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 

--- - 

David A. Peters, Professor and Chairnan 

and 

Timothy KO, Research Assistant 

Department of Mechanical Eragineerinq 

Washinaton University, Camy.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 

flASA-CB- 176076) OESICN OP BELICOPTSB BCXCR Y85-3 1044 

BLADES FOB OPT1 80 C CY bAflIC ILE16dCTEBISTICS 

Final i??~Ort, 17 Jan. IS82 - 17 Jan, 1985 

(gaskington Univ,) 116 p EC AC6/flP 801 Unc1.a~ 

CSCL OlC 63/05 15'121 
TABLE OF CONTENTS 

Pagg 

1. Introduction .................................................. 1 

2. Importance of the Research .e................................ .1 

3. Summary of the Project ........................................ 4 

3.1 Optimizati~n Problems Which Were Solved ................. 4 

3.2 Findings Related to Optilaization of Rotor Blades.....,.l6 

4. References.......................=......,............,.......21 

- 9 Appendix 1: Thesis of Timcthy KO ............................... LL 

'3 i !

I.

-. I- -' 

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i 

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i 

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i 
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 forcing 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 RPM may vary 

over a significant range throughout the flight envelope, thus reducing even further the area of acceptable natural frequencies. 
Third, the natural modes of the rotor blade are often coupled 

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

center and elastic axis, and large aerodynamic damping, These 

couplings complicate the calculation of natural frequencies. In 

fact, the dependence on pitch angle makes frequencies a function 

of loading condition, since loading affects collective pitch. 

Fourth, the 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- 3 

t; 

ment of the XH-17 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 helicopter 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 I\ ! both main rotors and tail rotors are now being fabricated from . . 

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

! designer can choose, with limited restrictions, the exact EI 

,e, 

,, distribution desired. Furthermore, the lightness of composite 

blades for the main rotor usually necessitates the additi-on of 

i 

weight to give sufficient autorotational blade inertia. Thus, 

-i 

. i tbsre is a considerable amount of flexibility as to how this ~. 

-i weight may be distributed. Second, the methods of structural 

g. 

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

. . .1 .-.I 

'2 

.. , -! point where they can be efficiently applied to the blade struc- 

..? 

L-. 

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

.. . . . 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, 

a1 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 constraints 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 maintaining the first natural frequency within a "window" (e.g., 2.4 < pl < 3.0 per rev). 

All optimization problems were solved with the CONMIN computer program [6]. CONMIN is based on the mathematical nonlinearprogramming method of feasible directions. 

The list of problems solved follows. 

-- Case 1. Cantilever beam 

Rotating: no 

Objective function: weight 
! ~esign 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 

-----.------------------------

-- Case 2. Cantilever beam with tip mass 

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 

------.-------------------------

-- Case 3. Wind-turbine blade 

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 

i 

\ Stress constraint: no I 

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

................................................................. 
-- Case 4. Hingeless rotor-blade 

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 tw~ frequency constraints 

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--pp---pppp- ~ ppp--p--p--p-p-p-p--ppp ~ -pp- - 

-- Case 6. Cantilever beam (similar to Case 5, except for three 

rather than two frequency constraints) 

Rotating: no 

Objective function: weight 

Design variables: area moments of inertia 

Bolndary condition (s) at root: fixed 

1 

Frequency Constraints: windows on first, second and third flapping I 

Autorotation constraint: no 
Stress constraint: no 

References: Second Semi-Annual Report pp.17-18 

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

-- Case 7. Cantilever beam (similar to Case 6, except for addition 

of lumped weights as 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 Semi-~nnual Report pp.17-21 

- - - - 

-- Case 8. Cantilever beam (similar t o C a s e 7, except for addition 

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 whole beam) 

Stress constraint: no 

References: Second Semi-Annual Report pp.17-22 

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

-- Case 9. Cantilever beam (similar to Case 5, except beam is 

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 

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: fixed I 

> 

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 

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 cyclic 

flapping 

Autorotation constraint: yes 

Stress constraint: no 

References: Third Semi-Annual Report pp.7, Thesis, pp. 50-51 

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

-- Case 12. Teetering rotor (similar to Cases 10 and 11, except that 

both cyclic and collective 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 ,

I conditions, another anaiysis perfomed with fixed conditions 

1 

\ Frequency Constraints: windows on first, second and third 

coll2ctive flapping and also on first, second and third 

cyclic flapping 

Autorotation constraint: yes 

Stress constraint: no 

References: Third Semi-Annual Report pp.7-9, Thesis, pp. 50-54 

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

Case -- 13. Teetering rotor (similar to Cases 10-12, except that 

col1ective and cyclic flapping and inplane and also 

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 

-------------..------------------------------------------------- k 

Case -- 14. T~etering rotor (similar to Case 13, except that box- . beam dimensicns are fixed) Y, 

Rotating: yes 

Objective function: weighted sum of squares of dirferences in 

frequencies 

Design variabies: lumped weights, stiffness of torsio~sl spring 

at root 

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

performed with pinned conditions, another ana~ysis perfomed 
with fixed conditions; b) inplane -- one analysis performed 

with pinned conditions, another analysis perfomed 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; and window on first 

torsional 

Autoratation constraint: yes 

Stress constraint: yes 

References: Third Semi-Annual Report pp.13-14, Thesis, pp. 60-61 

-- Case 15. Teetering rotor (similar to Case 14, except that 

I stiffness of blade cross-section at root is a design 

-1 

..I variable) 

Rotating: yes 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, all 

i other dimensions of the box cross-section are fixed) 

i Boun~ary condition(s) at root: a) flapping -- one analysis ! 

performed with pinned conditions, another analysis perfomed 

with fixed c~nditions; b) inplane -- one analysis performed 

with pinned conditions, another analysis perforned with fixed 

conditions; c) torsion -- fixed conditions 

I Frequency Constraints: windows on first, second and third - collective and cyclic flapping; windows on fjrst, second and I 
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 

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 

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 L i collective and cyclic flapping; windows on first, second and 

. I i third collective and cyclic inplane; window on first 

I 

torsional 

1 

-! Autorotation constraint: yes s 

I Stress constraint: yes 

i 

! 

I References: Thesis, p. 61, 64 

% I 

4 -- Case 18. Articulated rotor 

I Rotating: yes 

i Pretwisted Blade: yes 

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. . k 1 

Design variables: flange thicknesses, wall thicknesses of 4 
both sides of box cross-section, lumped weights, stiffness 

of torsional spring at root 

Boundary condition(s) at root: a) flapping -- ~inned; b) inplane 

-- pinned at same radial location as in the case 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 boxbeam 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 

, 1 

-- pinned at same radial locaclon as in the case of i 

i 

flapping; c) torsion -- fixed conditions i 

! 

Frequency Constraints: windows on first, second and third 

flapping; windows on first, second and third inplane; and 

;Y /... window on first torsional k 

Wi 

. 14 *$- 

. . F Q a ,q a w. r. ,i. Om . 
~utorotation constraint: yes 

Stress constraint: yes 

References: Fifth Semi-Annual Status Report, pp. 9-10 

Thesis, pp. 67-70 

Case 20. Articulated 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 

Objective function: initially the weighted sum of squares 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; / 

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

? 

from root ': 

Frequency constraints: windows on all first, second and third frequencies (flapping, inplane, torsion) ! 

Autorotation constraint: yes i 

Stress constraint: yes 

References: Sixth Semi-Annual Status Report, pp. 3-4 f 

3 

Case 21. Articulated rotor (similar to case 20 except that box beam 

dimensions are also design variables) 

Rotating: yes 

Pretwisted Blade: yes 

- 
Objective function: initially the weighted sum of squares of differences in frequencies; after a feasible design is found, the I 

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 a1 1 first, second and third frequencies (flapping, inplane, torsion) 

Auto rotation constraint; yes 

Stress constraint: yes t 

ReFcrences: Sixth Semi-Annual Status Reports, p. 5 

. .: . 

'i

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. 4 

F 

23 $ 

In addition to these general findings, the project estabi . ~ 

1 

.- 

I . 16 i 

I *- 
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 calculated by analytical formulas rather than by finite 

differences. However, finite differences serve as a 

useful check on the possibility of errors in the computer 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 frequency). If eigenvalues are used, then scaling should be 

employed in the constraint equations to ensure we1 lbehaved gradients for use in CONMIN. 

Reference: Second Semi-Annua 1 Status Report, pp. 10-12; 

Thesis, pp. 25 

3. The £01 lowing values of CONMIN parameters were adequate 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 

~hesis, 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 optimization studies, since errors in eigenvalue calculations 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 greatly 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 frequency minus desired value) may be used initially. In 

the process of minimizing this objective functio~, 

CONMIN is often able to find a design which satisfies 

the frequency constraints. If this occurs, the objective 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, 58 

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. 1 

13. Fini te-element model ling errors caused by neglecting Fe7: 

.',+ i .. - 

secondary structural items such as shear deformation, " p 

! . - ',. 

'. +. ! 

restraint of warping during twist, and filler stiffness y.. .T* . r 

are small. However, accurate f i 1 ler properties, dimensions 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 computational burden in rotor-blade optimization, an efficient 

eigenvalue routine should be used. For example, determ1 .:i 

1: 

28 I 
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 11 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 TMX62.282, August, 1973. 
APPENDIX 1 

Thesis of Timothy KO 
SEVER INSTITUTE OF TECHNOLOGY 

Doctor of Science Degree 

WGR 854, 853 

3ISSEKTATION ACCEPTANCE 

(To be submitted by the graduation approval deadiinej 

STUDENT'S NAME: Timothy Wai Hung KO 

E.R.S. CODE: 

This atudert 'a dissertation, entitled "Design of He1 icopter Rotor 

81 ades for Optinum Dynamic Characteristics" 

has beem examined by the undersigned committee of three faculty members 

and has received full approval for acceptance in partial fulfillment of 

the requirements for the degree Doctor of Science. 

// 

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

5 - D?seertatlon copies 

1 - Candidate 

1 - Department , 1 - Dean's Office 

1 - Registrar 
WASHINGTON UNIVERSITY 

SEVER INSTITUTE OF TECHNOLO1~Y 

DESIGN OF HELICOPTER ROMR BLJC ,R 

OPTMJM DYNAMIC CHARACTERISTIc.1 

b Y 

Thothy W H KO 

Prepared under the direction of Professor D. Peters 

A thesis presented to the Sever Institute of 

Washington University in Partial fulfillment 

of the requirements for the degree of 

DOCTOR OF SCIENCE 

Deoember, 1984 

Saint LOU~J, Yissouri 
YASHINGMH UNIVERSITY 

SEVER flOSTITTJTE GF TeCIRJOLOGY 

ABSTRACT 

DESIGN OF HELICOPTER ROTOR BLADE% FOR 

OPTIMUM DYIVAPIIC CRARACTERISTICS 

by Timothy W H KO 

ADVISOR : Professor D. Peters 

Saint Louis, Missouri 

The mess and stlftnsss distributions for helicopter 

rotor blades axe to be tailored in such a way to give a 

predetermined p lac em ant of blade natural frequencies. The 

optma1 design is pursusd with respect of rninirm~ weight, 

sufficient inertia, and resonable dynamic characteristics. 

Finite element technique wlll be used as a tool. Rotor 

types include hingeless, articulatsd, and teetering. 
TABLE OF CON'PENTS 

Paqe 

I. Intrduction .................................................. 1 

1.1 Hellcoptor Design ........................................ 1 

2 Prerious Work ............................................ 4 

1.3 Scope .................................................... 5 

1.4 Overview of Optimal Structural Pesign .................... 7 

2 . Backgrand .................................................... 9 

2.1 Foundation of Problem.. .................................. 9 

2.2 Finite Elemant aethod.. .................................. 12 

3 . Illustrativa Examples of Optimization ......................... 18 

3.1 Cantilever Beam with Given Frequency ..................... 18 

3.2 Ca;ttilever with Tip Mass ................................. 18 

4 . Numerical Btperiments ......................................... 23 

4.1 3esign Variables ......................................... 23 

4.2 Convewance ............................................. 25 

5 . Preliminary Calculation for Rotors ............................ 34 

5.1 Wind Turbine Blade ...................................... 34 

5.2 Hal:?opter Blade ........................................ 35 

6 . Teetering Rotors .............................................. 42 

6.1 Definition ............,............................... 42 

6.2 Flappinq Zesponse ........................................ 44 

6.2.1 Collective Modes .................................. 47 

6 ..? .2 Cyelic Modes ...................................... 50 

6.2.3 Combined Collective and Cyclic .................... 50 

6.3 Simult. wous Flapping. Inplane and Torsion ............... 54 

6.3.1 Variable Box Dimensions ........................... 58 
TABLE OF COTlTeWTS 

(continued) 

5.3.2 Pixed Box Dimmasions ............................. 60 

6.3.3 Variable Root Stiffnsss ....... .....,............... 61 

6.4.1 Variable BOX Beam ....................................... 64 

6.4.2 Fixed dax Be- .......................................... 64 

7 . Artioulated Rotors ........................................... 67 

7.1 Definition ........................................... 67 

7.2 Variable Ba ham Dimension .............................. 67 

7.3 FfxQd Box Beam Dimension ...................,............ 67 

8 . Relation Between Vibration and Fraquencg Placement ............ 71 

8.1 Fomlation .............................................. 71 

8.2 Response versus Praquencp ................................. 71 

8.3 Respotme versus Placement Prequeney ...................... 83 

11 . Appendices .................................................. 38 

1.2 Calculation of Torsional Stiffness . GJ .................. 102 

11.3 Derivation of Pass and Stiffnes~ Matrix as Function of 

Natural Frequencies ............... .................... 106 

12 . Bibliography .................................................. 108 

13 . Vita .......................................................... 1l2 
No. 

LIST OF TABLES 

1. Comparison of Cantilever Beam with Concent?ated mass......... 22 

2. Optimization of Wind TUrbine Blade........................... 36 

3. Uind Turbine Blade with Added Mass........................... 37 

4. Helicopter Blade............................................. 39 

5. Heltcopter Blade with Stiff Flapping..... .................... 40 

6. Iaitial and Flml Design for Collective and Flapping Hodes 

(Teetering Rot~r)....~..........~.......................... 49 

7. Initial and Final Design for Cyclic Flapping Modes 

(Teetering Rotor).......................................... 51 

8. Initial and Final Design for Coliective and C;elic Flapping 

Hodes (Teetering Rotor).................................... 53 

9. Data for Teetering Rotor Blades.............................. 56 

10. Initial and Final Design for Flappi=, Inplane, and Torsional 

Modes of Teetering Rotor Blade with Variable Box Beam 

Dmmion............-...................................... 59 

11. Initial and Final Design for Flapping, Inplane, and Torsional 

Modes of Teetering Rotor Blade with Box Beam Dimensions 

FIXd................................................... 62 

12. Initial and Final Design for Flzpping, Inplane, and Torsional 

Modes of Teetering Rotor Blade with Box Beam Dimensions 

Fixed Except Root Stiffness................................ 63 . . 

13. Initial and Final Design for Flapping, Inplane, and Torsional 

Modes of Teetering Rotor (Pretwist) Blade with Variable Box 

Beam Dimension......................................... 65 

14. Initial and Flnal Design for Flapping, Inplane, and Torsional 

Modes of Teetering Rotor (Pretwist) Blade Box Beam 

Dimensions Fixed.........................................., 56 

15. Data for Articulated Rotor Blade............................. 68 

16. Initial and Final Design for Flapping, Inpla~e, and Torsional 

Modes of A-ticulated Rotor Blade with Variaole Box Beam 

Dimensions............................................... 69 
100. ' 84 1 

17 .Initial and Finai Des ign for Flapping, Inplane, and Torsi~nt 1 

Kadea of Ariticulatd Rotor Blade vith Variable Box BL~ 

Dimensions Fired....,...................................... Yf 
LIST OF FIGURE? 

No. Paqe 

Typical Blade Cross-section...................... .,.-...... 11 

Area Moment of Inertia for Optimum Beam in Present Work ,..... 19 

Area Hoslent of Inertia for Optimum Beam from Rel ll..., . -, ,19 

Cantilever beam with concentrbted mass..........,............ 21 

Box Beam cr6r;-section ...................................... 24 

Cocvergence of Weight of Objective F~lnction.................. 28 

Weight Objective Function versus Number of Elements.......... 29 

Area Moment of Inertia versus Number f Elements............. 29 

Weight versus Number of Elements............................. 30 

Opt-1 Distributions for Various Meshes..................... 31 

Alternative Optimum... ....................................... 32 

Blade Cross-section with Variable Box Dimnzion.............. 43 

Dimensions and definition of Design Variable for Box Beam 

Cross-Section.............................................. 46 

Radial Variation of Forcing Function........................, 74 

Tip Respcnae bersus Forcing Frequency for Both Initial and 

Final Designs Without Damping. ............................. 76 

Tip Response Versus Forcing Frequency for Initiai Design Both 

With and Withcut Dan~ing..... .............................. 77 

Tip Response Versus Forcing Freqaency for Both Initial and 

Final Designs With Dampirrg............... .................. 78 

Sum of Squares of Shears Versus Forcing Frequency for Initial 

Design With and Without Damping. ........................... 80 

Sun of !?;xz-es of Shears Versus Forcing Frequency for Final 

Design With and Without Damping.... ........................ 81 

Sum of Squares of Shears Ver- Forcing Frequency for Both 

Initial and Final Design Wi,n Damping... ................... 82 

Sum of Squares of Shears Versus Second Natural Frequency for 

Initial Design Both With and Without Dampinq.. ......,...... 85 
LIST OF FIGURES 

icontinued 1 

No. Page 

21. Sum of Squares of Shears Versu~ Third Natural Frequency tor 

Initial Design Both With and Wlthout Damping............... 86 

22. Sum of Squares of Shears Versus Second Natural Frequency for 

Final Design Both With and Without Damping...........,..... 88 

23. Sum of Squares of Shears Versus Third Natural Frequency fcr 

Final Design Both With and Without Damping.......... ....... 89 

24. Sun of Squares of Shsars Second Natural Frequency for Initial 

Design Both With and Without Damping (where Forcing 

Function is Even Integer Multiple).... ..................... 91 

25. Sun of Squares of Shears Versus Third Natural Frequency for 

Initaal Design Both With and Without Damping (whex-e Forcrng 

Function is Even Itteger Multiple). ........................ 92 

26. Sum of Squares of Shears Second Natural Frequency for Final 

Design Ecth With and Without Damping (where Forcing 

Function is Even 1r;fagar Multiple).......... ................ 94 

27. SUB of Squares of Shears Versus Third Natural Frequency for 

Final Design Both With and Without Damping (where Forcing 

Function is Even Integer Multiple)......................... 95 

28. Ideal Two-cell Model for Calculation of Torsionzl Stiffness..lOS 
DESIGN OF HELICOPTER RQmR BLADES FOR 

GPTIHUH DYIVJMIC CRARACTERISTZCS 

1. IIPTRODUCTION 

1.1 FiELICOPTeR DESIGN 

The design of helicopter pator blades involws not adly considerations of strength, survivability, fatigue, and cost, but also requires 

that blade natural frequencies be significant1 y separated from the 

fundamental aerodynamic foreing frequencies (e.g. Ref. 1) . A proper 

placement of blade fmquencies is a difficult task for several reasons, 

First, there are my forcing frequencies (at all integer-multiples of 

the rotor RPU) uhich oocur at rather closelp-spaced intervals. For 

example, 51rev and 6lrev are less than 20 A apart. Second, the rotor RPM 

may vary over a sLgnificant range through the flight envelope, thus 

reducing even further the area of acceptable natural frequencie3. Third, 

the naturml maxies of the rotor blade are often coupled because of pitrh 

angle, blade twist, offset between the mass center and elastic axis, and 

large aerodynamle damping. These couplings complicate the calculation of 

natural frequencies, in fact, the dependence on pitch angle makes frequencies a nanction of loading condition. since loading affects collective pitch. kurth, the centrifigal stiffness often dcminates the lower 

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

stiffness or aas5. 

In the enply stages of the developaent of the helicopter, it was 
believed that heliccpter vibrations could be reduced (and even eliminabed) by ths corract choice of struatural coupling and mass stiffness 

distribution, Howsver. it is ewy to imagine how difficult it is to find 

just the proper parameters suah that the desired natural frequencies can 

be obtained. The difficulties in placement of natural frequencies have 

led. in many cases. to prellmimry designs vhich ignore irequency plazement. Then, after the struotura is 'finalized' (either on paper or in a 

prototype blade), the fisquemies ape calculated ( or measured) and 

final adjustments made, Refemace [21 describes the development of the 

XH-17 helicopter in tihioh a 30blb wight was added to each blade in 

order to change the spanulse and chordvise &s distribution and therby 

move the first flapwise frequency away from 3/rev. The authors were confident that similar adjustments to the mass distribution (and thus to 

the frequencies and 0068s of the blades) could greatly reduce rotor 

vibration on other mtors. An analytic study in Refemnee [31 predicts 

that chordwise emss distribution could also be used to lower overall 

helicopter vibrations. In particular, a forward shift of mass is shown 

to be useiul because it places torsion in resonance with a particular 

harmonic. The torsion loads car, then be tuned to cancel undesirable 

blade loads. The study also shows, however, that such mass changes may 

have an adverse effect an stability; and thus stability and vibration 

must be studied together. Similar benefits of inertia pitch-flap coupling are also obsemed in shaker 2ests in Reference [41. 

These poaitive results, and other like them. w.we at least partially responsible for the optimi3tic outlook so aptly presented in Ref [51. 

In that; reference. six helicopter pioneers express their belief that 

helicopter vibrations can be reduced through proper blade and fuselage 
design. This optinian of the 50's was soamwhat eroded in the 60's and 

70'3 as the true complications of rotarpring dynamics became bettor 

kncnn. Nevertheless. the belief is still held by most dynamicists that 

simple concepts (such as frequewy plaaawnt) can go a long way toward 

inproving rotor design. For example, in Reference (61 six helicopter 

pioneers (sow of the authors of Ref. 41, reminfac:, on the early days of 

rotary wing and the recent advances in our understanding of helicopter. 

Yet, they still contend that much can be learned from simple principles. 

Presently, helicopter blades are not tailored to give a set of 

desired natural frequewies. Instead. blades are designed based an other 

consldepatioru (including the desired aemdyuamlc characteristics and 

the cuarlative experienae of the designeP3). Then. after the design is 

analyzed (ritner by computer program or by fabrication and testing), the 

designer checks for frequemies that am poorly placed. These are then 

adjusted by judicious application of lumped inertias at crucial spanwise 

looatiom. These after-thefact alterations. however, can be detrimental 

to blade wight. blade cost, and the development time of the aircraft. 

Somatlmes, the problem are unsolvable, and a helicopter is left with a 

noticeable resonance problem. 

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

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

during preliml.nary design stages. There are several reasons lor this. 

Firs:. helicopter rotor blades for both main ~otors and tail rc,tors are 

now being fabricated from composite materials (Refs. 7 and 8). This 

implies that the designer can choose, with limited restrir~tions, tSe 

exact EI distribution desired. Furthermore, the lightness of composite 

blades for the main rotor usually necessitates the addit--n of weight to 
give sutiicie3t autorotational blade inertia. Thus. there is a consider 

able amount of flexibility as to how this weight may be dis*,ributed. 

Second. th* methods of' struatural optimlzat:..on and parameter identification are now mflned to the point where they can be efficiently applied 

to the blade trueturn. Sow elearntq technique3 have already been 

used for the destgn of rotor fiselages (Ref. 9). It follows that the 

time is right for the use of structv-a1 optlmlzation in helicopter blade 

design, Some work on this is already under development. a~d, although 

not published. sow companies are already experhent uith the oigtimum 

way to add'ueiqht to an existing blade in order to i~~~prove vibrations. 

1.2 PREVIOUS WOfflc 

In this light, M would like to mention a few mcsnt attempts at 

application of optimization techniques to rotor blade deslgn. In Ref 

[lo?, an optimization procedure is applied in order to reduce blade 

loads consistent with aeroolastic stability, The procedure is not cow 

pletsly automated, however. and the designer amst make the design increment at each itemtion based on numerical ~ensitivity parameters. The 

biggest needs (as identified in this work) are the complete automation 

of the optimization and the formulation of realis2ic design constraints. 

In Reference [Ill, an optimization package 13 applied to an aemelastic 

response praqmm, The results mirror the earlier conclusions of Reference 2-4. In parficular, minimization of vibrations tends to drive some 

natural frequencies close to integers in order to cancel loads. (The 

rotor becomes an isolator.) Although this :urns out to be a good mathematical sclution, stability analyses in Reference [Ill, aj in References 

[2-41, show that the coalascence of frequencies to supp:*ess vibration 

tends to introduce aeroolastic instabilities. Thus, flutter sargins tend 
to becam the domlnant constraints. Furthemre, lainlmization of loads 

at one flight coadit:on nmy not at all minimize l~ads at others. 

Another investigation into vibmtion mduction by alteration of 

wss and stiffisas distribution is given in Reference tlfl. In that 

refemme, a tip wight is used to ohange tho aode shape. It is hypothesizd that changing the mode shape suoh that it is orthogonal to the 

forcing f'unotion is a way to lower vibrations. However, ths conclusions 

are uncertain since the frequewies also are changed by this added 

weight (e.g. the seuond flap arode wves away from 5.06 to 5.19 per rev). 

One also notiaes that the loading distribution changes with flight 

condition so that modal shaping may help one condition but hurt others. 

Other related previous mrk is found in Refemme [131. That paper shows 

that design to minisam loads can result in a disjoint solution. FortutWtei7, in helicopter problems we generally begin with an zdequate (but 

not perfect) blade desen. Thus, many questions such as thi~ (1"s. 

disjoint in the design space) am automatically avoided. We already have 

a good first guess and merely wish to refine it. 

1.3 SC'?PE 

In this paper we undertake a much less ambitious aim than the minimization of hub loads. Instead ve look at the problem of using optimization techniques in older to place natural frequencies. Even within this 

roduced problem there are varying levels of complexity. For example, one 

could consider the retrofit problem: 

* Gl~en a blade design find :he amount and location of added 

masses required to move fraquemies away *om integer 

!'930CianC08.' 

One could also consider thc basic design problem in which both stiffness 
and mass distributions may be chosen. In this paper, we treat this 

latter design problem in the ;om of three uncoupled problems: flap, inplane, and torsion. 

The scope of this present work is not just to find a mass and 

stiffness distribution to give desired frequencies. It is also to determine meaningfil constraints and objeotfve functions that will result in 

realistic designs. In this area, several items are noteworthy. First, 

there is the airfoil envelop. Whatever the structural engineer designs 

mat lie within the airfsil cross-section. Second, there is mass balancing. TLe center af mass of each section should be forward of the onequarter chord. Third, there is the auto~otational constraint. The blade 

must have sufficient ass moment-of-inertia to insure a safe autorotational capability. Fourth, there is strength. The blade must be strong 

enough to endure the centrifu~al loads as well as the oscillatory bend- 

!-! loads. This last criteria is the most elusive of the four. Desig~ars 

know how to make a very soft section (hinge or flexure) which nevertheless can withstand high centrffugal and bending lsads. Such flgxures 

generally do not fall. within an airfoil envelope, however, and are 

placed near the root. Therefore, in the work to follow, we first obtain 

'optimum' designs and then check to see if the required EI distpibution 

has unrea:istically soft spots. 3imilarly, we check the final designs 

for axial stresses. 

In summary, we work with simple (bu; realistic) rotor-blade designs 

and simply experiment with constraints and objective functions in order 

to determine the feasibility of designing to a dotired 3et of frequencies. 
1.4 OVERVIEW OF OPTIMAL STRUCTURAL DESIGN 

Host appmaahes to optimal structural strustural design may be 

classified into three categories. (POI . ecent review articles see Refs. 

14 ad 15.1 One such aategory Is 'variational methods.' These generally 

rely on teohniques from the mathematical, thsory of the c.iiculus of 

variatioas, and, when app1ic.-ble, often provide useiul physical irsiqht 

into the nature of an optbal design. Unfortunatctly, only relativeiy 

slmple probleas can be solved by this approach, since tho mathematics 

baoasa Lntraotable when complex enqiner.ring structures am considered. 

A seoond catwary of struotural optmzation techniques consists of 

the application of mathematical prcgramming methods toqethe;' with the 

Glscretization of the strscture bp finite element tec ~nic,ues. This 

appr-mch to optinization was foundei in 1960 (Ref. 16) with the hore 

that more complex structures could be analyzed than were possible when 

using the analytical tecnniques of the calculus of variations. '3owever, 

in the late 60's it became apparent that mathematical programming mothod 

had limitatioa? of their m, namely, unacceptably Long computation 

tlms occurring when the numk3r of design variables becoma large (over 

20-100, depending on the type of structure). Fortunately, several 

improvements developed over the last few years appear to have 

siqnificantly extended the capability of the mathematical programming 

approach, and, as a result, it is this approact, ve intend to draw upon 

for solution technique3 in this research. 

A third catagory .? structural optimization approaches is the 

'optimality criterion' approach in which an equation expressing some 

necessary condition of optimality is used as the basis ?or constructinq 

an iterative (successive redesign) procedure. Origizall y developed \ 
because of di3sat:sfaction with aathenu8tical proqramning techniques, tho 

optFoPlltp-oriterion approach initially relied on ht~itive optimrlitp 

criterion suah w corm taut s tress-ratio and uniform s train-energy 

density conditions. More mently, optSPPlity ariterlon (and asmiated 

rsdosign equations) have bwn derived from the Kuhn-Tucker conditiomi 

(sw. e.g. Ref. 17) for a coastrained mlrddzation problem, 

me opt.iaalitp criterion appkoach seema- especially well-su: t ed to 

problas wih'? a 1-0 nunbsr af design variables. Sinaa our design 

problea will have a moderate numbsr of vartables and since deriving 

efflqiont rs-dsaign equations for our problem is not immediately 

straightforward, ua initially prefer t;s mathe6atical programming appmoh ocrw the optimalitr-ariteri~n approaah. 

A structural optimization computer program, called CONMIN (Ref. 

27) ,is available from NASA. It is +.his program that is used in our 

present work. CONMIN is based 3n the matkaaiical nonlinear programming 

method of feasible directions. 
3ACICGRQUNb 

2.1 POWELATIOH OF PROBLEM 

Because nuasrically-bassd optiplization is best carried out with discrete variables, the finite elemnt technit.48 stands as the most logical choices for the blade meidel. A recent research project ;Ref. 18) has 

resulted in a flnite-elemont computer program that. is ilgally suited t~ 

the work here. The program allows for tapered, tuisted finite elements 

in a rotati= envimnmont. The existing code can calculate natural frequen?ies, (with arl without, aerodynamic terms) and foms response. 

Another wortant aspect of the retor blade optimization problem is 

the salection of the optimalit>- critaria and constraints to be imposti. 

Cur design problam ha^ certain ieatures which are unusual compared to 

typical problem occurring in tire structurHl optimization literature. 

There are basically three catagories of criteria. In the first class,one 

would IPinMze weight given constpaints on the natural frequencies (i.e. 

frequency 'windows'). In this case, a constraint on rotary inertia is 

also implied since a roi;or must have sufficient inertia to autorotate. 

The advantage of this approach fs that it is directly related to the 

phystcal ~salitias of design. The disadvantage, however, is that the 

first guess will probably not 3e feasikile! ithat is wiii not nave frequercies that fail in the 'windows' 1 . This can be a stumbling block to 

convergence. A second type of criteria is one in which the objective 3 

to minimize the discreganoles between desired frequencies ar?d actual 

frequencies. The constraint then becomes a windaw or, autoro ta tionnl 

inertia. Although this avoids unfeasible solutions, it does not directly minimize weight (although we.!ght is limited by tke autorotational 
conntraint). An objective htnction can be constructed that combined combined blade mass and frequency placement, but the relative weigttings of 

the tno components is not obvious. Tne third category of constraint is 

to minimize vibrations directly without regard to frequency placement. 

Although this appears on the surface to be the perfecL solution, there 

are problems. First, calculation of vibrations is an order-of-magnitude 

more difficult than the ca.lculation of frequencies. Second, past efforts 

at this have resulted in strange designs, incompatible with standard 

helicopter practice. Third, there is still the problem of the weightvibration trade-off. In this work, we intend to concentrate on the first 

two categories with some attention to the third. 

Another type of cqnstraint involved in the problem is the limitation on structural properties. The blade planform, airfoil, and twist 

are chosen by the aerodynamicist on the basis of performance. The 

structural engineer must choose his design to 'it in the aerodynamic 

envelope given. There are five structural parameters to be chosen: 1) 

flapplrq stiffress, 2)inplane stiffness, 3) torsional stiffness, 4) 

mass, and 5) torsional moment of inertia. In practice, these cannot be 

chosen completely independently. Figure 1 shows the envelope of a 

typical blade section. All cstiffness is assumed to reside in a box-beam 

of dimension b x h with thicknesses t,dl,d2. This beam is placed as far 

forward as possible (to keep the elastic axis near the 1/4 chord). Mass 

properties are due to the box-beam, skin, honeycomb, and two lumped 

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

used to keep the mass center forward of the aerodynamic center. A second 

mass is included to allow independent choice of mass and mass-moment. 

The constraints of this construction are clear and are listed on the 
(I, 

- aa 

- - 

(I, 

C 

Q) 

0 
figure . 

In addition, there 're minimum con~traints on t,dl,d2 to hold 

centrifugal loads and to reaain withir! manufacturable imts. For 

example a simple minimum constraint on area could come from the centriiugal constraint (not considering bending stress). Thus, if & is the 

maximum stress and if f is a safety factor, then 

(la) 

Of oourse, when wr, @ate th6 vIbratorpraspou3e pha8e of the uorlc. bending stma8es will be imluded. 

OPT work Vill neve~theless inolude flutter criteria in a simplified 

EWUI~?. First, H can ohmzre fFeyuemy placement such that no coalasoeme mews betuwn fiap-?.ag, flag-torsion, or lag-torsion. Secoad, ws 

can constmn the five parameters in Figure 1 such that the mass ce~!ter 

is alnays foPwud of the 1/4-ahord, a comaon design practice to prevent 

torsion-fluttar Fn rotor :lades. 

2 F~ttMlemont Mode 

Although tapered, tvisted elements are within our capabilities, ue 

introduoe here a simple case which is also of value. The stifihesses GJ. 

gIzzD Exn are sad to be constant along the length of the eleaent. 
The lumped mass weight is assumed to be evenly distributed on the two 

nodes. 

Let the deflection of an element In the y a. ' z directions at 3 

distanee x be denoted as w(x) and v(x), for which the displacement 

models are assumed to be polynomials of third degree. The expmssions 

are given a8 

&ere vl, v3, v6 and vg represent the bending degrees of freedom in the 

n plane and u2, u,. u and u9 represent the degrees of freedom in the 

yx plane 
-1*- 

i) The strain energy due to bending deformation can be expressed as 

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

which is equivalent to the negative of kinetic energy due to radial displacement, is given by 

where T, termion force, is assumsd to be constant along each eipment . 

iif) The kinetic energy due to inplane displacement is given by 

which is equivalent to u = T 
Segr2es of Freedom of an El enen t 

Meanwhile, the pretwist angle C$ (XI, and the torsianal deformation 

6 (XI am assumed to be polpnomlals of first degree, and can be expressed as 

where fi , Ql, represent the prstwist angle at node I and 2, and u5 ,uI0 

represent the elzwtic torsional degree of f-eedom at each end. 
iv) The torsional energy. due to elastic deforwations and c8ntriiugal 

tern. can be expressed aa 

2 = /;r y2 dydz = 1 ka 2 z z 

V) The 'torsion-rotation' energy under the effect of rotation is 

given by 

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

Total displacement energy now can be used to form the stiffness matrix 

fioar 
where u is the vsotor of nodal displacomnts, in the order as ul, u6, 

u 3. us, u2. u,. u4, u9 ,us ,uI0 ; [Kl is the elemntal stiffness matrix 

of order 10. 

vi) The ams matrix will be obtained by the kinacic energy of an 

element, whiuh is given by 

Written la matrix forn. the kinetic enem can be expressed as 

where [HI is the mass matrix. 
3. ILLUSTRATIVE EXAHPLES OF STRUCTURAL OPTIMIZATION 

3 .1 CANTILEVER BEAH WITH GIVEN FREQUENCY 

Some simple example3 will be examined and discussed before the 

utilization of the program CONUIN. In each oaae the results will be compard with thme obtained by previous msearchea, if it is available. 

The first llmlting example is the problem of determining the optimal design of an elastio Cantilever beam, such that with a specific 

natursl frequency,the uelght of the structure attains the minimum value. 

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

length of 1C inohes, E = 1.0 1bi2 , EI = 10 lb-ln2. demity = 0.042 

3 lb/ln , and a speaified first lowest natural frequency = 0.6489 rad/soc. 

We obtain the final stiffness profile shown in Flgure 2. Figure 2 is the 

preuent result with ten elements. 

A ralated problem has also been treated by 0 1 ho f f [ 19 1 . He 

seeks the dsslgn of a cantilever beam that yields a mxlmum value cP a 

partiaular higher natural f'roquena~ wn (1.0. of specified order, n) 

with the volume and length of the beam specified. His work is the dual 

problem of the example shown in Figure 2. Optimization with respect to 

the frequency under the constraint of volume is similar to the one of 

mlnlmlzing weight (or v~lume) under the constraint of specified natural 

frequency. F~rs 3 gives the profile of the optimal cantilever for n = 

1 by Olhoff. One can see that the shapes in Figure I21 and [31 are very 

sinillax- in that thay give a nonlinear taper. 

3.2 CANTILEVER WITA TIP MASS 

Another example problem is to minlmize the weight of a cantilevar 

carrying a maas at the tip, subject to the constraint that the fundamental natural frsquenoy must be greater than or equal to a specified 
I '1 7 

-I I I I I I bt 

2 4 6 8 10 

Figure 2. Area merit of 1ner:ia for Opciamm Beam in 

Present Fork. 

Figure 3. Area 3oment of Inetzia for Opzimt~m Beam 

from ?-eference 11. 
value. The problem was originally fcmulated by Turner [201, &h- ana 

Willmsrt [211 ~sod an optimality criterion method to solvc, Tu~ner's 

prob:em. In hi^ example, four R:'..:Lcs elements are used, with the areas 

of each as the design variablee, aa illustrated in Figure 4. The specitid natural frequenay is 17.792 md/sso. The other initial data are 

Modulus of elasticity = 10.3 x 10 psi 

Mass density 2 = 2.5 x I0 lb-s /in 

Radius of gyration (AI 1 = 2.0 in 

Radius of gyration (A2 ) = 1.5 in 

Radius of gyration (A3 1 - 1 in 

Radius of gyration (A, 1 = 0.5 in 

Concentrated mass 2 = 1. lb-a /in 

Length of each eiement = 60 in 

2 where I = Area (radius of gyrdbion) . 
F~gure 4 Cantilever bcaa ulth concentrated mass 
The results of the optimization am show11 in Table 1. The ieasiblc 

starting deaQn is describd by A1 = 200, Lt = T5C. A3 = 60 ad A, = 35. 

Table 1 

Ref. [l21 Ref. !I3 1 This Eager 

- 

Iteration - 23 10 

A4 (in, , . 34.43 34.61 34.89 

It aan be seen that excall-nt results have bean obcained csing the 

present CORNIN optimization program. 
4. NTMX~ICAL ExPER~MENTS 

4.1 DESIGN VARIABLE 

Despite the str~nq documentation and intensive development that has 

sone into optimization programs, it is alwys advisable to do some experimentation with these proq~~~ for the pa~ticuia~ class of problems 

to which they are to be applied. m.'.s has been done in detail for the 

representative box beam shown in Figure 5. Tne parameters (see Fig 2; 

for this car. are: h = 2.5 in, s = 0.1 in, s = 0.1 in. b = 4 in, t 

variable. Thi3 beac has been analyzed for various values of the C3NEIIfi 

parameters and for v?.rious c~mbinationz of constraints. The fi~at 

studies are performed for vertical vibrations. a noarotatinq beam, np 

lumped mass, and with two frequency constraints. Each of the CO?PiiU 

optioas is then exercised, and several concll~sions Cr2un. 

Fii st, we ficd that the use of analytic. gradients (the derivative 

sf objective function and constraints in closed form) is 3reatly co be 

desired. For the particalar case in Figure 5, arza, weight, and moment 

of inertia can be expressed in terms of the siwle variables, t 

Therefore, analytic derivatives am straightforcrard. Where analytic 

gradients are not available, however, we find that finite diffsreace 

gradients still work albeit at a higher coffiputationa~ cost. Secand. wa 
Element Length 

SECTION A-A 

Figure 5 Box Beam Crgss-Sectlon 
find that the optimization is best behaved when frequency constraints 

are provided in Hz. A constraint (if not scaled) on eigenvalues (w2) is 

mere difficult for the program when default values are ssed. Third, we 

find that initial designs outside of the desired constraints 

(infeasible) sometimes can lead to convergence. Since this is not always 

the case, however, alternati-re strategies are necessary. Fourth, we 

find that the default values for the CaNMIN program worked reasonably 

well (although they are not always the most efficient values). An 

example is the number of iterations. Sometimes 40 iterations were 

required for convergence, although the defsult value is 10. 

In tams of Y-ious mdes of application. we also have come to 

severa4 conclusions. Firat, w could find optImam designs no matter how 

tightly we cloaod the wxindous on money (i.8. the f'requency const- 

-ta). Thus. we am able to essentiallp *zero8 an objective function 

based on frequencies (for vertical vibrations alone). Second, wt can 

handle a large rmmber of simultaneous fmquency constraints. (We have 

successC~iIy gone fiam 2 up to 5 constraints.) The optimization also rcr 

wll-bo&aved when we add blade rotation.lumped mass. and the autorotational constm.int. 

4.2 COIWERCENCE 

We have studied the co;.zvergence of the final design as a function I 

of the number of elements used in tte finite-element frequericy calculation. Tc study how ths optimal design changes as the number of elements 

increases, a cantilever hem with 'n' elemerts and with lumped weights 

added at. the nodes but otherwise similar to the beam in Figure 1 is 

considered. The density. and fhe constraints on the natu~al fl equency, 

lumed weights, and moments of inertia are 
aud the initial value of OBJ (the total weight) is 30.2249 lbf. 

Results of the study am shown in Pigums [6-Ili. In all cases. 

the active fmquency constraints were found to be 

f2 = ll*? (Hz) 

Figure 161 demonstrates, as one mid expect, that the optimum 

weight dwa La faat docrema aa aom element^ are added to the mesh. The 

c!mags Fn optiasPn wight is quits small (note that the saale of ths 

rartiaal axis b.gins at. 20.0). 

Figure [71 and [81 ghow the variation of the lmped weight and the 

nomint of inertia (of the cross-sectional area) at the froe end ve?sus 

the total number of elements In the mesh. It appears that these quantities do not converge. The resulc can be explained. however. by referring 

to Fiqure [91, in which the Lpper curve represents the total weight at 

the free end. ( m(n) is the non-structuralnor, lumped weigkt ~:n) is 

the structural weight associated with the mass distribute6 ct,rouuh-'15 
element ' It can be seen from the figure that the total weight 

appears to converge smoothly as the mash is refined. The explanation for 

the apparent non-aonvergenoe shown in Figr~res [71 and [81 and for the 

convsrgenoe shm in the top curve of Figure [91 is that the 'structural 

~Qht' at the free end of the cantilever iz not really structural, 

since there is no portion of the beam beyold the free end which needs to 

be supported, Thus, the optimization routine is indifferent to whether 

st~uctural or non-structural weight ia p~esent at the fV93 end - the 

only thing that counts is the total weight at that end. 

Figure t91 also ahon the wr'?tioa of the lumped weig3t slightiy 

beyond the middle of the beam. (All optimal designs have non-zero lumped 

weights there and at the free end of the beam.) The weight can be 

decrease smoothly as the mesh is refined, although no asymptote apbears 

to be p?.eaent, Tk_a explanation for this behavior is that, as the mesh is 

refined, the weight in the middle is being place more efficiently - aad 

thus less is needed. 

The various sketches in Figure 102 show t.he distribution of mass and 

stiffness aloriq the beam for inareas?.ng numbers of elements. It is interesting to observe that 2he optimization routine finds it most efficient to meet the constraints on frequency by varying the lumped weiqht 

rather than by varying the stiffness (moment of inertia), since this 

latter quantity is at its lower bound every. re except near the end of 

the beam. Another interestinq aspect of F,gure 10 is the manner in which 

the lumped mass at the center alternates between: 1) being all on one 

element. and 2) being split between tuo elements. This phenomenon is 3 

result of the fact that the minimum weight struct.ure would have all the 

mass at a single point (node or antinode). When this siryle point lies 
TOTAL NUMBER OF ELEMEMS IN MESH 

Figure 6 Convcrqence of Ueight of Objective Functior? 
1/ Weight At Free End (m,) 

n, TOTAL NUMBER OF ELEMEN73 IN MESH 

F~gurr 7 Uerght 0bjectL Function Versus Nuuocr '01 &laments 

I at Free Ebd (I,) 

C) 3.0 

2 4 6 8 10 12 14 16 18 20 22 

n, TOTAL NUMBER OF ELEMENTS IN MESH 

Figure 8 Rrea Moment of inertia Versus Nuaaer of Element; 
n Weight At Free End (m, * ?ii 1 

- - .. -- - ~ ~ - - 

Weight Near Middle (m 14. I • mn12+2 1 

1 

n, TOTAL NUMBER OF ELEMENTS IN MESH 

Fiqurr 9 Ueight Versus Number of Elements 
Figur e 10a 0 ptimal Distributicn fcr Varilsus M csh s 
Fig-lr e lob Alt gnativ e 0 ptimum 

n*lO >m,= a a 'ian 

# 

08J r 21.1457 

ma* 1.6010 (Now: OBJ < 

-US O8JT0) 

Ilo Q98311 
near a structural node the mass is place4i there. On the other hand, when 

the mesh causes the point to be between nodes, the mass iu accordingly 

divided between the two closest nodes. 

As was pointed out previously in reference to Figures [71 and 181, 

the optimization algorithm appears to tseat the struc'cural and nonstructural mass at the end of the beam as interchangable. To test this 

hypothesis iurther, the optimal design problem statement was altered 

slightly by decreasing the upper bound consti-aint on the moment of inertia from 5.2083 to 2.0. The resulting optimum design is shown in Figure 

[lob], and should be compared with the design (for n = 10) show11 in 

Figure [1L?al,, Note that the constraint on the moment of inertia for 

element 10 is not active in the optimal design of Figure llObl (the 

constraint was active during the CONMIN iteratlons leading to this 

optimal design). Thus, the effect of the constraint is to lead the 

optimization algorithm along a different path than that followed when 

the constraint value was 5.2083. The design found, however, has about 

the same total weight at the free end (= 9.9705 lbf) as ths previous 

ten-elemnt optimm (= 9.9222 lbf). This result confirms the hypothesis 

that CONMIN increases the moment of inertia at the free end only as a 

means of increasing the mass there. Once that option is closed (that is. 

the upper bound constraint is reduced to a value of 2.01, COWIN simply 

increases the :':aped weight at the beam tip. This findlng suggests 

that,in future optimization studies, a tight constraint be imposed on 

the moment of inertia at the free end, since little structural capability is needed there, and necessary end mass can be adequately represented by the lumped weight design variables. 
5, DRELIMINARY CALCULATION FOR ROTORS 

5.1 WIND TURBINE BLADE 

The first example in this section is the optimization of a wind 

turbine rotor blade at 30 rpm. Initial data is taken from Ref.[221. A 

ten-element model is used. Only the flappinq is considered. The area 

moment of inertia, I, and the lumped weight of each element are taken as 

the desQn variables (see Fig 1 for blade area cross-section) . Young's 

6 2 3 Modulus, E = 0.2 x 10 lb-in , and density = 0.0334 lb/in are assumed 

to be constant. Blade radius, R = 750 inches. Table 2 shows the profile. 

of moment of inertia and the di3tribution of added weight for the 

initial and final configurations. The final profiie of the area moment 

of inertia along the blade is similar as the one in the previous 

example. The optimization procedure has removed material from the 

inboard sections and placed it more outboard. The lumped mass is 

concentrated at the tip of the blade as might be expected. We also note 

that moat of the originally-pqstulated lumped mass is removed so that 

only the mass inherent in the stiffness elaments or necessary for the 

au torotational constraint is maintained. ( A1 though wind turbine have no 

autorotational constraint, a certain moment of inertia is still usefil 

to smooth out wind vibration. 

An important aspect of the optimization problem is the existence 

(or lack of it) of a feasible solution. A 'feasible solution' is defined 

as any set of design variables that satisfy the constraints (whether or 

not that particular solution is an optimum). Tt is possible that, if the 

problem is poorly formulated, no feasible solution exists. What is 

more often the case, however, is that there are feasible solutions but 
that the optimization scheme may not be able to find them. Thus. it is 

advantageous to have a feasible initial guasv so that one is assured 

that at lsast a local optimum is possibie. 

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

(wl> 2.82 no./rev). Here we found that CONMIN was able to move from this 

first guess through che space of feasible solutions. In other cases, 

however, when the first guess is not feasible we have found that COWIN 

is not able to reach a solution. In such cases, one must add or remove 

some weight (or add or remove EI) from the first guess to move from inta 

the feasible space; or, alternatively, we must begin with frequencyplacement as the objective mnction 2nd then switch to weight when the 

fequencies are within tolerance. 

For example, Table 3 represents data for the sr- e wind turbine as 

i~ Table 2, but the constraint on the first natural frequency has been 

lowered to remove it from the dangerous 31rev range. This implies that 

the first guess in Ta!>le 2 is no longer feasible. In order to overcome 

this, a lumped mass is added to ~tation 9(225.4 vs 49.50). This lowers 

w below 2.621 rev but also lowers w2 to 8,251rev. This could be alle- 1 

viated in one of two ways: 1) move the mass to the node of the second 

mode. or 2) simply widen the w2 window. We have done the latter. It is 

interesting that the added welght is ultimately rearranged to other 

places and other weight removed such that the new design 1.3 no heavier 

than the optimum in Table 2. Furthermore, wZ is raised to 8.57 so that 

the "idaned window' had nc effect on tne sol~tian. 
rr 4 t- 

\Q 

0 ma m lnco . . a0 ch L 

0 

P- m rn mlc h m nn 

C4 

0 

u br 

I m 503 . . os m n c.9 rc on d m 

9 ".? y1 ~n m ON r- m mu d m 

m SN 

. . u m (Y mh 

0 u "St * 

0 QI 0 

n m o &I4 h rD mul u r? 

0 ro* " . . O, d hh 

m nln 

9 V 

CL, - 1 c9 C,i* 

m Lm 

2 2 

r ,F C I = i I 

- 

I 
Tal~lc. 3. !J_!!d Turbit~e wttl~ Added Class 

-- 

NA'I'UHAI. : FROEI TO 

FRfdlIJENCY 2.4 2.62 PerfRev 

COilS'KRh 1 N 1's SECOND 4. tl 5.0 ll~ 

DENSITY 

AREA 2 I 65.88 57.4 52.8 47.2 42.0 36.89 30.51 22.86 

in 

111 SI'It I !iII'l'I:I) I 65.61 57.17 131.64 117.5 104.66 91.8 75.9 56.93 

0b.l ( I 11) w llz wl /R w2 llz w2/11 Elon1c111 of In

1 --- ---.--.--------- 

IHI'I'IAI. 1787.') 1.48 2.55 4.81 8.25 3.21 x 10 

(11)) - . ... . -~ ----.-------------------- 

1s.I flA1, 1017. 1 1.48 2.55 5 .OO 8.57 2.5 x 1C8 . . . - - -. - - -. - - - - . - - - -- - - - -. - .--- - -- - - - -- - -- - - - - - --- - - - -. - - - - - - 
3.2 HELICOPTER BLADE 

The deslgn and analysis of a representative helicopter olade is 

discussed. Similar to Section 3, only flapping is considered and a te.1- 

element model is used. The initial configuration is modeled after the 

rotor in Reference 23. Density is constant along the blade and equal to 

8 2 0.17 x slugslin3 . Young's Hodulus is equal to 0.49 x 1G lb-in at 

8 the root and is equal to 0.585 x 10 lb-in elsewhere. Blade radius is 

equal to 193 inches. Results are given in Table 4 an* 5. In Table 4. wl 

is in the desired range but w is too small. Furthermore, the autorota- 2 

tional inertia is larger than necessary. In this case, the ZONMIN program is able to remove mz~s and stiffness in such a way to raise w2 and 

lower wl. The minimum bending inertia set as a constraint (0.4) is 

reached at every point except the root. The root remained high to keep 

w1 ) 1 - 05. The new blade is one-third the original mass. In Tale 5, a 

stiffer initial design is used and the frequency wl is forced to be 

very high. In this case, the program CONMIN would 'like' tc aecrease EI 

and m, buf any removai of material could lower w beyond its lover 1 

bound of 1.24/rev. To counter this, the ~ptiuizatioo zheme adds EI near 

the root (to malntain wl > 1.24Irev). Furthermore, the lumped mss 

necessary to maintain ;iutorctational constraint is moved a:.ight:p 'nboard to have leas effect on w (keep it high) but more effect on u 1 2 

(keep it low!. %is exarcpie illustrates the physical soundne3s cf this 

optimLzation scheme. It dt-es the same things that a designer would do 

(given appropriate conat~aints) but in a more systemaZic manner. Thus. 

with proper constraints, optimization can prove a ;aluable tooi for 

frequency placement. 
To_hlo 4. HeL1cnlctc.r Blndo --. 

Fl RST 7.5 9.5 Hz ROTA'I'IM: SI'EED: 425 R

NATNRAI. r FROn TO 

FREQUENCY 1.05 1.34 Pur/Rev 

YoUIW'S WUU1,US: Eleme

Rest r 

CONSTRAINTS SkCOND 18.5 20.5 Hz 

NAT URAI, : FROM bnslty 8 o.~,I x 

FRLQlltENCY 2.61 2.89 PcriWev 

-- a - 0 L - 3.15 r - .I 

5 2 Q - 0 I - 2.5 t - varia

'ONEN' > 0.7 x 10 lb-ll, OF INLHTIA - 

0-0 

ELEMENT NO I 2 3 4 5 6 1 8 9 

-~ - - 

LENCTH In 5 10 20 15 15 25 30 3 5 30 .- - L -- 

tlOtlENTOF in 4 1 14.28 3.42 10.25 18.8 10.25 0.512 0.406 0.406 

INERT1 A F 0.87 0.4 0.4 0.4 0.4 04 0.4 0.4 0.4 

-- 

AREA ) 1 20.6 In' 3.127 10.7 22.19 18.7 0.664 0.59 0.59 0.59 

F 0.91 0.59 0.59 0.59 0.59 0.59 0.59 U.59 0.59 

-- 

DISTRIBUTED lbs I 1.2L 2.32 16.17 23.29 19.61 1.16 1.24 1.04 1.24 

nASS F 0.318 0.413 0.826 0.619 0.619 1.03 1.24 1.03 1.24 

- 

LUMP to 1b5 I 0 3.04 1.67 0 6.4 7.46 10.75 5.21 6.55 

MASS F 0 U 0 0 0 0 11.17 fi 0 

- - - - - - - - - --- - -- 

OIJ(lb) w llz ul /fiI ,I Ilz w2/Li Hwe~~t of lner

- - 1 ----- 2- 

131.5/1 8.3 1.17 17.66 2.49 9.76 x 10 5 INITIAL ---- -- ---~-- -- -------- 

20.411 2.89 8.71 r 10 5 

Ft NAI. 41. 1 J? 7.47 1.05 I- __________ _____--- ---- -- 

We have also examined the designs in Tables 3-5 with respect to 

axial stress due to centrifugal loads. In each case, the maximum stresses after optimization are equal to or only slightly higher than the 

original stressee. 
6. TEETERING ROTORS 

6.1 Definition 

In this section we attempt our first optimization of a realistic 

cross-section (see Figure ll), one which is little different from our 

first design pattern, (Figure 1). Therefore, it is scfficientiy general 

that both the bending and torsional stiffneases of some currently existing blades can be matched. Uslng this generic cross-section and starting 

froffi an actural rotor blade deslgn. (Ref.I241), we have studied the 

pcssibility of moving natural frequencies away from resonances while 

simultaneously satisfying constraints on the foll.owing: stress, the size 

of lumped weights to be added, capability for autorotation, and thickness of the main structural member (the box beam). Because a teetering 

blade was considered, cyclic and collective modes 02 vibration were 

calculated independently by a change in the boundary condition at the 

blade root. In the initial phase of the study, we considered collective 

flapping modes first, then cyclic flapping, and finally combined 

collective and cyclic flapping. The results of the~e studies were 

favorable (i.8. we were able to cha.lge the frequencies in the desired 

manner and still satisfy the constraints). Buildi~lg on these results, in 

the secard phase of thia section, we consider a more challongiry problem 

which involves combined modes of collective flapping, cyclic flapping, 

collective inplane, cyclic inplane and torsional vibrations. 

In this section, the primary design variables are: 1) the wall 

thickness of the box baam and 2) lumpc:! weights, that can be added at 

specified stakjons along the beam. In the final problem studied, the 
GENERIC BLADE SECTION CONS1 RAINTS 

FirceO M:lss 

BOX Beam I 

Trolling Edge Honeycomb 

_-&*I----- 

+- a -* * b b 

~ui~loeb 

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 ol lnerlla 

Figure 11 Blade cross-section with varid~le tnx dbnension 
wall thicknesses are taken as fixed, and only the lumped weights are 

allowed to vary. This situation corresponds to that encountered in 

practice when a blade has actually been designed and manufactured. but 

then found to have poorly placed frequencies - thus lumped weights are 

added at va.rious positions along the beam to change the frequencies. We 

found that our optimization routine was able to handle this problem 

adequately, although the total weight of the beam could not be used as 

the objaotive function, as had been done previously. Instead, a 

'frequency placement' objective function was used. 

6.2 Flapping Frequencies 

The first set of optimization problems in the section is concerned 

with flapping response only. The stzrtinq design for the cptimizaticn 

procedure in each of the three cases studied is a typical metal-bladed 

teetering rotor with a diameter of approximately 24 feet. Ten finite 

elements are used to model the rotor; their lengths are given in Table 

6. The objective of the opsimization is to minimize the total weight of 

the blade. The design variables are the wall tiiickness, ti, of the 

finite el,ement representation sf the box btan (the structu~al member in 

the rotor - see Fig. 1 and the lumped weight wi, associated with each 

finite element. The lumped weight is the sum of two components, a fixed 

component (representing the weight cf the l~ading and trailing edge 

strips, honeycomb, skin and nose weight,) and a variable component 

(representinq additional non-sts\lcfural mass which may be added at 

various positions alaq the lrngth cf the rotor) to modify the dynamic 

behavior in a desired rnacrer. rhe side conditions on the element thicknesses are, in units of inches, 
0.00044 < ti < 0.730 

The ~ide conditians on the lumped weights consist of a lower bound only, 

which represents the fixed component of weight for each element and is 

given in Table 6 under the headin# 'wmint. Nc:e fhat tho element thicknesses ti are not given in Table 6; instead, the area moment of inertia, 

I, is represented. Using the dimensions given in Fig. 12, we can show 

that I related to the thickness by the equation 

The moment of inertia is given, rather than the thickness, to facilitate 

comparison with Ion the portion of the moment of inertia which is contributed by those parts of the cross-section other than the box beam. 

(Thus I. remains fixed as ti is varied.) Table 6 also contains the 

values sf the box weight, which are calculated by multiplying the weight 

density of the box beam material by the cross-sectional area of the box. 

Thus, the box weight is not an independent design variable, but depends 

on the thickness ti. The box weight is included in the table to facllitate comparison with the distribution of lumped weight. The constraints 

for the optimization are both the autorotation constraint, 

sum (wi .)ri2 > 0.5567~10~ lb-in2 

(where wit is the total weight of element i, and ri is the distance from 

the root to the center of the i-th finite element). The frequency constraints will be described in subsequent sections of this thesis.Some 
F ig~-,:e 12 Slmensions 3nd Def init ion of Design Var iable for 

Box Beam Cross-Sectaon 
additional data which complete the problem description are the values of 

2 the elastic aodulur, 0.105~10~ lblin, the radius, 288.8 in, the 

rotational speed, 324 rpm, and the mare density of the box beam 

3 material, 0.000262 -aps/ in . 6.2.1 Collective Modes 

The initial problem to be considered ia the opticgization of the 

blade vith respect to collective flapping modes only. Because we are 

studying a teetering zotor, the collective mode of flapping may be 

analyzed by hoeing a fixed boundary condition at the root cf the 

rotor. The imposed frequency conrtraints are 

in which pl, p2 and p3 are the first three collective flapping mode 

frequene iea non-d hnr iona lizd by d ivid ing by the rotor speed. 

The starting design for the optimization algorithm is given in 

Table 6 under the beading 'initial'. This initial design was chosen to 

correspond t lorely with an actual rotor blade; thus, it is not surprising 

to find that the design is infeanible vith respect to the frequency 

constraintr we have impored. The optimization algorithm used in this 

study, CORMIN, supposedly permits an infeasible starti~g point and attempts to proceed from this starting point to s feasible point, However, 

for our design problems, this feature of CONHIN failed to produce a 

f car ible design after many iterations. As an a lternar ive approach, we 
formulated a preliminary optimization problem in which che previous 

objective function (veight) was replaced by a 'frequency-placement' 

ob jactive: 

The numbers 3.50 and 6.50 are the average of the bounds of the frequency 

constraint inequalities which are violated by thn initial design. The 

remaiader of the optimization problem is the same as the original problem, except that the constraints on those frequencies which appear in 

the frequency placmnt objective are omitted. CONHIN vas applied to 

this preliminary problem. In the process of minimizing the preliminary 

objective, COmR was able to drive the frequencies sufficiently close 

to their bound8 that a feasible design (vith respect to the original 

probled var obtained. At this point, the orCqinal objective function 

was reinstated and COFMIR applied once again. 

Table 6 giver the optimized design obtained by this two-stage 

opt imiution procedure, vith the corresponding frequtnc ies aad the total 

weight. Prom the point of view of helicopter vibrations, the initial 

design of this blade is acceptable, since (except for the third mode) the 

numbtr/rev is far away fr m even integer values. The third mode is, 

however, near 6.0/rw. The frequency of the second mode does not satisfy 

the inequality constraints, but is not near an even integer multiple. 

Note that the final design moves the third frequency to 6.53, while 

keeping the other frequencies witkin the constraints. At the same time, 

the weight of the blade drops from 344.5 lb to 265.6 lb. 

6.2.2 Cyclic Modes 

The next prsblem to be studied is the optimization of the blade 

with respect :P cyc lic wdes of flapping. The nondimcnsir~alized frequency constraints are nor 

in which pl, p2 ,\nd p3 are the first three cyclic non-dimanslonalized 

f lappimaode frequencies. For cy; lic flapping modes, the boundary condition at the root corresponds to a pinned support. T3e frequeilcyplacement objhctive vr8 agein chosen by noting vhich frequency conrtraints were violr?.ed by the initial design. Noting the initial ftequency values given in Table 7, we define 

Table 7 gives the optimal design found by the two-stage optimizati~n 

procedure with the corresponding frequencits. The final vcight of the 

blade in shown to drop from 344.5 lb to 295.2 lb. 

6.2.3 Combined Collective and Cyclic Moder 

Next ve consider the optimization of the beam wit' ;:'"ect to 

coubined crl.lective and cyclic modes. TP.c., in each itcrdrto:- dn analyrim mar: be performed to find the frecuencies correspond.'-ng lo a fixed 

boundary conditiod; and then anether analysis mst be pes formed t~ iir.d 

the frequencies for a pinned boundary condition. All other aspects of 

the design problem remain the same as before. The constraints on the 

collective flapping modes are 

fie conrtraints on the cyclic modes are 

. For this problem, weighting factors are introduced into the frequencyplacement objective, 

The results of the optimization are given in Tab1.e~ 8. The cyclic 

modes of the initial design are vell-placed in the sense that they are 

aot near odd integers/rev, but the third collective mode is near 6.0/rev 

(the same as in the first example). Note that the final design noves the 
Table B lnltlal and Fln~l derl~n fnr collective nr~d cyclic flnp(blon mollrc 

Rotat III~ Speed : 324 RPH 8 

Y,.~I~cL n~,~;~~l~ta a 0. I05 x 10 Ib/ln 2 

Ax 1.11 Strcns ( 20,000 prl 

aasa of aoment 

Inert la 

) 0.5429~1

denrlty of box bran : O.000261 

E II-IUCII~ NO. 1 2 3 4 5 6 7 4 

___ ._-_.___________---------------------------------------------------------------------

I.I-II~: 111 (in) 14.4 14.4 14.4 43.2 28.10 28.80 28.8 2R.RO 4

___._-_-__----_____----------------------------------

Arvn tktment lo(unchanged) 48.6 93.38 64.8 5.76 1.76 1.47 1.03 I. 

nl lr~crt la 

((st4 ) Cox 8ems Inlt. 1.86 1.86 1.86 1.16 1.16 I. 6 1.16 

I (I) Flnal 1.36 0.51 2.17 3.05 0.51 9.51 0.51 2.89 

tlln. I,~mped uelght 4 86.02 3.06 9.19 6.13 6.13 6.13 6.13 

(Ib) 

lornlu.d ucalgl~t (I) Inlt. 42.72 16.02 41.38 20.46 9.23 6.46 6.13 6.40 1

Final 52-72 86.02 21.95 31.15 6.13 6.13 6.13 6.13 

----------------------------.-------------------------------------------------------------.---

Ann Hem Urlgl~t Inlt. 1.98 4.68 4.68 14.05 9.36 9.36 9.36 9.36 

(111) Flnnl 1.53 2.21 5.51 31.51 4.43 4.43 4.43 17.34 

Natural Frequoicler (NO ./rev) 

liner: (I) -- dealgn varlable 

Collective Flapping Cycllc FIal.?lng 

(flred boundary) (plnned houndary ) 

1.15 < p1 < 1.60 0.9 < pl < 1.14 

3.4: : 02 < 3.60 2.4 < pa < 2.60 

6.40 < p3 < 6.60 4.4 < p3 < 4.60 

Blade Vel~l~t PI PZ ~3 PI PI PJ 

(Ih) 

Init la1 344 .) 9 32 6.00 1.02 2.64 4.67 

Flnal 338.5 1.20 3.40 6.33 ' .02 2.64 4.58 

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

where t fir the top tl~lckneacl of box hear 

c#.oon44 < t < 0.732n 
third collective frequency to 6.33 while keeping the other frequencies 

in the 'safe* range. However, the weight of the blade only decreased 

from 314.5 to 333.5 lb, in contrast to the previous example in which the 

weight decreased to 255.6. The difference is caused by the larger number 

cf frequency constraints in this example compared to the previp~s 

example. 

6.3 SzWaTMgOUS WPIRC IXVPUNE AHD TORSLOB 

The next problem to be studied is that of optimiziry a teeteriq 

rotor blade subject to the following simultaneous constraints on frequencies (non-dimensionalized by dividing by the rotor rotational 

speed : 

1 collective flapping modes, 

2. cyclic flapping modes, 

3. collective inplane modes, 
4. cyclic inplane mder and 

5. tors iom 1 mode 

(The first lover bod for cyclic inplane aodes , 0 .lo, was later rcpalced by 1.0 in the problem fornolationr of the folloving sections). Because we are considering a teetering blade, the collective-flapping and 

cyclic-inplane -des can be ~odeld by cl-ed boundary conditions at 

the root, while the cyclic-€ lapping and c.ollect ivcinplane modes can be I 

mde led by pinnd boundary coad it ions. 

The elartic modulus, blade length, tpeed of rotation, and density 

of the box-beam material are unchanged from the values used before. In 

5,. 

addition to frequency snd autorotation constraints, the uial stress is 

constrained to be less than 20,000 poi. The value of the bound in the 

ratorotatiotul constraint has been chsnged slightly to 0.5429~10' ibIn the problem dercribed in section 3.. the thicknesses, ti, d. . 11 

ad dZi, of ixth the vertical and horizontal nlls of the box beam (see 

Fig. 2) arc 8llod to vary - that is, are also desig3 variables, with 

the following side conrtraints (in unitb of inches); 
Table 9 Data for Teetering Rotor blade 

~otet ins Speed t 324 RPH 8 Youngs bdulua I 0.105 x 10 lb/ln 2 

~~iol Stress ,( 20,W pol 

mans of moment 

Inertla 2 0.542911

denalty at hox haam t 0.000263 

Element No. 1 2 3 4 5 6 7 8 

------------"""-'"'-"-""'-----"---r----------------------------------------------------------------- 

Lu116t h (in) 14.4 14.4 14.4 43.2 28.80 28.80 28.8 28.flO 

t (In) 

Don Ream 

l)lmr~wion dl (In) 

________________^______________--__----------------------------------------------------------------------

Ihx (ln4) 2.93 2.94 2.92 2.62 2.24 2.09 2.03 2.03 

Arva and 

mas Y lox (1114) 4754 92.30 64.05 5.38 1.53 1.30 0.85 0.85 

rec~tneot of -3 i~~ert ia fix (mu8-ln)xl~ 0.760 0.770 0.765 O.685 0.586 0.547 0.532 0.532 

c.C y-axls 

bx (ru8-ln)xl~-2 0.273 0.273 0.270 0.200 0.OAO 0.040 0.022 0.022 

-----------------------------------------------------------------------------------------------

Ihy (1114) 12.79 12.AO 12.50 10.54 8.98 8.57 8.40 8.40 

Ared dnd 

III~SS loy (Id) 48.11 160.50 312.90 294.22 210.66 167.62 132.50 102.70 

muc.~c~t of 

inert!a Hbr(m1~~-ln)x10-~ 0.33) 0.335 0.327 0.276 0.236 0.225 0.225 0.225 

of x-arls -2 

Hoy (m116-1n)xlO 0.015 0.015 4.378 11.146 5.690 4.490 3.576 3.280 

-----------------------------.-------------------------------------.*------------------------------------------- 

nln. lumped uelght (lb) 96.54 55.77 2.69 .R.01 5.38 5.3 5.38 5.31 

u -- lumped uel~ht (lb) 96.54 55.77 6.95 11-63 7.68 5.76 5.38 6.l2 
The initial values of there variables are given in Table 9. In the 

problem8 direursed in section 3 -2 and 3 -3, the box beam dirnaions t 

dli and dZi are fixed at there initial values. 

Tabler 9 also giver data defining both fixed and initial stiffness 

ad inertia valuer of the blade. In the tables, 'Iox' and 'I ' repre0 Y 

sent the portionr of the flapping and inplane area moaents of inertia of 

the blade sec~ion which are independent of the design variables. *Ia' 

ad 'I ' are the area PoPants of inertia of the box be= - thus funcBY 

tions of the (initial) values of the design variables, wi, dli, dti, and 

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

a7 "' -a= ru~rry inertias of the box beam with respect to 

flapping and inplane and are calculated simply by multiplying the mass 

density of the box-beam material by the area .ornt of inertias. Wox and 

are the contributiotls to the rota- inertia of the section which are 

OY 

independent of the doign variables. Rote that since these contributions 

 cop^ from itm with different densities, a single uniform value of 

density cannot be defined for the Mo tern. 

3ther initial inertia and stiffness properties are also defined in 

Table 9. As belore, the lumped weights associated with the finite elements are taken as design variables; but, to permit greater latituds in 

placing torcional frequencieo and to mtch more closely the behavior of 

r true helicopter blade, a torsionaL spring is introduced at the blade 

root; and it. stiffness is taken as a design variable. The side 
constraints on the lumped weights are given in table 9 under the beading 

'v '. the side constraint on the torsional stiffneos consists of the min 

requitc~rnt that the stiffaerr be non-negative. 

Since o torsional made is involved, special treatment is given to 

GJ, the torsional rigidity, vhich is a function of all variables including t , dl, d2, the area of the trailing edge, and the lumped mass. The 

procaduzc for calculating CJ ir described in Append- A. 

Table 9 8180 containa the contribution of the 1-ed wight to the 

2 

rota- inertia. which equals the impd +.a8 (wi/g) tirs (bf2) . This 

expression ha8 been chosen to match the behavior of the true blade. It 

ir assttocd that the lumped weights cf the first tn, elemnt, contribute 

nothing to the rota- inertia. 

6.3.1 Variable Box Dknr ion 

As war done with the optimization involving f la2ping only, a twoStep opt initat ion procr dore is used, vhich involves a frequencyplacamnt objective followed by a weight objective. The frequencyplacment objective has tho general form 

in which tk ya is taken over thore f-equencies which are to be changed 

from their ... itial values to the desired values p, Values oi p,i and i 

the nighting factors wfi are given in Table 10 with the results of the 

optimiution. Examination of the values of lumped weight given in Table 

10 stow that wight is concentrated at the tip of the blade because of 

the autorotational constraint. , ~e ntremr at the f irrt element is clore 

to the stress constraint value of 26.000 psi. Thi total weight changes 
- 

I I 0 CI h I*CII*-l,n+ll -.I .- . -0 00 iZ2lZSI2~i~y~q A 2 "Cn = 9". yy A 0 v nn un 2 I d& I c'd I c'c j ems I co I c -m I 

I I 

I I I -C I I re I I. .I 

;--! 

I 

I I 

1 use 1 1 <? I 

l rr I 

I I 

I I 

I I a* I 1??1 

1-1 

! ! 

Olf 

lhhh 

... 

-a= 

U 0- rt CDVVV rl a uC)*rNa 

a&- &a& 

w5:v"v 

i?? 0-2 

... : m~clnm , C w v v 

&--No 

mu aaP =or 

,,I VVV 

I 

! .., 

IN- 

; "ir 

100 

I 

N OI 

"i a,, rn n 
insignificantly. from 345 to 339 lb. how eve^, the root spring shows a 

significant effect on the torsional mode. The spring constant changes 

5 5 from 6.5~10 to 4.10~10 in-lblradian. 

The placement of the frequencies is shown in Table 9,also. Note 

that the third collective mode of flapping, which was near 6.Olpeve 

moves to 5.67/rev; and the second cyclic mode of inplane, which was near 

7lrev. moves to 6.7/rev. Finally, the torsion mode, which was 3.871rev. 

moves to 3.4/rev. 

6.3.2 Fixed Box Dimensions 

We next consider the oroblem of modifying a blade which has already 

been csnstructed, but which has been subsequently found to have inappropriate natural frequencies. Since the blade is already built, the ocly 

way its dynamic behavior can he modified is through the addition of 

lumped mass and also through chmging the root spring. Thus, in contrast 

to the problem of section 3.1, here the box-beam weight. the flapping 

and inplane-bending inertia, and the torsional rigidity are constant. 

All scarti- values and fixed parameters are the same for this problem 

as in the previous section. Finding a starting design which satisfies 

all of the frequency constraints is a difficult task for this problem, 

and thus the frequency-placement objective is the only objective 

iunction used; the second pha~e (wit9 weight as the objective function) 

is never reached. Values of the weighting factors and frequency bounds 

which appear in the objective f'unction are given in Table 11. 

The final values of the lumped weights and axial strsases are shown 

in same Table 11, with frequencies, root-spring stiffnes3, and total 

weight, The most significant frequencies are the third colloccive flapping mode and the second cyclic inpJ.ane mode, which are seen to move far 
away from the undesirable integerlrev values. However, the first cyclic 

inplane frequency is near llrev. 

6.3.3 Variable Root Bending Stiffness 

The problem formulated in the previous section presented computational difficulties in that it was found difficult to find a design 

which satisfied all the frequency constraints simultaneously when only 

lumped aass and the root spring were used as the design variables. This 

difficulty p.ay be caused by the dominating influence of the root bending 

inertias. Thus. it seems reasonable to include the values of 'I at 

OY 

the root as one of the design variables. The frequency-placement 

objective is used throughout the optimization (the weight is not used as 

the objective), and all starting data and fixed parameters are given the 

same values as in Section 3.2. Vahes of the weightinq factors and 

frequency bou~ds which appear in the objective function are given in 

Table 12 with the results of the optimization. 

The results of Tables 12 differ from those of Tables 10 and 11. The 

lumped weight changes at the first, fourth, and Pifth eleuents. The 

stifhess of the root spring mores fro. 6.51~10~ to 4.32~10~ inlb/radian. However, the most significant effect is the chanqe of I 

OY 

4 (the bending moment of inertia at the root from 48.11 to 290.29 in . Table 12 shows that all frequencies are placed in tne safe range. 

6.4 Effect of Pretwist 

In this section, we will do the optimization of a beam which is 

pretwisted. (Tretwisted blade implies that the mot' 4s of flapping and 

inplane are ccupled). The procedure of analysis will be the same as in 

sectiorl 6.3. Data are idontical to those of secfion 6.3 save that the \ 

*Ir. 8 
I 

1 omI !'??? 

I W. h d - I 

I 

1 hhh 

I mbn I ... I Imm" 

I mmm 
ma- 

??? 

mna L 

I i i 100 1 : ?? I I d-. I 

! I 

I I I 

I I I 

I iWb' I 

I I I 

I 1 

I j I I 

I* I IP. I I I- I ! -1 

mln I 

52 UKI ! +r 

we4 I * U 

'rn10C 

\S .I.AY 

CIU I c& 
prqtwisted angle is new inoluJwl. 

6.4.1 Variable Box Dmim 

A two-step optlaization pr*oaedure is still used. The box bean 

dhen~ions and Xmpod waights as (well 8s tha root spring) are considarad as design variables, Tbls results 8- shown in Table 13. In a simliar 

fashion as results in se~tion 6.3, the weight is ooncentrated at the tip 

of the blade because of the autorotational constraint. Boot stress is 

close to the stress sunst,raint which ts 20,000 psi. The total weight 

doe8 nat decrease. Imtraad, it in2reases from 7Sq I3 to 354 lb, The 

5 .* 

spring constpn~ ahangeq5 f'rom 6.5~10 to 4.2110' fr-lb/rad. The ~econd 

ao11eutive mode of flapping movus from 5.89 to 5.67 noirev while second 

cyclio mode of inplane moves from 7.08 to 6.54 no/rev. 

6.4.2. Fixed Box Dimensions 

For a blade of existing constmction, only the iumped weights am 

oonaidered as deal& vwiables. '!'h~ restilts of sptlmizatlon are snom in 

Table 14. The total waiqht increases from 345 lb to 372 lh, The thi.Td 

oollective mode of flapping and the ssoobd cmXic mode of i~plane move 

into the safe range. 


7. ARTICULATED ROTORS 

7.1 Definition 

btioulated rotor blade uili be the subject of design blade in this 

seation. RefC231 The primary design variables am saw as before: the 

wjll thickness of box bear ud lumped weights. Because the blade is 

artioulated with a rigid hub, them is no distinction between collective 

or cyclio modes for flapping aad inplane. i%e blade is pmtwisted. The 

boundary condition for flapping is a hinge at the root. 'hem is root 

spring for to~iordl motions and an offset for inplane. Table 13 gives 

data for both the initial (and mlnimm) blade stiffnesses and inertiaa 

as well as for the initial variables such am box beam diamasions, lumped 

ueQhts, etc . 7.2.1 Variable Box Dlmansion 

Bax beam dimensions, l-ed weahts and root sping are taken as 

design variables. Tables 16 shorn the fntial and final results of the 

optimization procedure. The inplans frequenoy mves fio~ 4.84 to 4.69 

no/mv while the torsional mode moves from 4.25 to 4.43. The total 

weight drops slightly from 96.53 to 93.48 lb. The root spring changes 

m 2.41110~ to 2.8~10~. 

7.3. Fixed Box Boa8 D~osioas 

Bolt beam dimensions will be considered to be fixed in this section. 

Only Lumped weights and the root spring are taken as design variables. 

The final results are shown in Table 17. 
d djni rl I; 

I 

OIQ, a *I- m elm 9 

old 0 

I 

I 

I I 

'Old I N 

I 

I 

N -1- 

m 413lm Nlh a h

I 

31% 3 *I- N qln 9 

ojo 0 

I 

f-4 410 

n m10 m d Nlh N 

I 

el* 0 914 m 01" 0 

ojo 0 

I 

I 

h m1m c elm 

ct 1:: 

0 Old I 

I 

d djni A 

4 I 

a 

s x 

a 0 

aoo , .- IS s CF 0 w S u. Pi- 

-IC C Y ~IC C ' 31 cI* - I el- - Z E~Y *I- - - -1- - - -1g 

Table 17 Initial and final desigr: for flapping, inplane and torsional 

of Articulated rotor blade with box bean dimension fixe

El enent - NO. 1 2 3 4 5 6 7 8 

----------------------------- _____-----------------------------------------------------

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 ( d2 < 2.75 F 

___________________----------------------------------------------------------

Hin. 0.89 1.435 1.435 1.435 1.435 1435 1.435 1.43

lumped weiqht # I o. 718 9.088 1.978 1.435 2.352 5.852 6.342 6.57

4 lbl F 6.015 8.164 2.401 1.901 3.779 1.745 4.288 4.06

----------------.----------------------..---------------------------------------------------------------

~tress x lo4 1 1.39 1.32 1-64 2.54 2.84 2.60 2.12 1.55 

(psi) F 1.37 1.30 1.61 2.49 2.74 2.4 1 1.91 1.47 

-----------------------------------------------------------------------------------------

<t<<<<(< Natural Frequencies (No/rev) ))

Flapping 

t hinged ) 

Inplane 

(offset) 

T

t s

0.01 < p1 < 1.50 0.01 < pl < 2.50 

'I'i~ta 1 mass Root-Spring 2.23 < p2 < 2.63 4.30 < p2 ( 4.70 < 

We iqht Stif fness(W) 4.26 ( p3 < 4.67 12.30 < p3.< 12.70 

6 

9h. 55 2.41~10~ 1.03 2.70 4.51 0.24 4.64 12.30 4

94.89 2.70~10 1.03 2.57 4.67 0.24 !.43 12.30 4

. . . . - - - - - - - - - - - - - - - - - - - - - - . Weiqhtinq Factor A A A A L & 

Desired Frequency & A & A A A 4

NoLe : (Y) -- Desiqn Varlable (A) -- Zero 
8 .I ?O~TIOI 

In thir section, w wuld like to sbor whether or not the forced 

response of the blade can be adequately controlled, u we have usuwd, 

by our approach of 'frequency placmnt', that is, of restricting the 

natural frequencies of the blade to lie within nrrrov iaterpuls located 

away from certain irtqer ~lrltipler of the rotor speed. Also ve wuld 

exuine whether or not aerodynamic duping substmtially raduct~ the 

resonant peaks, in which care concern about avoiding resonances through 

proper selection of frequency rind- wuld be unnecesrary. Finally, the 

sensitivity of the optiul design to the choice of frequency window 

rill be studied. 

This investigation is carried out through tw, sowwhat overlappa, prbblem. First, the forced response of an initial (i.e., nonopttized) design is corpar . to the reaponre of a final design; carer 

wisb and without aeradynamic damping are considered. Hext, the responre 

of initial and f in81 derigns are waluted 8s a single natural frequency 

is varied (the others being held fixed). In each care, a forcing 

function containing brronicr of the rotor speed is applied. Again, 

cases with and without aerodbic damping wire conridered. The general 

finding 'WE these rtudiaa is that frequency placclaant is a viable mans 

of reducing vibration, although it is by no means the only method and 

sbould be ured in con juact ion with others. 

a .2 ~sponsz vtwm FRIZQDL~~CY 

The eqwtionr of, ootioa for the finitulenmnt representation of a 

r 

rotor blade, subjected to an external excitation, may be vritten in 
vbere 

[MI - mar8 rrtrix, 

{X(t)} = colum rector of nodal dirplacmnt, 

IcJ - dupiag uttix, 

1x1 = rtiffnur matrix, and 

{P(t)l - forcing function colum vector. 

The forcing function may in turn be apre~red u 

(F(t)} = {v ) ti= 

0 , 

*ere 

w fortzing frequency, d 

V, = forcing .rplitude. 

After oolc calculation, it can be rho- that the mlitude of the 

rerponre - written u {XI, independent of t - cur be given as 

In thar rection, the rerponre of both the initial and final 

(optimal) d3rignr to an external forcing function ir rtudied 88 the 

frequency of the forcing function ir varied. Blades both 4th and without aerodynamic damping are conridered. To formulate these problems, 

conrider tbe forced bet~avior of a rotor-blade. On17 the flapping respow in oonaiderrd, Tho Uplane response is inferred !ME the results 
without damping, sinae there is ltttle aerodyn8mio damping in tho inplane direation. 

Fig. I131 shnn a plot, ~f the forcing amplitude Po [Ref. 251 urud 

a the Study. GiVOn tho f0XU- ~litude, ~d0ulatO the -8pOM8 

of each node of tho finitnlemnt representation of tho blade .s the 

value of the forcing frequency, w. ia varied. The tip if?~lite-uleasnt 

node farthest fmm the hub) respotme ia of special interest. Before the 

results obtalnd from this study am presented, it is usaiul to aramiae 

the froquanay placement results which am described in cha~ter 6 (808 

Table 10). The results for the frequencies {in units of cycles/rev) are, 

for flapping mode only, 

MODE XrJITIAL DESIGN FINAL DESIGN 

Blade dime?wions are given in Table 9. 

h; 

The frequencies in the above table cormspond to the symwtric 

modes of a teetering rotor. Thus, only h.rw!kios of the rotor spoed 

have been aonsidered aa forcing frequencies. As a result, the optimized 

blade (Final design) finds the third mode moved am7 from the critiaal 

6.0/rev (Ih.oa 5.89 to 5.67). S~arlj, the movement of the seaond mode 

to 3.09/ret remvos it frm 2.3 and 4.OIrev. In the comparison study to 

follow, howaver, we will ~rpply the entin spectrum of frequencies to 

thqs blade (not Just eves hmaonias) . Thus, the 'Final Design ' our no f 

lower be oonaidered optimum. A c~arison of the two blades, however, 
3.0 0.: 0.2 0.3 c.4 0.5 0.6 0.7 3.8 0.9 1.3 

Udr Statian :&'It) 

Figl~rr 13 Raaair Vari. t ron of Forcinq Function 
doas indloate the ntrong effeet of lusoa8noe beaauae each aaae hrs a 

distinat resonanas (6 .nd 31rev). 

We shall now aonsider the results of the present study. Fig. 14 

8h- the tip W8POM08 of both the initirl desun and fi~l (i.~., 

optimlzd) deaign rs iunationa of tha foming fmqueaoy. Aerodym~'~ 

damping h.s been neglaotsd (Alternatively, the mmrlts cur be interpreted as giving the inplane response). It aan be seen that now 1.18 

oyales/rev, the mspomes of the twc designs are very slmllar. However, 

the responses corresponding to the second and third modes differ signiflaantly. For cbx.ngle,ln the second mode, the peak af 3.22 oycledrev 

(initial design) mwes to 3.09 oycleslrev (final design)/ Slmllarly, the 

peak of the third mode moves from 5.89 ayclealmv to 5.67 oycle/rev, 

whlilh is espeaially important sinoe it is higS1y desirable to keep the 

irequenay am9 frm the integer frequenay of 6 cycles/rev. We conclude 

from thase results that the frequenay plaaement approach does have a 

signifioant effeet on the forowl tip msponae when damping is not considered . 

Raxt, the offeat of aerodynamio damping is oonsidered. The presence 

of the damping Implies that results to be presented aorn.~pond to flap 

ping. hthe~8tia.l details of the damping forwtlation are available 

tRef.181. The effect of aerodynamla damping on reduoiag the resonant 

peaks of the tip re8ponse of the initial blades is shown in Pig. 15. 

Fig, 16 ahow the dampod rosponaes of both the initial and optimized 

bladea so that the efieet of frequency placement can be studied. It is 

intersstitq to obssme hem that when damping is included, gg apparent 

advantage is gainad by cptimizing the blade, at least in terms of 

reduaing the tip mapotme, oxoept in the 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 ) 

Fiyure 14 Ilp Response Uersvs Forclng Frequency for botn 

initial and Final Desaqns Witnout Dampins 
0.0 1.6 2.0 3.0 4.0 5.0 6 

a Forcing Frequency ( Nc. /Rev ) 

Figure 15 Tip Response Versus Forcing Freq-!n=y for Iflltidl 

Design Both With and Without wing 
Figure 16 Tip Respwe Verw Forcing Freq~~~~cy for Both 

Irutial and Final Design With Danping 

8.0 1 .O ,.O 3.0 4.0 S.0 - 6.0 0 

Forcing Frequency (No. /Rev 1 
which a thirtyfive percent reduction occurs. However, we must also 

txamina the effect of optimization vhen the response is measured by the 

average shear force existing in the blade. 

Consequentl~, the sheariog force in the blade is considered next. 

As a measure of the 8versge shear in the rotor, we consider the sum of 

the aquares of the shear force (abbreviated SSS) , 

In this section, Yi represents the shear force at node i in the (tenelement) finite-elanant mdel. !!ate that the root shear is necessarily 

included as one of the term8 on the right-hand side of the equation, so 

that a large value of root shear will cause SSS to also be large. 

Pig. 17 show the variatian of SSS with rcrpcct to the forcing 

frequency for the initial design with and without aerodynamic damping. 

Fig. 18 shows the sm quantities for the final (optimized) design. Pig. 

19 compares the quantity SSS corresponding to initial and final designs 

when aetodpsaics is considered. Inspect ion of there figures show that, 

in corrtrast to bebvior of the tip response, the shear response is 

signif ieant ly affected by changing blade frequency, even vhen 

aerodynamic damping in included. Tht 3/rev loads are increased by fifty 

percent due to the povment of w2 from 3.22 to 3.09/rw. Similarly, the 

6Irev loads are reduced by seventy percent due to the moveacnt of w3 

from 5.89 to 5.671rw. Thus, even vith damping, frequency placement is a 

powerful driver of loads. It follow8 that frequency placement can be 

jurt if iably considered an important part of blade optimization. 
0.0 1 .O 2.0 3.0 4 . 1.1 5 . it 6.0 

Forcing Frequency (No./Rev) 

Figure 17 Sue 3f Squares of Shears Versus Forcing Frequency 

for initial Design aotn Yiih ana Yitnout Damping 
FINflC OESlGN (UITH f3ND WIIHOUT flERO. 1 

.A 

Forcing Frequency ( fb ./Rev ) 

-. tl3ure 18 Sum of Squares of Shears Versus Forcins 

Frequency for Final Design 3atn Xi:?. and 

Without Damping 
0.0 1 .O 2.0 3.0 4.Q 5.0 6 0 f .O 

Forcing ~re~uency ( No. /Rev ) 

Fiqcce 13 Sun oi Squares ot Shears versus Forclng Frequency 

fcr 80th Initial and Frnal Deslyns With Damping 
8.3 RESPOUSE VEBSOS PUcEHERT 

In the study just described, the responre of the blade to changes 

is the forcing frequency was coasidered, llav we consider r different 

approach. In affect, ue exmine bar the blade tespondr to A forcing 

fttnctiot 'during the optimization procedure' -- in the sense that during 

optiaiut ion, the optimizatiaa algoritkm vazier the natural frequency of 

the blade (to force it to satisfy the frequency constraints). IE obtaining the result8 to be presented nut, w si~~lated the optimization 

procedure by vaqiag the mtoral frequency. Thus we can obrer~e what 

bppeur to the forced resDonse during frequency placement. 

The formlation of the approach is a8 follows. Through appropriate 

tr.nsfomatioms (described in Appendix 11.31, the system mass nutrix can 

be written 80 

and the ryrta rtiffnesr matrix as 

!KI - [nl IUI diagl (wiz) I tulT [HI (32) 

were wi are the natural frequeacies of the system, [u] is a matrix whose 

columm are eigcnoectorr , and the notation 'diag' indicates a diagonal 

mstrix (a11 offdugom1 terms vanirh). Prom examination of these 

uprerrions, it can be seen that the stiffness and MIS matrices caE oe 

conridered functionr of the natural frequencies. Thus ic becomes 

porsible to fix all frequencies but one, and then study the response of 

the rystem ar that one frequency is varied with mode shapes also held 

fixed. In particular, the rerponre to the following forcing function 

will be studied: 
r - the rotor rpd, and 

where {Yo] was defined pmviously in Fig. 13. Since the arguments of the 

exponentials are integer multiples of w, resonance will occur at harmonics of the rotor speed. The particular forcing ~nctior given above 

is known from empirical observation to provide an approximate, but 

physically realistic representation of the radial qnd harmonic variation 

of the amplitud.3 of the load on a real blade. As mentioned in this 

section, the blade response will be defined by the tip displacement and 

the sum of the scpares of the shears (ucept that, here, the n = 1 term 

has been omitted from the expresslans for calculating tip displacement 

and shears because this term reprisents p, tip-path plane tilt that is 

controlled by the pilot Or:;. trimming purposes. It is not part of the 

true vibratory loads we are considering. 

Results for the pro3lem just formulated are shown in Fig. 20, where 

the sum of fhe .squares of the shears is plotted as a function of w2, the 

second natural frequency, with the other natsral frequencies being 

fixed. This figure corresponds to the initial blade design (blade dimensions are given in Table 9. Fig. 21. ~hows the same quantity for thz 

case where the third natural frequrncy is varied. It is interesting to 

note that the response curve '.or the dampad case in Fig. 21 lacks 

resonant peaks - apparently '.he damped response is so completely 

dominated by the resonance of the second natural frequency, which is 

fixed near 3/rev, that the (damped) resonant peaks for the third frequency are negligible by comparison. It should be noted that the 
AND UlTClOUT AERO. J 

Yithout Aerodynamic 

4 I 

I 

LO. . I 

I 

I 

With Aerodynamic I 

I 

I 

I 

I 

1 

I 

W2 (No./Rev) W1 and W3 am fixed. 

Figure 20 Sun of Squares of Snears Versus Second Natur.31 Frequency 

for Initial Desi3n both With and Wi thol~t nbrprr~g 
W3 (No. /Rev) W1 and W2 ar*e fixed. 

Figure 21 SIJ~ of Squares of Shears Vers.rs Third Natural Frequency 

for Initial Design both With ~nd kithout Dampiny 
response of the original design is reapresented by only one point on 

Figure 20 or Figure 21. This value can be found by taking w, = 3.22 or 

w3 = 5.89 on those f~ures, since these am the initial design values 

(Table 10). 

For the final (optimal) design, the analogous quantities are plotted in Flgs. 22 and 23. Again, no resonant peaks are present in the 

damped response when the third natural frequency is varied. Compnrison 

of magnitudes of ordinates in Figs. 20 and 22 (no damping) shows that 

the overall shear measure is reduced in the final design in the regions 

away from resonance. Also, the choice of scale OP the vertical axis in 

Fig. 22 highlights the effect of frequency placement. Note that by 

inspection of Flgs. 20-23. a designer may select the design frequency 

which minimizes the average shear as measured by the SSS. 

One of the most interesting results of Fig. 22 is information about 

the width of valleys and peaks, since this gives design informatioa. # 

First, let us examine the nondamping curve (inplane response). Here, 

the minimum points are nearly at the centers of the regions (2.551rev) 

and (3. Sr /rev) . The frequency windows to maintain no more than thirty 

t, 

pement incre%+e in loads are 2.48 - 2.701rev and 3.40 - 3.701rev (plus 

or minus 0.lSlrev) - a fairly narrow window. For the damped curves 

(flapping response), mimima are also near the one-half points, but the 

window for thirty-percent increases are much wiuer: 2.20 to 2.90lrev and 

3.20 to 3.801~s~ (plus or minus 0.30Irev). Stated another way, inplane 

frequencies should be no closer thac a 0.4lrev from integers, but flapping frequencies may be as close as 0.2 from an integer. It should be 

amyhasized that these observations apply to this particular example and 

z,ay not be generalized for other frequency constraints. 
c. U2 (NO./R~V) Wl and W3 are fixed. 

Figl~rt 22 Sum of Squares of Sncars Versus Second natural Frequency 

fcr Final Design both Uitn and Yithout Daa~ing I 
I 

Without Aerodynamic I 

I 

I 

W3 (No./Rev) W1 and W2 are fixed. 

20.0, 

-. Figure 23 Sum sf Squares of Shears t'ersus .nFra Sdtural 

ior Fin~i 3esi~n Borh Xith 3nd '.;;:nca: Zl.:~p;z 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

With Aerodynamic I 

-- 

I 

1 

I 
Another conoluaion to be drawn from the above reaults is that the 

undamped responts curve has very flat-bottomed lvalley' when one of the 

fixed frequoncies is near an integer vrclue (of. Flga. 22 and 23). 

8.4 RESPONSE DUE TO EVEN-INTEGER HWNICS 

In this section, we still study the response of the blade due to 

the change of foming h.equenay. However, the fcming frequemp xi11 be 

a little different from the one in section 8.3.1, Fe will consider, only 

the even integer foroing frequenaies for whioh the collective ADC ' . ard 

optimized. Thua, the forcing finotion may be written as 

i2wt + v4e + v6e i6wr + v8e i%wt [F(t)) = V2e 

where 

w = rotor speed 

Vn = lln Vo 

Since the argursrants of the exponentlala are even integer multiples of w 

only, the resonance will expectedly oocur at an even integer of 

harmonics of the rotor speed. The response of shear stress will be 

studied in this section. 

Results for this case are shm in Flg 24, where the sum of the 

squares of the shear harmonics is plotted aa a finction of w2 (wl md w3 

are fixed; wl = 1.18 and wg = 5.89). The figure corresponds to the 

initial blade design. Resonance peaks oocur at 2 and 41rev. Thus, for 

the shear response, a second frequencies around 3.0 would be the 

suitable choioe for w2 as design frequncy. Pig 25 is a similar 

comparison for a variable value of w3 ( wl = 1.18 and w2 = 3.22 are 

flxed) . Here* the rssomnoe is at 6.01rev, lad w3 = 5 or w3 = 7 would be 

ideal. 
! \ 

Without Aerodvr-amic 

I 

I 

I 

i 

. I 

\ I Initial Design 

I 

With Aerodvnarnic 

I 

,*,? 

\ 

\ 

\ 

/ \ 

\ 

/ 

i 

/ A'' I 2.5 lb2 

Ui(NO/RN) Ul(tl.l& J AND W3iSa99) AUE FIXED 

Figure 24 Sum of Squares of Sheprs Versus Second Natural Frequent 

fcr Initial Design Both With &id Without Daaping 

(where Forcing Function is Even Insegsr Multiple) 
k ~nitial Design 

3 

[NITlAL DESIGN (UITH NU WITHOUT CIEPOOYruAMICS) 

3,- ------------ --7 ---- ------------ ---- 

I 7 

I 

I I 

I 

Figure i5 Sum of Squares of Shears Versus Third Natural Frequency 

for Initial Des' -. Both with and without Damping 

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

0 

I 

1 

I 

I 

I 

I 

I I I 

I i I 

, 

I 

! 

i 

I 

0 .. I 

'Jithout Xerodvnamic I 

I 

-: 

I 

I 

I 

I 

I i I With Aetodyaacic I 

! 

I 

I 

1 

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

I I 
ideal. 

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

Again, it is 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 final derign. In either case, hovever, we see 

the senritivity of vibrations to frequency, even vith aerodynamics. 
Flyvre ?,h SIJ~ of Squares ol Snears Versus Second N3tl.lrii Frquency 

Tar Frnal Desrgr~ botn wit,n .and Yttnol~t bamorng ' 

(unere Forcing F~~rrctlon 1s Even Integer nuitip~el 
Figure 27 Sum of Squares of Shears Versus Third Natural Frquency 

for Flr131 Design both Uith and Without Dinpin3 

cuhors Forctnq Function 1s Even lnteger nultiple) 
9. SUMMARY AND CONCLUSION 

The optimizatior technique works very successfully on the design 

of rotor blades even when there are as many as 55 constraints. The most 

efficient optimiz?.tion orocedure involves 2 steps. In the first step, 

the objective function is based on frequeccy placemect with appropriate 

3tructura.L const-aints. In the seond step, the objective Function is 

weight with frequency wis~dows as constraints. 

Ar fi%r as the optimization of helicopter blades is concerned, the 

appropriate constraints include autorotational inertia, axfal stress, 

geometric limitations of the cross-section, and the placement of mass 

center forwad of the quarter chard. 

Proper choice of input de2a can ensure the optimization runs 

smoothly and converges faster. Although we have up to 15 full constraints and 40 side constraints at one time, the program works very 

well. The reason may be due to the fact that the input data are 

practical encugh to meet (or to be close to) most of the constraints 

evep before the optimization starts. However, if we start the optimization with random input data, the results may not be as good as expected. 

The forced response of the blade can be adequately controlled, as 

expected, by the approach of 'frequency placement'. 

The optbization tachniqc~s result3 in realistic designs by placement mass at antinodes or nodeq, by adding stiffness at antinodes os 

nodes, and by placing mass near tne tip to achieve autorotational 

inertia at minimum weight. 
10, ACKNOWLEDGEMENTS 

The author would U.ke to take this opportunity to express his 

gratitude to Dr David A Peters for his guidance and ericcuragement 

throughout this work. A special acknowledgement is owed to Dr. Mark 

Rassow and Dr. Alfred Rorn for their discussion. This research was 

supported by NASA Grant # NAG 1-250, 
7 

11. APPENDICES 
size of tip ausr 

area of croas-section of box beam 

width of box bem 

blade chord 

dasuping matrix 

width of lumped mars 

wall thickness of box beam 

weighting function 

flapping stiffness 

inp lane st if fnes a 

safety factor 

finoi design 

fort ing f unc t ion 

gravity 

torsional stiffness 

height af box bean; 

initial desi~n 

area ~nt of inertia of box beam 

cotst=: area moment of inertia of blade 

maas momcnt of inertia of box beam 

constant 1388s moment of inertia of bLade 

stiffnear matrix 

area moment of inertia 

mass momcnt of inertia 
length of an element 

mass matrix 

blade first three frequencies (nolrev) 

distance from rotat ion axis 

length of the blade 

box beam thickness 

SSS sum of squares of shear fcrce 

box beam thickness 

tens ion force ' 

kinetic energy 

total- energy 

displacement of degree of freedom of element 

unit matrix 

forcing amplitude 

displacement in yx plane' 

displacement in u plane 

blade frequencies 

weighting factor 

coordinate axis and length parameter 

blade twist 

pretvist anqle 

mass density of the blade 

sma 11 parameter 

nt 

rotating speed 

maximum stre-? 

mass of skin 
density of lumped mass, box beam, honeycomb 
Calculation 0% Torsional Stiffness, GJ 

The blade cross section is shown in Fig. 1 and is idealized into a 

two cell torsion ~QX in Fig.28. Although the bending and torsional 

inertia inolude the contribcltion of all masses (including the filler 

elements), the torsional stiffness is based only on the structural bcx 

and thin-skin elements. The torasional stiffness, GJ, is based on classical thin-walled closed section theory.(Ref.26) The effects of warping re 

and distortion of the cross section are neglected. The equations are 

briefly summarized as follows: 

Considered cell i, having an enclosed area of Ai, and thickneaa t. 

The length along the circumference is maasrrred by c, and the shear 

stress is denoted by s..l%e torsion constant is 

where dc is the line int~gral along the entire closed box, and T 

represents the total torque applied to the entiye system. We also have 
vhere c is the conrtant shear flow in the i-th box. Denoting the single 

adajcent box as k, there will be a ctnmmn wall, ik, betweeu the two 

boxer. The sheor f Low in the c-n -11 will be the difference of sheor 

flow qi and qk. Thus, for the i-th box, 

and for the k-th box 

Defining q =Q.(Z?/J) and n = I(dc/t), we can reform j 3 jm jm 

Eqn.3 to read 

a. .Q. 11 1 - 'i 

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

Thus, the torsional constant, J, is given by 

3pecifically, for the tuo cell box shovn in Fig.28, cal bcz i the 

box beam and box 2 the equivalent trailing edge. Denote the 

sirtumfermee ot box 2 by C. The areas, A1 and At are given. 
Then, from the other dimenrioar, 

Then, solving Eqr . 4. .ad 4b giver 

Thar, all qaurtitier in Eq.6 are known and the torsional constant, J, 

c.n be ev8luated. 
- * 

h 

* 

Ce! I I : Constant* thicknesses for sides of box beam 

represent weighted effects of variable 

tt lckness elements. 

Cell 2: This cell represent3 canfiguration cf 

skin and tisiling edge. 

Fig. 28 Idealirsd tvo-cell model for calculation oi torsional s:iffxeso 
3erivation of Wasr and Stiffiess Watricer 

a. Functionr of Natural Frequencies 

Define 

IK*] = IMI-~/~IKI IHI-~~~. 

* * 

and constmct a rquare matrix [U 1 by using the eigen~ectors of [R 1 as 

columa. If the eigenvectora rre nomlized to the identity matrir, that 

is, if 

r$Pr~*i = r11, (47) 

i, then fo'.lorr that 

2 tdiT[t~[$l - diag i (w. 1 I I, 

where r.' are the eigcnvalues of K I. 

1 

Next, Pet 

from which it folluw ehat 

-112 forT - [a*lT [MI . u - [~*l*[.l'fZ. 

~ulT~wl~cl - [I?. 
Finally, then, the etiffness and mse matrices can be written as 

functions of the eiganvalues, r 

1 

'- - 

and 

Note that the eigenmtors. [a], and eigenvaloes, viZ, appearing on 

the right-hand aide vere originally calculated from the stiffness and 

nus# matrices, [K] and [MI. If ws connider only relatively small changes 

in the frequencies, w. then the eigmvectors should relatively 1 ' 

unchanged. Thur the last two equations for EM1 an6 [K! with [u] held 

fired can be considered ar erpredsing the mas and stiffness matrices aa 

uplic it functions of the natnral frequencies . 
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25. Private information from a Helicopter Company. 

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Biographical item on the author of the thesis, Hr Tbthy W H KO 

1) Born Septder 16, 1953 in Bong hug. 

2) Attended National Cheng fang University in Taiwan, R 0 C from 

1973 to 1978. Received Bachelor of Science Degree in Mechanical 

Engineering in M.7 1977. 

3) Attended Wuhington University in St. Louis, Missouri from 

August, 1978 to present. Recieved Master of Science Degree in 

Mechanical Engineering in EI.7 1980. Working on Docotor of 

Science Degree in Machincal Engineering. Avardcd a Graduate 

Traineeship and Research Assistmtship from September to by 

1986. 

4) Member of American Helicopter Society. 
Short Title : Optirman Design of Rots; Blades , KO, D.Sc. 1984