 

Case 17: This 300 RPM run converges in 39 iterations. The convergence 

criteria is satisfied by the DABFUN requirement (3 consecutive 

iterations with absolute change in objective funct~Jn ~ 0.01). 

Again, a large tip mass is needed. but that may be required 

to satisfy the more demanding auto-rotational constraint. 

The use of l~ped madS appears to be the more efficient ~ay 

of controlling the fequency placement. In particular, masses 

are placed at the t~p and near the zero points of the 2nd 

and 3rd mode shapes. In contrast, changes in thickness modify 

both stiffness and mass, and are less effective in ~~rturbing 

the frequencies. 
· . 

f>:',:"-·::;f ).f'; '.\-'j%~~:.'~~i ,~·t,t:",:~~",'!\)~'~~'P P/f.:;->¥'¥i:{ ;t·~{$.:t·L~ t'!'~~jOt!!~.t;t::(t::,"5~?'%'i7,":~,,"4.}~,,!i?:~Sf;'N~;;~ ,\~5":\ !! ~,,~~#.l{'!i.Je-i;t~,":'::0\ ~';"";''''J;~'f..;., 

.... y •• 

-25-

5. NUHERICAL ASPECTS OF THE PROBLEM 

The results reported in Sections 2, 3 and 4 are encouraging. since 

they demonstrate that. given a mathematical representation of a cantilever 

beam under various conditions, the CONMIN program can produce an improved 

design. That is. at least for tho problems considered thus far, nuoerical 

optimization is possible. Clearly. however, the design found through the 

use of r.ONMIN will be of no usc if the mathematical representation of the 

structure is at fault. Thus, in the present section. several numerical 

aspects of the accuracy of the analysis model are considered. 

5.1 Convergence with Increasing Number of Elements 

One aspect of the mathematical representation of a structure with the 

use of finite elements is the question of how many elements arc required 

to obtain an acceptable accuracy. For the present optimal design studies, 

this questio~ takes two forcs: 1) are enough elements used to predict the 

frequencies accurately. and 2) are enough elements used to describe the 

optimal deSign (that is. will essentially the same optimal design result 

if the mesh is refined)? The first question is addressed in Section 5.1a; the 

second in Section S.lb. 

5.la Convergence of Frequency 

To study convergence of frequency with increasing number of elements. 

a vibrating cantilever beam i9 considered. The beam is non-rotating. The 

a:talyticul solution for the frequency in cps is known to be (Ref. 9) 

? 

N~ iF-EI f·--')~w 2nL-

(15) 

'-.:t., 

.~. " 

1 

1 

I 

I 

j 
,~~~ .... '. : . 

-"-.- . 

-26- '" 

. :'. 

" 

where (NL)2 • 3.515 (first mode) .-

\ 

.... 

,i: • 22.4 (second mode) 

L • length of beam 

I 

g • acceleration of gravity 1\ 

I 

I , 

I ~ 

E • elastic modulus 

I • moment of inertia of cross-sectional area 

I 

i W - weight per length of beam I 

'\ 

.~ 

For the example under consideration. 

I • 20 ft 

.:: • 107 Ibf/ft2 

I - 1.3555 inches4 

W - 1.5120 lbf/ft 

The results of the finite clement analysis for v~rious numbers of 

elenlents are shown in Table 12. It can be seen that a~ few as six elements 

gives a good approximation (less than one percent error) for the first 

frequency. As would be expected. the approximations for the second frequency 

are not as accurate, but the error is only about two percent when ten ~lements 

are used. In general, these results indicate that the choice of ten elements 

in the optimization studies described in Sections 2, 3 and 4 is justified. 

S.lb Convergence of Optimal Design 

To study how the optimal design changes as the numb~r of element!] 

increases, a cantilever beam with "N" elements and with lumped weights added 

at the nodes but otherwise similar to the beam of SEction S.la and Flgure 1 

is considered. The density, and the constraints on the natural frequencies, 

lumped weights. and moments of inertia are 
1""', i P,", ,LP. ,." ........ ,P, €I . ) ,."' '!$I . ", .. ~tt:._ ',' If', .-

The initial design is 

. 1"_,, 

-27-

y - 0.05 lbf/in3 

1.0 ~ ~ ~ 1.3 (Hz) 

10.0 ~ f2 ~ 11.2 (Hz) 

(lbf) 

Ii - 1. 3555 

ORIGIWU .• P:.-::: !~ 

OF POOR QUALIlY 

Hi - l5.l20/N; I • 1.2 ••••• N 

which implies an initial value of OBJ (- the objective function - the 

total weight) of 30.2249 lbf. 

Results of the study are shown in Figures 3-8. In all cases, the 

active frequency constraints were found to be 

fl - 1.3 (Hz) 

f2 - 11.2 (Hz) 

Figure 3 demonstrates. as one would expect. that the optimum weight 

(16) 

does in fact decrease as more elements are added to the mesh. The change 

in optimum weight is quite small (note that the scale of the vertical 

axis begins at 20.0). 

Figures 4 and 5 show the variation of the lumped weight and the moment 

of inertia (of the cross-sectional area) at the free end versus the total 

number of elements in the mesh. It appears that these quantities do not 

converge. This result can be explained. however, by referring to Figure 6, 

in which the upper curve represents the total weight at the free end. 

. , ..... ~: 

. , 

'; 

. ::' 

I \ 

I 

! 
-28-

(U(N) is the non-structural, or, lumped weighti M(N) is the structural 

weight associated with the mass distributed throughout element "N"). It 

can be seen from the figure that the total weight appears to converge 

smoothly as tho mesh is refined. The explanation for the apparent nonconvergence shown in figures 4 and 5 and the convergence shown in the top 

curve of Figure 6 is that the "structural weight" at the free end of the 

cantilever is not really structural, since there is no portion of the beam 

beyond the free end which needs to be supported. Thus, the optimization 

routine is indifferent to whether structural or non-structural weight is 

present at the free end - the only thing that counts is the total weight 

at that end. 

Figure 6 also shows the variation of the lumped weight slightly beyond 

the middle of the beam. (All optimal designs had non-zero lumped weights 

there and at the free end of the beam.) The weight can be seen to decrease 

smoothly as the mesh is refined, although no asymptote appears present. 

A possible explanation for this behavior is that as the mesh is refined, the 

weicht in the middle is being placed more efficiently - and thus less is 

needed. 

The various sketches in Figure 7 show the distribution of mass and 

stiffness along the beu~ for increasing numbers of elements. It is 

interesting t observe thut the optimi~ation routine finds it most efficient 

to meet the contruints on frequency by varying the lumped weight rather than 

by varying the stiffness (moment of inl!rtia), since this latter quantity is 

at it9 lower bound everywhere except neur the end of the beam. 

As was pointed out previously in reference to FiRures 4 and 5, the 

optimization al~orithm .1ppl.!llrS to tt'l!llt the structural and non-9tructural 

mass at the end of the b~nrn as int~rchan~ellble. To tcst this hVpotllcsis 
,' •. -~.-.-~ , .. ..-. ·--c- ~-..,.~~ ........ ~ ..... ".""' .. --...... "" .... -.\~.'.- ... ;. .. -. 

-29-

further, the optimal design problem statCQcnt was altered nlishtly bv 

decreasing the upper bound constraint on tho moment of inertia from 

5.2083 to 2.0. The resulting optimum design is dhown in Figure 8, and 

should b~ compared with the design (for N - 10) shown in Figure 7. Note 

that the constraint on the moment of inertia for element 10 is not active 

in the opti~~l design of figure 8 (the constraint waB active durin[ the 

CONMIN iterations leading to this optimal design). Thus, the effect of 

the constraint is to lead the optimization algorithm along a different 

path than that followed when the constraint value was 5.2083. The design 

found, however, has about the same total weight at the free end (- 9.9705 lbf) 

as the previous ten-element opticum (- 9.9222 lhf). This result confirms the 

hypothesis that COhYIN increases the moment of inertia at the free end only 

as a meall!'J of increasing the mass there. Once that option is closed (that 

is, the upper bcund constraint is reduced to a value of 2.0), CONMIN simply 

increases the lumped weight at the beam tip. This findinr. susgests thnt. 

in future optimization studies, a tight constraint be imposed on the moment 

of inertia at the free end. since little structural capl~ility is needed 

there, and necessary end mass cen be adequately represented by the lumped 

weight design variable. 

5.2 Accuracy of Eigenvalue Calculaticns 

Among finite element analysts, the problem of the static analysis 

of a cantilever beam subjected to an end load is notorious for being 

numerically ill-conditioned. This ill-conditioning also becomes apparent 

in the eigenvalue calculations associated with the present optimal design 

studies. Table 13 illustrates the maRnitude of the errors ari:>ing in the 

eiKcnvalue calculaticns for a non-rotating cantilever like that of Figure I, 

I· 

I 

I 

] 

I 

! 

j 

1 

l 

j 

! 

j 

1 
" ......... ,~'. 

-30-

with thickness 0.1 inches, density 0.1' lbf/in3, and no lumped m4SS. Column 1 

in the table contains the first twenty frequencies of the beam, "hich were 

obtained by a double-precision version of a code based on the Sturm-sequence 

method with inverse iteration. The eigenvalue is defined as the square of ~he 

frequency W (in rad/sec) in the equation 

([K) - W

2 [H»{Y} - 0 • (17) 

Here, K is the stiffness matrix, M the mass matrix, and Y the eigenvector. 

Columns 2 and 3 contain the e1.genvalues for the same problem, but found by 

the IBM scientific subroutine program "NROOT" (based on the Jacobi method) 

in a single-precision versicn (column 2) and a double-precision version 

(column 3). The difference in the first entries in columns 1 and 2 is about 

three percE:nt. 

An alternative manner of formulating the eigenvalue problem is to 

write it as 

1 ([M] - 2" [K]){Y} - 0 • (18) 

w 

The eigl!:lvalue is now eeHned to be the reciprocal of the square of the 

circular freqcency. For thi3 formulation of the problem. column 4 gives 

the frequencies found by the Sturm-sequence method, and column 5 gives the 

frequencies found by the single-precision routine "NROOT". The two methods 

now give essentially the same frequencies. The improved performance of the 

single-precision "NROOT" routine is attributable to the fact that the 

accuracy of the Jacobi algorithm is dependent on the order (in terms of size) 

in which the eignevalucs are found. By contrast. the Sturm-sequence method 

is independent of the "largeness" or "smallness" of the eigenvalues. 

I 

I , ! --. 

I ." 

1 
." -......,.. ... .". 

-31-

The significance of these erroru in the eigenvalue calculations can 

be seen by inspection of Table 14, in which are given optical designs found 

by CONMIN using both the inaccurate eigenvalue calculation and the accurate 

eigenvalue calculation. The example corresponds to Case 6 of Table 6. The 

designs are seen to differ appreciably. 

5.3 Sensitivity of Frequency to Small Changes in Thickness 

Because rotor blades can be manufactured only to within certain 

dimensional tolerances, the question naturally arises as to the sensitivity 

of the natural frequencies of the blade to small changes in blade dimensions. 

Clearly, if small changes in blade dimensions produce large changes in 

natural frequencies, then designing a theoretically optimum blade is futile: 

the small variations in blade dimensions introduced during manufacturing would 

destroy the optimally designed vibratory behavior. To study this question, 

the data of Table 15 were generated for the cantilever beam described in 

Section 4 (Case 14 of Table llA). The first column in the table contains 

the derivative of the fundamental frequency with respect to the thickness 

of the first, second, third, ••• , and tenth (free end) element. If 0.01 inches 

is taken as a representative manufacturing 

show that the maximum corresponding change 

is only about five percent (=0.01 + 4.791). 

tivity study is illustrated by the data in 

contains the derivative of the fundamental 

weight of the individual elements. Since 

and 

diiii -..,- '" 9.12 

oti 

tolerance, then the data in column 

in the first fundacental frequency 

Another aspect of this sensicolumns Z and 3. Column 2 

frequency with respect to the 

(19) 

(ti is the thickness; see Fig. 1), 

1 
-32-

the entries in column 2 are derived by dividing the entries of column 1 by 

9.12. Column 3 contains the derivative of the fundamental frequency with 

respect to the lumped weight of the individual elecents. If 0.01 lbs is 

taken as a representative manufacturing tolerance. then the data in columns 

2 and 3 show that the maximum corresponding change in the first fundamental 

frequency is about half a percent (-0.01 + 0.525). 

It is also interesting to compare corresponding entries in columns 2 and 

3 and to observe that they disagree near the blade root, with the discrepancy 

diminishing appreciably as the free end is approached. TIe explanation for 

this behavior lies in the fa~t thet as the distributed weight of an element 

is increased, the cross-sectional area must of course also increase. Thus the 

second column in the table represents changes in stiffness as well as in mass. 

It follows that a unit increase in distributed weight (as represented bV 

column 2) will produce a larger increase in frequency than is caused by a 

unit decrease of lumped weight (column 3), since, qualitatively speaking, 

stiffness appears in the numerator and mass in the denominator of the 

frequency expression, 1KlM. Thus 

The inequality is large near the root, because increasing element 

stiffness has a large effect on structure stiffness. At the tip, the 

structure stiffness is barely changed (as witnessed by the last two 

entries in columns 2 and 3: - 0.227 and - 0.229). 

t 

j 

j 

."j 

f 
-33-

6. CONCLUSIONS 

For the types of beam vibration problems described in this report, 

the following conclusions may be draw~. 

1. Either analytical or finite-difference gradients can be used. The 

finite-difference approach will of course require many more function 

evaluations, but it is easily implemented. Furthermore, the problems 

considered are not especially sensitive to numerical error in gradient 

calculations, therefore, we use analytic gradients; but we reserve the 

finite-difference gradients as a viable d1tcrnative should they be needed. 

2. The frequency constraints shoul~ be formulated directly in terms of 

frequencies, rather than eigenvalues (frequency squared). 

3. CONMIN can find an optimum design, after starting with an initially 

infeasible design, at least for the problems considered. 

4. The default values for CONMIN appear adequate, with the exception 

of DAB FUN ITMAX, for which a value of 40-80 works well. 

5. Optimum designs can be found even for relatively tight constraints 

on frequencies. 

6. The objective function is relatively Elat near the optimum, and 

appreciably different distributions of stiffness may Yield essentially the 

same value of the objective function. 

7. Two and three frequency constraints can be handled. 

8. Using both stiffness and lumped !MSS as decision variables presents 

no special difficulties. Indeed, the admittedly limited experience gained 

thus far with these types of problems indicates that the optimization seems 

to proceed ~ore rapidly (fewer iterations to obtain convergcnc) if CONMIN 

can add lumped mass rathe~ than just add mass in the form of structural mass. 

Thus the greater complexity of the problem, i.e., the increased number of 

• ··_· .......... 1. _.a~ 

'I 

-I 

·-1 

I 

! 

; 

i 

1 

, 

! 

j 

I 

i 
-34-

decision variables. appears to be Clore than compensated by the greater 

freedom in choosing a design. 

9. Optimal designs for rotating beams subject to frequency and autorotational constraints present no difficulties. Again. the use of lucped 

mass (rather than structural mass) appears to be a Clore efficient way of 

controlling frequencies. 

\ 

.. " 

1 

I 

-\ . 1 

1 

I 

! 

I 

. \' .,. ...... ~. __ ."',_. __ ~ __ , ..... __ .... _ .. _, ___ ... _____ ~_ ... _~~ _______ .-~_ .. ~ .. ,,-_~ i r,s ..;, i .... . " J 
'1 

j 

-35- .J 

1 

7. NEAR FUTURE PLANS I 

At this writing. current research effort is concentrated on developing 1 

1 

a generic cross-section sufficiently general that both the bending and J 

" 

torsional stiffnesses of currently existing blades can be matched. When a \ ' ! 

I 

1 

I 

suitable generic section has been attained. it will be used in studies of ; 

! 

the opticization of a non-rotating cantilever experiencing coupled flap. lag. -' 

and torsional vibrations. Some preliminary studies of torsional and in-plane 

vibration have already been done. 

Preliminary work has also been done on the feasibility of using an 

objective function involving the sum of the squares of frequencies, rather 

than the weight. The results look promising. and this approach will be 

pursued. especially if difficulties arise with the weight-objective function 

approach, as more complex problems are analyzed. 

Another topic of research in the immediate future will be the inclusion 

of a stress constraint in the problem of the optimization ot a non-rotating 

cantilever. Preliminary. exmaination of present optimum designs show no 

particular stress problems; but, in principle, the stress constraint should 

be necessary to prevent elicination of too much material from blade designs. 

"'8b?,h-' 
-36-

8. REFERENCES 

1. Niebank, C. and Girvin, W., "Sil~orsky S-76 Analysis, Design and Development 

for Successful Dynamic Characteristics." Proceedings of the 34th Annual 

Natural Forum of the American Helicopter Society, May 1978, pp 78-23-1 

through 78-23-17. 

2. Hirsh, Harold; Dutton, Robert E., and Rasumoff, Abner, "Effect of Spanwlse 

and Chordwise MDss Distribution on Rotor Blade Cyclic Stresses," Journal 

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

3. Ellis, C. W. et a!., "Design, Development, and Testing of the Boeing 

Vertol/ Army YUH-6lA," Proceedings of the 32nd Annllal National Forum of the 

American Helicopter Society, Preprint 1010, }my 1976. 

4. Fenanghty, Ronald R. and Noehren, William L., "Composite Bearing1ess Tail 

Rotor for UTTAS," Journal of the American Helicopter Society, Vol. 22, 

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

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

Vibration Reduction Using Structural Optimization Techniques," Proceedings 

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

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

6. Vanderplaats, G.N., CONMIN - A Fortran Program for Constrained Function 

Minimization. User's ~~nual, NASA TMX-62.282, August, 1973. 

7. Zoutendijk, G., Method of Feasible Directions, Elsevier Publishing Co., 

Amsterdam, 1960. 

8. Peters, D.A. An Approximate Solution for the Free Vibrations of Rotating 

Uniform Cantilever Beams, NASA TMX-62.299, August, 1978. 

9. Thomson, W. T., Mechanical Vibrations, Prentice-Hall, Inc., Neu York, 

1953, pp 36 and ~17. 

\ 

.- I 

, 
-37-

9. APPENDICES 
-38-

APPENDIX I 

ORIG!NI'\L PAC:':: tJ 

OF POOR QUALITY 

GRADIENTS OF OBJECTIVE FUlICTION AND CONSTRAINTS 

When analytical gradients are ut:f lized in COmfIN t the following 

derivations are necessary. 

Let Z - objective flmction 

th Sj • j frequency constraint function 

ti - ith element thickness 

Ii - ith element moment of inertia 

th ~i - i eigenvalue of the vibration problem 

1) To find dZ ar- we employ the chain rule 

i 

10 

Since z· r ti 

iLil 

and, 

Then, 

ilZ 1 2 -1 - - 1{24(285 - 456ti + l82.4ti } aIi 

Furthermore, 

Finallv, 

2) To find 

ilg .. 

~ we ~lso employ the chain rule arK 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

..• '.1" 

, . ;. ; '. " ,'- i.~ -; ;'..:>;:';'$ :.,,-

'] 

.-
........ , 

-39- ORIGINAL F/\G~ fJ 

OF POOR QUALITY 

For con~trsint8 that are linear functions of frequency, 

Since 

aSi ~f • +1, -1, or 0 

O[i 

3fi 1 

~. 4n~ 

a>'i 

--- is obtained as a function of 3IK 

the eigenvectors, mass matrix, and 

stiffness matrix of the probl~m (1) 

R~adlnR. PAt 1971. 

(7) 

\ 

(6) 
_ .• ..,. .-.. ~' ..... - .... ~ - - -~. . ••• "_., • ..-o;-'''-~'- ,. " ...... - • •• ... .. - ~.- ,- ~"'-~ -- -' ... \.-...,. - ~--'" ~ ... '1 ........... ,.~··-··-: .--. --.--,-.,,-,. r .~-".-r"'-1I __ "" ".., 

o· .......... 0_ • 

-40-

APPEtIDIX II 

OHIG!NAL r;\'..:,: ~J 

OF POGR QW~UN 

GRADIENTS OF TIlE AUTO-ROTATATIONAL COnSTRAUIT 

For a system of uniform el~ents and lucped wei3htG, the mass moment 

of inertia, I , can be written as: 

m 

where, 

I - m 

p - mnss rlensity 

Ai - area of ith element - O.S + 7.6 ti 

Wi - weight of ith l~p 

g - gravitational constant 

rU,rL - radius from beam root to the upoer and lover end 

of the element respectively. (Note that half of 

(1) 

the lumped veight has been placed at the lower end 

1) 

but, 

and upper end of the element). 

aS7 To find --- wc employ the chain rule 

ali 

ag7 aI 

1 (--..!:l) at .. - -1- ali 

i min 

(2) 

,-

.. I 

I 

1 

i 

j 

.. 

i oj 

, 

01 , 

, 

, 
.' -.... ..... -. 

-41-

aS7 2) To find aw • direct differentiation can be employed 

i 

ORk.iNAL P;.G:: m 

OF POOR QUI''\LITY 

(4) 

.-
· -..-... 

-42-

APPEtIDIX III 

ORtGINAL Ft.;?·:: t~ 

OF POOR QUALIrV 

MODIFICATIONS TO ACCm~ODATE ELEMENTS IN TENSION 

For an element under constant tension, T, the potenti.al enorgy is 

given by 

where W represents the vertical deflection coordinate in the Y direction, 

as shown in Fig. 2. 

From Ref. 2, the displacement function for toJ is given by: 

(2) 

where, the nodal translations are Vl and V3, and the nodal rotations are 

V2 and V4• 

Substituting the derivatives of (2) into (1). performing the integration 

1 T { over the length. and comparing the results to U - 2{V} IK(T)1 V} leada to 

the identification of the added contribution to the element stiffness 

matrix, IK(T)] 1021L 0.1 -1. 2/L 0.1 ............ 

IK(T)] - T 2L/15 -0.1 -L/30 

Symm. "'-

1. 2/1. -0.1 

"-

2L/15 (3) 

The element tensions arc easily defined by computing the centriKual force 

for each lumped mass. and accumulating the t.of.al tension from tip to root of 

(2)Peters. D.A. Ko, T., Korn, A., and Rossow, M.P., First Semi-Annual Status 

Report on DeRi~n of Heliconter Rotor Bl~des for Optimum Dvnamic CharacteriRtics,:-lASA-Lan~ll'~' (~rant ~o. ~'\r.-1-250, Sept. 15, ICl82. 
-43-

the beam. To keep a constant tension in e~ch e1eQent, the distributed and 

lumped weights were divided by two and placed at the element nodes. 

To compute analytical gradients, the pSI'till1 derivatives of KeT) 

with respect to the decision variables are needed. 

ilK(T' 1) To find ~ aIK 

for Ilny element, the chain rule is used: 

aK aK aT at 

at - aT x at x at 

The first term represents the matrix elements of Eqn. 3. 

(4) 

The middle term can be obtained by writing the element tension as the 

2 cumulated sum of min r i terms, where, n - rotational speed in red/s~c. Since 

each mass is a function of thickness, the derivative of tension with respect 

to thickness can be obtained. 

The final term has been defined in Eqn. 3 of Appendix II. 

aK(T) 2) To find aw for nny element, we usc: 

K 

(5) 

The second term can be obtained by writinr, the elcmenL tensions caused 

by the lumped masses, and then taking the appropriate partial derivatives. 

\ 

,J 

I 

l 

; 

.----.-
10. Tab\es 

(a) 

(b) 

(c) 

-44-

Initinl Vnlue:J 

Constant t inchcs: 

4 Conctnnt I inchQs : 

ORIGINl\l r ... ~:~::: . ~ 

OF POOR QUALI fY 

0.05 

0.831 

Objoctive Function,inches: 0.500 

Weight Ibf: 21.12 

f

1,1U: 1.86 

f 2,&:: ll.6 

0I!timtil Solution 

(c) 

AnalItieal Gradients Finit~ Difference 

Gradients 

Notes: 

15 

0.540 

21.85 

2.00 

12.20 

1. 250(0. 0896) 

No. of Iterntions 

Ob.1octive Function l.uches 

Weight lbf 

fl Hz 

(d) f2 Hz 

I1inches4(t~nChCs) 

(Elements 2 through 10 were essentially 

all at the lower bound value of side 

constraint on thickness: 

t • 0.05 inches, I - 0.831 inehcs4

) 

10 

(a) Ob~eetive Function,inehns. E ti 

i-I 

10 

(b) Weight Ibf " 12 + 18.24 ~ ti 

i-l 

(c) Convergence criteria were the same. except that IT~~ was raised 

from 3 to 8 cycles when finite differences were used. 

(d) Both solutions have an active frequencY constraint: f~ is at 

its lower bound value. 

21 

0.539 

21.83 

1.99 

12.20 

1.253(0.0899) 

Table L\: Comparison of Annlvtic:ll Gr~dient vs Finite Difference Solution 

j 

I 

I 

I 
(c) 

-45-

Initial.Values 

Constant t inches 

Constant I inches4 

(a) Objective Function inches 

(b) Weight 1bf 

f1 Hz 

Optimal Solution 

Analytical Gradients 

15 No. of Iterations 

ORIGlt:r,L F{,~~::: n 

OF poon QU'~UT'; 

n.10 

1.355 

1.000 

30.24 

1.98 

12.4 

(c) 

Finite Difference Gradients 

0.541 Objective Function inches 

18 

0.543 

21.90 

1.94 

12.20 

21.87 Wei~ht Ibf 

I 

1.94 f

1 ,HZ 

12.20 (d) f 2,Hz 

1. 262{0. 0908) I i h 4{ inches) 1 nc es tl 1. 214(0. 0861) 

0.331(0.0500) I6inches4{t6inChes) 0.852(0.0519) 

0.831(0.0500) 17inches4(t7inChes) 0.880(0.0545) 

(Elements 2-5 and 8-10 were all at the lower bound 

value of si~e constraint{tl.ic~,esn): t - 0.05 inches, 

I - 0.831 inches4

) 

Notes: 10 

(a) Ob.1ective function. inches· E ti 

i-I 

10 

(b) Weight.lbf ~ 12 + 18.24 E ti 

i-I 

(c) Convergence criteria for the two runs were identical. 

(d) Both Solutions have an active frequency constraint: f2 is at 

it lower bound value. 

Table lB: Comparison of Analvtical r.radient vs Finite Di~fercnce Solution 

._' 

"1 

J 

I 

'f 

I 

.j 

\ . 

; 

1 

I 

~ 
(c) 

-46-

Initial Values 

Conotant t inches 

Constant I incheo 4 

(a) Objective function inches 

(b) Weight 1bf 

f1 Hz 

f2 Hz 

Opt~l Solution 

Eisenvaluo Con9troint~ 

No. of Iterations 

0.05 

0.831 

0.500 

21.12 

1.86 

1l.6 

12 

0.551 Objective function inches 

22.05 

1.94 

12.20 

1.150(0.0799) 

0.846(0.0514) 

0.910(0.0573) 

0.922(0.0584) 

0.875(0.0541) 

Weight 1bf 

f

1 ,Hz 

(d) f

21Rz 

4 inches 1

1 , inct-.es (t1 ) 

4 inches I

5 ,inches (t5 ) 

161inChes4(t6inches) 

17,inChes4(t7inches) 

T i h 4(t inches) -8' nc l!S 8 

(e) 

!rcguency Constraints 

21 

0.539 

21.83 

1.99 

12.20 

1.253(0.0899) 

0.831(0.0500) 

0.831(0.0500) 

0.831(0.0500) 

0.831(0.0500) 

(£lecento 2-4, 9, 10 were all at the lower bound value of 

side constraint on thickness: t - 0.05 inches. 1 • 0.831 inche04

) 

Notes: 

10 

(a) Objective function,inches. E ti 

i"l 

10 

(b) Weight.lbf - 12 + 18.24 E ti 

i-1 

(c) Convergence criteria for the two runs were identical.' Gradients were 

computed by analytical technique~ for the eigenvalue constrained run, 

"whereas finite difference techniques were used for the frequency 

constrained run. 

(~) !lot~ 901u!:ions ~avc a~ active "'rec:uencv const:-aint: f2 io at its 

lowar bound value. 

Table 2: Comparison of Eigenvalue and Frequency Constraints 

.. 
.... ,t 

Initial Values 

Constant tlinches 

Constant I.inchcs4 

(a) Objective function inches 

(b) Weight.lbf 

f

1 , Hz 

f

2, Hz 

Optimal Solution (c) 

No. of Iterations 

Objective function inches 

Weight lbf 

f1' Hz 

(d) f.." Hz 

I~ inches4(tlinches) 

-47-

0.05 

0.8:31 

0.500 

21.12 

1.86 

11./i (e) 

15 

0.540 

21.85 

2.00 

12.20 

1.250(0.0896) 

0.10 

1.355 

1.000 

30.24 

1.98 

12.4 

15 

0.541 

21.87 

1.94 

12.20 

1.262(0.0908) 

(f\ Case j 

1.101 

5.200 

11.013 

212.87 

1.46 (e) 

9.17 (e) 

18 

0.547 

21.98 

1.92 

12.26 

1.327(0.0972) 

(Elements 2 through 10 were essentially all at thp. 10w~~ bound 

value of the side constraint on thic~~ess: t • G.OS inche~. 

I • 0.831 inches4

) 

Notes: 

(a) Objective function. inches' • 

10 

(b) Weight,lbf • l~ + 18.24 L ti 

i"l -

(c) All gradients were computed by analytical techniques. 

(d) All solutions have active frequency contraint: f2 is at its 

lower bound value. 

(e) Initial frequency violates the constraint. 

(f) In this case, a high value of the parameter DABFL~ led to premature 

designation of convergence. The solution is nearly optimal, but the 

objective function was still changing in the 3rd significant figure. 

Table 3: Comparison of Solutions Obtained for Various Initial Designs 

I 

I : \ .. 

.-
(a) 

-48-

Initial Values 

Constant t.inchcs 

4 Con9tant I. inches 

Ob.1active function, incho9 

We1ght.lbf 

f

l , Hz 

f

2, Hz 

ObjectivQ Funct10n 

0.10 

1.355 

1.000 

30.24 

1.98 

12.4 

(b) 

Using Default Pnr~atars Usin8 Tightor Cotlvcrgance Criteria 

0.54279 Iteration 10 (c) 

0.54212 

0.54076 

0.54076 (No channel 

0.54076 (No chango) 

0.54076 (No change) 

15 

0.54076 

21.87 

1.94 

12.20 

1.2619(0.09076) 

Notes: 

I tcarat ion 11 

Iteration 12 

Iteration 13 

Iteration 14 

Iteration 15 

Iteration 16 

Iteration 17 

Optimal Solution 

No. of iter~tions 

Objactive function inches 

Weight1lbf 

f1t Hz 

f2' It:: 

I1inches4(t1inches) 

(a) Default values: ITRM· 3 

DELFUN - 0.0001 

DABFUN • 0.001 

(b) Tighter Criteria: IT~~· 5 

DELFUN • 0.00005 

DABFUN • 0.0005 

(c) The first ten iterations were identical. 

0.54279 

0.54279 (No change) 

0.54083 

0.540133 (No change) 

0.54083 (No ChangC1) 

0.54012 

0.54012 (No change) 

0.54072 (No change) 

17 

0.54072 

31.87 

1.96 

12.20 

1.2615(0.09073) 

Tabl~ 4: Conver~enc~ HiDtorv for Two Different Cv~vergence Criteria 
-49-

(a) No. of Final Objoctive 11 inchas4 

flt Hz (b) 

...t.. Iterations Function ( t

1

1nchets) 

1 15 0.540 1.262(0.0908) 

10 19 0.546 1.315(0.0960) 

100 13 0.552 1.308(0.0953) 

700 26 0.549 1.190(0.0838) 

Notas: 

(.) For all runs, Default Parnmctcrs were used: 

PHI • 5.0 

ITR.'i • 3 

DELFUN • 0.0001 

DABFUN • 0.001 

1.94 

2.01 

2.03 

2.06 

(b) The second frequ~ncYt f

2, was at the lower bound constraint 

(f2 • 12.2 H%). 

(c) The 1nit1al design was based on con~tant thickness, constant moment 

4 of inertia of 0.10 inches and 1.355 inches respectively. 

Table 5: Tho Effect of Pnracetcr THETA on the Opti~~l Solution 
-50-

Initial Values 

Constant t,inches 

Constant I t inChcs4 

0.25 

2.675 

2.50 

ORIGt~!f;'- r"'·F.~ ~3 OF POOR Q'j;"~.lT·t 

(a) .Objective function,inchas 

(b) . Weight Ilbf 

f

l , 11: 

f

2, Hz 

57.60 

2.02 

12.6 

Qptical Solutions (c) 

No. of iterations 

Objective function, inches 

Weight,lbf 

f1' Hz 

f

2, Hz 

t

1 ,inches 

t 2·inches 

t 3.inches 

t 41 inches 

t 5·inches 

t 6 ·inches 

t 7.inches 

t 8.inches 

t

9,inches 

t1o,inches 

Notes: 

Case 4 

flL " 1.9 

flU • 2.1 

f2L • 12.3 

f 2U • 12.5 

32 

0.552 

22.07 

Hz 

2.09 (e) 

12.32 (d) 

0.1022 

0.0500 (f) 

0.0500 (f) 

(a) Objective function,inches • 

Case 5 

flL • 1.5 Hz 

flU • 1.9 

f2L • 12.8 

f 2U • 13.2 

29 

0.662 

24.08 

1.87 

12.80 (d) 

0.1122 

0.0500 (f) 

1 

0.0500 (f) 

0.0623 

0.0791 

0.1059 

0.0515 

0.0514 

Case 6 

flL - 2.1 Hz 

flU - 2.5 

f2L - 11.6 

fZU - 12.0 

28 

0.593 

22.81 

2.11 (d) 

11.96 

0.0753 

0.0703 

0.0886 

0.0586 

0.0500 (f) 

0.0500 (f) 

10 

(b) Weight,lbf - 12 + 18.24 I ti 

i-1 

(c) ITI1AX - 40, ITR.'1 • 5, DELFUN .. 0.0001, DABFUN - 0.0025 for case 5 and.6. 

For Case 4, IT~~ .. 3. 

(d) Active lower bound contraint. (e) Active upper bound constraint. 

(f) Active side constraint. 

Table 6: ~lacement of Frequencies 

\ . 

: 

l ". i> " .• ,..... ". ........ .-oj .... . " c· ."!' ,:'. . ...... ,:,.~\>,.~.:j.o .. ~\ ..... ,::/.j .. ;.: ... ~ ........ ·.,L, .. ~.~.:i .. · ... :~ .. ' ..•... ;.:~. L .. ~,:.~j ... ',< 1.:: ~j...w. \"'~ ~~~~L.{-,*;:~:~o&,.~~~~.i~~ .... ,-","~",,,,,,,=,,,,,,,,~~,,,,,=~ ~ __ ---=:. 
URlGmr,L P.".2:: ~3 

-51- OF. poon QUi'\U1Y 

Initial Values 

Constant t.inches 

4 Constant I,inches 

Case 7 

(a) O,.,jective functionlinches 

(b) Yeight 11bf 

0.10 

1.355 

1.000 

30.24 

f

1 , Hz 

f

2, Hz 

f

3

, Hz 

ITMAX 

ITRM 

DELFUN 

DABFUN 

flL,flU, Hz 

f 2L,f2U' Hz 

f 3L,f3U' Hz 

(c)Optimal Solution 

No. of iterations 

Objective function inches 

lieight Ibf 

f1' Hz 

f

2, Hz 

f

3, Hz 

tl inches 

t2 inches 

t3 inches 

t4 inches 

t5 inches 

t6 inches 

t7 inches 

ts inches 

t9 inches 

t

lO inches 

1.98 

12.4 

34.8 

40 

3 

0.0001 

.001 

1.8,2.2 

12.2,12.6 

34.6,35.0 

20 

0.579 

22.56 

2.02 

12.51 

(d) 34.60 

0.117S 

(f) 0.0500 

1 

(f) 0.0500 

0.0501 

0.0608 

(f) 0.0500 

(f) 0.0500 

Case SA 

0.10 

1.355 

1.000 

30.24 

1.98 

12.4 

34.8 

80. 

5. 

0.0001 

0.0005 

2.1,2.5 

11.6,12.0 

33.6,34.0 

57 

0.623 

23.37 

2.11 

12.00 (e) 

33.62 

0.0906 

0.0560 

0.0771 

0.0995 

0.0500 (f) 

I 

0.0500 (f) 

,,:... - -'<~"-----'---"'-- -" "' '·i",·:" •. " ~:. : ~". ~ r~" .. :~~:,.'.~,,;- "'_ '~.".' - t~ 

Case SB 

0.05 

0.831 

0.500 

21.12 

1.86 

11.6 

32.7 

44 

0.611 

23.13 

2.10 (d) 

12.00 (e) 

33.60 (d) 

0.0884 

0.0526 

0.0744 

0.0951 

0.0500 (f) 

I 

0.0500 (f) 
L

····'·:· •. ~ ..• ~ • .z 

-52-

Notes: 10 

(a) Objective function.inchas. :.ti i-1 

10 

(b) Weight,lbf • 12 + 18.24 t ti 

1-1 

(c) All gradients were computed by &1la.l..ftica1 techni'lues 

(d) Activo lower bound constraint 

(e) Active upper bound constraint 

(f) Active side constraint 

Table 7A: Optimization uith Three Freque~cy Constraints 

; ~---;'-::~'~---."7"'-~ .. , '.> '.' J... ..:-. -------...... . . ';.:"'.~~. ·,·:cc-: .... :'}i: ': .. :.... ',<>. ':.::.: ,::.I.'.:~ 

·rt •• ..-.~_ ..... ',",. ~ ......; ...... , ;..:..,.. -->'<"",~ ..>... .... ."i ......... ',;,.:• ..i' • ...;:L.,' .. ;. :;. ;-_ .. ;;~~," .... ', «,,';;';'., ,.;\ .• ~:. : .• h ..,~~ ........ .tLa.'.'-. ""'~~-......: .,.;~ .... _~ 2",.',·~_J 
""".' , 
-54-

NotaG: 

(a) Objactive functlon,inchol -

10 

(b) Woisht,lbf - 12 + 18.24 t tl 

i-l 

(c) All gradient. wor~ computed by analtyical te~hnique. 

(d) Active lover bound constraint 

(e) Activo uppGr bound conotraint 

(f) Active side constraint 

Table 7B: Optici:ation with Three Frequency Con~traints 

. 1 

I 
F.W~,t,·, "}I.f.; ... :",.:J.if';£1!.if.\~k1'\·!M.\':bie#!jii!~%~.?f'N,?'Ij &$ ~:~;:;"':'lI('f:""~~~~·M",':':·~:jMfh'!M;; . ,tl . .t-:§ifr*t'f'!t.sy;¥"'t'~""""~<"'\:i"~":~3-~ . .... -.. ~ 

Initial Vllluaa 

Constant tlinches 

4 Constant Ilinches 

Donsity, y,lbwVinches3 

Conatant Wi1lbf 

(a)Objectivo functlon)lbf 

(b)Woieht,lbf 

fl' Hz 

f

2, Hz 

f

3

, Hz 

flL,flU ' E: 

f 2L,f2U' Hz 

f 3L,f3C' H.: 

(c)Optimal Solution 

No. of H't:r.1t!ons 

Objective function Ibf 

Weight Ibf 

f

l , H: 

f2' Hz 

f

3

, Hz 

tlinches and WI Ibf 

t

2inches and W2 Ibf 

t 3inches and W3 Ibf 

t

4inches and W4 Ibf 

tsinches and t"s 1bf 

t 6inches and W6 Ibf 

t 7inches nnd \"7 1bf 

tsinches and 118 1bf 

t

9inches and 1"9 Ibf 

t 101nchcs and WIOlbf 

-55-

Cllee 10 

< 

~ 

0.05 

1.512 

24.24 

30.24 

1.98 

12.4 

34.8 

< 

< 

~ 

53 

9.252 

15.25 

2.12 

12.60(p) 

3S.QO(e) 

0.0500(f) & O(f) 

O(f) 

0.056 

2.214 

0.001 

O(f) 

1.194 

0(0 

O(f) 

0.0500(0 1. 227 

CtIIlO 11A 

0.10 

1.355 

0.10 

1.000 

28.24 

40.24 

1.66 

10.7 

29.9 

1.8,2.2 

12.2,12.6 

34.6,35.0 

38 

10.557 

22.56 

1.96 

12.51 

34.60(d) 

0.1150 & O(f) 

0.0500(f) 

I 

0.0500(f) 

0.0535 

0.0618 

O. 0500( f) 

0.05(;0(0 &0(0 

Caso UB 

~ 

> 

0.10 

0.000 

18.24 

30.24 

1.98 

12.4 

34.8 

• 

) 

~ 

19 

10.557 

22.56 

2.03 

12.51 

34.60(d) 

0.1147 & O(f) 

0.0500(f) 

t 

O.OSOO(£) 

0.05:5 

0.0616 

0.0500(0 'V 

0.0500(0 5. 0(0 

• \ . I 
,,~ Y;iQ"'l(.4i ;", j", .if· •. ,F. "", ,M'+''' 14 $'1' 4 ...... Iif* i'-V #4 Pi, ·.,:sqS +,. .1j'4 ',' < ,: .... ,+i. ! An .. 1 :PSi ,,,,,," "AS,_, 

....... -" 

-56-

Notes: 

10 10 

(a) Objective function,1bf - 132.4 Y t ti + E Wi 

i-l i-l 

(b) Weisht,1bf - 120y + objective function 

(c) All gradients wero computed by analytical techniques 

For all cases, Ill~X - 80, ITRM • 5, DELFUN • 0.0001, DADFUN • 0.001 

(d) Active lower bound constraint 

(e) Active upper bound constraint 

(f) Active side constraint 

Table SA: Opt~zation Including Lumped Weights 

,* , ... i: •• _,"- ..... ,,~ I.,. i,\;" " i """" 

! 

i 

I 

j 

I 

~ 

• \ 

, 

I 

I 

; 

, 
_~_~,.",.;.~ .•• "". - ........... - '. ,.-..-_. __ .••.. ~~-~ _r· ... · -", ,.,. 

Initial VllluM 

ConstAnt t1inehu 

4 Con.t~nt Ilinelle. 

Dcnalty, Yllb~/inehQo3 

Conlltllnt W1,lbf 

..... ' .. 

<.) Objective funetion,lbf 

(b) '~Gight,lbf 

f

l

, HE 

f2' Hz 

f3' Hz 

f

lL, flU' H: 

f

2L, f:.!U' Hz 

f

3L, f 3U ' Hz 

(c) Optimal Solution 

No. of iturntions 

Objoetive fllnction,lbf 

'~e1ght,lbf 

f

l

, H: 

f:, H: 

f 3• H: 

tl inch(!s :md Ill' 1M 

t2 inches :lnd I~ 2' 1M 

t3 inchu :tndl~3,lbf 

t~ inchu :lnd Il. ,1bf .. ts inchas ,'md '''5' lbf 

t6 inchns And '''6,lbf 

t] inches ;tnd 1"],lM 

ts il\.:h~~ :md "'S,1M 

til in.:has ,md "'~ ,1M 

t 10 in.:hc!( :\nJ \~ ll), the 

.~. "'.' 4 r _.~ .... - .... >'~ ......... ~.~., ••••• ,-.; .... --,,-___ "t" ... .-~.~ •• --.: ., - .. ~'-.. - .... - .• ~ ............ -'~ .. __ ". __ ~ • ._._..., 

-57-

0.1360 

0.17~,) 

0.1976 

0.=053 

0.0769 

0.10 

1.355 

0.10 

1.000 

28.24 

40.24 

1.66 

10.7 

29.9 

2.6,3.0 

10.5,11.0 

31.6,32.0 

46 

23.697 

35.70 

2.60 (d) 

10.99 

31.99 

0.0500 «(, 

1 

O.5~7 

0.~35 

0.426 

0.339 

1.395 

1.135 

o (f) 

1 

(0 

l 

.1 
',-. '. , 

-saNotos: 

10' 10 

(a) Objcctlv~ f~~ction.lbf • l82.4y E tl + t Wl 1"1 lal 

(b) Wc1Sht,lbf • 120., + ObjQctlvCl function 

(c) All gradients vera cocputed by analytical technlquos 

lniAX • 80. ITRH • .5. nnFUU • 0.0001, DADFUN • 0.001 

(d) Active louar bound con3traint 

(e) Activo uppi:!r bound constr.l1int 

(f) Active aldo constrdint 

Table an: Opticizstion Including L~ped Weights 
-59-

Initin1 Vn1ul!s 

Constant t inches, I inches4, Y~b~/inchea3 

ConstDJlt Wi I Ibf. 

(a) Objective function.1bf 

(b) Weight 11bf 

fl' Hz 

f2' Hz 

f3' Hz 

f lL, flU' Hz 

f 2L, f 2U' Hz 

f 3L, f 3U' Hz 

ORIGIN;~L F:~u: t3 

OF POOR QUALITY 

CaSQ 10 Cnoe 13 

0.10, 1.355, 0.05 

1.512 

24.24 

30.24 

1.98 

12.4 

34.8 

1.8,2.2 

12.2,12.6 

34.6,35.0 

2 lminlbf inches sec o. ..1100 

(c) Optical Solution 

No. of iterations 

Objective function lbf 

Weight Ibf 

f

l , Hz 

f2' Hz 

f3' Hz 

2 I lbf inches sec 

m 

tl inches and Wl lbf 

t2 inches and W2 lbf 

t3 inches and W3 lbf 

t4 inches and W4 lbf 

t5 inches and Ws lbf 

t6 inches and W6 lbf 

t7 inches and W7 Ibf 

ts inches and Ws Ibf 

t9 inches and W9 Ibf 

t

lO inches and W10 Ibf 

53. 

9.252 

15.25 

2.12 

12.60 (O!) 

35.00 (a) 

807.7 

0.0500 (f) ~ 0 

" 

0,0500(f) 

o 

0.056 

2.214 

0.001 

o 

1.194 

o 

o 

1.227 

(f) 

(f) 

(f) 

(f) 

(f) 

80 

11.596 

17.60 

1. 80 (d) 

12.59 

35.00 (e) 

1144 (g) 

0.0500 (f) & O(f) 

0.0869 O(f) 

0.0757 

O.OsOO(f) 

O.OsOO(£) 

0.1094 

0.131 

0.S16 

0.235 

0.142 

0.719 

1.175 

1.174 

1.833 

\ -
-60-

Notes: 

10 10 

(a) Objective function,1bf - l82.4y E ti + E Wi 

i-1 i-1 

(b) Weight lbf • l20y + objective function 

(c) All gradients were computed by analytical techniques 

For all casco, ITMAX - 80, ITRM - 5, DELFUH • 0.0001, DABFUN - 0.001 

(d) Active lower bound constraint 

(e) Active upper bound constraint 

(f) Active side constraint 

(8) Very close to being active auto-rotational constraint 

Table 9: Optimization Including Auto-Rotational Constraint 

• ," '\ -: ~ ct-: .~:-r~ 

'j 

\ . 
. ' 

-61-

ORIGINAL PAGa t~iI 

OF POOR QUALlTiJ 

y • 0.05 lbm/inchas3, t • 0.10 inchon n. 0 RPM n • 300 RPM 

(a) Wi • 1.5 1bf f1 • 5.7 Hz 

(a) Wi • 100 lbf 

r • 0.05 3 lb.,..1 inches I 

(a) W • i 1.5 1bf 

(a) Wi • 100 lbf 

t = 1.25 inchc!J 

f1 • 2.0 Hz 

f1 • 0.3 Hz 

n • o RPM 

f 1 • 1.4 Hz 

f1 • 0.7 Hz 

f1 • 5.1 Hz 

n • 300 RPM 

f1 • 5.5 Hz 

f1 • 5.2 Hz 

Note: (a) Lumped veights vere aasumed to be uniformly distributed 

for purposes of computation. 

Table 10: Influence of Rotational Speed, ~ASS, and Stiffness on the 

Fundamental Frequency of a Unifo~ Cantilever 

. .~. . . 

\ 

~' 
-

[nitinl Va1uns 

:onstAnC, t, inches, 4 I inches , 

Constlmt Wi ,lbf 

(4) Objoctiva function, Ibf. 

(b) Weight, Ibf 

flL' flU' Hz: 

f 2L, f 2U ' n: 

f 3L, f 3U ' Hz: 

2 l

min,1bf inches sac 

S1,RPM 

(c) Optin~1 Solution 

No. of iterations 

ObjectivQ function,lbf 

Woight,1bf 

fl· Hz 

f 2• 

R% 

f 3• H: 

2 I

m,lbf inches sac 

t

1,inches and W1,lbf 

t 2,inchas and W2 ,lbf 

t

3,inches and W3,lbf 

t 4 ,inchcs and W4,lbf 

ts,inchcs nnd W5,lbf 

t 6,inches nnd W6,lbf 

t

7

,inches and W7,lbf 

ts,inches nnd WS,lhf 

t

9 ,inches and W9 ,lbf 

t

10• inches and W10,lbf 

ORlGI:'I!!~' j .. " .• r. .... .'\~ .. , 

OF POOR Q:.lf'.Lrrv 

-62-

CSSQ 14 

y 1bf/cu.in ..;: 

0( 

" 

~ 

~ 

~ 

0( 

.: 

0 

68 

9.252 

15.25 

2.12 

12.60(e) 

35.00(0) 

796.7 

0.0500(f) & 0 (f) 

o (n 

0.022 

2.377 

o (0 

o (f) 

1.145 

o (f) 

o (0 

O.OSOO(f) 1.149 

CIl~e 15 

0.10,1.355,0.05 

1.512 

24.24 

30.24 

1.0,2.2 

12.2,1:!.6 

34.6,35.0 

500 

30 

76 

9.303 

15.30 

2.16 

12.60(e) 

35.00(e) 

811.7 

0.0500(f) & 0 

o 

o 

(f) 

(f) 

(f) 

2.345 

o (f) 

o (f) 

1.121 

o (0 

Case 16 

~ 

') 

::,. 

,. 

~ 

~ 

l" 

)r 

100 

80 

16.718 

22.72 

2.20(e) 

12.59 

34.97 

19S7 

O. OSOO(!) &O(f) 

L 

0(0 

1.00: 

o (C 

0.40( 

1.44: 

o (f) O.OSOO(f) O(f) 

0.0500(f) 1.277 0.3685 6.39~ 

\ 

. ---.. -- ..... ---~~- ... ~-~. --- .~,. -' .... , .. ~--_\ .... 
-63-

Notes: 

10 10 

(a) Obj active function Ibf - 182. 4 ~ I: tl + I: Wl 1-1 10 1 

(b) Weight Ibf • 120y + objoctive function 

(c) All g~adlenta ware computed by analytical techniques 

Eigonvalues were cozputed with double procision routinGs 

For all cases, ITMAX - 80, ITRM • 5, DELFUN • 0.0001, DABFUN • 0.001 

(d) Activo lower bound const~aint 

(e) Active upper bound constraint 

(f) Active side constraint 

(8) Active auto-rotational constraint 

Table IlA: Optici:ation of Rotating Beam 

\ 

-- . -,,--.. ---- '~--'-' .. ~- .. ----.. -~ -............ _~ .. __ .... _;o.._ ----.... ~"- ~~'-'--~--...---...... --....-... - ... ____ .. _-...........i.._ ... .wJ.t-'_,~ 
-64-

Initial Values Case 17 

4 Constant t,inches, I inches , Ylb~/cu.in. 0.10,1. 355,0.05 

Constant Wi11bf 

(a) Objective function J Ibf. 

(b) Weight J Ib-f 

f

1L, flU' Hz 

f 2L, f 2U' Hz 

f 3L, f 3U' Hz 

2 

1mn 1bf inches sec 

n RPM 

(c) Optim4l Solution 

No. of iterations 

Objective function 1bf 

Weight lbf 

f

1, Hz 

f2' Hz 

f

3

, Hz 

2 I Ibf inches sec 

m 

t

1 ,inches and W1 ,1bf 

t 21 inches and W2 .1bf 

t 3 .inches and tl3 ,1hf 

t

4 .inches and W4,lbf 

ts I inches and tIs \ 1bf 

t 6 ,inches and W6 ,1bf 

t 7 ,inches and W7 ,lbf 

ts,inches and WS,1bf 

t

9,inches and W9,1bf 

t 10,inches and WIOtbf 

1.512 

24.24 

30.24 

5.2,5.6 

18.0,18.6 

42.0,43.0 

1100 

300 

39. 

10.539 

16.54 

5.59 (e) 

18.59 (e) 

42.99 (e) 

1113. (g) 

0.0500 (f) & 0 (f) 

o (f) 

0.002 

0.591 

0.712 

o (f) 

0.74S 

0.438 

o (f) 

0.0500 (f) 3.492 

I 

I 

! 

I 

I 

I , 

\ 

"~""",,,,,,,,_,,,,""_-,-,,~~~,", ...::0......1&'.'&8-'-- '>d~ •• " .. ....k A

• - .. >4.·6 ·a""...' ....... ·r .... ,:;- .t,i:,':4 
-65-

Notes! 

10 10 

(a) Objective function,1bf - 182.4y E ti + E Wi 

i-l i-l 

(b) Weighttlbf - 120y + objective function 

(c) All gradients were computed by analytical techniques 

Eigenvalu~9 were computed with double precision routines 

For all cases, l~~ - 80, ITRM ~ 3, DELFUN - 0.0001, DABFUN - 0.01 

(d) Active lower bound constraint 

(e) Active upper bound constraint 

(f) Active side constraint 

(g) Active auto-rotational constraint 

Table ltB: Optimization of Rotating Beam 

I \ J 

1 

I 

1 
-66-

Frequency 

No. of First Second 

Mode Error Hode Errol 

Elements (Hz) ~%~ ~Hz) ~%2 

6 1.9678 0.6 12.142 3.6 

8 1.9733 0.3 12.258 2.7 \ 

10 1.9761 0.2 12.312 2.3 

12 1.9780 0.1 12.342 2.0 

14 1.97G9 0.06 12.360 1.9 

20 1.9805 0.03 12.385 1.7 

Analytical 1.980 12.6 

Table 12: Frequency VB. Number of Elements 
._--------- F.,e..E .. h: 249 $4f1" .. g\i<.?I"'ZU"iU',~.,h ".£.4~"""·l.£"'''*''''''(¥¥';iW.xQ··, .... ""!I\"", .~"*.,.-'" se«t cqeI.QII,J.;;\;oilM¥t'.'-*W 4'" M<';, ..... 'y;;': ..... i"\""""'., (,,*';8C'£.I'~ 

-67-

F'l"equency 

(c:'e) 

Mode No. (1) (~) rn 

1 2.01622 1.95462 2.01614 

2 12.6354 12.6354 12.635S 

3 35.3872 35.3916 

4 69.3934 

5 114.892 

6 172.121 Rest are essentially 

7 241.495 iQentica1 to (1) 

8 323.537 

9 418.329 

10 520.042 

11 692.152 

12 836.210 

13 1013.68 

14 1223.45 

15 1470.15 

16 1758.30 

17 2086.84 

18 2436.29 

19 2743.53 

20 3433.47 

Table 13: Frequencic~ Ca1cul~tcd bv Various M~thQds 

\ 

~ 

~4) (5) 

2.01614 2.01511 

35.3875 

69.3937 

114.892 

172.122 reat are 

241.495 essontially 

323.536 identical 

418.329 

520.043 to (1) 

692.155 

836.~lO 

1013.68 

1223.44 

14;0.15 

1758.31 

2086.85 

2436.30 

2743.53 

3433.43 

, 

I 

I 

• I 

I 

I 

l 

I ! " 

I 

, 

i 

-. 

-

, ." 

\ 

\ 

"-
'f>. * ",*,,"",.<;. If ... '. . '" . 

-68-

B1GmAnt Thicl~ .. '\oa ... (inchos) 

Element Singlo-Preciaion Double-Precision 

No. Calculation Calculation 

1 0.0753 0.0724 

2 0.0703 0.0881 

3 0.0886 0.0821 

4 0.0856 0.0564 

5 0.0500 0.0500 

6 0.0500 0.\)500 

1 0.0500 0.0500 

8 0.0500 0.0500 

9 0.0500 0.0500 

10 0.0500 0.0500 

ObjectivQ 

Function 0.593 0.599 

(LBF) 

Table 14: Difference in Optimal Designs Cau~~d bv Errors in 

Eig6~valuo ~alcul&tion 
....... r--;~""""~-""""7 .-.~.''''''- , ............... ~~ - .. _ .... ~ ,-- ... ~.- .~- ~- ..... - -~ ..... -. ; • -. . -.-- -----. ,--_., ······c·.,,.·- -'-".-.-~' _ .• -.. ....... _--....... -".-~ -._ •. -~--. · .. ~-...-.·..,.-·l 

EleClont 

No. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

(1) 

3fl13t1 

(cpal1n. ) 

4.791 

3.492 

2.388 

1.472 

0.729 

0.105 

-0.443 

-0.957 

-1.489 

-2.074 

-69-

(2) (3) 

3f1/dfil1 3fl 1311l1 

( cps/lhE) (cpa/lhE) 

0.525 0.000 

0.383 -0.001 

0.262 -0.003 

0.161 -0.009 

0.080 -0.022 

0.012 -0.043 

-0.049 -0.073 

-0.105 -0.114 

-0.163 -0.166 

-0.227 -0.229 

T"b1a 15: Sensitiv1t\· of Frequoncy to Changes in Thicknass :.nd '~e1~ht 

1 

i 

'1 
.' ...... -... '. .. - .. -... .. ,~. ~ 

,. ,"t 

; 

-70-

; 

, I. FIGURES 
- .---,-. ~-.~"".' 

-72- ORIG1"'(l" " ...... "";.,..- r, ._ .... r~' 1o,'.,J 

OF POOR Qut-urt p' 

~I' t

V

3 

T • /ot:S;:oi==_=* := :::::'[=.=: ='=:::=::=:::~7~" 

V2 V4 

/z 

Figures 2: Illemcnt undor Constant Tension. 

22.0 

-fi: 2 i.5 

om 

-I 

-.., 

en 

o 20.5 

20.0"-~-~-~~~.....!:::-~121=----1~4-...l16-----1~8-2~!O~~22 

TOT.!:\L NUMBER OF ELEMENTS IN MESH 

Figure 3: Obj vs. No. of Elements. 
-73- ORIGiN;;.:" -;:-r/;i,~ ;;: 

OF. POOR Ql'l~Ll1Y 

10.0 Weight At Froa End (mn) 

\ ' 

", TOTAL NUMBER OF ELEMENTS IN MESH 

Figure 4: Lumped Weight Versus No. of Elements. 

4 

n, TOTAL NUMBER OF ELEMENTS IN r,-~ESH 

Figure 5: Moment of Inertia at Free End Versus No. of Ele~ents in Mesh. 
u= 10.0 

m 

...I 

~ 

s: 

~ 5.0- ~ :;... 

2.0 

-74-

ORIGINAL PAGE rs 

OF POOR QUALITV 

Waight Near Middlo (m'2.' + mn/2~2) 

~LQ .. e <> ,. 

", TOTAL NUMBER OF ELE~viEN1S IN MESH 

Figure 6: Coo:h1nedt~eight va. No. of Elcmento. 

\ 
"=10 

ILl?Wr. 

"7:10.0022 

-75- ORIGIN;\L PP.G~ t~ 

OF. POOR QUALITY 

611 9.8510 

08Ja- 21.5436 

maD 5.n4G 

09Jo: 21.6110 

(Note: OBJa>08J6 ) 

loa 3.3512 

"'o+rua

ll 10.4906 

08J10:1 21.24'05 

Figure 7: O~timal ~e9igns for Various Values of n. 

, .... , 
-76-

ma=0.82884 

n=I2 

mg= 0.925\9 

n = 14 ma = 0.78826 

Figure 7: (Continued) 

ORIGIN~.L P/\GZ tS 

OF POOR QUALITY 

m'2= 4.8994 

OBJ'2 = 21.0201 

1'2= 4.3145 

m'2 + ffi'2 = 9.5243 

m'4 = 5.\901 

OBJ'4 = 20.8\54 

1'4 = 4.4000 

m'4 + ffi l4 = 9.3064 

\ 

I 

I 

! 

I 

I 

; , t-
n=15 

n::20 

Figure 7: (Continued) 

~,," . ...,. ....... 

-77- C~IG!f!AL PAGE If! 

OF POOR QUALITY 

116= 4.5334 

mlS + miG = 8.9647 

m20= 7.9819 

OBJ20 = 20.3768 

m20 + ffi 20 = 8.7980 

I20= 1.4881 

\ 

l ita . " . t 'ei 1 .~u4iL: t'V '~&t9·,*1;"·"""~1VW.··ff,..,,)"":~.""'>itf0'·';:'ri,.cc."';I<:l}j;··e...;i,~.!...!~~~~'4I:!.J 
-78-

n = 10 

l..BM 

08J = 21.1457 I-BJ= 

(Note: OBJ < 

Previous OBJ,o) 

Figure 8: Alternative Opti~um - Found by Imposing Constraint 0.83073 ~ I 10 ~ 2.0. 
" 

~, '~- \ -' .-~ 

.' : 

.'.' 

....... " 

GK[]) .: -, ~>' .> 

'. -./ 

,.~. 

~, ~'-

..•. :!l.;i;~) •. T-\lf5r···':i,·· ~>~.;.> . . 

.... ~~ ..• ~> ::c· ". 

'-,.- . , ~. 

-. " 

."'--

'," . 

. ;.-

-:." 

> .. 

.~ .> 

.. :: 

....... -

:' . 

' .. ~ 

I 

.. -.~' ) 

., » 
End of Document