Design of helicopter rotor blades for optimum dynamic characteristics

FINAL REPORT NASA RESEARCH GRANT NC. NAG-1-258 for Period Covering 17 January 1382 through 17 January 1985 "DESIGN OF HELICOPTER ROTOR BLADES FOR OPTIMUM DYNAMIC CHARACTERISTICSn

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David A . Peters, Professor and Chairnan and Timothy KO, Research Assistant Department of Mechanical Eragineerinq Washinaton University, C a m y . s Box 1185 St. Louis, MO 63130 Alfred Korn, Professor and :lark P. Rossow, Profsssor Department of Civil Engineering Southern Illinois University Edwardsville, IL 62026 January 17, 1935 f l A S A - C B - 176076) OESICN OP B E L I C O P T S B BCXCR Y85-3 1044 BLADES FOB O P T 1 80 C CY b A f l I C ILE16dCTEBISTICS

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TABLE OF CONTENTS Pagg

2. I m p o r t a n c e o f t h e R e s e a r c h .e................................ .1

........................................ 3 . S u m m a r y o f t h e P r o j e c t 4

3.2 Findings Related to Optilaization of Rotor B l a d e s . . . . . , . l 6 4. R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . = . . . . . . , . . . . . . . . . . . . , . . . . . . . 2 1

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Appendix 1: Thesis of Timcthy KO L L ' 3

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1. INTRODUCTION This document is the final report of a research project concerned with the optimal design of helicopter rotor blades. The report contains three main parts: 1 . a discussion of the reasons for which the research was undertaken; 2. a summary of project accomplishments, presented in the form of a list of optimization problems which have been solved and a list and brief description of findings related to optimization of rotor blades; 3 . the doctoral thesis of Timothy KO, which contains many detai 1s af the computations performed during the project.

2 . IMPORTANCE OF THE RESEARCH The design of helicgpter rotor blades involves not only considerations of strength, survivabi 1 ity, fatigue, and cost, but also requires that blade natural frequencies be ;ignificantly separated from the fundamenta 1 aerodynamic forcing frequencies

(egg. Ref. 1) . A proper placement of blade frequencies is a

difficult task for several reasons. First, there are many forc- ing frequencies (at all integer-multiples of the rotor RPM) which occur at rather closely-spaced intervals. For example, S/rev and 6/rev are less than 20% apart. Second, the rotor R P M may vary over a significant range throughout the flight envelope, thus re- ducing even further the area of acceptable natural frequencies.

Third, the natural modes of the rotor blade are often coupled because of pitch angle, blade twist, offset between the mass center and elastic axis, and large aerodynamic damping, These couplings complicate the calculation of natural frequencies. In fact, the dependence on pitch angle makes frequencies a function of loading condition, since loading affects collective pitch.

Fourth, the centrifugal stiffness of ten dominates the lower modes, making it difficult to alter frequencies by simple changes in stiffness.

In the early stages of the development of the helicopter, it was believed that helicopter vibrations could be reduced (and even eliminated) by the correct choice of structural coupling and mass 'stiffness distributions. However, it is easy to imagine how difficult it is to find just the proper parameters such that the desired natural frequencies can be obtained. The difficulties in placement of natural frequencies have led, in many caseg, to preliminary designs which ignore frequency placement. Then, 1 after the structure is "finalized" (either on paper or in a prototype blade), the frequencies are calculated (or measured) and final adjustments made. Reference (2) describes the develop- t; ment of the X H - 1 7 helicopter in which a 300-lb weight was added to each blade in order to change the spanwise and chordwise xass distiribution and thereby move the first flapwise frequency away fromn 3/rev. However, these types of a1 terations are detrimental to blade wight, aircraft development time, and blade cost. In addition, corrections usually are not satisfactory, and the heli- copter is often left with a noticeable vibration problem.

The state-of-the-art in he1 icopter technology is now to the point, however, that it should be possible to correctly place rotor frequencies during preliminary design stages. There are i several reasons for this. First, helicopter rotor blades for . i i . !

I both main rotors and tail rotors are now being fabricated from I\ !

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I composite materials (Refs. 3 and 4). This implies that the !

designer can choose, with limited restrictions, the exact E I ,e, , , distribution desired. Furthermore, the lightness of composite blades for the main rotor usually necessitates the additi-on of i weight to g i v e sufficient autorotational blade inertia. Thus, -i . i tbsre is a considerable amount of flexibility as to how this ~.

weight may be distributed. Second, the methods of structural

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% . i optimization and parameter identizication are now refined to the .?

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. ture. Some elementary techniques have already been used for the 7 : .. .

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design of rotor fuselages (Ref. 5 ) . It follows that the time is right for the cse of structural optimization in helicopter blade design. Some work on this is already under development, and, a 1 though not pub1 ished, some companies are already experimenting with the optimum way to add weight ot an existing blade in order to improve vibrations.

The purpose of the research project described in this report was to investigate the possibilities (as well as the limitations) of tailoring blade mass and stiffness distributions to give an optimum blade design in terms of weight, inertia, and dynamic characteristics. The work has focused on aonf igurations that are

simple snough to yield clear, fundamental insights into the structural mechanism but which are sufficiently complex to result in a realistic result for an optimum rotor blade.

3 . 0 SUMMARY OF THE PROJECT 3.1 OPTIMIZATION PROBLEMS WHICH WERE SOLVED The basic structure optimized was a beam free'at one end and supported at the other. Various support conditions and con- straints on natural freqencies were used. The behavior of the beam was computed by using a 10-element f inite-element aodel.

Quantities associated with the finite-element model, such as the thickness or area moment of inertia of each element, served as desig~l variables in the the optimization procedure. A typical formulation of an optimization problem was Find the flange and wall thicknesses of a box-bean cross-section (three variables per finite element) which minimize the weight of the beam, while main- taining the first natural frequency within a "win- dow" (e.g., 2 . 4 < p l < 3 . 0 per rev).

All optimization problems were solved with the CONMIN com- puter program [ 6 ] . CONMIN is based on the mathematical nonlinear- programming method of feasible directions.

The list of problems solved follows.

Case 1. Cantilever beam

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Rotating: no Objective function: weight

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~ e s i g n variables: area moments of inertia Boundary condition(s1 at root: fixed \ i Frequency Constraints: first flapping specified through equality !

! constraint Autorotation constraint: no Stress constraint: no References: First Semi-Annual Report pp.17-18, Thesis, pp. 18-19

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Case 2 . Cantilever beam with tip mass

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Rotating: no Objective function: weight Design variables: cross-sectional areas Boundary condition (s) at root: fixed Frequency Constraints: lower bound on first flapping Autorotation constraint: no Stress constraint: no References: First Semi-Annual Report pp.17-19, Thesis, pp. 18-21

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Case 3. Wind-turbine blade

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Rotating: yes Objective function: weight Design variables: area moments of inertia, lumped weights Boundary condition (s) at root: fixed Frequency Constraints: windows on first and second flapping i Autorotation constraint: yes

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\ Stress constraint: no

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References: First Semi-Annual Report pp.21-24, Thesis, pp. 34-37 .................................................................

Case 4 . Hingeless rotor-blade

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Rotating: yes Objective function: weight Design variables: area moments of inertia, lumped weights Boundary condition (s) at root: fixed Frequency Constraints: windows on first and second flapsing Autorotation constraint: yes Stress constraint: no References: First Semi-Annual Report pp.23-27, Thesis, pp. 38-41 .................................................................

Case 5.eCantilever beam with t w ~ frequency constraints

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Rotating: co Objective function: weight Design variables: area moments of inertia Boundary condition (s) at Loot: fixed Frequency Constraints: windows on first and second flapping Autorotation constraint: no Stress constraint: no References: Second Semi-Annual Report pp.7-16, Thesis, pp. 23-33 - - p - - p p - - - p p p p - ~ p p p - - p - - p - - p - p - p - p - - p p p ~ - p p - - Case 6. Cantilever beam (similar to Case 5, except for three

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rather than two frequency constraints) Rotating: no Objective function: weight Design variables: area moments of inertia Bolndary condition (s) at root: fixed Frequency Constraints: windows on first, second and third flapping I Autorotation constraint: no

Stress constraint: no References: Second Semi-Annual Report pp.17-18 ..................................................................

C a s e 7 . Cantilever beam (similar to C a s e 6, except for addition

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of lumped weights a s design variables.)

Rotating: no Objective function: weight Design variables: area moments of inertia, lumped weights Boundary condition (s) at root: fixed Frequency Constraints: windows on first, second and third flapping Autorotation constraint: no Stress constraint: no References: Second S e m i - ~ n n u a l Report pp.17-21 - - - - C a s e 8 . Cantilever beam (similar t o C a s e 7, except for addition

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of autorotation constraint) Rotating: no Objective function: weight Design variables: area moments of inertia, lumped weights Boundary condition(s) at root: fixed Frequency Constraints: windows on first, second and third flappin5 Autorotation constraint: yes (constraint applied to mass momect of inertia of w h o l e beam) Stress constraint: no References: Second Semi-Annual Report pp.17-22 ...............................................................

C a s e 9 . Cantilever beam (similar to Case 5, except beam is

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rotating) Rotating: yes

Objective function: weight Design variables: area moments of inertia, lumped weights Boundary condition (s) at root: fixed Frequency Constraints: windows on first, second and third flapping Autorotation constraint: yes (constraint applied to mass moment of inertia of whole beam) Stress constraint: no References: Second Semi-Annual Report pp.17-24 - - - - - - Case 10. Teetering rotor

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Rotating: yes Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, lumped weights I Boundary condition(s) at root: fixed > Frequency Constraints: windows on first, second and third collective flapping Autorotation constraint: yes Stress constraint: no References: Third Semi-Annual Report pp.5-6, Thesis, pp. 42-49 ...............................................................

Case 11. Teetering rotor (similar to Case 10, except cyclic

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flapping modes considered, instead of collective modes) Rotating: yes Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, lumped weights

Boundary coodition(s) at root: pinned Frequency Constraints: windows on first, second and third c y c l i c flapping Autorotation constraint: yes Stress constraint: no References: Third Semi-Annual Report pp.7, Thesis, pp. 50-51 ...............................................................

C a s e 12. Teetering rotor (similar to Cases 10 and 11, except that

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both c y c l i c and c o l l e c t i v e flapping modes are considered) Rotating: yes Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, lumped weights Boundary condition(s) at root: one analysis performed with pinned I , conditions, another anaiysis perfomed with fixed conditions I Frequency Constraints: windows on first, second and third \ c o l l 2 c t i v e flapping and a l s o on first, second and third c y c l i c flapping Autorotation constraint: yes Stress constraint: no References: Third Semi-Annual Report pp.7-9, Thesis, pp. 50-54 ...............................................................

C a s e 13. Teetering rotor (similar to Cases 10-12, except that

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c o l 1 e c t i v e and c y c l i c flapping and inplane and a l s o torsional modes considered) Rotating: yes

Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, wall thicknesses cf both sides of box cross-section, lumped weights, stiffness of torsional spring at root

Boundary condition(s) at root: a) flapping -- one analysis

performed with pinned conditions, another analysis perfomed

with fixed conditions; b) inplane -- one analysis performed

with pinned conditions, another analysis perfomed with fixed

conditions; c) torsion -- fixed conditions

Frequency Constraints: winCows on first, second and third collective and cyclic flapping; windows on first, second and third collective and cyclic inplane; and window on first torsional Autorotation constraint: yes Stress constraint: yes References: Third Semi-Annual Report pp.12-13, Thesis, pp. 58-60

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Case -- 1 4 . T ~ e t e r i n g rotor (similar to Case 13, except that box-

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beam dimensicns are fixed) Y , Rotating: yes Objective function: weighted sum of squares of dirferences in frequencies Design variabies: lumped weights, stiffness of t o r s i o ~ s l spring at root

Boundary condition(s) at root: a) flapping -- one ana1y:;is

performed with pinned conditions, another a n a ~ y s i s perfomed

with fixed conditions; b) inplane -- one a n a l y s i s performed

with pinned conditions, another analysis perfomed with fixed

conditions; c) torsion -- fixed conditions

Frequency Constraints: windows on first, second and third c o l l e c t i v e and c y c l i c flapping; windows on first, second and third c o l l e c t i v e and c y c l i c inplane; and window on first torsional Autoratation constraint: yes Stress constraint: yes References: Third Semi-Annual Report pp.13-14, Thesis, pp. 60-61 C a s e 15. Teetering rotor (similar to Case 14, except that

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I stiffness of b l a d e cross-section at root is a design -1 . . I variable) Rotating: y e s j -.

Objective function: weighted sum of squares of differences in . I - i frequencies I Design variables: lumg.ad weights, stiffness of tor:sional spring at root, variable root-stiffness (but except at root, a l l i other dimensions of the box cross-section are fixed)

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B o u n ~ a r y condition(s) at root: a) flapping -- one analysis

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performed with pinned conditions, another analysis perfomed

with fixed c ~ n d i t i o n s ; b) inplane -- one analysis performed

with pinned conditions, another analysis perforned with fixed

conditions; c) torsion -- fixed conditions

Frequency Constraints: windows on first, second and third

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c o l l e c t i v e and c y c l i c flapping; windows o n fjrst, second and

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third collective and cyclic inplane; and window on first torsional Autorotation constraint: yes Stress constraint: yes Re~2rences: Third Semi-Annual Report pp.14, Thesis, pp. 61 .................................................................

Case 16. Teetering rotor (similar to Case I?, except that blade

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pretwist is included.

Rotating: yes Pretwisted Blade: yes Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, wall thicknesses of both sides of box cross-section, lumped weights, stiffness of torsional spring at root

Boundary condition(s) at root: a ) flapping -- one a!:alysis

performed with pinned conditions, another analysis performed

with fixed conditions; b ) inplane -- one analysis performed

with pinned conditions, another analysis performed with

fixed conditions; c) torsion -- fixed conditions

Frequency Constraints: windows on first, second and third collective and cyclic flapping; windows on first, second and third collective and cyclic inplane; window on first torsions: Autorotation constraint: yes Stress constraint: yes References: Thesis, p. 61, 64

Case 17. Teetering rotor (similar to Case 14, except that blade

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pretwist is included Rotating: yes Pretwisted Blade: yes Objective function: weighted sum of squares of differences in frequencies Design variables: lumped weights, stiffness of torsional spring at root

Boundary condition(s) at root: a) flapping -- one analysis

performed with pinned conditions, another analysis performed

with fixed conditions; b) inplane -- one anlaysis performecl

with pinned conditions, another analysis performed with

fixed conditions; c) ccrsion -- fixed conditions

1 Frequency constraints: windows on first, second and third r collective and cyclic flapping; windows on first, second and L i . I

i third collective and cyclic inplane; window on first

I torsional - ! Autorotation constraint: yes s Stress constraint: yes I i !

References: Thesis, p . 61, 64 I

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Case 18. Articulated rotor

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Rotating: yes

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Pretwisted Blade: yes

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1 Objective function: initially the weighted sum. of squares of

differences in frequencies; after a feqsible design is found, the objective is changed to the weight.

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Design variables: flange thicknesses, wall thicknesses of

both sides of box cross-section, lumped weights, stiffness of torsional spring at root

Boundary condition(s) at root: a) flapping -- ~ i n n e d ; b) inplane

-- pinned at s a m e r a d i a l l o c a t i o n a s in t h e c a s e of

flapping; c) torsion -- fixed conditions

Frequency Constraints: windows on first, second and third flapping; windows on first, second and third inplane; and window on first torsional Autorotation constraint: yes Stress constraint: yes References: Fifth Semi-annual Status Report pp. 8-9, Thesis, pp. 57-69 Case 19. Articulated rotor (similar to Case 18, except that box-

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beam dimensions are fixed) Rotating: yes Pretwisted blade: yes Objective function: initially the weighted sum of squares of differences in frequencies; after 3 feasible design is found, the objective is changed to the weight.

Design variables: lumped weights, stiffness of torsional spring at root

Boundary condition(s) at root: a) flapping -- pinned; b) inplane i

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-- pinned at s a m e r a d i a l l o c a c l o n a s in t h e c a s e of

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flapping; c) torsion -- fixed conditions i

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Frequency Constraints: windows on first, second and third flapping; windows on first, second and third inplane; and ;Y /...

window on first torsional k Wi .

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~ u t o r o t a t i o n constraint: yes Stress constraint: yes References: Fifth Semi-Annual Status Report, pp. 9-10 Thesis, pp. 67-70 Case 20. A r t i c u l a t e d rotor ( a r t i c u l a t i o n a t d i f f e r e n t s t a t i o n s for flapping and inplane motion) Rotating: yes Pretwisted Blade: yes O b j e c t i v e function: i n i t i a l l y t h e w e i g h t e d s u m o f s q u a r e s of difference in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: lumped weights, stiffness of torsional spring at root.

Boundary conditions (s) at root: a) Flapping -- pinned;

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b) Inplane -- pinned, but pin location is another few feet away t

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from root ' : F r e q u e n c y constraints: w i n d o w s o n a l l first, second and third fre- quencies (flapping, inplane, torsion) !

Autorotation constraint: yes

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Stress constraint: yes

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References: Sixth Semi-Annual Status Report, pp. 3-4 Case 21. Articulated rotor (similar to case 20 except that box beam dimensions are also design variables) Rotating: yes Pretwisted Blade: yes

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Objective function: initially the weighted sum of squares of differ- I ences in frequencies; after a feasible design is found, the objective is changed to the weight.

Design variables: flange thicknesses, wall thicknesses of both sides I of box cross-section, lumped weights, stiffness of torsional spring at root.

Boundary conditions(s) at root: a) Flapping -- pinned;

b) inplane -- pinned, but pin location is another few feet awal

from root Frequency constraints: windows on a 1 1 first, second and third fre- quencies (flapping, inplane, torsion) Auto rotation constraint; yes t Stress constraint: yes ReFcrences: Sixth Semi-Annual Status Reports, p. 5 . .: .

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3 . 2 FINDINGS RELATED TO OPTIMIZATION OF ROTOR BLADES - 1 The most important general finding of the project is that it t is possible to use an optimization routine such as CONMIN to tailor blade mass and stiffness distributions in an optimal manner. Furthermore, formulating the optimization problem in i !

terms of frequency placement (that is, restricting the natural ' !

frequencies of the blade to lie within narrcw intervals located I r away from certain integer multiples of the rotor speed) has been shown to be a useful approach for reducing vibrations.

F 2 3 $ In addition to these general findings, the project estab- ~ .- i .

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lished a number of specific results, knowledge of which would be useful to anyone intending to apply or extend the optimization approach developed during the project. A list of these results follows.

In applying CONMIN to rotor-blade design, gradients of the objective and constraint functions should be calcu- lated by analytical formulas rather than by finite differences. However, finite differences serve as a useful check on the possibility of errors in the com- puter implementation of the analytical formulas.

Reference: Second Semi-Annual Status Report, pp. 9-10; Thesis, pp. 23.

2 . Frequency constraints may be formulated directly in

. terms of the frequency - in -' Hz rather than in terms of

eigenvalues e . , the square of the circular frequen- cy). If eigenvalues are used, then scaling should be employed in the constraint equations to ensure we1 l- behaved gradients for use in C O N M I N .

Reference: Second Semi-Annua 1 Status Report, pp. 10-12; Thesis, pp. 25 3. The £01 lowing values of C O N M I N parameters were ade- quate for most of the optimization studies: ITMAX = 48-80 ITRM = 3 DELFUN = 0.00B1 (for cantilever beams) = 0.00001 (for rotor blades)

DABFUN = 0.0025 (for cantilever beams) = 0.00001 (for rotor blades) THETA = 1 . 0 PHI = 5.0 Reference: Second Semi-Annual Status Report, pp. 13-16; 4 . More efficient designs can be achieved if lumped weights are included as design variables (along with dimensions of the cross-section of the blade).

Reference: Second Semi-Annual Status Report, pp. 18-21; i ~ h e s i s , pp. 27.

i 5 . Because of the stiffening effsct of the centrifugal forces in a rotating blade, frequency placement is much less dependent on stiffness and mass distributions than in s non-rotating blade. Thus, the rotational speed has a strong influence on what can be achieved in the optimization process.

Reference: Second Semi-Annual Status Report, pp. 23-24;

6. Use of - ten finite-elements appears adequate to model a

rotor-blade for optimization studies, a 1 though if many frequencies must be calculated, more elements must be used. Empirical rules which have been suggested are a) Use 4n degrees-of-freedom; and b) use n2 degrees, where n is the number of frequencies to be found.

Reference: Second Semi-Annual Status Report, pp. 25-29;

7 . An accurate eigenvalue routine should be used in opti- mization studies, since errors in eigenvalue calcula- tions can appreciably affect the optimal design.

Reference: Second .Semi-Annual Status Report, pp. 29-31; 8. The natural frequencies of a rotor blade are not great- ly affected by small changes in dimensions of the blade cross-section.

Reference: Second Semi-Annua 1 Status Report, pp. 31-32; 9. For tight frequency-constraint windows, CONMIN is often unable to find a feasible design. In such cases, an objective function consisting of the weighted sum of squares of the differences in frequencies (actual fre- quency minus desired value) may be used initially. In the process of minimizing this objective f u n c t i o ~ , CONMIN is often a b l e to find a design which satisfies the frequency constraints. If this occurs, the objec- tive function may then be switched to the weight of the blade.

Reference: Third Semi-Annual Status Report, pp. 5-9; Thesis, p . 44, 48, 50, 52, 5 8 10. The natural frequencies of a blade which has already been built can be modified in a rational (rather than trial-and-error) manner by using CONMIN to specify where lumped mass should be added. It appears best to either raise all undesirable frequencies or lowex all unzesirable frequencies (a mixture of raised and

lowered frequencies is much more difficult to attain).

References: Third Semi-Annual Status Report, pp. 13-14; Thesis, pp. 60-61; Sixth Semi-Annual Status Report, p . 4.

11. The forced response of a rotor blade can be adequately controlled through the approach of frequency placement.

Reference: Fourth Semi-Annual Status Report, pp. 2-18; Thesis, pp. 71-95.

12. Aerodynamic damping substantial ly reduces resonant peaks, but even in the presence of damping, frequency placement is a powerful driver of loads, and, as a result, frequency placement can be justifiably con- i i sidered an iniportant part of blade optimization in the presence of damping.

Reference: Fourth Semi-Annua 1 Status Report, pp. 5-7; I ; f Thesis, pp. 75-82.

13. Fini te-element model ling errors caused by neglecting F e 7 :

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restraint of warping during twist, and filler stiffness y.. .T* . r are small. However, accurate f i 1 ler properties, dimen- sions and locations are required in order to model the mass distribution properly.

Reference: Fourth Semi-Annual Status Report, pp. 11-12; 14. Since calculating eigenvalues is the major computa- tional burden in rotor-blade optimization, an efficient eigenvalue routine should be used. For example, determ-

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insnt search or subspace iteration can be used to calculate only the needed first few frequencies.

Reference: Fifth Semi-Annual Status Report, pp. 2-5 4. REFERENCES 1 . Uiebank, C . and Girvin, W . , 'Sikorsky S-76 Analysis, Design and Development for Successful Dynamic Characteristics", Proceedings of the 34th Annual National Forum of the American Helicopter Society, May 1978, pp. 78-23-i through 78-23-17.

2 . Hirsh, Harold, Dutton, Robert E . , and Rasumoff, Abner, "Effect of Spanwise and Chordwise Mass Distribution on Rotor Blade Cyclic Stresses, ' I Journal of the American Helicopter Society, Vol. 1, No. 2, April 1956, 3 . Ellis, C . W . et al., " Design, Development, and Testing of the Boeing Vertol/Army YUH-61A, "Proceedings of the 32nd Annual National Forum of the American Helicopter Society, Preprint 1010, May 1976.

4 . Fenanghty, Ronald, R . and Noehren, Wi 1 1 iam L . , ttComposi te Bearingless Tailor Rotor for UTTAS," Journal of the American Helicopter Society, Vol, 22, No. 3, July 1977, pp. 19-26.

5 . Hanson, H . W . and Calapodas, M . J . , "Evaluation of the Practical Aspects of Vibration Reduction Using Structural Optimzation 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

Constt-ained Function Minimization, User's Manual, NASA TMX- 62.282, August, 1973.

APPENDIX 1 Thesis of Timothy KO

W G R 8 5 4 , 853 SEVER INSTITUTE OF TECHNOLOGY Doctor of Science Degree 3ISSEKTATION ACCEPTANCE (To be submitted by the graduation approval deadiinej Timothy Wai Hung KO STUDENT'S NAME: E . R . S . CODE: This atudert ' a dissertation, entitled " D e s i g n of He1 i c o p t e r R o t o r 81 ades f o r Optinum Dynamic C h a r a c t e r i s t i c s " has beem examined by the undersigned committee of three faculty members and has received full approval for acceptance i n partial fulfillment of the requirements for the degree Doctor of Science.

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Signatures : A ,- , G;~~t.f//";~<<Lj;~~e - - a & -

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WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLO1~Y DESIGN O F HELICOPTER R O M R BLJC ,R OPTMJM DYNAMIC CHARACTERISTIc.1 b Y T h o t h y W H K O Prepared under t h e d i r e c t i o n o f P r o f e s s o r D . P e t e r s A t h e s i s presented t o the Sever I n s t i t u t e o f Washington U n i v e r s i t y i n P a r t i a l f u l f i l l m e n t of the requirements for t h e degree o f DOCTOR OF SCIENCE Deoember, 1984 S a i n t L O U ~ J , Y i s s o u r i

YASHINGMH UNIVERSITY SEVER flOSTITTJTE GF TeCIRJOLOGY ABSTRACT D E S I G N OF HELICOPTER ROTOR BLADE% FOR OPTIMUM DYIVAPIIC CRARACTERISTICS by Timothy W H K O ADVISOR : P r o f e s s o r D. P e t e r s S a i n t L o u i s , M i s s o u r i The mess and stlftnsss d i s t r i b u t i o n s f o r h e l i c o p t e r r o t o r b l a d e s axe t o be t a i l o r e d i n such a way t o g i v e a predetermined p lac em ant of b l a d e n a t u r a l f r e q u e n c i e s . The o p t m a 1 d e s i g n is pursusd w i t h r e s p e c t o f r n i n i r m ~ w e i g h t , s u f f i c i e n t i n e r t i a , and r e s o n a b l e dynamic c h a r a c t e r i s t i c s .

F i n i t e element t e c h n i q u e w l l l b e used a s a t o o l . Rotor t y p e s i n c l u d e h i n g e l e s s , a r t i c u l a t s d , and t e e t e r i n g .

TABLE OF CON'PENTS Paqe ........................................

1.1 Hellcoptor Design 1 .................................................... 1.3 Scope 5 .................... 1.4 Overview of Optimal S t r u c t u r a l Pesign 7 ......................... 3 . I l l u s t r a t i v a Examples of Optimization 18 .....................

3.1 C a n t i l e v e r Beam with Given Frequency 18 ............................ 5 . Preliminary Calculation f o r Rotors 34 5 . 1 Wind Turbine Blade ...................................... 3 4 6 . Teetering Rotors .............................................. 4 2 .................... 6.2.3 Combined C o l l e c t i v e and Cyclic 50 ............... 6 . 3 Simult. w o u s Flapping. Inplane and Torsion 5 4 6.3.1 Variable Box Dimensions ........................... 5 8

TABLE OF COTlTeWTS (continued) ............................. 5 . 3 . 2 Pixed Box Dimmasions 60 6 . 4 . 1 Variable BOX Beam .......................................

6.4.2 Fixed dax Be- ..........................................

..............................

7 . 2 Variable B a ham Dimension 67 ............

8 . R e l a t i o n Between Vibration and Fraquencg Placement 71 ................................. 8.2 Response versus Praquencp 71 . .................. 1 . 2 C a l c u l a t i o n of Torsional Stiffness GJ 102 11.3 Derivation o f Pass and S t i f f n e s ~ Matrix as Function of 13 . Vita .......................................................... 1 l 2

LIST O F TABLES No.

1 .

2.

4. Helicopter B l a d e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ....................

5. Heltcopter Blade with S t i f f Flapping..... 4 0 I a i t i a l and F l m l Design f o r C o l l e c t i v e and Flapping Hodes 6 .

(Teetering R o t ~ r ) . . . . ~ . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . 49 I n i t i a l and F i n a l Design f o r Cyclic Flapping Modes 7.

I n i t i a l and F i n a l Design f o r C o l i e c t i v e and C;elic Flapping 8.

10. I n i t i a l and F i n a l Design f o r Flappi=, Inplane, and Torsional Modes of Teetering Rotor Blade with Variable Box Beam D m m i o n . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 11. I n i t i a l and F i n a l Design f o r Flapping, Inplane, and Torsional Modes of Teetering Rotor Blade with Box Beam Dimensions F I X d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 12. I n i t i a l and F i n a l Design f o r Flzpping, Inplane, and Torsional Modes of Teetering Rotor Blade with Box Beam Dimensions Fixed Except Root Stiffness................................ 63 . .

13. I n i t i a l and Final Design f o r Flapping, Inplane, and Torsional Modes of Teetering Rotor ( P r e t w i s t ) Blade with Variable Box Beam D i m e n s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 14. I n i t i a l and F l n a l Design f o r Flapping, Inplane, and Torsional Modes of Teetering Rotor ( P r e t w i s t ) Blade Box Beam Dimensions F i x e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 5 6 16. I n i t i a l and F i n a l Design f o r Flapping, I n p l a ~ e , and Torsional Modes of A-ticulated Rotor Blade w i t h Variaole Box Beam D i m e n s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

1 0 0 . ' 84 1

17 . I n i t i a l and Finai Des ign for Flapping, Inplane, and T o r s i ~ n t 1 Kadea of A r i t i c u l a t d Rotor Blade v i t h Variable Box B L ~ Dimensions F i r e d . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y f

LIST O F FIGURE?

Paqe No.

f o r Optimum Beam i n P r e s e n t Work ,..... 1 9

Area Moment o f I n e r t i a Area Hoslent of I n e r t i a f o r Optimum Beam from R e l ll..., . -, , 19 C a n t i l e v e r beam with concentrbted mass..........,............ 2 1 Cocvergence of Weight of Objective F ~ l n c t i o n . . . . . . . . . . . . . . . . . . 28

A l t e r n a t i v e Optimum... ....................................... 3 2

f o r Box Beam Dimensions and d e f i n i t i o n of Design Variable Radial V a r i a t i o n of Forcing Function........................, 74 Tip Respcnae bersus Forcing Frequency f o r Both I n i t i a l and Tip Response Versus Forcing Frequency f o r I n i t i a i Design Both Tip Response Versus Forcing Freqaency f o r Both I n i t i a l and Sum o f Squares of Shears Versus Forcing Frequency f o r I n i t i a l Sun of !?;xz-es of Shears Versus Forcing Frequency f o r F i n a l Sum of Squares of Shears Ver- Forcing Frequency f o r Both Sum of Squares of Shears Versus Second Natural Frequency f o r

LIST OF FIGURES icontinued 1 No. Page 21. Sum of Squares o f Shears Versu~ Third Natural Frequency t o r 22. Sum of Squares o f Shears Versus Second Natural Frequency f o r 23. Sum of Squares o f Shears Versus Third Natural Frequency f c r Sun of Squares of S h s a r s Second Natural Frequency f o r I n i t i a l 24.

Design Both With and Without Damping (where Forcing

..................... Function is Even Integer Multiple).... 9 1

Sun of Squares of Shears Versus Third Natural Frequency f o r 25.

Initaal Design Both With and Without Damping (whex-e Forcrng ........................ Function is Even Itteger Multiple).

Sum of Squares o f Shears Second Natural Frequency f o r F i n a l 26.

Design Ecth With and Without Damping (where Forcing S U B of Squares o f Shears Versus Third Natural Frequency f o r 27.

F i n a l Design Both With and Without Damping (where Forcing 28. I d e a l Two-cell Model f o r C a l c u l a t i o n o f Torsionzl Stiffness..lOS

DESIGN OF HELICOPTER R Q m R BLADES FOR

GPTIHUH DYIVJMIC CRARACTERISTZCS 1. IIPTRODUCTION 1.1 FiELICOPTeR DESIGN The design o f h e l i c o p t e r p a t o r b l a d e s i n v o l w s not a d l y considera- t i o n s o f s t r e n g t h , s u r v i v a b i l i t y , f a t i g u e , and c o s t , but a l s o r e q u i r e s t h a t blade n a t u r a l f r e q u e n c i e s be s i g n i f i c a n t 1 y s e p a r a t e d from t h e f u n d a m e n t a l aerodynamic f o r e i n g f r e q u e n c i e s (e.g. Ref. 1) . A proper placement o f b l a d e f m q u e n c i e s is a d i f f i c u l t t a s k f o r s e v e r a l reasons, F i r s t , t h e r e are m y f o r c i n g frequencies ( a t a l l i n t e g e r - m u l t i p l e s of t h e r o t o r RPU) uhich oocur a t r a t h e r c l o s e l p - s p a c e d i n t e r v a l s . F o r example, 51rev and 6 l r e v are l e s s than 20 A a p a r t . Second, t h e r o t o r RPM may vary over a s L g n i f i c a n t range through t h e f l i g h t envelope, t h u s reducing even f u r t h e r t h e area of a c c e p t a b l e n a t u r a l frequencie3. Third, t h e naturml maxies of t h e r o t o r blade a r e o f t e n coupled because of p i t r h angle, blade t w i s t , o f f s e t between t h e mass c e n t e r and e l a s t i c a x i s , and l a r g e aerodynamle damping. These couplings complicate t h e c a l c u l a t i o n o f n a t u r a l frequencies, i n f a c t , t h e dependence on p i t c h angle makes fre- quencies a nanction o f loading condition. s i n c e loading a f f e c t s c o l l e c - t i v e pitch. k u r t h , t h e c e n t r i f i g a l s t i f f n e s s o f t e n dcminates t h e lower modes, making it d i f f i c u l t t o a l t e r frequencies by simple changes i n s t i f f n e s s o r aas5.

I n t h e enply s t a g e s o f t h e developaent of t h e h e l i c o p t e r , i t was

believed t h a t h e l i c c p t e r v i b r a t i o n s could b e reduced (and even eliminab- ed) by t h s c o r r a c t choice o f s t r u a t u r a l coupling and mass s t i f f n e s s d i s t r i b u t i o n , Howsver. it is e w y t o imagine how d i f f i c u l t it is t o f i n d j u s t t h e proper parameters suah t h a t t h e d e s i r e d n a t u r a l frequencies can be obtained. The d i f f i c u l t i e s i n placement of n a t u r a l frequencies have led. i n many cases. t o p r e l l m i m r y d e s i g n s vhich ignore irequency plaze- ment. Then, a f t e r t h e s t r u o t u r a is ' f i n a l i z e d ' ( e i t h e r on paper o r i n a p r o t o t y p e b l a d e ) , t h e fisquemies ape c a l c u l a t e d ( o r measured) and f i n a l adjustments made, Refemace [21 d e s c r i b e s t h e development o f t h e XH-17 h e l i c o p t e r i n tihioh a 3 0 b l b w i g h t was added t o each blade i n o r d e r t o change t h e spanulse and chordvise &s d i s t r i b u t i o n and therby move t h e first flapwise frequency away from 3/rev. The a u t h o r s were con- f i d e n t t h a t similar adjustments t o t h e mass d i s t r i b u t i o n (and t h u s t o t h e frequencies and 0068s o f t h e b l a d e s ) could g r e a t l y reduce r o t o r v i b r a t i o n on o t h e r m t o r s . A n a n a l y t i c s t u d y i n Refemnee [ 3 1 p r e d i c t s t h a t chordwise emss d i s t r i b u t i o n could a l s o be used t o lower o v e r a l l h e l i c o p t e r v i b r a t i o n s . I n p a r t i c u l a r , a forward s h i f t o f mass is shown t o be u s e i u l because it p l a c e s t o r s i o n i n resonance with a p a r t i c u l a r harmonic. The t o r s i o n loads car, then be tuned t o cancel u n d e s i r a b l e blade loads. The s t u d y a l s o shows, however, t h a t such mass changes may have an adverse e f f e c t an s t a b i l i t y ; and t h u s s t a b i l i t y and v i b r a t i o n must be s t u d i e d together. S i m i l a r b e n e f i t s o f i n e r t i a pitch-flap coupl- ing a r e a l s o obsemed i n shaker 2 e s t s i n Reference [ 4 1 .

These p o a i t i v e r e s u l t s , and o t h e r l i k e them. w.we a t l e a s t p a r t i a l - l y r e s p o n s i b l e f o r t h e o p t i m i 3 t i c outlook s o a p t l y presented i n Ref [ 5 1 .

I n that; reference. six h e l i c o p t e r pioneers express t h e i r b e l i e f t h a t h e l i c o p t e r v i b r a t i o n s can be reduced through proper blade and fuselage

design. This o p t i n i a n of the 50's w a s soamwhat eroded i n t h e 60's and 70'3 a s t h e t r u e complications of r o t a r p r i n g dynamics became b e t t o r kncnn. Nevertheless. t h e b e l i e f is still held by most dynamicists t h a t simple concepts (such as f r e q u e w y p l a a a w n t ) can go a long way toward inproving r o t o r design. For example, i n Reference (61 six h e l i c o p t e r pioneers ( s o w of t h e authors o f Ref. 41, reminfac:, on the e a r l y days of r o t a r y wing and t h e recent advances i n our understanding of h e l i c o p t e r .

Yet, they s t i l l contend t h a t much can be learned from simple p r i n c i p l e s .

Presently, h e l i c o p t e r blades a r e not t a i l o r e d t o g i v e a s e t of desired n a t u r a l f r e q u e w i e s . Instead. blades a r e designed based an o t h e r consldepatioru (including t h e desired aemdyuamlc c h a r a c t e r i s t i c s and t h e c u a r l a t i v e experienae of t h e designeP3). Then. a f t e r t h e design i s analyzed ( r i t n e r by computer program o r by f a b r i c a t i o n and t e s t i n g ) , t h e designer checks f o r f r e q u e m i e s t h a t a m poorly placed. These a r e then adjusted by judicious a p p l i c a t i o n of lumped i n e r t i a s a t c r u c i a l spanwise l o o a t i o m . These a f t e r - t h e f a c t a l t e r a t i o n s . however, can be detrimental t o blade w i g h t . blade c o s t , and t h e development time of t h e a i r c r a f t .

Somatlmes, t h e problem a r e unsolvable, and a h e l i c o p t e r is left w i t h a noticeable resonance problem.

The state-of-the-art i n h e l i c o p t e r technology is now t o the point, however, t h a t it should b e possible t o c o r r e c t l y place r o t o r frequencies during preliml.nary design stages. There a r e s e v e r a l reasons lor this.

Firs:. h e l i c o p t e r r o t o r blades f o r both main ~ o t o r s and t a i l rc,tors a r e now being fabricated from composite m a t e r i a l s (Refs. 7 and 8 ) . This implies t h a t t h e designer can choose, w i t h limited r e s t r i r ~ t i o n s , tSe exact E I d i s t r i b u t i o n desired. Furthermore, t h e l i g h t n e s s o f composite blades f o r t h e main r o t o r usually n e c e s s i t a t e s t h e a d d i t - - n of weight to

g i v e s u t i i c i e 3 t a u t o r o t a t i o n a l blade i n e r t i a . Thus. t h e r e is a c o n s i d e r a b l e amount of f l e x i b i l i t y a s t o how t h i s weight may b e dis*,ributed.

Second. th* methods of' s t r u a t u r a l optimlzat:..on and parameter i d e n t i f i c a - t i o n a r e now m f l n e d t o t h e p o i n t where they can be e f f i c i e n t l y a p p l i e d t o t h e blade t r u e t u r n . S o w e l e a r n t q technique3 have a l r e a d y been used f o r t h e d e s t g n of r o t o r f i s e l a g e s (Ref. 9 ) . It follows t h a t t h e time is r i g h t f o r t h e u s e o f s t r u c t v - a 1 o p t l m l z a t i o n i n h e l i c o p t e r b l a d e design, Some work on t h i s is a l r e a d y under development. a ~ d , although n o t published. sow companies are a l r e a d y e x p e r h e n t u i t h t h e oigtimum way t o a d d ' u e i q h t t o an e x i s t i n g b l a d e i n o r d e r t o i ~ ~ ~ p r o v e v i b r a t i o n s .

1.2 PREVIOUS WOfflc I n t h i s l i g h t , M would l i k e t o mention a few m c s n t a t t e m p t s a t a p p l i c a t i o n o f o p t i m i z a t i o n techniques t o r o t o r b l a d e deslgn. I n Ref [ l o ? , an o p t i m i z a t i o n procedure is a p p l i e d i n o r d e r t o reduce b l a d e l o a d s c o n s i s t e n t with a e r o o l a s t i c s t a b i l i t y , The procedure is not c o w p l e t s l y automated, however. and t h e d e s i g n e r amst make t h e d e s i g n incre- ment a t each i t e m t i o n b a s e d on numerical ~ e n s i t i v i t y parameters. The b i g g e s t needs (as i d e n t i f i e d i n t h i s work) are t h e complete automation o f t h e o p t i m i z a t i o n and t h e formulation o f r e a l i s 2 i c d e s i g n c o n s t r a i n t s .

In Reference [ I l l , an o p t i m i z a t i o n package 13 a p p l i e d t o an a e m e l a s t i c response p r a q m m , The r e s u l t s m i r r o r t h e e a r l i e r c o n c l u s i o n s o f Refer- ence 2-4. I n p a r f i c u l a r , minimization o f v i b r a t i o n s tends t o d r i v e some n a t u r a l f r e q u e n c i e s c l o s e t o i n t e g e r s i n o r d e r t o c a n c e l loads. (The r o t o r becomes an i s o l a t o r . ) Although t h i s :urns out t o b e a good mathe- matical s c l u t i o n , s t a b i l i t y a n a l y s e s i n Reference [ I l l , aj i n References [2-41, show t h a t t h e coalascence o f f r e q u e n c i e s t o supp:*ess vibration t e n d s t o introduce a e r o o l a s t i c i n s t a b i l i t i e s . Thus, f l u t t e r s a r g i n s tend

t o b e c a m t h e domlnant c o n s t r a i n t s . F u r t h e m r e , lainlmization of l o a d s a t one f l i g h t coadit:on nmy not a t a l l minimize l ~ a d s a t o t h e r s .

Another i n v e s t i g a t i o n i n t o v i b m t i o n m d u c t i o n by a l t e r a t i o n o f w s s and stiffisas d i s t r i b u t i o n is g i v e n i n Reference t l f l . I n t h a t refemme, a t i p w i g h t is used t o ohange tho aode shape. It is hypothe- s i z d t h a t changing t h e mode shape suoh t h a t it is orthogonal t o t h e forcing f'unotion is a way t o lower v i b r a t i o n s . However, t h s conclusions are u n c e r t a i n since t h e f r e q u e w i e s also a r e changed by t h i s added weight (e.g. t h e seuond f l a p arode w v e s away from 5.06 t o 5.19 p e r r e v ) .

One a l s o n o t i a e s t h a t t h e loading d i s t r i b u t i o n changes w i t h f l i g h t c o n d i t i o n s o t h a t modal shaping may h e l p one c o n d i t i o n b u t h u r t o t h e r s .

Other r e l a t e d previous mrk is found i n R e f e m m e [131. That p a p e r shows t h a t design t o m i n i s a m loads c a n r e s u l t i n a d i s j o i n t s o l u t i o n . Fortu- tWtei7, i n h e l i c o p t e r problems w e g e n e r a l l y begin with an zdequate ( b u t n o t perfect) b l a d e d e s e n . Thus, many q u e s t i o n s such a s t h i ~ ( 1 " s .

d i s j o i n t in t h e design space) a m a u t o m a t i c a l l y avoided. W e a l r e a d y have a good first guess and merely w i s h t o r e f i n e it.

1.3 SC'?PE I n t h i s paper w e undertake a much l e s s ambitious aim than t h e mini- mization o f hub loads. Instead ve look a t t h e problem of using optimiza- t i o n techniques i n o l d e r t o place n a t u r a l frequencies. Even within t h i s roduced problem t h e r e are varying l e v e l s of complexity. For example, one could consider t h e r e t r o f i t problem: * G l ~ e n a blade design f i n d :he amount and l o c a t i o n of added masses required t o move f r a q u e m i e s away *om i n t e g e r !'930CianC08.'

One could a l s o consider thc b a s i c design problem i n which both s t i f f n e s s

and mass d i s t r i b u t i o n s may be chosen. I n t h i s paper, w e treat t h i s l a t t e r design problem i n t h e ;om o f t h r e e uncoupled problems: f l a p , in- plane, and t o r s i o n .

The scope o f t h i s p r e s e n t work is n o t j u s t t o f i n d a mass and s t i f f n e s s d i s t r i b u t i o n t o g i v e d e s i r e d f r e q u e n c i e s . It is a l s o t o d e t e r - mine meaningfil c o n s t r a i n t s and o b j e o t f v e f u n c t i o n s t h a t w i l l r e s u l t i n r e a l i s t i c designs. I n t h i s a r e a , s e v e r a l items a r e noteworthy. F i r s t , t h e r e is t h e a i r f o i l envelop. Whatever t h e s t r u c t u r a l engineer d e s i g n s m a t l i e w i t h i n t h e airfsil cross-section. Second, t h e r e is mass balanc- ing. T L e c e n t e r a f mass o f each s e c t i o n should b e forward of t h e one- q u a r t e r chord. Third, t h e r e is t h e a u t o ~ o t a t i o n a l c o n s t r a i n t . The b l a d e must have s u f f i c i e n t a s s moment-of-inertia t o i n s u r e a safe a u t o r o t a - t i o n a l c a p a b i l i t y . Fourth, t h e r e is s t r e n g t h . The blade must be s t r o n g enough t o endure t h e c e n t r i f u ~ a l l o a d s a s well a s t h e o s c i l l a t o r y bend- !-! loads. This last c r i t e r i a is t h e most e l u s i v e o f t h e f o u r . D e s i g ~ a r s know how t o make a very s o f t s e c t i o n (hinge o r f l e x u r e ) which neverthe- less can withstand high c e n t r f f u g a l and bending l s a d s . Such f l g x u r e s g e n e r a l l y d o n o t fall. w i t h i n an a i r f o i l envelope, however, and a r e placed near t h e r o o t . Therefore, i n t h e work t o follow, w e first o b t a i n 'optimum' d e s i g n s and then check t o see i f t h e r e q u i r e d E I d i s t p i b u t i o n has unrea:istically s o f t s p o t s . 3 i m i l a r l y , w e check t h e f i n a l d e s i g n s f o r a x i a l stresses.

I n summary, w e work with simple (bu; r e a l i s t i c ) rotor-blade d e s i g n s and simply experiment w i t h c o n s t r a i n t s and o b j e c t i v e f u n c t i o n s i n o r d e r t o determine t h e f e a s i b i l i t y o f designing t o a d o t i r e d 3et of frequen- c i e s .

1.4 OVERVIEW OF OPTIMAL STRUCTURAL DESIGN Host appmaahes t o optimal s t r u c t u r a l s t r u s t u r a l d e s i g n may b e c l a s s i f i e d i n t o t h r e e c a t e g o r i e s . (POI . e c e n t review a r t i c l e s see Refs.

14 a d 15.1 One such a a t e g o r y Is ' v a r i a t i o n a l methods.' These g e n e r a l l y r e l y on t e o h n i q u e s from t h e mathematical, t h s o r y o f t h e c.iiculus o f v a r i a t i o a s , and, when app1ic.-ble, o f t e n provide u s e i u l p h y s i c a l i r s i q h t i n t o t h e n a t u r e o f an o p t b a l design. Unfortunatctly, only r e l a t i v e i y slmple probleas can b e solved by t h i s approach, s i n c e t h o mathematics b a o a s a L n t r a o t a b l e when complex enqiner.ring s t r u c t u r e s a m considered.

A seoond c a t w a r y o f s t r u o t u r a l o p t m z a t i o n techniques c o n s i s t s of t h e a p p l i c a t i o n o f mathematical prcgramming methods toqethe;' with t h e G l s c r e t i z a t i o n o f t h e s t r s c t u r e bp f i n i t e element tec ~ n i c , u e s . This appr-mch t o o p t i n i z a t i o n was foundei i n 1960 (Ref. 16) with t h e hore t h a t more complex s t r u c t u r e s could be analyzed than were p o s s i b l e when u s i n g t h e a n a l y t i c a l tecnniques of t h e c a l c u l u s o f v a r i a t i o n s . '3owever, i n t h e l a t e 60's it became apparent t h a t mathematical programming mothod had l i m i t a t i o a ? of t h e i r m, namely, unacceptably Long computation tlms occurring when t h e numk3r o f d e s i g n v a r i a b l e s becoma l a r g e (over 20-100, d e p e n d i n g on t h e t y p e o f s t r u c t u r e ) . F o r t u n a t e l y , s e v e r a l improvements d e v e l o p e d o v e r t h e l a s t f e w y e a r s a p p e a r t o h a v e s i q n i f i c a n t l y extended t h e c a p a b i l i t y o f t h e mathematical programming approach, and, as a r e s u l t , it is t h i s approact, v e intend t o d r a w upon f o r s o l u t i o n technique3 i n t h i s research.

A t h i r d c a t a g o r y . ? s t r u c t u r a l o p t i m i z a t i o n approaches is the ' o p t i m a l i t y c r i t e r i o n ' approach i n which an equation expressing some necessary c o n d i t i o n of o p t i m a l i t y is used as t h e b a s i s ?or c o n s t r u c t i n q an i t e r a t i v e ( s u c c e s s i v e r e d e s i g n ) procedure. O r i g i z a l l y d e v e l o p e d \

because of di3sat:sfaction with aathenu8tical proqramning techniques, t h o o p t F o P l l t p - o r i t e r i o n approach i n i t i a l l y r e l i e d on h t ~ i t i v e o p t i m r l i t p c r i t e r i o n suah w corm t a u t s t r e s s - r a t i o and u n i f o r m s t r a i n - e n e r g y d e n s i t y conditions. More m e n t l y , optSPPlity a r i t e r l o n (and a s m i a t e d r s d o s i g n equations) have b w n derived from t h e Kuhn-Tucker conditiomi (sw. e.g. Ref. 17) f o r a coastrained m l r d d z a t i o n problem, me o p t . i a a l i t p c r i t e r i o n appkoach seema- e s p e c i a l l y well-su: t ed t o p r o b l a s wih'? a 1 - 0 nunbsr a f design variables. Sinaa o u r d e s i g n problea w i l l have a moderate numbsr of v a r t a b l e s and s i n c e d e r i v i n g e f f l q i o n t r s - d s a i g n e q u a t i o n s f o r o u r problem i s n o t i m m e d i a t e l y straightforward, ua i n i t i a l l y prefer t;s mathe6atical programming ap- p m o h o c r w t h e o p t i m a l i t r - a r i t e r i ~ n approaah.

A structural optimization computer program, c a l l e d CONMIN (Ref.

27) , i s a v a i l a b l e from NASA. It is +.his program t h a t is used i n our present work. CONMIN is based 3n t h e m a t k a a i i c a l nonlinear programming method of f e a s i b l e d i r e c t i o n s .

3ACICGRQUNb 2 . 1 POWELATIOH OF PROBLEM Because nuasrically-bassd optiplization is b e s t c a r r i e d out with d i s - c r e t e v a r i a b l e s , t h e f i n i t e e l e m n t technit.48 s t a n d s as t h e most l o g i - c a l choices f o r t h e blade meidel. A r e c e n t r e s e a r c h p r o j e c t ;Ref. 18) has r e s u l t e d i n a flnite-elemont computer program t h a t . is i l g a l l y s u i t e d t~ t h e work here. The program allows f o r tapered, t u i s t e d f i n i t e elements in a r o t a t i = envimnmont. The e x i s t i n g code can c a l c u l a t e n a t u r a l fre- quen?ies, (with arl without, aerodynamic terms) and f o m s response.

Another w o r t a n t a s p e c t of t h e retor blade o p t i m i z a t i o n problem is t h e s a l e c t i o n o f t h e optimalit>- c r i t a r i a and c o n s t r a i n t s t o b e imposti.

Cur design problam ha^ c e r t a i n i e a t u r e s which are unusual compared t o t y p i c a l p r o b l e m occurring i n tire s t r u c t u r H l o p t i m i z a t i o n l i t e r a t u r e .

There a r e b a s i c a l l y t h r e e c a t a g o r i e s of c r i t e r i a . I n the first c l a s s , o n e would I P i n M z e weight given c o n s t p a i n t s on t h e n a t u r a l f r e q u e n c i e s ( i . e .

frequency 'windows'). I n t h i s c a s e , a c o n s t r a i n t on r o t a r y i n e r t i a is a l s o implied s i n c e a roi;or must have s u f f i c i e n t i n e r t i a t o a u t o r o t a t e .

The advantage of t h i s approach fs t h a t it is d i r e c t l y r e l a t e d t o t h e phystcal ~ s a l i t i a s of design. The disadvantage, however, is t h a t t h e first guess will probably not 3e feasikile! i t h a t is w i i i not nave fre-

q u e r c i e s t h a t f a i l i n t h e 'windows' 1 . This can be a stumbling block t o

convergence. A second type o f c r i t e r i a is one i n which t h e o b j e c t i v e 3 t o minimize t h e d i s c r e g a n o l e s between desired f r e q u e n c i e s ar?d a c t u a l frequencies. The c o n s t r a i n t then becomes a windaw or, a u t o r o t a t i o n n l i n e r t i a . Although t h i s avoids u n f e a s i b l e s o l u t i o n s , i t does not d i r e c t - l y minimize weight (although we.!ght is l i m i t e d by t k e a u t o r o t a t i o n a l

c o n n t r a i n t ) . A n o b j e c t i v e h t n c t i o n can be c o n s t r u c t e d t h a t combined com- bined blade mass and frequency placement, b u t t h e r e l a t i v e w e i g t t i n g s o f t h e tno components is not obvious. Tne t h i r d c a t e g o r y of c o n s t r a i n t is t o minimize v i b r a t i o n s d i r e c t l y without regard t o frequency placement.

Although t h i s appears on t h e s u r f a c e t o be t h e p e r f e c L s o l u t i o n , t h e r e a r e problems. F i r s t , c a l c u l a t i o n of v i b r a t i o n s is a n order-of-magnitude more d i f f i c u l t than t h e c a . l c u l a t i o n o f frequencies. Second, p a s t e f f o r t s a t t h i s have r e s u l t e d i n s t r a n g e designs, incompatible with standard h e l i c o p t e r p r a c t i c e . Third, t h e r e is still t h e problem o f t h e weight- v i b r a t i o n t r a d e - o f f . I n t h i s work, w e intend t o c o n c e n t r a t e on t h e first two c a t e g o r i e s with some a t t e n t i o n t o t h e t h i r d .

Another type o f c q n s t r a i n t involved i n t h e problem is t h e limita- t i o n on s t r u c t u r a l p r o p e r t i e s . The blade planform, a i r f o i l , and t w i s t a r e c h o s e n by t h e a e r o d y n a m i c i s t on t h e b a s i s o f performance. The s t r u c t u r a l engineer must choose h i s d e s i g n t o 'it i n t h e aerodynamic envelope given. There are f i v e s t r u c t u r a l parameters t o be chosen: 1) f l a p p l r q stiffress, 2 ) i n p l a n e s t i f f n e s s , 3 ) t o r s i o n a l s t i f f n e s s , 4 ) mass, and 5 ) t o r s i o n a l moment o f i n e r t i a . I n p r a c t i c e , t h e s e cannot be chosen completely independently. Figure 1 shows t h e e n v e l o p e o f a t y p i c a l blade s e c t i o n . A l l cstiffness is assumed t o r e s i d e i n a box-beam of dimension b x h with t h i c k n e s s e s t , d l , d 2 . This beam is placed a s far forward a s p o s s i b l e ( t o keep t h e e l a s t i c a x i s near t h e 1 / 4 chord). Mass p r o p e r t i e s a r e due t o t h e box-beam, s k i n , honeycomb, and two lumped masses. The lumped mass i n t h e t i p is t y p i c a l o f r o t o r b l a d e s and is used t o keep t h e mass c e n t e r forward of t h e aerodynamic c e n t e r . A second mass is included t o allow independent choice of mass and mass-moment.

The c o n s t r a i n t s o f t h i s c o n s t r u c t i o n are c l e a r and a r e l i s t e d on t h e

(I, a a

-

-

-

(I, C Q)

figure .

I n addition, t h e r e ' r e minimum c o n ~ t r a i n t s on t , d l , d 2 t o hold c e n t r i f u g a l l o a d s and t o r e a a i n w i t h i r ! manufacturable imts. For example a simple minimum c o n s t r a i n t on a r e a could come from t h e c e n t r i -

iugal c o n s t r a i n t (not considering bending stress). Thus, i f & is t h e

maximum s t r e s s and if f is a s a f e t y f a c t o r , then (la) O f oourse, when wr, @ a t e th6 v I b r a t o r p r a s p o u 3 e pha8e of t h e uorlc. bend- ing s t m a 8 e s w i l l be imluded.

OPT work V i l l n e v e ~ t h e l e s s inolude f l u t t e r c r i t e r i a i n a simplified EWUI~?. F i r s t , H can ohmzre fFeyuemy placement such t h a t no coalas- o e m e m e w s b e t u w n fiap-?.ag, flag-torsion, or lag-torsion. Secoad, ws can c o n s t m n t h e f i v e parameters i n Figure 1 such t h a t t h e mass c e ~ ! t e r is alnays foPwud of t h e 1/4-ahord, a comaon design p r a c t i c e t o prevent t o r s i o n - f l u t t a r Fn r o t o r :lades.

2 F ~ t t M l e m o n t Mode Although tapered, t v i s t e d elements a r e within our c a p a b i l i t i e s , ue introduoe here a simple case which is also of value. The s t i f i h e s s e s G J .

e l e a e n t .

Exn are sad t o b e constant along t h e length of t h e gIzzD

The lumped mass weight is assumed t o be evenly d i s t r i b u t e d on t h e two nodes.

Let the deflection of an element In the y a . ' z directions a t 3 d i s t a n e e x b e denoted as w(x) and v ( x ) , for which the displacement models are assumed t o be polynomials o f third degree. The expmssions are given a8 &ere vl, v3, v6 and vg represent the bending degrees of freedom i n the u and u9 represent the degrees o f freedom i n the n plane and u2, u , .

yx plane

-1*- i) The s t r a i n energy due t o bending deformation can be expressed as i f ) The p o t e n t i a l energy i n t e n s i o n from t h e c e n t r i f u g a l f o r c e f i e l d , which is e q u i v a l e n t t o t h e negative o f k i n e t i c energy due t o r a d i a l d i s - placement, is g i v e n by where T, termion f o r c e , is assumsd t o be c o n s t a n t along each eip- ment .

i i f ) The k i n e t i c energy due t o inplane displacement is given by which is equivalent t o u = T

Segr2es o f Freedom o f an El enen t Meanwhile, t h e p r e t w i s t angle C$ (XI, and t h e t o r s i a n a l deformation

6 (XI a m assumed t o b e polpnomlals of first d e g r e e , and can b e express-

ed as

where fi , Ql, r e p r e s e n t t h e p r s t w i s t angle a t node I and 2 , and

u 5 , u I 0 r e p r e s e n t t h e e l z w t i c t o r s i o n a l degree of f-eedom a t each end.

iv) The torsional energy. due to elastic deforwations and c8ntriiugal t e r n . can be expressed aa

2 = /;r y 2 dydz = 1

ka 2 z z V ) The 'torsion-rotation' energy under the effect of rotation is given by

where . kmZ2 am isass mmnt of inertia which can be expressed as

Total displacement energy now can b e used to form the stiffness matrix fioar

where u is the vsotor o f nodal d i s p l a c o m n t s , in the order as ul, u6, us, u2. u,. u4, u9 ,us ,uI0 ; [Kl is the e l e m n t a l stiffness matrix u 3.

o f order 10.

v i ) The ams matrix w i l l be obtained by the kinacic energy o f an element, whiuh is given by Written l a matrix forn. the k i n e t i c e n e m can be expressed a s where [HI is the mass matrix.

3 . ILLUSTRATIVE EXAHPLES OF STRUCTURAL OPTIMIZATION 3 .1 CANTILEVER BEAH WITH GIVEN FREQUENCY Some simple example3 w i l l b e examined and discussed before t h e u t i l i z a t i o n of t h e program CONUIN. I n each oaae t h e r e s u l t s w i l l be com- p a r d with t h m e obtained by previous msearchea, if it is a v a i l a b l e .

The first l l m l t i n g example is t h e problem o f determining t h e o p t i - mal d e s i g n o f an e l a s t i o C a n t i l e v e r beam, such t h a t with a s p e c i f i c n a t u r s l frequency,the u e l g h t of t h e s t r u c t u r e a t t a i n s t h e m i n i m u m value.

W e start with a uniform beam, modeled by t e n elements, with a g i v e n l e n g t h of 1C inohes, E = 1.0 1 b i 2 , EI = 10 lb-ln2. d e m i t y = 0.042 l b / l n , and a s p e a i f i e d first lowest n a t u r a l frequency = 0.6489 rad/soc.

W e o b t a i n t h e final stiffness p r o f i l e shown i n Flgure 2. Figure 2 is t h e preuent r e s u l t with t e n elements.

A ralated problem has also been t r e a t e d by 0 1 ho f f [ 19 1 . H e

seeks t h e dsslgn of a c a n t i l e v e r beam t h a t y i e l d s a mxlmum value cP a p a r t i a u l a r h i g h e r n a t u r a l f'roquena~ wn (1.0. of s p e c i f i e d o r d e r , n) with t h e volume and l e n g t h of t h e beam s p e c i f i e d . H i s work is t h e d u a l problem o f t h e example shown i n Figure 2. Optimization with r e s p e c t t o t h e frequency under t h e c o n s t r a i n t o f volume is s i m i l a r t o t h e one o f mlnlmlzing weight ( o r v ~ l u m e ) under t h e c o n s t r a i n t of s p e c i f i e d n a t u r a l frequency. F ~ r s 3 g i v e s t h e p r o f i l e of t h e optimal cantilever f o r n = 1 by Olhoff. One can see t h a t t h e shapes i n Figure I 2 1 and [ 3 1 are very sinillax- in t h a t thay g i v e a nonlinear t a p e r .

3.2 CANTILEVER W I T A TIP MASS Another example problem is to minlmize t h e weight of a c a n t i l e v a r c a r r y i n g a maas a t t h e t i p , s u b j e c t t o t h e constraint t h a t t h e fundamen- tal n a t u r a l frsquenoy must be g r e a t e r than o r equal t o a s p e c i f i e d

I '1 7

- I I I I I I b t 2 4 6 8 10 Figure 2 .

Area merit of 1ner:ia for Opciamm Beam in Present Fork.

Figure 3 . A r e a 3oment of I n e t z i a f o r O p z i m t ~ m Beam from ?-eference 11.

value. The p r o b l e m was o r i g i n a l l y f c m u l a t e d by Turner [201, &h- ana Willmsrt [211 ~ s o d an o p t i m a l i t y c r i t e r i o n method t o solvc, T u ~ n e r ' s prob:em. In hi^ example, four R:'..:Lcs elements are used, w i t h t h e areas of each a s t h e design v a r i a b l e e , aa i l l u s t r a t e d in F i g u r e 4. The spec- itid n a t u r a l frequenay is 17.792 m d / s s o . The o t h e r i n i t i a l data are Modulus of e l a s t i c i t y = 10.3 x 10 p s i M a s s d e n s i t y = 2 . 5 x I0 l b - s / i n Radius of g y r a t i o n (AI 1 = 2 . 0 in Radius of g y r a t i o n (A2 ) = 1 . 5 i n

Radius of g y r a t i o n ( A 3 1 - 1 i n

Radius of g y r a t i o n (A, 1 = 0.5 i n Concentrated mass = 1 . lb-a / i n Length of each eiement = 60 i n where I = Area ( r a d i u s of gyrdbion) .

F ~ g u r e 4 C a n t i l e v e r b c a a u l t h c o n c e n t r a t e d mass

The r e s u l t s o f t h e o p t i m i z a t i o n am show11 i n Table 1. The i e a s i b l c starting deaQn is d e s c r i b d by A1 = 200, Lt = T5C. A 3 = 60 a d A, = 3 5 .

Table 1 Ref. [l21 Ref. !I3 1 This Eager

-

I t e r a t i o n - 23 10

34.43 3 4 . 6 1 34.89 A4 (in, , .

It aan b e s e e n t h a t e x c a l l - n t r e s u l t s have bean obcained csing the p r e s e n t CORNIN o p t i m i z a t i o n program.

4. N T M X ~ I C A L ExPER~MENTS 4.1 D E S I G N VARIABLE Despite t h e s t r ~ n q documentation and intensive development t h a t h a s sone i n t o o p t i m i z a t i o n programs, i t is a l w y s a d v i s a b l e t o do some ex- perimentation w i t h t h e s e p r o q ~ ~ ~ f o r t h e p a ~ t i c u i a ~ c l a s s of problems t o which they a r e t o be a p p l i e d . m.'.s has been done i n d e t a i l f o r t h e r e p r e s e n t a t i v e box beam shown i n F i g u r e 5. Tne parameters ( s e e Fig 2; f o r t h i s c a r . a r e : h = 2.5 i n , s = 0.1 i n , s = 0.1 i n . b = 4 i n , t v a r i a b l e . Thi3 beac has been analyzed f o r v a r i o u s v a l u e s of t h e C 3 N E I I f i parameters and f o r v?.rious c ~ m b i n a t i o n z o f c o n s t r a i n t s . T h e f i ~ a t s t u d i e s are performed f o r v e r t i c a l v i b r a t i o n s . a n o a r o t a t i n q beam, np lumped mass, and w i t h two frequency c o n s t r a i n t s . Each of t h e C O ? P i i U o p t i o a s is t h e n e x e r c i s e d , and s e v e r a l c o n c l l ~ s i o n s Cr2un.

F i i st, w e f i c d t h a t t h e u s e of a n a l y t i c . g r a d i e n t s ( t h e d e r i v a t i v e sf o b j e c t i v e f u n c t i o n and c o n s t r a i n t s i n c l o s e d form) is 3 r e a t l y co be d e s i r e d . For t h e p a r t i c a l a r c a s e i n F i g u r e 5, arza, weight, and moment of i n e r t i a can b e expressed i n terms o f the s i w l e v a r i a b l e s , t Therefore, a n a l y t i c d e r i v a t i v e s am straightforcrard. Where analytic g r a d i e n t s a r e not a v a i l a b l e , however, w e find t h a t f i n i t e diffsreace g r a d i e n t s still work a l b e i t a t a higher coffiputationa~ c o s t . Secand. wa

Element Length SECTION A-A F i g u r e 5 Box Beam C r g s s - S e c t l o n

find t h a t t h e optimization is b e s t behaved when frequency c o n s t r a i n t s are provided i n Hz. A c o n s t r a i n t ( i f not s c a l e d ) on eigenvalues (w2) is mere d i f f i c u l t f o r t h e program when d e f a u l t v a l u e s a r e ssed. Third, w e f i n d t h a t i n i t i a l d e s i g n s o u t s i d e o f t h e d e s i r e d c o n s t r a i n t s ( i n f e a s i b l e ) sometimes can lead t o convergence. Since t h i s is not always t h e c a s e , however, a l t e r n a t i - r e s t r a t e g i e s a r e necessary. Fourth, w e find t h a t t h e d e f a u l t values f o r t h e CaNMIN program worked reasonably w e l l (although t h e y a r e not always t h e most e f f i c i e n t v a l u e s ) . An example i s t h e number o f i t e r a t i o n s . Sometimes 40 i t e r a t i o n s were required f o r convergence, although t h e d e f s u l t value i s 10.

In tams o f Y-ious mdes o f a p p l i c a t i o n . w e a l s o have come t o s e v e r a 4 conclusions. F i r a t , w could find optImam d e s i g n s no matter how t i g h t l y we cloaod t h e wxindous on m o n e y (i.8. t h e f'requency const- - t a ) . Thus. w e a m a b l e t o e s s e n t i a l l p *zero8 an o b j e c t i v e f u n c t i o n based on frequencies ( f o r v e r t i c a l v i b r a t i o n s a l o n e ) . Second, wt c a n handle a large rmmber o f simultaneous f m q u e n c y c o n s t r a i n t s . ( W e have s u c c e s s C ~ i I y gone fiam 2 up t o 5 c o n s t r a i n t s . ) The o p t i m i z a t i o n a l s o rcr wll-bo&aved when we add b l a d e rotation.lumped mass. and t h e auto- r o t a t i o n a l constm.int.

4.2 COIWERCENCE W e have s t u d i e d t h e co;.zvergence o f t h e f i n a l d e s i g n as a f u n c t i o n I of t h e number o f elements used i n t t e f i n i t e - e l e m e n t frequericy c a l c u l a - t i o n . Tc s t u d y how t h s o p t i m a l d e s i g n changes as t h e number o f elements i n c r e a s e s , a c a n t i l e v e r hem w i t h ' n ' e l e m e r t s and w i t h lumped w e i g h t s added at. t h e nodes b u t o t h e r w i s e s i m i l a r t o t h e beam i n Figure 1 is considered. The d e n s i t y . and f h e c o n s t r a i n t s on t h e n a t u ~ a l f l equency, lumed weights, and moments o f i n e r t i a a r e

aud t h e initial v a l u e o f OBJ ( t h e t o t a l weight) is 30.2249 l b f .

R e s u l t s of t h e study a m shown in Pigums [6-Ili. In all c a s e s .

t h e a c t i v e fmquency c o n s t r a i n t s were found t o be f 2 = ll*? (Hz) Figure 161 demonstrates, as one mid expect, t h a t t h e optimum weight d w a La f a a t d o c r e m a aa aom element^ are added t o t h e mesh. The c!mags Fn optiasPn w i g h t is q u i t s small ( n o t e t h a t t h e saale of ths r a r t i a a l axis b . g i n s at. 20.0).

F i g u r e [71 and [ 8 1 ghow t h e v a r i a t i o n of t h e l m p e d weight and t h e nomint o f i n e r t i a ( o f t h e c r o s s - s e c t i o n a l area) at t h e froe end ve?sus t h e t o t a l number of elements I n t h e mesh. It a p p e a r s t h a t t h e s e q u a n t i - t i e s do not converge. The r e s u l c can be e x p l a i n e d . however. by r e f e r r i n g t o F i q u r e [ 9 1 , i n which t h e Lpper c u r v e r e p r e s e n t s t h e t o t a l weight a t t h e free end. ( m(n) is t h e n o n - s t r u c t u r a l n o r , lumped weigkt ~ : n ) is t h e s t r u c t u r a l weight a s s o c i a t e d with t h e mass d i s t r i b u t e 6 ct,rouuh-'15

element ' It c a n be s e e n from t h e figure t h a t t h e t o t a l weight a p p e a r s t o converge smoothly a s t h e mash is r e f i n e d . The e x p l a n a t i o n f o r t h e a p p a r e n t non-aonvergenoe shown i n F i g r ~ r e s [71 and [81 and f o r t h e convsrgenoe s h m i n t h e t o p c u r v e o f Figure [91 is t h a t t h e ' s t r u c t u r a l ~ Q h t ' a t t h e free end o f t h e c a n t i l e v e r i z n o t r e a l l y structural, s i n c e t h e r e is no p o r t i o n o f t h e beam beyold t h e free end which needs t o b e s u p p o r t e d , Thus, t h e optimization r o u t i n e is i n d i f f e r e n t t o whether s t ~ u c t u r a l o r n o n - s t r u c t u r a l weight i a p ~ e s e n t a t t h e fV93 end - t h e o n l y t h i n g t h a t c o u n t s is t h e t o t a l weight a t t h a t end.

Figure t 9 1 a l s o a h o n t h e w r ' ? t i o a o f t h e lumped weig3t s l i g h t i y beyond t h e middle o f t h e beam. ( A l l o p t i m a l d e s i g n s have non-zero lumped w e i g h t s t h e r e and a t t h e f r e e end o f t h e beam.) The weight c a n be d e c r e a s e smoothly as t h e mesh is r e f i n e d , a l t h o u g h no asymptote a p b e a r s t o be p?.eaent, Tk_a e x p l a n a t i o n f o r t h i s b e h a v i o r is t h a t , a s t h e mesh is r e f i n e d , t h e weight i n t h e middle is being p l a c e more e f f i c i e n t l y - aad t h u s less is needed.

The v a r i o u s s k e t c h e s i n F i g u r e 102 show t.he d i s t r i b u t i o n of mass and stiffness aloriq t h e beam f o r inareas?.ng numbers o f elements. It is in- t e r e s t i n g t o observe t h a t 2he o p t i m i z a t i o n r o u t i n e f i n d s it most e f f i c i - e n t t o meet t h e c o n s t r a i n t s on frequency by v a r y i n g t h e lumped weiqht r a t h e r t h a n by v a r y i n g t h e s t i f f n e s s (moment o f i n e r t i a ) , s i n c e t h i s latter q u a n t i t y is a t its lower bound every. re e x c e p t n e a r t h e end of t h e beam. Another i n t e r e s t i n q a s p e c t o f F,gure 10 is t h e manner i n which t h e lumped mass a t t h e c e n t e r a l t e r n a t e s between: 1) b e i n g all on one element. and 2 ) being s p l i t between t u o elements. T h i s phenomenon is 3 r e s u l t of t h e f a c t t h a t t h e minimum weight s t r u c t . u r e would have all t h e mass a t a s i n g l e p o i n t (node o r a n t i n o d e ) . When t h i s s i r y l e p o i n t l i e s

TOTAL NUMBER OF ELEMEMS IN MESH Figure 6 Convcrqence of U e i g h t of Objective Functior?

Weight At Free End (m,)

1 /

n , TOTAL NUMBER OF ELEMEN73 I N MESH F ~ g u r r 7 Uerght 0 b j e c t L Function Versus Nuuocr ' 0 1 & l a m e n t s I at Free Ebd ( I , ) 3.0 C) 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 22 n, TOTAL NUMBER OF ELEMENTS IN MESH F i g u r e 8 Rrea Moment o f i n e r t i a Versus Nuaaer o f E l e m e n t ;

n Weight At Free End (m, * ?ii 1 - - ..

- - - ~ ~ - - Weight Near Middle (m 1 4 . I • mn12+2 1 n , TOTAL NUMBER OF ELEMENTS IN MESH Fiqurr 9 U e i g h t Versus Number of E l e m e n t s

Figur e 10a 0 p t i m a l D i s t r i b u t i c n f c r Varilsus M csh s

n * l O >m,= a a 'ian # 08J r 21.1457 ma* 1.6010 (Now: OBJ < -US O8JT0) Ilo Q98311 F i g - l r e lob Alt g n a t i v e 0 ptimum

n e a r a s t r u c t u r a l node t h e mass is p l a c e 4 i t h e r e . On t h e o t h e r hand, when t h e mesh c a u s e s t h e p o i n t t o b e between nodes, t h e mass i u a c c o r d i n g l y d i v i d e d between t h e two c l o s e s t nodes.

A s was p o i n t e d o u t p r e v i o u s l y i n r e f e r e n c e t o F i g u r e s [71 and 181, t h e o p t i m i z a t i o n a l g o r i t h m a p p e a r s t o tseat t h e s t r u c ' c u r a l and non- s t r u c t u r a l mass a t t h e end o f t h e beam as i n t e r c h a n g a b l e . To test t h i s h y p o t h e s i s i u r t h e r , t h e o p t i m a l d e s i g n problem s t a t e m e n t was a l t e r e d s l i g h t l y by d e c r e a s i n g t h e u p p e r bound c o n s t i - a i n t on t h e moment o f i n e r - t i a from 5.2083 t o 2.0. The r e s u l t i n g optimum d e s i g n is shown i n F i g u r e [ l o b ] , and s h o u l d b e compared w i t h t h e d e s i g n ( f o r n = 1 0 ) show11 i n F i g u r e [1L?al,, Note t h a t t h e c o n s t r a i n t on t h e moment o f i n e r t i a f o r e l e m e n t 10 is n o t a c t i v e i n t h e o p t i m a l d e s i g n o f F i g u r e l l O b l ( t h e c o n s t r a i n t was a c t i v e d u r i n g t h e CONMIN i t e r a t l o n s l e a d i n g t o t h i s o p t i m a l d e s i g n ) . Thus, t h e effect o f t h e c o n s t r a i n t is t o l e a d t h e o p t i m i z a t i o n a l g o r i t h m a l o n g a d i f f e r e n t p a t h t h a n t h a t f o l l o w e d when t h e c o n s t r a i n t v a l u e was 5.2083. The d e s i g n found, however, h a s a b o u t t h e same t o t a l w e i g h t a t t h e free end (= 9.9705 l b f ) as t h s p r e v i o u s t e n - e l e m n t o p t i m m (= 9.9222 l b f ) . This r e s u l t c o n f i r m s t h e h y p o t h e s i s t h a t CONMIN i n c r e a s e s t h e moment of i n e r t i a a t t h e free end o n l y as a means of i n c r e a s i n g t h e mass t h e r e . Once t h a t o p t i o n is c l o s e d ( t h a t is.

t h e u p p e r bound c o n s t r a i n t is reduced t o a v a l u e o f 2.01, C O W I N s i m p l y i n c r e a s e s t h e :':aped w e i g h t a t t h e beam t i p . T h i s f i n d l n g s u g g e s t s t h a t , i n f u t u r e o p t i m i z a t i o n s t u d i e s , a t i g h t c o n s t r a i n t b e imposed on t h e moment o f i n e r t i a a t t h e f r e e end, s i n c e l i t t l e s t r u c t u r a l capa- b i l i t y is needed t h e r e , and n e c e s s a r y end mass c a n be a d e q u a t e l y r e p r e - s e n t e d by t h e lumped w e i g h t d e s i g n v a r i a b l e s .

5 , DRELIMINARY CALCULATION FOR ROTORS 5.1 WIND TURBINE BLADE The first example i n this s e c t i o n is t h e o p t i m i z a t i o n of a wind t u r b i n e r o t o r b l a d e a t 30 rpm. I n i t i a l d a t a is taken from Ref.[221. A ten-element model is used. Only t h e f l a p p i n q is considered. The a r e a moment o f i n e r t i a , I, and t h e lumped weight o f each element a r e taken a s t h e d e s Q n v a r i a b l e s (see Fig 1 f o r b l a d e a r e a c r o s s - s e c t i o n ) . Young's 6 2 3 Modulus, E = 0.2 x 10 lb-in , and d e n s i t y = 0.0334 l b / i n a r e assumed t o b e c o n s t a n t . Blade r a d i u s , R = 750 i n c h e s . Table 2 shows t h e p r o f i l e .

o f moment o f i n e r t i a and t h e d i 3 t r i b u t i o n o f added weight f o r t h e i n i t i a l and f i n a l c o n f i g u r a t i o n s . The f i n a l p r o f i i e o f t h e a r e a moment o f i n e r t i a a l o n g t h e b l a d e i s s i m i l a r a s t h e one i n t h e previous example. The o p t i m i z a t i o n p r o c e d u r e h a s removed m a t e r i a l from t h e i n b o a r d s e c t i o n s and p l a c e d i t more o u t b o a r d . The lumped mass is c o n c e n t r a t e d a t t h e t i p o f t h e b l a d e a s might be expected. W e a l s o n o t e that moat o f t h e o r i g i n a l l y - p q s t u l a t e d lumped mass is removed s o t h a t o n l y t h e mass i n h e r e n t i n t h e s t i f f n e s s elaments o r n e c e s s a r y for t h e au t o r o t a t i o n a l c o n s t r a i n t is maintained. ( A 1 though wind t u r b i n e have no a u t o r o t a t i o n a l c o n s t r a i n t , a c e r t a i n moment o f i n e r t i a is s t i l l u s e f i l t o smooth o u t wind v i b r a t i o n .

A n important a s p e c t o f t h e o p t i m i z a t i o n problem is t h e e x i s t e n c e ( o r l a c k of it) of a f e a s i b l e s o l u t i o n . A 'feasible s o l u t i o n ' is d e f i n e d a s any set o f d e s i g n v a r i a b l e s t h a t s a t i s f y t h e c o n s t r a i n t s (whether o r n o t t h a t p a r t i c u l a r s o l u t i o n is an optimum). T t is p o s s i b l e t h a t , if t h e problem is poorly formulated, n o feasible s o l u t i o n e x i s t s . What is more o f t e n t h e c a s e , however, is t h a t t h e r e a r e f e a s i b l e s o l u t i o n s but

t h a t t h e o p t i m i z a t i o n scheme may not be a b l e t o f i n d them. Thus. it is advantageous t o have a f e a s i b l e i n i t i a l guasv s o t h a t one is assured t h a t a t l s a s t a l o c a l optimum is p o s s i b i e .

For example, Table 2 i l l u s t r a t e s t h a t t h e first guess is f e a s i b l e (wl> 2.82 no./rev). Here w e found t h a t CONMIN w a s a b l e t o move from t h i s first guess through che space o f f e a s i b l e s o l u t i o n s . I n o t h e r c a s e s , however, when t h e first g u e s s is not f e a s i b l e w e have found t h a t COWIN is not a b l e t o reach a s o l u t i o n . I n such c a s e s , one must add o r remove some weight ( o r add o r remove EI) from t h e f i r s t g u e s s t o move from i n t a t h e f e a s i b l e space; o r , a l t e r n a t i v e l y , w e must begin with frequency- placement a s t h e o b j e c t i v e m n c t i o n 2nd t h e n switch t o weight when t h e f e q u e n c i e s a r e w i t h i n t o l e r a n c e .

For example, Table 3 r e p r e s e n t s d a t a f o r t h e s r - e wind t u r b i n e a s i~ Table 2, b u t t h e c o n s t r a i n t on t h e f i r s t n a t u r a l frequency h a s been lowered t o remove it from t h e dangerous 31rev range. This i m p l i e s t h a t t h e f i r s t guess i n Ta!>le 2 is no l o n g e r f e a s i b l e . I n o r d e r t o overcome t h i s , a lumped mass is added t o ~ t a t i o n 9 (225.4 v s 49.50). This lowers w below 2.621 r e v b u t a l s o lowers w2 t o 8,251rev. This could be a l l e - v i a t e d i n one o f two ways: 1) move t h e mass t o t h e node of t h e second W e have done t h e l a t t e r . It is mode. o r 2) simply widen t h e w2 window.

i n t e r e s t i n g t h a t t h e added welght is u l t i m a t e l y rearranged t o o t h e r p l a c e s and o t h e r weight removed such t h a t t h e new d e s i g n 1.3 no h e a v i e r than t h e optimum i n Table 2. Furthermore, wZ is r a i s e d t o 8.57 s o t h a t t h e "idaned window' had nc effect on tne s o l ~ t i a n .

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' T a l ~ l c . 3. !J_!!d T u r b i t ~ e w t t l ~ Added Class -- P NA'I'UHAI. : FROEI TO FRfdlIJENCY 2 . 4 2 . 6 2 PerfRev DENSITY COilS'KRh 1 N 1's SECOND 4 . tl 5.0 l l ~ A R E A 2 I 6 5 . 8 8 5 7 . 4 5 2 . 8 4 7 . 2 4 2 . 0 3 6 . 8 9 30.51 22.86 19.56 19.715 i n 111 SI'It I !iII'l'I:I) I 65.61 5 7 . 1 7 131.64 117.5 104.66 91.8 7 5 . 9 56.93 48.69 49 - 0 f l 0b.l ( I 11) w llz wl /R w 2 l l z w2/11 Elon1c111 o f I n e r t i a - - - - - ---.--.--------- e I H I ' I ' I A I . 1787.') 1.48 2 . 5 5 4 . 8 1 8.25 3.21 x 10 ( 1 1 ) )

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1s.I flA1, 1 0 1 7 . 1 1 . 4 8 2 . 5 5 5 .OO 8 . 5 7 2 . 5 x 1C8 . . . - - -. - - -. - - - - . - - - -- - - - -. - . - - - - -- - - - -- - -- - - - - - - -- - - - -. - - - - - - - - - - - - - - - - . . . -

3.2 HELICOPTER BLADE The deslgn and a n a l y s i s of a r e p r e s e n t a t i v e h e l i c o p t e r olade is discussed. Similar t o Section 3, only flapping is considered and a te.1- element model is used. The i n i t i a l configuration is modeled a f t e r t h e r o t o r i n Reference 23. Density is constant along t h e blade and equal t o 8 2

0.17 x s l u g s l i n 3 . Young's Hodulus is equal t o 0.49 x 1 G lb-in a t

t h e r o o t and is equal t o 0.585 x 10 lb-in elsewhere. Blade r a d i u s is equal t o 193 inches. Results are given i n Table 4 an* 5. I n Table 4. wl is i n t h e desired range b u t w is too small. Furthermore, t h e autorota- t i o n a l i n e r t i a is larger than necessary. I n t h i s case, t h e Z O N M I N pro- is a b l e t o remove m z ~ s and s t i f f n e s s i n such a way t o raise w2 and gram lower wl. The minimum bending i n e r t i a set a s a c o n s t r a i n t ( 0 . 4 ) is reached a t every point except t h e root. The r o o t remained high t o keep a t h e o r i g i n a l mass. In T a l e 5, w1 ) 1 - 05. The new blade is one-third stiffer initial design is used and t h e frequency wl is forced t o be very high. I n t h i s c a s e , t h e program CONMIN would ' l i k e ' tc aecrease E I a n d m, buf any removai of m a t e r i a l could lower w beyond its l o v e r bound of 1.24/rev. To counter t h i s , t h e ~ p t i u i z a t i o o zheme adds E I near

t h e r o o t ( t o malntain wl > 1.24Irev). Furthermore, t h e lumped mss

necessary t o maintain ;iutorctational c o n s t r a i n t is moved a:.ight:p 'n- board t o have l e a s effect on w (keep it high) but more e f f e c t on u 1 2 (keep it low!. %is exarcpie i l l u s t r a t e s t h e physical soundne3s cf t h i s optimLzation scheme. It dt-es t h e same t h i n g s t h a t a designer would do (given appropriate c o n a t ~ a i n t s ) but i n a more systemaZic manner. Thus.

with proper c o n s t r a i n t s , optimization can prove a ;aluable t o o i f o r frequency placement.

To_hlo 4. HeL1cnlctc.r Blndo - - .

F l RST 7.5 9.5 Hz ROTA'I'IM: SI'EED: 425 RPH NATNRAI. r FROn TO FREQUENCY 1.05 1.34 Pur/Rev YoUIW'S WUU1,US: Element I: 4 . 9 x 10 l b - i n Rest r 5.85 x 10 CONSTRAINTS SkCOND 18.5 20.5 Hz

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) 1 20.6 3.127 10.7 22.19 18.7 0.664 0.59 0.59 0.59 0 . 5 9 AREA In' F 0.91 0 . 5 9 0.59 0 . 5 9 0.59 0.59 0.59 U.59 0.59 0.59

-

-- DISTRIBUTED lbs I 1.2L 2.32 16.17 23.29 19.61 1.16 1.24 1.04 1.24 0.54 n A S S F 0.318 0.413 0.826 0 . 6 1 9 0.619 1.03 1.24 1.03 1.24 0.54 - LUMP t o 1b5 I 0 3.04 1.67 0 6.4 7.46 10.75 5.21 6.55 6.6 M A S S F 0 U 0 0 0 0 11.17 fi 0 16.08 - -- -- - - - - - - - - - --- O I J ( l b ) w llz ul /fiI ,I I l z w 2 / L i H w e ~ ~ t o f l n e r t l a -- - - - ----- 2 - 131.5/1 8 . 3 1.17 17.66 2.49 9.76 x 10 INITIAL

---- -- ---~-- -- --------

20.411 2.89 8.71 r 10 F t NAI. 41. 1 J ? 7.47 1 . 0 5

-

--

I - _ _ _ _ _ _ _ _ _ _ _____--- ----

W e have a l s o examined the designs i n Tables 3-5 with respect t o a x i a l s t r e s s due to centrifugal loads. In each c a s e , the maximum stress- es a f t e r optimization are equal t o or only s l i g h t l y higher than the original s t r e s s e e .

6 . TEETERING ROTORS 6.1 D e f i n i t i o n I n t h i s s e c t i o n w e attempt our first o p t i m i z a t i o n o f a r e a l i s t i c cross-section (see F i g u r e ll), one which is l i t t l e d i f f e r e n t from our first d e s i g n p a t t e r n , ( F i g u r e 1). Therefore, it is s c f f i c i e n t i y g e n e r a l t h a t both t h e bending and t o r s i o n a l s t i f f n e a s e s o f some c u r r e n t l y e x i s t - ing b l a d e s can be matched. Uslng t h i s g e n e r i c c r o s s - s e c t i o n and s t a r t i n g froffi a n a c t u r a l r o t o r b l a d e deslgn. (Ref.I241), w e have s t u d i e d t h e p c s s i b i l i t y o f moving n a t u r a l f r e q u e n c i e s away from resonances while simultaneously s a t i s f y i n g c o n s t r a i n t s on t h e foll.owing: stress, t h e s i z e o f lumped weights t o be added, c a p a b i l i t y f o r a u t o r o t a t i o n , and thick- n e s s o f t h e main s t r u c t u r a l member ( t h e box beam). Because a t e e t e r i n g b l a d e was considered, c y c l i c and c o l l e c t i v e modes 02 v i b r a t i o n were c a l c u l a t e d independently by a change i n t h e boundary c o n d i t i o n a t t h e blade r o o t . I n t h e i n i t i a l phase o f t h e study, w e considered c o l l e c t i v e f l a p p i n g modes f i r s t , t h e n c y c l i c f l a p p i n g , and f i n a l l y combined c o l l e c t i v e and c y c l i c f l a p p i n g . The r e s u l t s o f t h e ~ e s t u d i e s were f a v o r a b l e (i.8. w e were a b l e t o cha.lge t h e f r e q u e n c i e s i n t h e d e s i r e d manner and s t i l l s a t i s f y t h e c o n s t r a i n t s ) . B u i l d i ~ l g on t h e s e r e s u l t s , i n t h e secard phase o f t h i a s e c t i o n , w e c o n s i d e r a more c h a l l o n g i r y problem which i n v o l v e s combined modes o f c o l l e c t i v e f l a p p i n g , c y c l i c f l a p p i n g , c o l l e c t i v e inplane, c y c l i c i n p l a n e and t o r s i o n a l v i b r a t i o n s .

I n t h i s s e c t i o n , t h e primary d e s i g n v a r i a b l e s a r e : 1) t h e wall t h i c k n e s s o f t h e box baam and 2 ) lumpc:! weights, t h a t can be added a t s p e c i f i e d s t a k j o n s along t h e beam. I n t h e f i n a l problem s t u d i e d , t h e

GENERIC BLADE SECTION CONS1 RAINTS FirceO M:lss

BOX Beam I

Trolling Edge Honeycomb _ - & * I - - - - - +- a -* * b b ~ u i ~ l o e b Weight Design Variobler: I, dl and d2.

Fixed Poromelers: h 8 2.0 in b 4.65 in 1 , = 0.016 In Area Of Horteycomb = 25.2 in2 Circutnference Of Trolling Edge = 30.5 in Dlnlerlslon o was Ioken lo be zero for Ihe compu9allon of rnoss rnornent o l lnerlla Figure 11 B l a d e cross-section with varid~le t n x dbnension

wall t h i c k n e s s e s a r e t a k e n a s f i x e d , and o n l y t h e lumped w e i g h t s a r e a l l o w e d t o v a r y . T h i s s i t u a t i o n c o r r e s p o n d s t o t h a t e n c o u n t e r e d i n p r a c t i c e when a b l a d e h a s a c t u a l l y been d e s i g n e d and manufactured. b u t t h e n found t o have p o o r l y p l a c e d f r e q u e n c i e s - t h u s lumped w e i g h t s a r e added a t va.rious p o s i t i o n s a l o n g t h e beam t o change t h e f r e q u e n c i e s . W e found t h a t o u r o p t i m i z a t i o n r o u t i n e was a b l e t o h a n d l e t h i s problem a d e q u a t e l y , a l t h o u g h t h e t o t a l weight o f t h e beam c o u l d n o t be used a s t h e o b j a o t i v e f u n c t i o n , a s had b e e n d o n e p r e v i o u s l y . I n s t e a d , a ' f r e q u e n c y placement' o b j e c t i v e f u n c t i o n was u s e d .

6.2 F l a p p i n g F r e q u e n c i e s The f i r s t set o f o p t i m i z a t i o n problems i n t h e s e c t i o n is concerned w i t h f l a p p i n g r e s p o n s e o n l y . The s t z r t i n q d e s i g n f o r t h e c p t i m i z a t i c n p r o c e d u r e i n e a c h of t h e t h r e e c a s e s s t u d i e d is a t y p i c a l metal-bladed t e e t e r i n g r o t o r w i t h a d i a m e t e r o f a p p r o x i m a t e l y 24 f e e t . Ten f i n i t e e l e m e n t s are used t o model t h e r o t o r ; t h e i r l e n g t h s are g i v e n i n Table 6. The o b j e c t i v e o f t h e o p s i m i z a t i o n is t o minimize t h e t o t a l weight o f t h e b l a d e . The d e s i g n v a r i a b l e s a r e t h e wall t i i i c k n e s s , ti, of t h e f i n i t e el,ement r e p r e s e n t a t i o n sf t h e box b t a n ( t h e s t r u c t u ~ a l member i n

t h e r o t o r - see Fig. 1 and t h e lumped weight wi, a s s o c i a t e d w i t h each

f i n i t e element. The lumped weight is t h e sum o f two components, a f i x e d component ( r e p r e s e n t i n g t h e weight cf t h e l ~ a d i n g and t r a i l i n g e d g e s t r i p s , honeycomb, s k i n and nose w e i g h t , ) and a variable component ( r e p r e s e n t i n q a d d i t i o n a l n o n - s t s \ l c f u r a l mass w h i c h may b e a d d e d a t v a r i o u s p o s i t i o n s a l a q t h e l r n g t h cf t h e r o t o r ) t o modify t h e dynamic b e h a v i o r i n a d e s i r e d rnacrer. rhe s i d e c o n d i t i o n s on t h e element t h i c k - n e s s e s a r e , i n u n i t s o f i n c h e s ,

0.00044 < ti < 0.730

The ~ i d e c o n d i t i a n s on t h e lumped weights c o n s i s t o f a lower bound o n l y , which r e p r e s e n t s t h e fixed component o f weight f o r each element and is g i v e n i n Table 6 under t h e headin# 'wmint.

Nc:e f h a t t h o element t h i c k - n e s s e s ti are not g i v e n i n Table 6; i n s t e a d , t h e area moment o f i n e r t i a , I, is r e p r e s e n t e d . Using t h e dimensions g i v e n i n Fig. 1 2 , we can show t h a t I r e l a t e d t o t h e t h i c k n e s s by t h e e q u a t i o n The moment of i n e r t i a is given, r a t h e r t h a n t h e t h i c k n e s s , t o f a c i l i t a t e comparison w i t h Ion t h e p o r t i o n o f t h e moment o f i n e r t i a which is con- t r i b u t e d by t h o s e parts o f t h e c r o s s - s e c t i o n o t h e r t h a n t h e box beam.

(Thus I. remains f i x e d as ti is v a r i e d . ) Table 6 a l s o c o n t a i n s t h e v a l u e s sf t h e box weight, which are c a l c u l a t e d by m u l t i p l y i n g t h e weight d e n s i t y o f t h e box beam material by t h e c r o s s - s e c t i o n a l a r e a o f t h e box.

Thus, t h e box weight is not an independent d e s i g n v a r i a b l e , b u t depends on t h e t h i c k n e s s ti.

The box weight is included i n t h e t a b l e t o f a c l l i - tate comparison with t h e d i s t r i b u t i o n o f lumped weight. The c o n s t r a i n t s f o r t h e o p t i m i z a t i o n are both t h e a u t o r o t a t i o n c o n s t r a i n t , sum (wi .)ri2 > 0 . 5 5 6 7 ~ 1 0 ~ lb-in2 (where w i t is t h e t o t a l weight o f element i, and ri is t h e d i s t a n c e from t h e r o o t t o t h e c e n t e r o f t h e i - t h f i n i t e e l e m e n t ) . The frequency con- s t r a i n t s w i l l be d e s c r i b e d i n subsequent s e c t i o n s o f t h i s thesis.Some

F i g ~ - , : e 12 Slmensions 3nd Def i n i t i o n of Design Var i a b l e f o r Box Beam C r o s s - S e c t a o n

a d d i t i o n a l d a t a which complete t h e problem d e s c r i p t i o n a r e t h e v a l u e s of t h e e l a s t i c a o d u l u r , 0 . 1 0 5 ~ 1 0 ~ l b l i n , t h e r a d i u s , 288.8 i n , t h e r o t a t i o n a l s p e e d , 3 2 4 rpm, a n d t h e mare d e n s i t y o f t h e box beam m a t e r i a l , 0.000262 - a p s / i n .

6.2.1 C o l l e c t i v e Modes The i n i t i a l problem t o be c o n s i d e r e d i a t h e opticgization of the b l a d e v i t h r e s p e c t t o c o l l e c t i v e f l a p p i n g modes o n l y . Because w e a r e s t u d y i n g a t e e t e r i n g z o t o r , t h e c o l l e c t i v e mode of f l a p p i n g may b e a n a l y z e d by h o e i n g a f i x e d boundary c o n d i t i o n a t t h e r o o t cf t h e r o t o r . The imposed frequency c o n r t r a i n t s a r e i n which p l , p2 and p3 a r e t h e f i r s t t h r e e c o l l e c t i v e f l a p p i n g mode frequene i e a non-d h n r i o n a l i z d by d i v i d i n g by t h e r o t o r speed.

The s t a r t i n g d e s i g n f o r t h e o p t i m i z a t i o n a l g o r i t h m i s g i v e n i n T a b l e 6 under t h e beading ' i n i t i a l ' . T h i s i n i t i a l d e s i g n was chosen t o correspond t l o r e l y w i t h an a c t u a l r o t o r b l a d e ; t h u s , it is n o t s u r p r i s i n g t o f i n d t h a t t h e d e s i g n is i n f e a n i b l e v i t h r e s p e c t t o t h e frequency c o n s t r a i n t r we have impored. The o p t i m i z a t i o n a l g o r i t h m used i n t h i s s t u d y , CORMIN, supposedly p e r m i t s an i n f e a s i b l e s t a r t i ~ g p o i n t and a t - tempts t o proceed from t h i s s t a r t i n g p o i n t t o s f e a s i b l e p o i n t , However, f o r o u r d e s i g n problems, t h i s f e a t u r e of C O N H I N f a i l e d t o produce a f c a r i b l e d e s i g n a f t e r many i t e r a t i o n s . As an a l t e r n a r i v e approach, we

formulated a p r e l i m i n a r y o p t i m i z a t i o n problem i n which c h e p r e v i o u s o b j e c t i v e f u n c t i o n ( v e i g h t ) was r e p l a c e d by a 'frequency-placement' ob j a c t i v e : The numbers 3.50 and 6.50 a r e t h e a v e r a g e o f t h e bounds of t h e frequency c o n s t r a i n t i n e q u a l i t i e s which a r e v i o l a t e d by t h n i n i t i a l d e s i g n . The remaiader o f t h e o p t i m i z a t i o n problem is t h e same as t h e o r i g i n a l pro- blem, e x c e p t that t h e c o n s t r a i n t s on t h o s e f r e q u e n c i e s which a p p e a r i n t h e frequency p l a c m n t o b j e c t i v e a r e o m i t t e d . CONHIN vas a p p l i e d t o t h i s p r e l i m i n a r y problem. I n t h e p r o c e s s o f minimizing t h e p r e l i m i n a r y o b j e c t i v e , C O m R was a b l e t o d r i v e t h e f r e q u e n c i e s s u f f i c i e n t l y c l o s e t o t h e i r bound8 that a f e a s i b l e d e s i g n ( v i t h r e s p e c t t o t h e o r i g i n a l p r o b l e d v a r o b t a i n e d . A t t h i s p o i n t , t h e o r C q i n a l o b j e c t i v e f u n c t i o n was r e i n s t a t e d and COFMIR a p p l i e d once a g a i n .

T a b l e 6 g i v e r t h e o p t i m i z e d d e s i g n o b t a i n e d b y t h i s t w o - s t a g e o p t i m i u t i o n p r o c e d u r e , v i t h t h e c o r r e s p o n d i n g f r e q u t n c i e s aad t h e t o t a l weight. Prom t h e p o i n t o f view o f h e l i c o p t e r v i b r a t i o n s , t h e i n i t i a l d e s i g n o f t h i s blade is acceptable, s i n c e ( e x c e p t f o r t h e t h i r d mode) t h e numbtr/rev i s f a r away f r m even i n t e g e r v a l u e s . The t h i r d mode i s , however, n e a r 6 . 0 / r w . The f r e q u e n c y of t h e second mode does n o t s a t i s f y t h e i n e q u a l i t y c o n s t r a i n t s , b u t is n o t n e a r an even i n t e g e r m u l t i p l e .

Note that t h e f i n a l d e s i g n moves t h e t h i r d f r e q u e n c y t o 6.53, w h i l e keeping t h e o t h e r f r e q u e n c i e s w i t k i n t h e c o n s t r a i n t s . A t t h e same t i m e , t h e weight o f t h e b l a d e d r o p s from 344.5 l b t o 265.6 l b .

6 . 2 . 2 C y c l i c Modes The n e x t prsblem t o be s t u d i e d i s t h e o p t i m i z a t i o n of t h e b l a d e w i t h r e s p e c t : P cyc l i c w d e s of f l a p p i n g . The n o n d i m c n s i r ~ a l i z e d f r e - quency c o n s t r a i n t s a r e n o r i n which pl, p2 ,\nd p3 a r e t h e f i r s t t h r e e c y c l i c non-dimanslonalized f l a p p i m a o d e f r e q u e n c i e s . For cy; l i c f l a p p i n g modes, t h e boundary con- d i t i o n a t t h e r o o t corresponds t o a pinned s u p p o r t . T 3 e f r e q u e i l c y - placement o b j h c t i v e vr8 a g e i n chosen by n o t i n g vhich frequency con- r t r a i n t s were v i o l r ? . e d by t h e i n i t i a l d e s i g n . Noting t h e i n i t i a l f t e - quency v a l u e s g i v e n i n T a b l e 7 , we d e f i n e Table 7 g i v e s t h e o p t i m a l d e s i g n found by t h e two-stage o p t i m i z a t i ~ n procedure w i t h t h e corresponding f r e q u e n c i t s . The f i n a l vcight of t h e b l a d e i n shown t o drop from 344.5 l b t o 295.2 l b .

6 . 2 . 3 Combined C o l l e c t i v e and C y c l i c Moder Next v e c o n s i d e r t h e o p t i m i z a t i o n of t h e beam wit' ;:'"ect t o coubined c r l . l e c t i v e and c y c l i c modes. TP.c., i n each i t c r d r t o : - dn analy- rim mar: be performed t o f i n d t h e f r e c u e n c i e s correspond.'-ng l o a f i x e d boundary c o n d i t i o d ; and then a n e t h e r a n a l y s i s mst be pes formed t~ i i r . d

the frequencies f o r a pinned boundary condition. A l l o t h e r a s p e c t s of t h e design problem remain t h e same a s before. The c o n s t r a i n t s on t h e c o l l e c t i v e flapping modes a r e fie c o n r t r a i n t s on t h e c y c l i c modes a r e .

For t h i s problem, weighting f a c t o r s a r e introduced i n t o the frequency- placement o b j e c t i v e , The r e s u l t s of t h e optimization a r e given i n Tab1.e~ 8. The c y c l i c modes of t h e i n i t i a l design a r e vell-placed i n the sense t h a t they a r e a o t near odd i n t e g e r s / r e v , but the t h i r d c o l l e c t i v e mode is near 6.0/rev ( t h e same as in t h e f i r s t example). Note t h a t t h e f i n a l design noves the

T a b l e B l n l t l a l and F l n ~ l d e r l ~ n f n r c o l l e c t i v e n r ~ d cyclic f l n p ( b l o n m o l l r c R o t a t I I I ~ Speed : 324 RPH a a s a o f aoment I 8 2 ) 0 . 5 4 2 9 ~ 1 0 I h - I n .

I n e r t l a Y , . ~ I ~ c L n ~ , ~ ; ~ ~ l ~ t a a 0 . I 0 5 x 1 0 I b / l n d e n r l t y o f b o x b r a n : O.000261 a c ~ g s / l n A x 1 . 1 1 Strcns ( 20,000 p r l E II-IUCII~ NO. 1 2 3 4 5 6 7 4 9 1 0 _ _ _ ._-_.___________------------------------------------------------------------------------------------------ I.I-II~: 111 ( i n ) 14.4 14.4 14.4 43.2 28.10 28.80 28.8 2R.RO 41.2 43.2 ___._-_-__----_____--------------------------------------------------------- A r v n tktment lo(unchanged) 48.6 93.38 64.8 5.76 1.76 1.47 1.03 I . 1.03 1.03 n l l r ~ c r t l a ( ( s t 4 ) Cox 8ems I n l t . 1.86 1.86 1.86 1.16 1.16 I . 6 1.16 1 . 6 1 - 8 6 I ( I ) F l n a l 1.36 0.51 2.17 3.05 0.51 9.51 0.51 2.89 0.7A6 3.05 t l l n . I , ~ m p e d u e l g h t 4 86.02 3.06 9.19 6.13 6.13 6.13 6.13 9.19 9.19 ( I b ) lornlu.d u c a l g l ~ t ( I ) I n l t . 42.72 16.02 41.38 20.46 9.23 6.46 6.13 6.40 14.3 20.46 F i n a l 5 2 - 7 2 86.02 21.95 31.15 6.13 6.13 6.13 6.13 9.46 9.19 ----------------------------.-------------------------------------------------------------.------------------ Ann H e m U r l g l ~ t I n l t . 1 . 9 8 4.68 9.36 9.36 9.36 9.36 9.36 9.36 4.68 14.05 4.43 4.43 17.34 7.90 31.51 F l n n l 1.53 2.21 5.51 31.51 4.43 ( 1 1 1 ) N a t u r a l F r e q u o i c l e r (NO . / r e v ) Collective F l a p p i n g C y c l l c F I a l . ? l n g liner: (I) -- d e a l g n v a r l a b l e ( f l r e d b o u n d a r y ) ( p l n n e d h o u n d a r y )

1.15 < p 1 < 1.60 0 . 9 < p l < 1.14

3.4: : 02 < 3.60 2.4 < pa < 2.60

6.40 < p3 < 6.60 4.4 < p 3 < 4.60

Blade V e l ~ l ~ t PI PZ ~3 PI PI PJ ( I h ) I n i t l a 1

344 .) 9 3 2 6.00 1.02 2.64 4.67

F l n a l 338.5 1.20 2.64 4.58 3.40 6.33 ' .02

Area moment of Inert*. : 1 - 2.593t3 - 7.78Ut + 7.7ROt + 0.5067

where t fir t h e t o p t l ~ l c k n e a c l o f box h e a r

c#.oon44 < t < 0.732n

t h i r d c o l l e c t i v e frequency t o 6.33 w h i l e keeping t h e o t h e r f r e q u e n c i e s i n t h e ' s a f e * range. However, t h e weight of t h e blade o n l y decreased from 314.5 t o 333.5 l b , i n c o n t r a s t t o t h e previous example i n which t h e w e i g h t decreased t o 255.6. The d i f f e r e n c e is caused by t h e l a r g e r number c f frequency c o n s t r a i n t s i n t h i s example compared t o t h e p r e v i p ~ s example.

6.3 S z W a T M g O U S W P I R C IXVPUNE AHD TORSLOB The next problem t o be s t u d i e d is t h a t o f o p t i m i z i r y a t e e t e r i q r o t o r blade s u b j e c t t o t h e following simultaneous c o n s t r a i n t s on fre- q u e n c i e s ( n o n - d i m e n s i o n a l i z e d by d i v i d i n g by t h e r o t o r r o t a t i o n a l speed : 1 c o l l e c t i v e f l a p p i n g modes, 2. c y c l i c f l a p p i n g modes, 3. c o l l e c t i v e i n p l a n e modes,

4. c y c l i c inplane m d e r and 5. t o r s i o m 1 mode

(The f i r s t lover b o d f o r c y c l i c inplane aodes , 0 .lo, was l a t e r rcpalc-

Be- ed by 1.0 i n t h e problem f o r n o l a t i o n r of t h e folloving s e c t i o n s ) .

cause we are considering a t e e t e r i n g blade, t h e collective-flapping and cyclic-inplane -des can be ~ o d e l d by cl-ed boundary conditions a t I t h e r o o t , while t h e cyclic-€ lapping and c.ollect i v c i n p l a n e modes can be m d e led by p i n n d boundary coad it ions.

The e l a r t i c modulus, blade length, tpeed of r o t a t i o n , and d e n s i t y of t h e box-beam m a t e r i a l a r e unchanged from t h e values used before. In 5,.

addition t o frequency snd a u t o r o t a t i o n c o n s t r a i n t s , t h e u i a l s t r e s s i s constrained t o be l e s s than 20,000 poi. The value of t h e bound i n the r a t o r o t a t i o t u l c o n s t r a i n t has been chsnged s l i g h t l y t o 0.5429~10' ib- I n t h e problem dercribed i n s e c t i o n 3 . . t h e thicknesses, t i , d. .

1 1 of ixth t h e v e r t i c a l and horizontal n l l s of the box beam ( s e e a d dZi,

Fig. 2) a r c 8 l l o d t o vary - that is, a r e a l s o desig3 v a r i a b l e s , with

t h e following s i d e c o n r t r a i n t s (in unitb of inches);

T a b l e 9 D a t a for Teetering Rotor b l a d e ~ o t e t i n s Speed t 324 RPH mans o f moment 8 7 2 2 Youngs b d u l u a I 0 . 1 0 5 x 1 0 l b / l n I n e r t l a 2 0.54291110 I h - I n ~ ~ i o l S t r e s s ,( 2 0 , W p o l d e n a l t y a t hox haam t 0 . 0 0 0 2 6 3 m n R 2 / l n 3 Element No. 1 2 3 4 5 6 7 8 9 1 0 ------------"""-'"'-"-""'-----"---r----------------------------------------------------------------- Lu116t h ( i n ) 14.4 14.4 14.4 4 3 . 2 2 8 . 8 0 2 8 . 8 0 2 8 . 8 28.flO 4 3 . 2 4 1 . 1 t ( I n ) Don Ream l ) l m r ~ w i o n d l ( I n ) ________________^______________--__--------------------------------------------------------------------------- I h x ( l n 4 ) 2 . 9 3 2.94 2 . 9 2 2 . 6 2 2.24 2 . 0 9 2.03 2 . 0 3 2 . 0 3 2 . 0 3 Arva and mas Y l o x (1114) 4 7 5 4 9 2 . 3 0 6 4 . 0 5 5 . 3 8 1 . 5 3 1 . 3 0 0 . 8 5 0 . 8 5 0 . 8 5 0 . 8 5 rec~tneot o f -3 i ~ ~ e r t i a fix (mu8-ln)xl~ 0 . 7 6 0 0 . 7 7 0 0 . 7 6 5 O.685 0 . 5 8 6 0 . 5 4 7 0 . 5 3 2 0 . 5 3 2 0 . 5 3 2 0 . 5 3 2 c.C y - a x l s b x ( r u 8 - l n ) x l ~ - 2 0 . 2 7 3 0 . 2 7 3 0 . 2 7 0 0 . 2 0 0 0.OAO 0 . 0 4 0 0 . 0 2 2 0 . 0 2 2 - 0 . 0 2 2 (1.022 ------------------------------------------------------------------------------------------------------------- I h y (1114) 1 2 . 7 9 12.AO 1 2 . 5 0 10.54 8 . 9 8 8 . 5 7 8 . 4 0 8 . 4 0 8 . 4 0 8.40 Ared dnd 0 '0 III~SS l o y (Id) 4 8 . 1 1 1 6 0 . 5 0 3 1 2 . 9 0 2 9 4 . 2 2 210.66 167.62 1 3 2 . 5 0 1 0 2 . 7 0 8 4 . 9 0 8 1 . 8 8 m s

-ri n

m u c . ~ c ~ t o f 0 % i n e r t ! a H b r ( m 1 ~ ~ - l n ) x 1 0 - ~ 0 . 3 3 ) 0 . 3 3 5 0 . 3 2 7 0 . 2 7 6 0 . 2 3 6 0 . 2 2 5 0 . 2 2 5 0 . 2 2 5 0 . 2 2 5 0 . 2 2 5

z g

of x - a r l s -2 Hoy (m116-1n)xlO 0 . 0 1 5 0 . 0 1 5 4 . 3 7 8 11.146 5 . 6 9 0 4 . 4 9 0 3.576 3 . 2 8 0 3 . 1 9 0 2 . 9 1 0

g~

-----------------------------.-------------------------------------.*------------------------------------------- Y "..

nln. lumped u e l g h t ( l b ) 9 6 . 5 4 5 5 . 7 7 2.69 . R . 0 1 5 . 3 8 5 . 3 5 . 3 8 5 . 3 1 8 . 0 7 8 - 0 1 -- r- , .

- .'

' s 4 ~ f i u - - lumped u e l ~ h t ( l b ) 9 6 . 5 4 5 5 . 7 7 6 . 9 5 1 1 - 6 3 7 . 6 8 5 . 7 6 5 . 3 8 6 . l 2 1 1 . 8 1 19.98

The i n i t i a l values of t h e r e v a r i a b l e s a r e given i n Table 9. I n the problem8 direursed i n s e c t i o n 3 - 2 and 3 -3, t h e box beam d i r n a i o n s t dli and dZi a r e fixed a t t h e r e i n i t i a l values.

Tabler 9 a l s o giver d a t a defining both fixed and i n i t i a l s t i f f n e s s a d i n e r t i a valuer of t h e blade. I n t h e t a b l e s , 'Iox' and 'I ' repre- 0 Y sent t h e portionr of t h e flapping and inplane area moaents of i n e r t i a of *Ia' t h e blade s e c ~ i o n which a r e independent of t h e design variables.

a d 'I ' a r e t h e a r e a PoPants of i n e r t i a of t h e box be= - thus func-

BY t i o n s of t h e ( i n i t i a l ) values of t h e design v a r i a b l e s , wi, dli, dti, and

t:. K . and n --- *h- - - - ~

a7 "' -a= r u ~ r r y i n e r t i a s of t h e box beam with respect t o A flapping and inplane and a r e calculated simply by multiplying t h e mass density of t h e box-beam m a t e r i a l by t h e a r e a . o r n t of i n e r t i a s . Wox and a r e t h e contributiotls t o t h e rota- i n e r t i a of t h e s e c t i o n which a r e OY independent of t h e d o i g n variables. Rote t h a t s i n c e these contributions cop^ from itm with d i f f e r e n t d e n s i t i e s , a s i n g l e uniform value of density cannot be defined f o r the Mo t e r n .

3ther i n i t i a l i n e r t i a and s t i f f n e s s p r o p e r t i e s a r e a l s o defined i n Table 9. A s belore, t h e lumped weights associated with t h e f i n i t e ele- ments a r e taken a s design variables; b u t , t o permit g r e a t e r l a t i t u d s in placing t o r c i o n a l frequencieo and t o m t c h more c l o s e l y t h e behavior of r t r u e helicopter blade, a torsionaL spring is introduced a t t h e blade r o o t ; and it. s t i f f n e s s i s t a k e n a s a d e s i g n v a r i a b l e . The s i d e

c o n s t r a i n t s on t h e lumped weights a r e given i n t a b l e 9 under t h e beading 'v '. the s i d e c o n s t r a i n t on t h e t o r s i o n a l s t i f f n e o s c o n s i s t s o f the min r e q u i t c ~ r n t t h a t t h e s t i f f a e r r be non-negative.

Since o t o r s i o n a l made is involved, s p e c i a l treatment is given t o GJ, t h e t o r s i o n a l r i g i d i t y , vhich is a function of a l l v a r i a b l e s includ- d2, t h e area of t h e t r a i l i n g edge, and t h e lumped mass. The ing t , dl, procaduzc f o r c a l c u l a t i n g CJ i r described i n Append- A .

Table 9 8180 containa t h e contribution of t h e 1-ed w i g h t t o the This rota- i n e r t i a . which equals t h e i m p d +.a8 (wi/g) t i r s ( b f 2 ) .

expression ha8 been chosen t o match t h e behavior of t h e t r u e blade. It i r assttocd that t h e lumped weights c f t h e f i r s t tn, e l e m n t , c o n t r i b u t e nothing t o t h e rota- i n e r t i a .

6.3.1 Variable Box D k n r ion A s war done with t h e optimization involving f la2ping only, a two- S t e p o p t i n i t a t i o n procr d o r e is u s e d , v h i c h i n v o l v e s a frequency- p l a c a m n t o b j e c t i v e followed by a weight o b j e c t i v e . The frequency- p l a c m e n t o b j e c t i v e has tho general form i n which tk ya is taken over thore f-equencies which a r e t o be changed from t h e i r ... i t i a l values t o t h e desired values p, Values o i p,i and i t h e n i g h t i n g f a c t o r s wfi a r e given i n Table 10 with the r e s u l t s of the o p t i m i u t i o n . Examination of t h e values of lumped weight given i n Table 10 s t o w t h a t w i g h t is concentrated a t the t i p of t h e blade because o f t h e a u t o r o t a t i o n a l c o n s t r a i n t . , ~e ntremr a t t h e f i r r t element i s c l o r e to the stress constraint value of 26.000 psi. Thi total weight changes

- 0 CI h I I I * C I I * - l , n + l l -.I . - .

- 0 0 0

i Z 2 l Z S I 2 ~ i ~ y ~ q A 2 " C n = 9". y y

n n u n A 0 v

2 I d & I c'd I c'c j e m s I c o I

c

- m I I I I I I - C I I r e I I . . I

; - - !

I I I 1 use 1

1 < ? I

l r r I I I O l f I I I I a * I

1 ? ? 1

1 - 1

! !

lhhh . . .

- a =

U 0 - rt C D V V V rl a u C ) * r N a a & - & a &

w5:v"v

i??

0 - 2

. . .

: m ~ c l n m

, C I

w v v N OI & - - N o ! . . , "i a,, m u a a P I N - = o r rn n

,,I V V V ; "ir

1 0 0 I

i n s i g n i f i c a n t l y . from 345 t o 339 l b . how eve^, t h e r o o t s p r i n g shows a s i g n i f i c a n t e f f e c t on t h e t o r s i o n a l mode. The s p r i n g c o n s t a n t changes 5 5 from 6 . 5 ~ 1 0 t o 4 . 1 0 ~ 1 0 i n - l b l r a d i a n .

The placement o f t h e f r e q u e n c i e s is shown i n Table 9 , a l s o . Note t h a t t h e t h i r d c o l l e c t i v e mode o f f l a p p i n g , which was n e a r 6.Olpeve moves t o 5.67/rev; and t h e second c y c l i c mode o f i n p l a n e , which was n e a r 7 l r e v . moves t o 6.7/rev. F i n a l l y , t h e t o r s i o n mode, which was 3.871rev.

moves t o 3.4/rev.

6.3.2 Fixed Box Dimensions W e n e x t c o n s i d e r t h e oroblem o f modifying a b l a d e which has a l r e a d y been c s n s t r u c t e d , b u t which h a s been s u b s e q u e n t l y found t o have inappro- p r i a t e n a t u r a l f r e q u e n c i e s . S i n c e t h e b l a d e is a l r e a d y b u i l t , t h e o c l y way its dynamic behavior can h e modified is through t h e a d d i t i o n o f lumped mass and a l s o through c h m g i n g t h e r o o t s p r i n g . Thus, i n c o n t r a s t t o t h e problem o f s e c t i o n 3.1, h e r e t h e box-beam weight. t h e f l a p p i n g and inplane-bending i n e r t i a , and t h e t o r s i o n a l r i g i d i t y a r e c o n s t a n t .

A l l scarti- v a l u e s and f i x e d parameters are t h e same f o r t h i s problem a s i n t h e p r e v i o u s s e c t i o n . Finding a s t a r t i n g d e s i g n which satisfies a l l o f t h e frequency c o n s t r a i n t s is a d i f f i c u l t t a s k f o r t h i s problem, and t h u s t h e f r e q u e n c y - p l a c e m e n t o b j e c t i v e i s t h e o n l y o b j e c t i v e i u n c t i o n used; t h e second p h a ~ e ( w i t 9 weight a s t h e o b j e c t i v e f u n c t i o n ) is never reached. Values o f t h e weighting f a c t o r s and frequency bounds which appear i n t h e o b j e c t i v e f'unction are g i v e n i n Table 11.

The f i n a l v a l u e s o f t h e lumped weights and a x i a l s t r s a s e s a r e shown i n same Table 11, w i t h f r e q u e n c i e s , r o o t - s p r i n g s t i f f n e s 3 , and t o t a l weight, The most s i g n i f i c a n t frequencies a r e t h e t h i r d c o l l o c c i v e f l a p - ping mode and t h e second c y c l i c inpJ.ane mode, which are s e e n t o move f a r

away from t h e u n d e s i r a b l e i n t e g e r l r e v values. However, t h e first c y c l i c i n p l a n e frequency is n e a r l l r e v .

6.3.3 V a r i a b l e Root Bending S t i f f n e s s The problem formulated i n t h e p r e v i o u s s e c t i o n p r e s e n t e d computa- t i o n a l d i f f i c u l t i e s i n t h a t it was found d i f f i c u l t t o f i n d a d e s i g n which s a t i s f i e d a l l t h e frequency c o n s t r a i n t s s i m u l t a n e o u s l y when o n l y lumped aass and t h e r o o t s p r i n g were used as t h e d e s i g n v a r i a b l e s . T h i s d i f f i c u l t y p.ay be caused by t h e dominating i n f l u e n c e o f t h e r o o t bending i n e r t i a s . Thus. i t seems r e a s o n a b l e t o i n c l u d e t h e v a l u e s o f 'I a t OY t h e r o o t as one o f t h e d e s i g n v a r i a b l e s . The f r e q u e n c y - p l a c e m e n t o b j e c t i v e is used throughout t h e o p t i m i z a t i o n ( t h e weight is n o t used as t h e o b j e c t i v e ) , and a l l s t a r t i n g d a t a and f i x e d parameters a r e g i v e n t h e same v a l u e s a s i n S e c t i o n 3.2. Vahes o f t h e weightinq f a c t o r s and frequency b o u ~ d s which a p p e a r i n t h e o b j e c t i v e f u n c t i o n a r e g i v e n i n Table 12 w i t h t h e r e s u l t s o f t h e o p t i m i z a t i o n .

The r e s u l t s o f Tables 1 2 d i f f e r from t h o s e o f Tables 10 and 11. The lumped weight changes a t t h e first, f o u r t h , and P i f t h e l e u e n t s . The stifhess o f t h e r o o t s p r i n g mores fro. 6 . 5 1 ~ 1 0 ~ t o 4 . 3 2 ~ 1 0 ~ in- l b / r a d i a n . However, t h e most s i g n i f i c a n t e f f e c t is t h e chanqe o f I OY ( t h e bending moment o f i n e r t i a a t t h e r o o t from 48.11 t o 290.29 i n .

Table 1 2 shows t h a t a l l f r e q u e n c i e s are placed i n t n e s a f e range.

6.4 E f f e c t o f P r e t w i s t I n t h i s s e c t i o n , w e w i l l do t h e o p t i m i z a t i o n o f a beam which is p r e t w i s t e d . (Tretwisted b l a d e i m p l i e s t h a t t h e mot' 4s o f f l a p p i n g and i n p l a n e a r e c c u p l e d ) . The procedure o f a n a l y s i s w i l l be t h e same as i n s e c t i o r l 6.3. Data are i d o n t i c a l t o t h o s e o f s e c f i o n 6.3 s a v e t h a t t h e

\

*Ir. 8

I 1 o m -

I ! ' ? ? ?

I W. h d

-

I I 1 h h h I m b n I . . .

I I m m " I m m m

m a -

? ? ?

m n a L I

i i

1 0 0 1

: ? ? I

I d-. I I

!

I I I I I I I

I i W b '

I

I I

I m l n I

5 2 U K I ! +r

I j

I w e 4 I * U

I ' r n 1 0 C \S . I . A Y I* I C I U

I P . I

I c & I I - I ! - 1

prqtwisted angle is new inoluJwl.

6.4.1 Variable Box D m i m A two-step o p t l a i z a t i o n pr*oaedure i s s t i l l u s e d . The box bean d h e n ~ i o n s and Xmpod waights as (well 8s t h a r o o t s p r i n g ) are considar- ad as design v a r i a b l e s , Tbls r e s u l t s 8- shown i n Table 13. I n a simliar fashion a s r e s u l t s i n s e ~ t i o n 6.3, t h e weight is ooncentrated a t t h e t i p of t h e blade because of t h e a u t o r o t a t i o n a l c o n s t r a i n t . Boot s t r e s s is c l o s e t o t h e stress sunst,raint which ts 20,000 psi. The t o t a l weight doe8 n a t decrease. Imtraad, it i n 2 r e a s e s from 7 S q I3 t o 354 l b , The .* s p r i n g c o n s t p n ~ ahangeq5 f'rom 6 . 5 ~ 1 0 t o 4.2110' fr-lb/rad. The ~ e c o n d noirev while second ao11eutive mode of flapping movus from 5.89 t o 5.67 c y c l i o mode of inplane moves from 7.08 t o 6.54 no/rev.

6.4.2. Fixed Box Dimensions For a blade of existing c o n s t m c t i o n , only t h e iumped weights a m oonaidered as deal& v w i a b l e s . '!'h~ restilts of s p t l m i z a t l o n a r e s n o m i n Table 14. The t o t a l waiqht i n c r e a s e s from 345 l b t o 372 l h , The thi.Td o o l l e c t i v e mode of f l a p p i n g and t h e ssoobd cmXic mode of i ~ p l a n e move i n t o t h e safe range.

7. ARTICULATED ROTORS 7.1 Definition b t i o u l a t e d r o t o r blade uili be t h e subject of design blade i n t h i s seation. RefC231 The primary design v a r i a b l e s a m saw as before: t h e wjll thickness of box bear ud lumped weights. Because t h e blade is a r t i o u l a t e d with a r i g i d hub, t h e m is no d i s t i n c t i o n between c o l l e c t i v e or cyclio modes for flapping aad inplane. i%e blade is pmtwisted. The boundary condition f o r flapping is a hinge at t h e root. 'hem is root spring f o r t o ~ i o r d l motions and an o f f s e t f o r inplane. Table 13 gives d a t a f o r both t h e initial (and mlnimm) blade stiffnesses and i n e r t i a a as w e l l as for t h e i n i t i a l variables such am box beam diamasions, lumped ue Q h t s , etc .

7.2.1 Variable Box Dlmansion Bax beam dimensions, l-ed w e a h t s and root s p i n g are taken as design variables. Tables 16 shorn t h e f n t i a l and f i n a l r e s u l t s of t h e optimization procedure. The inplans frequenoy mves f i o ~ 4.84 t o 4.69 no/mv while t h e t o r s i o n a l mode moves from 4.25 t o 4.43. The t o t a l weight drops s l i g h t l y f r o m 96.53 t o 93.48 lb. The root spring changes m 2.41110~ t o 2 . 8 ~ 1 0 ~ .

7.3. Fixed Box Boa8 D ~ o s i o a s Bolt beam dimensions w i l l be considered t o be fixed i n t h i s section.

Only Lumped weights and t h e root spring are taken as design variables.

The final results are shown i n Table 17.

d d j n i rl I; I O I Q , a * I - m I I I

elm 9

' O l d

o l d 0

I N I I I N - 1 - m 4 1 3 l m N l h a h I I f-4 4 1 0 3 1 % 3 * I - N n m 1 0 m

q l n 9 d N l h N

o j o 0

I I e l * 0 h m 1 m 9 1 4 m c e l m 0 1 " 0 ct 1::

d d j n i A

o j o 0 0 O l d I I I I

a

s x

a 0

a o o

, . -

I S s CF 0 w S u. P i - - I C C Y ~ I C C ' 3 1

c I * - I e l - - Z E ~ Y

* I - - - - 1 - - - - 1 g

Table 17 Initial and final desigr: for flapping, inplane and torsional modes of Articulated rotor blade with box bean dimension fixed El enent - NO. 1 2 3 4 5 6 7 8 9 10 _ _ _ _ _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Box Beam t 1 - - - - - - - - - .-

0.01 ( t < 0.7 F _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - - - - - - - _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Box Bean dl 1 - - - - - - - - - - 0.01 < dl < 2.75 F _ _ _ _ _ _ _ _ - - - - _ - - - _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Box Bean dZ - I - - - - - - - - - -

0.01 ( d 2 < 2.75 F _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0.89 1.435 1.435 1.435 1.435 1 4 3 5 1.435 1.435 1.435 1.435 Hin.

lumped weiqht # I o . 718 9.088 1.978 1.435 2.352 5.852 6.342 6.573 6.372 5.962 8.164 2.401 1.901 3.779 1.745 4.288 4.067 4.525 7.595 4 lbl F 6.015 - - - - - - - - - - - - - - - - . - - - - - - - - - - - - - - - - - - - - - - . . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ t r e s s x l o 4 1 1.39 1.32 1 - 6 4 2.54 2.84 2.60 2.12 1.55 0.94 0.61 ( p s i ) F 1.37 1.30 1.61 2.49 2.74 2.4 1 1.91 1.47 1.00 0.75 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - < t < < < < ( < Natural Frequencies (No/rev) ) ) , ) ) ) ) Flapping Inplane Torsion 1 5 t hinged ) (offset) t spr i r r g ) . -P , , 0.01 < p1 < 1.50 0.01 < pl < 2.50 4.3 'I'i~ta 1 mass Root-Spring 2.23 < p2 < 2.63 4.30 < p2 ( 4.70 < pl < We iqht Stif fness(W) 4.26 ( p3 < 4.67 12.30 < p3.< 12.70 4.7 12.30 4.25 2 . 4 1 ~ 1 0 ~ 1.03 2.70 4.51 0.24 4.64 9 h . 55 1.03 2.57 4.67 0.24 !.43 12.30 4.39 94.89 2 . 7 0 ~ 1 0 . . . . - - - - - - - - - - - - - - - - - - - - - - A A A A L & 1.00 . Weiqhtinq Factor & A A A 4.50 Desired Frequency & A ( A ) - - Zero NoLe : ( Y ) - - Desiqn Varlable

8 .I ? O ~ T I O I I n t h i r s e c t i o n , w w u l d l i k e t o sbor whether o r not t h e forced response of t h e blade can be adequately c o n t r o l l e d , u w e have u s u w d , by our approach of 'frequency p l a c m n t ' , that is, of r e s t r i c t i n g t h e n a t u r a l frequencies of t h e blade t o l i e within n r r r o v i a t e r p u l s located away from c e r t a i n i r t q e r ~ l r l t i p l e r of t h e r o t o r speed. Also ve w u l d exuine whether o r not aerodynamic d u p i n g s u b s t m t i a l l y r a d u c t ~ t h e resonant peaks, in which c a r e concern about avoiding resonances through proper s e l e c t i o n of frequency rind- w u l d be unnecesrary. F i n a l l y , t h e s e n s i t i v i t y of t h e o p t i u l design t o t h e choice of frequency window rill be studied.

This investigation is c a r r i e d out through tw, sowwhat overlap- p a , prbblem. F i r s t , t h e forced response of an i n i t i a l (i.e., non- o p t t i z e d ) design is corpar . t o t h e reaponre of a f i n a l design; c a r e r wisb and without aeradynamic damping a r e considered. Hext, t h e responre of i n i t i a l and f in81 derigns a r e w a l u t e d 8s a s i n g l e n a t u r a l frequency is v a r i e d ( t h e o t h e r s b e i n g h e l d f i x e d ) . I n each c a r e , a forcing function containing b r r o n i c r of t h e r o t o r speed is applied. Again, cases with and without a e r o d b i c damping wire conridered. The general finding 'WE these rtudiaa is t h a t frequency placclaant is a viable m a n s of reducing v i b r a t i o n , although it is by no means t h e only method and sbould be ured i n con juact ion with others.

a .2 ~sponsz vtwm FRIZQDL~~CY

The e q w t i o n r of, o o t i o a f o r t h e f i n i t u l e n m n t representation of a r r o t o r blade, subjected t o an e x t e r n a l e x c i t a t i o n , may be v r i t t e n i n

vbere [MI - mar8 rrtrix, { X ( t ) } = c o l u m r e c t o r of nodal d i r p l a c m n t , IcJ - d u p i a g uttix,

1x1 = r t i f f n u r matrix, and

{ P ( t ) l - forcing function c o l u m vector.

The forcing function may in t u r n be a p r e ~ r e d u (F(t)} = {v ) ti= , *ere w fortzing frequency, d V , , = forcing . r p l i t u d e .

After o o l c c a l c u l a t i o n , it can be rho- that t h e m l i t u d e of the

rerponre - w r i t t e n u { X I , independent of t - cur be given as

In thar r e c t i o n , t h e r e r p o n r e of b o t h t h e i n i t i a l and f i n a l (optimal) d3rignr t o an e x t e r n a l forcing function i r rtudied 88 the frequency of the forcing function ir varied. Blades both 4 t h and with- out aerodynamic damping are conridered. To formulate these problems, conrider t b e forced bet~avior of a rotor-blade. On17 t h e flapping res- p o w in oonaiderrd, Tho U p l a n e response is i n f e r r e d !ME the r e s u l t s

without damping, s i n a e t h e r e is l t t t l e aerodyn8mio damping in t h o in- plane d i r e a t i o n .

I 1 3 1 s h n n a p l o t , ~f t h e forcing amplitude Po [Ref.

251 urud Fig.

a t h e Study. GiVOn t h o f0XU- ~ l i t u d e , ~ d 0 u l a t O the -8pOM8

o f each node of t h o f i n i t n l e m n t representation o f t h o blade .s t h e value o f t h e f o r c i n g frequency, w. i a varied. The t i p i f ? ~ l i t e - u l e a s n t node f a r t h e s t f m m t h e hub) respotme i a o f special interest. Before t h e results o b t a l n d from t h i s s t u d y a m presented, it is u s a i u l t o aramiae t h e froquanay placement r e s u l t s which a m described in cha~ter 6 (808 Table 10). The r e s u l t s f o r the frequencies { i n units of cycles/rev) are, f o r flapping mode only, MODE XrJITIAL DESIGN FINAL DESIGN Blade dime?wions are given i n Table 9.

h ; The f r e q u e n c i e s in t h e above t a b l e cormspond t o the symwtric modes of a teetering r o t o r . Thus, only h.rw!kios of t h e r o t o r spoed have been aonsidered aa f o r c i n g frequencies. A s a r e s u l t , t h e optimized blade ( F i n a l design) f i n d s t h e third mode moved am7 from t h e c r i t i a a l 6.0/rev ( I h . o a 5.89 t o 5.67). S ~ a r l j , t h e movement of the seaond mode t o 3.09/ret r e m v o s it frm 2.3 and 4.OIrev. I n t h e comparison study t o follow, howaver, w e w i l l ~ r p p l y t h e e n t i n spectrum o f frequencies t o

thqs blade (not Just eves hmaonias) . Thus, t h e 'Final Design ' our no

f l o w e r be oonaidered optimum. A c ~ a r i s o n o f t h e two blades, however,

3.0 0 . : 0.2 0 . 3 c.4 0 . 5 0 . 6 0 . 7 3 . 8 0 . 9 1.3 U d r Statian : & ' I t ) F i g l ~ r r 13 Raaair Vari. t ron o f Forcinq Function

doas i n d l o a t e t h e ntrong effeet of lusoa8noe beaauae each aaae h r s a d i s t i n a t resonanas (6 .nd 31rev).

W e shall now aonsider t h e r e s u l t s of t h e present study. Fig. 14 8h- t h e t i p W8POM08 of both t h e initirl d e s u n and f i ~ l (i.~., o p t i m l z d ) deaign rs iunationa of t h a f o m i n g fmqueaoy. Aerodym~'~ damping h . s been neglaotsd (Alternatively, t h e m m r l t s cur be i n t e r p r e t - ed a s g i v i n g t h e inplane response). It aan be seen t h a t now 1.18 oyales/rev, t h e m s p o m e s of t h e twc designs are very slmllar. However, t h e responses corresponding t o t h e second and t h i r d modes d i f f e r s i g n i - fla a n t l y . For cbx.ngle,ln the second mode, t h e peak a f 3.22 o y c l e d r e v (initial design) mwes t o 3.09 oycleslrev (final design)/ Slmllarly, the peak of the t h i r d mode moves from 5.89 a y c l e a l m v t o 5.67 oycle/rev, whlilh is e s p e a i a l l y important sinoe it is higS1y d e s i r a b l e t o keep t h e irequenay am9 frm t h e i n t e g e r frequenay of 6 cycles/rev. W e conclude from t h a s e r e s u l t s that the frequenay plaaement approach does have a s i g n i f i o a n t effeet on t h e forowl t i p msponae when damping is not consi- dered .

Raxt, t h e o f f e a t of aerodynamio damping is oonsidered. The presence of t h e damping Implies t h a t r e s u l t s t o be presented aorn.~pond t o f l a p ping. h t h e ~ 8 t i a . l d e t a i l s of t h e damping forwtlation are a v a i l a b l e tRef.181. The e f f e c t of aerodynamla damping on reduoiag t h e resonant peaks of t h e t i p re8ponse of t h e initial blades is shown i n P i g . 15.

Fig, 16 ahow the dampod rosponaes of both t h e i n i t i a l and optimized bladea s o that t h e efieet of frequency placement can be studied. It is i n t e r s s t i t q t o obssme h e m t h a t when damping is included, gg apparent adv a n t a g e is g a i n a d by cptimizing the blade, a t l e a s t i n terms of reduaing t h e t i p mapotme, oxoept i n t h e range of 3-4/rev,in whioh a

0 . 0 1 * O 2 . 0 3 . 0 4 . 0 5 . V 6 . 0 Forcing Frequency ( No./Rev ) F i y u r e 1 4 I l p Response U e r s v s F o r c l n g Frequency f o r b o t n initial and Final Desaqns Witnout Dampins

5 . 0 0 . 0 1.6 2 . 0 3 . 0 4 . 0 a Forcing Frequency ( Nc. /Rev ) F i g u r e 15 Tip Response Versus Forcing Freq-!n=y for Iflltidl Design Both With and Without w i n g

8 . 0 1 . O , . O 3 . 0 4 . 0 S.0 - 6 . 0

Forcing Frequency (No. /Rev 1 Figure 16 Tip Respwe V e r w Forcing F r e q ~ ~ ~ ~ c y for Both Irutial and Final Design W i t h Danping

which a t h i r t y f i v e percent reduction occurs. However, we must a l s o txamina t h e e f f e c t of optimization vhen t h e response i s measured by t h e average shear f o r c e e x i s t i n g i n t h e blade.

C o n s e q u e n t l ~ , t h e sheariog f o r c e i n t h e blade is considered next.

A s a measure of t h e 8versge shear i n t h e r o t o r , w e consider t h e sum of t h e aquares of t h e shear f o r c e (abbreviated SSS) , I n t h i s s e c t i o n , Yi r e p r e s e n t s t h e shear f o r c e a t node i i n t h e (ten- element) f i n i t e - e l a n a n t m d e l . ! ! a t e t h a t t h e root shear i s n e c e s s a r i l y included a s one of t h e term8 on t h e right-hand s i d e of t h e equation, so that a l a r g e value of r o o t shear w i l l cause SSS t o a l s o be l a r g e .

P i g . 17 s h o w t h e variatian of SSS w i t h r c r p c c t t o t h e forcing frequency f o r t h e i n i t i a l design with and without aerodynamic damping.

Fig. 18 shows t h e s m q u a n t i t i e s f o r t h e f i n a l (optimized) design. Pig.

19 compares t h e q u a n t i t y SSS corresponding t o i n i t i a l and f i n a l designs when a e t o d p s a i c s is considered. Inspect ion of t h e r e f i g u r e s show t h a t , i n c o r r t r a s t t o b e b v i o r of t h e t i p response, t h e shear response is s i g n i f i e a n t l y a f f e c t e d b y c h a n g i n g b l a d e f r e q u e n c y , e v e n v h e n aerodynamic damping i n included. T h t 3 / r e v loads a r e increased by f i f t y percent due t o t h e povment of w2 from 3.22 t o 3.09/rw.

S i m i l a r l y , the 6 I r e v loads a r e reduced by seventy percent due t o t h e moveacnt of w3 from 5.89 t o 5.671rw. Thus, even v i t h damping, frequency placement i s a powerful d r i v e r of loads. It follow8 t h a t frequency placement can be j u r t i f i a b l y considered an important p a r t o f blade optimization.

0 . 0 1 . O 2 . 0 3 . 0 4 . 1.1 5 . i t 6 . 0

Forcing Frequency (No./Rev) F i g u r e 17 Sue 3f Squares of Shears Versus F o r c i n g F r e q u e n c y f o r i n i t i a l D e s i g n a o t n Y i i h ana Y i t n o u t D a m p i n g

FINflC O E S l G N ( U I T H f3ND W I I H O U T flERO. 1 Forcing Frequency ( f b ./Rev ) .A - .

t l 3 u r e 18 Sum o f S q u a r e s o f S h e a r s V e r s u s F o r c i n s F r e q u e n c y f o r F i n a l D e s i g n 3 a t n Xi:?. a n d W i t h o u t D a m p i n g

0 . 0 1 . O 2 . 0 3 . 0 4 . Q 5 . 0 6 0 f . O Forcing ~ r e ~ u e n c y ( No. /Rev ) F i q c c e 13 Sun o i Squares o t S h e a r s versus F o r c l n g F r e q u e n c y f c r 8 0 t h I n i t i a l and F r n a l Deslyns With Damping

8.3 RESPOUSE VEBSOS PUcEHERT I n t h e study just described, t h e responre of t h e blade t o changes is t h e forcing frequency was coasidered, llav we consider r d i f f e r e n t approach. In a f f e c t , ue exmine bar t h e blade tespondr t o A forcing fttnctiot 'during t h e optimization procedure' -- i n t h e sense that during o p t i a i u t ion, t h e optimizatiaa algoritkm vazier t h e n a t u r a l frequency of t h e blade ( t o f o r c e it t o s a t i s f y t h e frequency c o n s t r a i n t s ) . IE obtain- ing t h e r e s u l t 8 t o be presented n u t , w s i ~ ~ l a t e d t h e optimization procedure by vaqiag t h e m t o r a l frequency. Thus we can o b r e r ~ e what b p p e u r to the forced resDonse during frequency placement.

The f o r m l a t i o n of t h e approach is a8 follows. Through appropriate tr.nsfomatioms (described in Appendix 11.31, t h e system mass n u t r i x can be w r i t t e n 80 and t h e r y r t a r t i f f n e s r matrix a s

!KI - [nl IUI diagl (wiz) I tulT [HI

( 3 2 ) were wi a r e t h e n a t u r a l frequeacies of t h e system, [u] is a matrix whose

columm a r e eigcnoectorr , and t h e n o t a t i o n 'diag' indicates a diagonal

mstrix (a11 o f f d u g o m 1 terms v a n i r h ) . Prom e x a m i n a t i o n of t h e s e u p r e r r i o n s , it can be seen t h a t t h e s t i f f n e s s and MIS matrices caE oe conridered f u n c t i o n r of t h e n a t u r a l f r e q u e n c i e s . Thus i c becomes p o r s i b l e t o f i x a l l frequencies but one, and then study the response of t h e rystem a r that one frequency is varied with mode shapes a l s o held fixed. I n p a r t i c u l a r , t h e rerponre t o the following forcing function w i l l be studied:

r - the rotor r p d , and

where {Yo] was defined p m v i o u s l y i n Fig. 13. S i n c e t h e arguments o f t h e e x p o n e n t i a l s are i n t e g e r m u l t i p l e s of w, resonance w i l l occur a t har- monics o f t h e r o t o r speed. The p a r t i c u l a r f o r c i n g ~ n c t i o r g i v e n above is known from e m p i r i c a l o b s e r v a t i o n t o p r o v i d e a n a p p r o x i m a t e , b u t p h y s i c a l l y realistic r e p r e s e n t a t i o n o f t h e r a d i a l qnd harmonic v a r i a t i o n o f t h e amplitud.3 o f t h e load on a real blade. A s mentioned i n t h i s s e c t i o n , t h e b l a d e response w i l l be d e f i n e d by t h e t i p displacement and t h e sum o f t h e scpares o f t h e s h e a r s (ucept t h a t , h e r e , t h e n = 1 term h a s been omitted from t h e e x p r e s s l a n s f o r c a l c u l a t i n g t i p displacement and s h e a r s because t h i s term r e p r i s e n t s p, tip-path plane tilt t h a t is c o n t r o l l e d by t h e p i l o t Or:;. trimming purposes. It is n o t p a r t of t h e true v i b r a t o r y l o a d s w e are considering.

R e s u l t s f o r t h e pro3lem j u s t formulated are shown i n Fig. 20, where t h e sum o f f h e .squares o f t h e s h e a r s is p l o t t e d a s a f u n c t i o n of w2, t h e s e c o n d n a t u r a l f r e q u e n c y , with t h e o t h e r n a t s r a l f r e q u e n c i e s being f i x e d . This figure corresponds t o t h e i n i t i a l b l a d e d e s i g n (blade dimen- s i o n s a r e g i v e n i n Table 9. F i g . 21. ~ h o w s t h e same q u a n t i t y f o r t h z case where t h e t h i r d n a t u r a l frequrncy is v a r i e d . It is i n t e r e s t i n g t o n o t e t h a t t h e r e s p o n s e curve '.or t h e dampad case i n Fig. 21 l a c k s resonant peaks - a p p a r e n t l y '.he damped r e s p o n s e i s s o c o m p l e t e l y dominated by t h e resonance o f t h e second n a t u r a l frequency, which is f i x e d near 3/rev, t h a t t h e (damped) resonant peaks f o r t h e t h i r d fre- quency a r e n e g l i g i b l e by c o m p a r i s o n . It s h o u l d b e n o t e d t h a t t h e

AND UlTClOUT AERO. J Yithout Aerodynamic I I

LO. . I

I I With Aerodynamic I I I I I I W 2 (No./Rev) W1 and W 3 am fixed.

F i g u r e 20 Sun of Squares of Snears Versus Second Natur.31 F r e q u e n c y f o r I n i t i a l Desi3n both With and W i t h o l ~ t n b r p r r ~ g

W 3 (No. /Rev) W 1 and W2 ar*e fixed.

F i g u r e 21 S I J ~ of S q u a r e s of S h e a r s Vers.rs T h i r d N a t u r a l F r e q u e n c y f o r I n i t i a l Design b o t h W i t h ~ n d kithout D a m p i n y

response of t h e o r i g i n a l d e s i g n is reapresented by only one p o i n t on F i g u r e 20 o r F i g u r e 21. This v a l u e can b e found by t a k i n g w, = 3.22 o r w3 = 5.89 on those f ~ u r e s , s i n c e t h e s e am t h e i n i t i a l d e s i g n v a l u e s (Table 1 0 ) .

For t h e f i n a l (optimal) design, t h e analogous q u a n t i t i e s are p l o t - t e d i n F l g s . 22 and 23. Again, no r e s o n a n t peaks are p r e s e n t i n t h e damped response when t h e t h i r d n a t u r a l frequency is v a r i e d . Compnrison of magnitudes of o r d i n a t e s i n Figs. 20 and 22 (no damping) shows t h a t t h e o v e r a l l s h e a r measure is reduced i n t h e f i n a l d e s i g n i n t h e r e g i o n s away from resonance. Also, t h e c h o i c e of scale OP t h e v e r t i c a l a x i s i n F i g . 22 h i g h l i g h t s t h e effect o f frequency placement. Note t h a t by i n s p e c t i o n o f F l g s . 20-23. a d e s i g n e r may s e l e c t t h e d e s i g n frequency which minimizes t h e average s h e a r as measured by t h e SSS.

One o f t h e most i n t e r e s t i n g r e s u l t s o f Fig. 22 is information about t h e width of v a l l e y s and peaks, s i n c e t h i s g i v e s d e s i g n informatioa. # F i r s t , l e t u s examine t h e nondamping curve ( i n p l a n e response). Here, t h e minimum p o i n t s are n e a r l y a t t h e c e n t e r s o f t h e r e g i o n s (2.551rev)

and (3. S r / r e v ) . The frequency windows t o maintain no more than t h i r t y

t,

p e m e n t i n c r e % + e i n l o a d s a r e 2.48 - 2.701rev and 3.40 - 3.701rev

( p l u s

o r minus 0 . l S l r e v ) - a f a i r l y narrow window. For t h e damped c u r v e s

( f l a p p i n g response), mimima a r e a l s o n e a r t h e one-half p o i n t s , b u t t h e window f o r t h i r t y - p e r c e n t i n c r e a s e s a r e much wiuer: 2.20 t o 2.90lrev and 3.20 t o 3 . 8 0 1 ~ s ~ ( p l u s o r minus 0.30Irev). S t a t e d a n o t h e r way, i n p l a n e f r e q u e n c i e s should be no c l o s e r t h a c a 0.4lrev from i n t e g e r s , b u t f l a p - ping f r e q u e n c i e s may be as c l o s e as 0.2 from an i n t e g e r . It should be amyhasized t h a t t h e s e o b s e r v a t i o n s apply t o t h i s p a r t i c u l a r example and z,ay n o t be g e n e r a l i z e d for o t h e r frequency c o n s t r a i n t s .

c. U2 (NO./R~V) W l and W 3 are f i x e d .

F i g l ~ r t 22 Sum of S q u a r e s o f S n c a r s V e r s u s S e c o n d n a t u r a l F r e q u e n c y f c r F i n a l D e s i g n b o t h U i t n and Y i t h o u t D a a ~ i n g I

I I Without Aerodynamic I I I I I I I I I 2 0 . 0 , I I I With Aerodynamic I I I

- -

W3 (No./Rev) W 1 and W2 are fixed.

- .

F i g u r e 2 3 S u m s f S q u a r e s o f S h e a r s t ' e r s u s . n F r a S d t u r a l i o r F i n ~ i 3 e s i ~ n B o r h X i t h 3 n d '.;;:nca: Z l . : ~ p ; z

Another conoluaion t o be drawn from t h e above r e a u l t s is t h a t t h e undamped responts curve has very flat-bottomed l v a l l e y ' when one of t h e fixed frequoncies is near an i n t e g e r vrclue (of. Flga. 22 and 23).

8.4 RESPONSE DUE TO EVEN-INTEGER H W N I C S I n t h i s s e c t i o n , we still study t h e response of t h e blade due t o t h e change of f o m i n g h.equenay. However, t h e f c m i n g frequemp x i 1 1 be a l i t t l e d i f f e r e n t from t h e one i n s e c t i o n 8.3.1, Fe w i l l consider, only t h e even i n t e g e r foroing frequenaies f o r whioh t h e c o l l e c t i v e ADC ' . ard optimized. Thua, t h e forcing f i n o t i o n may be w r i t t e n as i2wt i % w t + v4e + v6e i6wr + v8e [ F ( t ) ) = V2e where w = r o t o r speed Vn = l l n Vo Since t h e argursrants of t h e exponentlala a r e even i n t e g e r multiples of w only, t h e r e s o n a n c e w i l l e x p e c t e d l y o o c u r a t a n e v e n i n t e g e r o f harmonics of t h e r o t o r speed. The response of shear stress w i l l be studied i n t h i s section.

Results f o r t h i s case a r e s h m i n Flg 24, where t h e sum of t h e squares of t h e shear harmonics is p l o t t e d aa a f i n c t i o n of w2 (wl m d w3 a r e fixed; wl = 1.18 and wg = 5.89). The f i g u r e corresponds t o t h e i n i t i a l blade design. Resonance peaks oocur a t 2 and 41rev. Thus, f o r t h e shear response, a second f r e q u e n c i e s around 3.0 would b e t h e s u i t a b l e choioe f o r w2 as design frequncy. Pig 25 is a s i m i l a r comparison f o r a v a r i a b l e value of w3 ( wl = 1.18 and w2 = 3.22 a r e

flxed) . Here* t h e rssomnoe is a t 6.01rev, lad w3 = 5 o r w3 = 7 would be

ideal.

\ !

W i t h o u t Aerodvr-amic I I

I

i . I

\

I I n i t i a l Design

I With A e r o d v n a r n i c , * , ?

I \ \ \ \ \

/

i

/ 2.5 lb2 / A'' I U i ( N O / R N ) U l ( t l . l & J A N D W 3 i S a 9 9 ) AUE F I X E D F i g u r e 2 4 Sum of Squares of Sheprs Versus Second Natural Frequent f c r I n i t i a l Design Both With &id Without Daaping (where Forcing Function is Even Insegsr Multiple)

[ N I T l A L D E S I G N ( U I T H NU W I T H O U T C I E P O O Y r u A M I C S )

- - - - - - - - - - - - --7 ---- ------------ ----

3,- I I I I

I

I I I I I

I I I

I i I

, I !

i I I 0 ..

' J i t h o u t X e r o d v n a m i c I I - : I I I I I

i I

W i t h Aetodyaacic I !

I I

I ----.._ -.---

I I

k ~ n i t i a l Design

F i g u r e i 5 Sum of Squares of Shears Versus Third Natural Frequency

f o r Initial Des' -. Both with and without Damping

(where Forcing . -.fiction is Even Integer Multiple)

ideal.

Results are shorn in Pig 26 and 27 for the final desian blade.

Again, it i s noted that the resonance peaks appear at even integers. A carparison of Fig 24 and 25 with Fig 26 and 27 show the relatively lover vibration8 of the f i n a l derign. In either case, hovever, we see the senritivity of vibrations to frequency, even v i t h aerodynamics.

S I J ~ of Squares o l Snears Versus Second N3tl.lrii Frquency Flyvre ?,h ' Tar Frnal Desrgr~ botn wit,n .and Yttnol~t b a m o r n g (unere Forcing F~~rrctlon 1s Even Integer n u i t i p ~ e l

F i g u r e 27 Sum o f S q u a r e s of S h e a r s V e r s u s T h i r d N a t u r a l F r q u e n c y for F l r 1 3 1 Design b o t h U i t h and Without D i n p i n 3 cuhors F o r c t n q F u n c t i o n 1 s Even lnteger n u l t i p l e )

9 . SUMMARY AND CONCLUSION The o p t i m i z a t i o r technique works very s u c c e s s f u l l y on t h e d e s i g n of r o t o r b l a d e s even when t h e r e a r e a s many a s 55 c o n s t r a i n t s . The most e f f i c i e n t optimiz?.tion orocedure i n v o l v e s 2 s t e p s . I n t h e first s t e p , t h e o b j e c t i v e f u n c t i o n i s based on frequeccy placemect with a p p r o p r i a t e 3tructura.L c o n s t - a i n t s . I n t h e s e o n d s t e p , t h e o b j e c t i v e Function is weight with frequency wis~dows a s c o n s t r a i n t s .

A r fi%r a s t h e o p t i m i z a t i o n of h e l i c o p t e r b l a d e s is concerned, t h e a p p r o p r i a t e c o n s t r a i n t s i n c l u d e a u t o r o t a t i o n a l i n e r t i a , a x f a l stress, geometric l i m i t a t i o n s o f t h e cross-section, and t h e placement of mass c e n t e r f o r w a d o f t h e q u a r t e r chard.

P r o p e r c h o i c e o f i n p u t d e 2 a c a n e n s u r e t h e optimization r u n s smoothly and converges f a s t e r . Although w e have up t o 15 f u l l con- s t r a i n t s and 40 s i d e c o n s t r a i n t s a t one time, t h e program works very well. The reason may b e d u e t o t h e f a c t t h a t t h e i n p u t d a t a a r e p r a c t i c a l encugh t o meet ( o r t o be c l o s e t o ) most of t h e c o n s t r a i n t s evep b e f o r e t h e o p t i m i z a t i o n s t a r t s . However, i f w e s t a r t t h e optimiza- t i o n with random i n p u t d a t a , t h e r e s u l t s may n o t be a s good as expected.

The forced response o f t h e b l a d e can be adequately c o n t r o l l e d , as expected, by t h e approach o f 'frequency placement'.

The o p t b i z a t i o n t a c h n i q c ~ s r e s u l t 3 i n r e a l i s t i c d e s i g n s by place- ment mass a t a n t i n o d e s o r nodeq, by adding s t i f f n e s s a t a n t i n o d e s o s nodes, and by placing mass near tne t i p t o a c h i e v e a u t o r o t a t i o n a l i n e r t i a a t minimum weight.

10, A C K N O W L E D G E M E N T S The author would U.ke t o take t h i s opportunity t o express his gratitude t o D r David A Peters f o r h i s guidance and ericcuragement throughout t h i s work. A s p e c i a l acknowledgement is owed t o Dr. Mark Rassow and Dr. Alfred Rorn for t h e i r discussion. This research was supported by N A S A Grant # NAG 1-250,

11. A P P E N D I C E S

s i z e of t i p ausr area of croas-section of box beam width of box b e m blade chord dasuping matrix width of lumped mars wall thickness of box beam weighting function flapping s t i f f n e s s inp lane s t i f fnes a s a f e t y f a c t o r f i n o i design f o r t ing f unc t ion g r a v i t y t o r s i o n a l s t i f f n e s s height af box bean; i n i t i a l d e s i ~ n area ~ n t of i n e r t i a of box beam cotst=: area moment of i n e r t i a of blade maas momcnt of i n e r t i a of box beam constant 1388s moment of i n e r t i a of bLade s t i f f n e a r matrix area moment of i n e r t i a mass momcnt of i n e r t i a

l e n g t h of an element mass matrix b l a d e f i r s t t h r e e f r e q u e n c i e s ( n o l r e v ) d i s t a n c e from r o t a t i o n a x i s l e n g t h of t h e b l a d e box beam t h i c k n e s s SSS sum of s q u a r e s of s h e a r f c r c e box beam t h i c k n e s s t e n s i o n f o r c e ' k i n e t i c energy t o t a l - energy displacement of d e g r e e of freedom of element u n i t m a t r i x f o r c i n g amplitude displacement i n yx plane' displacement i n u p l a n e b l a d e f r e q u e n c i e s weighting f a c t o r c o o r d i n a t e axis and l e n g t h parameter b l a d e t w i s t p r e t v i s t a n q l e mass d e n s i t y of t h e b l a d e sma 11 parameter

nt

r o t a t i n g speed maximum s t r e - ?

mass of s k i n

density o f lumped m a s s , box beam, honeycomb

C a l c u l a t i o n 0% Torsional S t i f f n e s s , G J The blade c r o s s s e c t i o n is shown i n Fig. 1 and is i d e a l i z e d i n t o a two c e l l t o r s i o n ~ Q X i n Fig.28. Although t h e bending and t o r s i o n a l i n e r t i a inolude t h e contribcltion o f a l l masses ( i n c l u d i n g t h e f i l l e r elements), t h e t o r s i o n a l s t i f f n e s s is based only on t h e s t r u c t u r a l bcx and thin-skin elements. The torasional s t i f f n e s s , G J , is based on c l a s s i - c a l thin-walled closed s e c t i o n theory.(Ref.26) The e f f e c t s o f warping r e and d i s t o r t i o n o f t h e c r o s s s e c t i o n are neglected. The e q u a t i o n s are b r i e f l y summarized as follows: Considered c e l l i, having an enclosed area o f Ai, and thickneaa t.

The l e n g t h along t h e circumference is maasrrred by c, and t h e s h e a r s t r e s s is denoted by s..l%e t o r s i o n c o n s t a n t is where d c is t h e l i n e i n t ~ g r a l along t h e e n t i r e c l o s e d box, and T r e p r e s e n t s t h e t o t a l torque applied t o t h e e n t i y e system. W e a l s o have

vhere c is t h e c o n r t a n t shear flow i n t h e i-th box. Denoting t h e s i n g l e adajcent box a s k , t h e r e w i l l be a ctnmmn w a l l , ik, betweeu t h e two boxer. The sheor f Low i n t h e c - n -11 w i l l be t h e d i f f e r e n c e of sheor f l o w qi and qk. Thus, f o r t h e i-th box, and f o r t h e k-th box Defining q = Q . ( Z ? / J ) and n = I ( d c / t ) , we can reform j 3 j m jm Eqn.3 t o read a. .Q.

11 1 - 'i

T A + t4T/J)(QiAi + Qkbr) 1 7 .

Thus, t h e t o r s i o n a l c o n s t a n t , J , is given by 3 p e c i f i c a l l y , f o r t h e tuo c e l l box shovn i n Fig.28, c a l bcz i t h e b o x b e a m a n d box 2 t h e e q u i v a l e n t t r a i l i n g e d g e . D e n o t e t h e s i r t u m f e r m e e o t box 2 by C. The a r e a s , A1 and At are given.

Then, from the other dimenrioar,

Then, solving Eqr . 4 . . a d 4b giver

Thar, a l l qaurtitier in Eq.6 are known and the torsional constant, J , c.n b e ev8luated.

*

-

h * Ce! I I : Constant*thicknesses for sides of box beam represent weighted effects of variable t t lckness elements.

Cell 2: This cell represent3 canfiguration cf skin and tisiling edge.

F i g . 28 I d e a l i r s d tvo-cell model for calculation oi torsional s : i f f x e s o

3erivation of Wasr and S t i f f i e s s Watricer a . Functionr of Natural Frequencies Define IK*] = I M I - ~ / ~ I K I I H I - ~ ~ ~ .

* *

and constmct a rquare matrix [U 1 by using the e i g e n ~ e c t o r s of [R 1 as columa. I f the eigenvectora rre n o m l i z e d to the identity matrir, that is, i f

r$Pr~*i = r11, ( 4 7 )

i, then fo'.lorr that tdiT[t~[$l - diag i (w. 1 I I , where r.' are the eigcnvalues of K I .

Next, Pet from which it f o l l u w ehat -112 forT - [a*lT [MI .

u - [ ~ * l * [ . l ' f Z .

~ u l T ~ w l ~ c l - [I?.

F i n a l l y , then, t h e e t i f f n e s s and m s e matrices can be w r i t t e n as

functions of t h e eiganvalues, r '-

1 -

and Note that t h e e i g e n m t o r s . [a], and eigenvaloes, viZ, appearing on t h e right-hand a i d e v e r e o r i g i n a l l y c a l c u l a t e d from t h e s t i f f n e s s and nus# matrices, [K] and [MI. If ws connider o n l y r e l a t i v e l y small changes i n t h e frequencies, w. then t h e e i g m v e c t o r s should r e l a t i v e l y

1 '

unchanged.

Thur t h e l a s t two equations f o r EM1 an6 [K! with [u] h e l d fired can be considered a r erpredsing t h e m a s and s t i f f n e s s matrices aa u p l i c it functions of t h e n a t n r a l frequencies .

!Viebank, C . and Girvlh, W., 'Sikorskp S-76 Analps i s , D t s ign and

Developwnt f o r Successful D m i c Ch.racterit@tics,' i?roceedings gf t h e 34th Annual R a t u r a l Forum o f t h e hrrarican H e l i c o p t e r Society, ?I.y 1978, yp 78-23-1 through 78-23-17.

Hirsh, Iiarold, Dutton, l o b c r t E o s and Rasueoff, Abner, 'Effect of Spanwiae and Chordvise ! ! a s s D i s t r i b u t i o n on Rotor Blade Cyclic S t r e s s e r ,'Journal of t h e A m r i c m E e l i c o p t e r S o c i e t y , Vol. 1 , No.

2, A p r i l 1956.

U i l l e r , Bane 8. and E l l i s , Charles W . , 'Helicopter Blade Vibration and F l u t t e r , ' J o u n u l of t h e American H e l i c o p t e r S c c i e t y , Vol. A , !lo. 3 , J u l y 1556.

Daugbday, H., Duifaidt, F., and Gates, C . , ' I s v e r t i g a t i o n of Heli- c o p t e r Blade F l u t t e r and Load Amplification Probl.ema,' J o u r n a l o f t h e American H e l i c o p t e r S o c i e t y , Pol. 2, Ao. 3 , J u l y 1957.

C c r s t m b e r g e r , e t a l , , 'The Rotary Baunc! Table: How can H e l i c o p t e r Vibration be Minimized?@ Jcrurnal of American E c l i c ~ ~ ~ t e r S3c i e t y , Vol. 2 , H0.3, J u l y 1957.

A l u , c t a1. 'Pioueers Panel,' AHS 3 5 t h Anniversary Forum and Technology Display, Washington, DC, Hay 21-23, 1979.

E l l i s , C.W. c t a l . , 'Design, Developlmcnt, and Testing of t h e Boein ~ertol/Arm.jr 'rEE-6lA,' Proceedings of t h e 32nd Auauai National FON t h e hmzricur H e l i c o p t e r S o c i e t y , P r e p r i n c 1310, Hay 1976.

Fenanght y , Ronald B . and Noehrea, William L., 'Coapos i t e Bearing-

. l e s s T a i l o r Rotor f o r VTTAS,' J o u r n a l of t h e American H e l i c o p t e r S o c i e t y , Vol. 22, No. 3 , J u l y 1977, pp 19-26.

9. Banson, H.W. andCalapodas, M . J . , ' E v a l u a t i o n o f t h e p r a c t i c a l Aspects of Vibration Reduct ion Using S t r u c t u r a l 3pt imizat ion Tech- niques,' Proceediags of t h e 35th Annual National Forum o f t h e ApCricpt Helicopter Society, Kay 1979, pp 79-21 th.xmgt! 79-21-12.

10. B i e l m , Richard L., *Techniques f o r S t a b i l i t y Analysis and Design O p t i m i u t i o ~ v i t h Dynamic C o n s t r a i n t s of Nonconservative Linear

S y s t e m ,' A I M I A S ~ 12th S t r u c t u r e s , S t r u c t u r a l Dynamics, and

Materials Conference, Anaheim, C a l i f o r n i a , April 19-21, 1071. A I A A Paper b. 1 1-388.

11. Priedmann, P e r c t z , 'Response Studies of Rotors and Rotor Blades v i t h Application t o h e r o e l a s t i c Tailoring,' Semi-Annual Progregs Report on Grant PSG-1578, December 1981. ( h l s o unplubished U.S.C.

Doctoral Thesis, December 1982) .

12. Talyot, Robert B . , 'Helicopter Vibration Reduction by Emtor Blade Modal Shaping,' Proceedings of t h e 38th Annual Forum of t h e Ameri- can Helicopter Society, Anaheim, CA, Hay 1982.

13. Johnson, R. J., 'Disjoint Design Spaces i n t h e Optimization of Harmonically Excited S t r u c t u r e s , ' A I A A Journal, Vol. 14, No. 2 , February 1975, pp 259-261.

14. l i o r d s o n , 1 and Pederson, P., ' a Qcviev o f Optimal S t r u c t u r a l Design,' Proceedings of t h e 1 3 t h International Cangtess of Theore- t i c a l and Applied Mechanics, Springer-Verlag , Moscow (1973) .

1 5 Venkayya, V.B., ' S t r u c t u r a l Optimization: A Review and Soee Recout- mendations,' Int. J. f o r 8um. Merh. Engrg., Vol. 13, l o - 8 , pp 203- 228.

16. SoMpit, L.A., ' S t r u c t u r a l Design by Systematic Synthesis,' Proc,

-

2nd Nat. Conf. E l m t r o n i c Computation, AJCE, 105-132 (1960).

17. Fox. R.L., Optiacization Methods f o r Engineering Design, Addison- Wesley, Read'ng, P A 1971.

18. KO, Timothy, Use of Tapered, Twisted F i n i t e Elements, H.S. Thesis, Washington University, 1979.

19. Olhoff, Niels, 'Optimization of Vibration Beam with respec? t o

Hfgher Crder Natural Frequencies, ' J. S t r u c t . Mech. r!l), 87-122

(1976).

20. Turner, H.J., 'Desun of Minimum Hass S t r u c t u r e s with Specified Natural Fraquencies.' A I A A J. 5 ( 2 ) , 496-412 (1967;.

21. Khan, H.R. and Wiilmer, K , D . , 'An E f f i c i e n t Optimalit; C r i t e r i o n Method f o r Natural Frequency Constrained S t r u c t u r e s , ' Computers S t r u c t u r e s , Vol. 14, No. S 6 , pp 501-507, 1981.

22. Donham, Robert E . and Schmidt, Jaap, '100-W Hingeleas Metal Wind Turbine Blade: Design, Analysis, and F a b r i c a t i o n , ' P r e p r i n t No. S- Proceedings of t h e 31th Annual National Forum of t h e American 998, Helicopter Society, May 13-15, 1975.

23, Robinson, e t al., 'Design Variables f o r a Controllable Twist Rotor,' P r e p r i n t no. 914, Proceedings of t h e 31th Annual National F o r m of t h e American Helicopter Society, May 13-15, 1975.

P r i v a t e information from a Helicopter Company.

24.

P r i v a t e information from a Helicopter Company.

25.

26. Kollbrunner, C.F. and Basler,K.,Torsion i n Structures; an Engineering Approach. S p r i n g e r 7 e r l a g . Berlin, 1969.

27. Vanderplaats, G . N . ' CONMIN - A FORTRAN PROGRAM FOR C O N S T R A I N E D

F U N C T I O N MINIMIZATION'., A m e s Research Center and U.S. A r m y Air Mobility R D Laboratory, Koffett Field, Calif. 94035

. ~J

-ll.2- 13. VITA Biographical it ... on the author of the the•i•, Mr Timothy W H Ko 1) Born in .

2) Attended Rational Cheng tung On.iver1ity in Taiwan., It 0 C from 1973 to 1978. Received Bachelor of Science Degree in Mechanical Engineering in May 1977.

3) Attended Waahi.Jlgtou University in St. Loui1, Mis1ouri fro• Auguat, 1978 to preaent. lecieved Kaater of Science Degree in Mecbanical Engineering in May 1980. Working on nocotor of Science Degree in Mechaiucal Engineering. Awarded a Graduate TraiDH•hip aAd l.eaearcb AHiatautlhip fr01ll September to May 1984.

4) Member of American Helicopter Society.

December ' 19 84

Short T i t l e : Optirman Design of Rots; Blades , KO, D.Sc. 1984