Aeroelasticity and Structural Optimization of

I

NASA

Aeroelasticity of Composite Swept Tips K. A.

Yuan

and

P.

Contractor

Report

4665

and Structural Optimization Helicopter Rotor Blades With

P. Friedjnann

'qp

(NASA-CR-4665) STRUCTURAL COMPOSITE _TTH SWEPT _33 p

AEROELASTICITY OPTIMIZATION OF HELrCOPTER ROTOR TIPS (California

AN_

N95-20262

_LACES Univ.)

Unclas

Hi/05

0050098

Grant Prepared

for

Langley

NAG1-833

Research

May

Center

1995

NASA

Aeroelasticity of Composite Swept Tips K. A. Yuan University

Contractor

Report

4665

and Structural Optimization Helicopter Rotor Blades With

and P. P. Friedmann of California

• Los Angeles,

California

National Aeronautics and Space Administration Langley Research Center • Hampton, Virginia 23681-0001

Prepared

for Langley Research Center under Grant NAG1-833

May

1995

Primed

NASA

copies

Center

800 Elkridge Linthicum

available

for AeroSpace Landing

Heights,

(301) 621-0390

t¥om

the

following:

Information

Road MD 21090-2934

National 5285

Port

Springfield, (703)

Technical Royal

Information Road

VA 22161-2171

487-4650

Service

(NTIS)

PREFACE

This

report

onstrating

the

tailoring that

structure

The Aerospace

and

by

Grant

NASA

the grant monitor

The principal

sertation; prove

Finally, during

comments

this

the

certain

was

Dr.

hereby

It is shown in the

primary

in the vibration

carried

out

in the

at UCLA,

and

express

appreciation

essentially were

made

report.

authors

gratefully

acknowledge

Professor

orientation

from

it into this

from

aeroelastic

blades.

H. Adelman,

turning

research,

and

at dem-

levels

their

NASA

Mechanical, it was

funded

Langley,

as

to the grant

suggestions.

constitutes changes

rotor

reductions

for this sponsored

This

optimization

ply

Department

and

aimed

be achieved.

report

with

authors

investigator

however,

it, before

1-833

The

P. Friedmann.

can

Engineering

NAG

for his useful

Peretz

in this

Nuclear

monitor.

tip; remarkable hub,

research

helicopter

of composite

blade

described

analytical

for structural

composite

the swept

at the

research

tip

combination

and

flight,

innovative

potential

in swept

a judicious

in forward

in detail

remarkable

present

by

blade

describes

L.A.

iii

Schmit,

research the to the

activity first

was

author's

dissertation,

the

help

and

Jr.

and

Dr.

Professor Ph.D.

dis-

so as to im-

advice

received

C. Venkatesan.

CONTENTS

List of Figures

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

List of Tables

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

Nomenclature

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

SUMMARY

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

ix xvii xviii xxx

Chapter I.

pa_e Introduction and Literature Review Introduction .................................. Literature Review ..............................

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

Structural Modeling of lsotropic Rotor Blades Structural Modeling of Composite Rotor Blades Structural Modeling of Swept-tip Blades ........... Structural Optimization for Vibration Reduction Objectives of the Research ....................... II.

_:'RECEDING

1 ! 3 ...... ..... ....

Model Description and Coordinate Systems ............. Basic Assumptions ............................. Ordering Scheme .............................. Coordinate Systems ............................ Nonrotating, Hub-fixed Coordinate System ........ Rotating, Hub-fixed Coordinate System ........... Preconed, Pitched, Blade-fixed Coordinate System Undeformed Element Coordinate System .......... Undeformed Curvilinear Coordinate System ........ Deformed Curvilinear Coordinate System .......... Preconed, Blade-fixed Coordinate System .......... Coordinate Transformations ......................

...

3 6 13 16 19 21 21 24 25 26 26 26 27 27 28 28 29

Rotating to Nonrotating Transformation .......... Blade-fixed to Hub-fixed Transformation .......... Element to Blade Transformation ................ Undeformed Curvilinear to Undeformed Element Transformation ......................... Deformed to Undeformed Curvilinear Transformation

31 32

Deformed Curvilinear Transformation

33

PAGE BLANK;NOT

FiLM_

to Undeformed .........................

29 30 30

Element

V

INTENT!O,%,\LLyBLt'_,r'_V

Preconed, Blade-fixed Transformation

111.

to Preconed, Pitched, .........................

Structural Modeling of the Composite Rotor Kinematics of Deformation .......................

Blade-fixed 33

Blade

........

34 35

Strain Components ............................. Strain Components in Curvilinear Coordinates ...... Strain Componcnts in Local Cartesian Coordinates ... Explicit Strain-Displacement Relations ............ Constitutive Relations ...........................

IV.

Formulation of the Finite Element Strain Energy Contributions Kinetic Energy Contributions External Work Contributions

Equations of Motion ...................... ..................... .....................

....

Summary of the Partial Differential Equations of Motion Finite Element Discretization of the Equations of Motion Element Matrices Associated with the Strain Energy Variation ............................ Element

Matrices Associated with the Variation ............................ Element Matrices Associated with the External Loads ........................ Summary of the Beam Finite Motion .............................. Local-to-global V°

VI.

Coordinate

Element

Kinetic

Energy

Virtual

Work

39 39 40 45 51 55 56 74 97 101 105 109 1 !0

of 1 !0

Equations

Transformation

of ! 11 .......

112

Incorporation of Aerodynamics in the Equations of Motion Aerodynamic Lift and Pitching Moment ............ Blade Velocity Relative to Air .................... Blade Pitch Angle with respect to Free Stream ........ Aerodynamic Forces and Moments in the Undeformed Element Coordinate System ................. Treatment of Reverse Flow ......................

118 118 121 124

Method

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

128

Treatment of the Axial Degree of Freedom .......... Free Vibration Analysis ........................ Modal Coordinate Transformation and Assembly Procedure ..............................

128 132

Hover Analysis ............................... Forward Flight Analysis ........................ Trim Analysis ............................. Distributed Loads on the blade ................

135 139 140 145

of Solution

Inertial

Loads

Aerodynamic Rotor Hub Loads

125 126

133

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

145

Loads ...................... ..........................

146 147

vi

Coupled

Trim Harmonic

Vibratory Stability VII.

and Aeroelastic Response Balance .....................

Hub shears and in Forward Flight

Function

153

Moments ............. ...................

156 158

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

Formulation of Approximate Detailed Description of the

Problem Optimization

Model Verification ............................... Validation for the Case of Hover lsotropic

Blade

Blade Response Blade Stability Vibratory Hub Free

Vibration

and

.............. Process

......

171 174 177 181

...........

............................ ............................. Loads ....................... Behavior

in Hover

182 ! 83 185 ........

Free

Vibration Analysis ........................ Influence of Ply Orientation ................... Single-cell Composite Blade ................ Two-cell Composite Blade .................. Effects of Tip Sweep and Anhedral ............. Aeroelastic Stability in Hover .................... Effects of Swept Tip ........................ Single-cell Composite Blade ................... Two-cell Composite Blade .................... X,

Aeroelastic Behavior Blade Response Trim Variables Vibratory

Hub

in Forward Flight .............................. ............................... Loads

Blade Stability Combined Effect XI.

Structural

XIl.

Concluding

References

Remarks

200 203 205

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

Results

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

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

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

187 187 187 188 189 191 192 193 196 197

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

............................... of Sweep and Ply Orientation

Optimization

166 167 170 171

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

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

Aeroelastic

161 161 162 163 165

Single-cell Composite Blade ................... Validation for the Case of Forward Flight Trim Results ..............................

IX.

Using

Structural Optimization for Vibration Reduction ......... Statement of the Optimization Problem ............. Design Variables ........................... Constraints ............................... Objective

VIII.

Solution

206 .......

207 210 212 223

228

vii

Figures

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

238

oa_e

Appendix A.

Comparison

of the Transformation

Deformed B°

Finite

and

Element

Element Variation

Finite

Element

for the

Local-To-Global

Matrices

Between

Coordinate

Systems

Composite

Beam

...... Model

Matrices Associated ...............................

with

the

Strain

Matrices

with

the

Kinetic

......................... Associated with

the

Virtual

Energy Variation Element Matrices of the

C°

Matrices

Finite

Finite

Undeformed

External

Associated

Loads

Transformation

...

370

Energy 370 381 Work

...................... Matrices

Transformation

for Rotational

Transformation Freedom

for the Vector of Nodal ...............................

viii

365

Degrees

394 .............

396

of Freedom Degrees

....

396

of 398

LIST

OF

FIGURES

oa_e

_ure

1. I

Rotor

2.

Nonrotating, coordinate

1

blade

with

tip sweep

hub-fixed system

2.2

Preconed,

2.3

Undeformed

element

2.4

Undeformed

curvilinear sequence

pitched,

3.

1

Deformation

4.

1

Motion

4.2 5.

Finite 1

blade-fixed

nodal

and

degrees

Components

of aerodynamic

5.3

Reverse

6.2 7.

Forces 1

7.2 8.

8.2

flow region

Finite

element

Organization 1

helicopter model

240 ...........

241

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

system angles

242

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

243

.................. elastic

244

axis

during

the 245

of freedom

velocity

relative force

.................. to the

acting

air

246 ............

on the

blade

247 .......

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

of a four-bladed on the

hub-fixed

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

5.2

Schematic

Euler

239

rotating,

system

on the deformed

of blade

1

and

system

coordinate

Components

6.

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

coordinate

coordinate

displacement

element

anhedral

coordinate system ................................

of an element

virtual

and

helicopter in steady,

for two-cell

of the optimization

level

249 ...................

250

flight

251

composite process

.............. cross

section

.....

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

Nonlinear function

equilibrium position of blade collective

of isotropic blade pitch ......................

Imaginary function

part of hover eigenvalues of isotropic of blade collective pitch ......................

ix

248

252 253

in hover,

as a 254

blade,

as a 255

8.3

8.4

Real part of hover eigenvalues of isotropic blade collective pitch .............................. Effect

of axial

isotropic 8.5

Effect

of axial

of isotropic 8.6

Effect

isotropic

mode

mode

blade. composite

on the nonlinear

in hover.

blade.

of axial

Single-cell

mode

blade

on the

Analysis

on the

equilibrium

Analysis

part

with

rectangular

position

substitution

part

8.10

8.11

Trim variables pitch setting Trim variables inflow and Trim

variables

of hover

pitch 8.12

setting

Trim variables inflow and

257

eigenvalues 258

eigenvalues

of

substitution

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

259

box beam

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

260

for soft-in-plane isotropic .....................................

blade

isotropic

blade

in forward

as a 261

flight; 262

for soft-in-plane isotropic blade rotor angle of attack ...................... for stiff-in-plane

of

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

Real part of hover eigenvalues for single-cell composite function of (thrust coefficient[solidity) .................. 8.9

of

.......

of hover

with substitution

real

as a function

256

with

imaginary

Analysis

blade,

in forward

flight; 263

blade

in forward

flight;

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

264

for stiff-in-plane isotropic blade rotor angle of attack ......................

in

forward

flight; 265

8.13

Blade

tip response

for soft-in-plane

isotropic

blade

(/a = 0.30)...

266

8.14

Blade

tip response

for stiff-in-plane

isotropic

blade

(/t =0.30)...

267

8.15

Blade damping for soft-in-plane isotropic blade first and second lag modes ..........................

in forward

Blade damping for soft-in-plane first flap and torsion modes

isotropic blade .........................

in forward

isotropic blade .........................

in forward

Blade damping for stiff-in-plane isotropic blade first and second lag modes ..........................

in forward

Blade

in forward

8.16

8.17

8.18

8.19

Blade damping second and

first

damping flap

for soft-in-plane third flap modes

for stiff-in-plane

isotropic

mode ...................................

X

blade

flight; 268 flight; 269 flight; 270 flight; 271 flight; 272

8.20

8.21

Blade damping first torsion, The

4/rev

hub

flight; 8.22

8.23

The

The

8.24

8.26

9.

9.2

9.3

9.4

9.5

9.6

The

1

9.8

and

rolling

isotropic

blade

moment

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

273

in forward 274 in fi)rward

4/rev hub loads for soft-in-plane flight; vertical shear and yawing

isotropic blade moment ...............

in forward

4/'rev

hub

loads

for stiff-in-plane

isotropic

blade

shear

moment

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

logitudinal

and

rolling

275

276 in forward 277

4/rev hub loads for stiff-in-plane flight; lateral shear and pitching


in forward

4/rev hub loads for stiff-in-plane flight; vertical shear and yawing


in forward

278

279

Natural frequencies as a function of ply angle single-cell composite blade ..........................

in vertical

Natural frequencies as a function for single-cell composite blade

in horizontal

Natural frequencies two-cell composite

Natural frequencies composite blade Natural

frequencies

of ply angle ........................

blade

in vertical

zero

of tip anhedral

Effect

of tip sweep

isotropic

blade,

the

on the modified

284 for two-cell

ply angle ...................

285

real

on

wall

for two-cell

angle

Effect

baseline

for

283

imaginary part of hover configuration ..................

blade,

wall

in horizontal

Effect of tip sweep on the isotropic blade, baseline of tip sweep

wall

282

as a function of tip sweep angle with zero ply angle ...................

with

for

281

as a function of ply angle blade ............................

as a function

wall

280

Natural frequencies as a function of ply angle for two-cell composite blade .........................

isotropic 9.9

shear

flight;


composite 9.7

for soft-in-plane

logitudinal

flight; The

loads

isotropic blade in forward flap modes ...............

4/rev hub loads for soft-in-plane flight; lateral shear and pitching

The

8.25

for stiff-in-planc second and third

part

of hover

configuration imaginary torsional

xi

eigenvalues

of 286

eigenvalues

of

.................. part

of hover

frequency

287 eigenvalues

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

of 288

9.10

Effect

of tip sweep

isotropic 9.11

9.12

9.13

9.14

9.15

9.16

9.17

9.18

9.19

9.20

9.21

9.22

9.23

9.24

9.25

on the

blade,

modified

Effect of tip anhedral of isotropic blade,

real

part

of hover

torsional

of

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

on the imaginary part of hover baseline configuration ................

Effect of tip anhedral on the real part isotropic blade, baseline configuration Effect of tip anhedral of isotropic blade,

eigenvalues

frequcncy

289 eigenvalues 290

of hover eigenvalues ..................

on the imaginary modified torsional

of 291

part of hover eigcnvalues frequency ...........

Effect of tip anhedral on the real part of hover isotropic blade, modified torsional frequency

cigenvalues .............

292

of 293

Root locus of first lag mode cigcnvalues as a function of ply angle in vertical wall for single-cell composite blade in hover .....

294

Root locus of first flap mode eigenvalues in vertical wall for single-cell composite

295

Root locus of first angle in vertical

as a function of ply angle blade in hover .....

torsion mode eigenvalues as a function of ply wall for single-cell composite blade in hover.

296

Root locus of first lag mode eigenvalues as a function of ply angle in horizontal wall for single-cell composite blade in hover. ..

297

Root locus of first flap mode eigenvalues as a function of ply angle in horizontal wall for single-cell composite blade in hover. ..

298

Root locus of first torsion mode eigenvalues as a function angle in horizontal wall for single-cell composite blade

299

Effect of tip sweep on the imaginary two-cell composite blade, baseline

part of hover configuration

of ply in hover.

eigenvalues ..........

Effect of tip sweep on the real part of hover eigenvalues composite blade, baseline configuration .................

of 300

of two-cell 301

Root locus of first lag mode eigen calues in vertical wall for two-cell composite

as a function of ply angle blade in hover .......

302

Root locus of first flap mode eigenvalues in vertical wall for two-cell composite

as a function of ply angle blade in hover .......

303

Root locus of first angle in vertical

torsion mode eigenvalues as a function of ply wall for two-cell composite blade in hover.

xll

304

9.26

9.27

9.28

9.29

10.

1

10.2

10.

3

10.5

10.6

10.

Root locus of first flap mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover .....

306

Root locus of first torsion mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover.

307

Root locus of second angle in horizontal

308

Effect of horizontal wall ply angle (#=0.30) .......................................

10.12

on blade

7

8

tip response;

on blade

tip response;

on blade

Effect of vertical wall ply angle (p =0.30) .......................................

on blade

tip response;

lag mode

tip response;

flap

mode 313

tip response;

Effect of tip sweep angle on blade (#=0.30) .......................................

tip response;


tip response;

torsion 314

lag mode 315 flap

mode 316

on blade

torsion

mode 317

tip response;

lag mode

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

318

Effect of tip anhedral angle on blade (#=0.30) .......................................

tip response;

Effect

tip response;

of tip anhedral

(#=0.30)

torsion

312

tip response;

angle

mode

311

tip response;


of tip anhedral

flap

310

Effect of vertical wall ply angle (#=0.30) .......................................

Effect

lag mode 309

Effect of horizontal wall ply angle on blade mode (# = 0.30) ..................................

(#=0.30) 10.11

flap mode eigenvalues as a function of ply wall for two-cell composite blade in hover.

Effect of vertical wall ply angle on blade mode (# = 0.30) ..................................

10. 9

10.10

305

Effect of horizontal wall ply angle (U =0.30) .......................................

10. 4

10.

Root locus of first lag mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover .....

angle

on blade

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

flap mode 319 torsion

mode 320

xiii

10.13

Effect

of horizontal

(U =0.30) 10.14

Effect

10.16

angle

10.19

10.20

wall

of attack

ply angle

on trim

pitch

setting

on trim

of vertical

wall

of attack

variables:

inflow

and

(/_ = 0.30) .......................

Effect

Effect

variables;

321

on trim

322

variables;

pitch

setting 323

ply angle

variables;

inflow

and

rotor

(p = 0.30) ...........................

of tip sweep

(g =0.30) 10.18

on trim

Effect of vertical wall ply angle (g = 0.30) .......................................

angle 10.17

ply angle

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

of horizontal

rotor 10.15

wall

angle

on trim

324

variables;

pitch

setting

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

325

Effect of tip sweep angle on trim of attack (/_ = 0.30) ...........

variables; inflow .....................

Effect of tip anhedral angle on trim (/_ =0.30) .......................................

variables;

and

rotor

326

pitch

setting 327

Effect of tip anhedral angle on trim variables; angle of attack (/_ = 0.30) ...........................

inflow

and

Effect

ply angle

on 4/rev

hub

shears

10.22

Effect of horizontal wall ply angle (_ =0.30) .......................................

on 4/rev

hub

moments

wall

rotor 328

10.21

of horizontal

angle

(/a = 0.30).

329

330

10.23

Effect

of vertical

wall

ply angle

on 4/rev

hub

shears

! 0.24

Effcct

of vertical

wall

ply angle

on 4/rev

hub

moments

(g =0.30).

332

10.25

Effect

of tip sweep

angle

on 4/rev

hub

shears

(/_ =0.30)

.......

333

10.26

Effect

of tip sweep

angle

on 4/rev

hub

moments

10.27

Effect

of tip anhedral

angle

on 4/rev

hub

shears

10.28

Effect

of tip anhedral

angle

on 4/rev

hub

moments

10.29

Effect

of horizontal

(_ =0.30) 10.30

wall

ply angle

on blade

6u =0.30).

(kt = 0.30) .....

334

(# =0.30)

.....

335

(# =0.30)...

336

stability;

first

lag mode

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

Effect of horizontal mode (/_ =0.30)

331

337

wall ply angle on blade ..................................

xiv

stability;

first

flap 338

I0.31

Effect

of horizontal

and 10.32

Effect

second

10.33

Effect

Effect

10.35

Effect

10.36

Effect

stability:

The

10.39

ply angle

modes

of tip anhedral

on blade

stability:

first

340 on blade

on blade

angle

of horizontal flap

wall

modes,

longitudinal

combined

effect

4/rev

The

lateral

stability;

first,

4/rev

The

vertical

4/rev

effect hub

combined 10.42

The

stability

on

blade

ply angle

4/rev

hub

The

4/rev

hub

10.44

Real

The

with hub

for the

first

six modes

for the

first

six 343

on stability

of first

frequency

with

ply orientation

with

part

of characteristic of tip sweep

345

of tip sweep

angle, 346

angle,

of tip sweep

as a function

347 angle,

of tip sweep

(/_ =0.30)

as a function

ply orientation

as a function

348 angle,

...........

of tip sweep

(# =0.30)

ply orientation exponent

six modes,

(/_ = 0.30) ...........

ply orientation

angle,

first

349 angle,

...........

of tip sweep

350 angle,

(_t = 0.30) ........... of blade

combined

first

effect

lag mode

with

(soft-in-plane)

shears and

to first

first

function

XV

as a 352

corresponding objective

351

ply

(g. = 0.30) .............................. hub

344

(p = 0.30) ...........

ply orientation

moment

with

and

(t_ =0.30).

of tip sweep

as a function

moment

torsion

(# = 0.30) ...........

as a function

moment

with

as a function

ply orientation

shear

function

4/rev

flap 341

stability

torsional

shear

shear

yawing effect

orientation 1

with

pitching effect

combined

hub

rolling

effect

combined 10.43

second

342

modified

hub

effect

combined 10.41

lag mode

(it = 0.30) ..................................

combined 10.40

torsion

(/x = 0.30) .....................

angle

4/rev

The

first

339

Effect of tip sweep angle on blade stability for the modified torsional frequency (_ =0.30) .................

10.38

11.

blade

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

second 10.37

on

(t_ = 0.30) .....................

wall ply angle

torsion

of tip sweep

modes

angle

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

first

(/1=0.30)

ply

modes wall

of vertical

and 10.34

flap

of vertical

(l_ =0.30)

wall

blade

configuration

(/x = 0.30) .......

353

11.2

The

4/rev

hub

(soft-in-plane) 11.3

The

4/rev

hub


The

4/rev

hub


The

4/rev

hub


The

4/rev

hub

configuration

moments and shears and moments and shears and moments

corresponding first

objective

corresponding second

to first function

blade

function


corresponding

to second

first

function

objective

corresponding

(soft-in-plane)

and

blade

function

.....

blade

The

4/rcv

hub


The

4/rev

hub

shears and moments


objective

corresponding

configuration (soft-in-plane) 0_ =0.30) ....................................... 11.9

11.10

The

The

4/rev hub (stiff-in-plane) 4/rev

hub

(stiff-in-plane) 11.11

The

4/rev

hub

(stiff-in-plane) 11.12

The

4/rev

hub

(stiff-in-plane)

and

(/_ = 0.30) .......

and shears and moments and

corresponding first

objective


objective


357

blade

objective

function

358 to second

blade

function to second second

configuration

(p = 0.30) .....

359

blade

objective

function 360

shears corresponding and first objective moments

356

configuration

0.30)....................................... 11.7

355

configuration

(p = 0.30) .....

to second first

354

configuration (/1 =0.30)

to first

objective

configuration

(/1 = 0.30) .......

to first

objective

blade

objective

xvi

to third function to third function to third function to third function

blade configuration (_ =0.30) ....... blade

configuration

(_ =0.30) blade

.......

362

configuration

(/_ = 0.30) ..... blade

361

363

configuration

(/u = 0.30) .....

364

LIST OF TABLES

Table

ap_9_g_e_

8. I

Baseline

configuration

for isotropic

8.2

Baseline

configuration

for single-cell

8.3

Baseline

configuration

forward

8.4

Basclinc forward

8.5

9.

11.1

composite

for soft-in-plane

in hover rotor

isotropic

1

1

......

blade

rotor

blade

172 ....

175

in

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

configuration flight

blade

179

for stiff-in-plane

isotropic

rotor

blade

in

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

180

Frequency comparison for isotropic rotor blade used in forward flight analysis .......................

9.2 10.

flight

rotor

configurations 181

Baseline

configuration

for the two-cell

composite

Baseline

configuration

for the isotropic

Baseline rotor

configuration for the two-cell blade .....................................

soft-in-plane

Baseline rotor

configuration for the two-cell blade .....................................

stiff-in-plane

rotor

blade

blade

..

..........

191 193

composite 203 composite 213

11.2

Summary

of optimization

results

for the

11.3

Summary

of optimization

results

for the second

11.4

Summary

of optimization

results

for the

xvii

rotor

first configuration

third

....

configuration configuration

218 ..

...

220 221

NOMENCLATURE a

lift curve

A

cross-sectional

b

blade

b(U),

b(T),

b(We)

slope area

of beam

semichord

boundary terms in the variations kinetic energy and external work

B

number

C

blade

chord,

c = 2b

Cdo

blade

profile

drag

Cmo

blade

moment

c_j(i, j = l, ..., 6)

coefficients

CT

thrust

coefficient

Cw

weight

coefficient

[c]

system

damping

[c]

damping

[Cbb'],

[-Cbs'],

[c j,

[c_,] damping

D

profile

Df

parasite

D

vector

Ct

blade

A

A

A

ex, Cy, e z

unit

energy,

of blades

coefficient

matrix

vectors

material

per unit

space,

Eq. (6.12)

system stiffness

of i-th element,

of blade

matrix

Eq. (4.87) span

of the fuselage

of design root

in modal

of linearized

matrix

drag

matrix

of helicopter

of the

drag

stiffness

of rotor

matrix

coordinate

coefficient

of material

sub-matrices

[c,]

of strain of beam

variables

offset

from

associated system

xviii

center with

of rotation the undefromed

element

unit

vectors

associated

curvilincar unit

with

coordinate

vectors

with

base vectors associated elastic axis of the blade

E;

with

longitudinal

ET

transverse

EA

modulus weighted Eq. (4.13a)

EAB0EABIs, EAB3' , EAB s'

anisotropic material coupling cross section, Eqs. (4. ! 5a-r)

EAC o - EAC 3

modulus weighted Eqs. (4.13g-j)

EA D O - EA DT,

modulus section,

weighted warping Eqs. (4.14a-o)

modulus section,

weighted first Eqs. (4.13b-c)

-EADT'

EAr/_,

EA(_

EI_,

f

EI_c,

EI_

Young's

a virtual

EL

EAD0'

Young's

strain

defined tensor

a point

on the

deformed

modulus modulus

cross

area

of the

constants

section

constants

moments

moments (4.13d-f)

beam,

of the

integrals

of the

of inertia

beam

of the

beam,

beam

of the beam

of the

cross

beam

in Eq. (6.1) in the curvilinear

drag

area

coordinate

fCdf

parasite

f

vector

of blade

equations,

Eq. (6.25)

vector

of blade

equations,

Eq.

vector

of trim

distributed

curvilinear

motion,

cross-sectional

modulus weighted cross-section, Eqs. symbol

the deformed

system

the triad (?_'_,^' ^' after %, e_) Eqs. (4.68a-c) E x, E_,

undeformed

system

associated

coordinate

the

of fuselage

equations,

aerodynamic

xix

system

Eq. force

(6.35) (6.37) vector,

Eq.

(6.61)

cross

fl

distributed

incrtial

F

symbol

defined

F

system

load

Fi

load

Fll

vector

of total

hub

FRk

vector

of root

force

{Fc''}

clement

centrifugal

clement

applied

vector

gq

q-th constraint

G x, G,.

G;

deformed

Eq. (6.14)

(4.87)

force

for k-th

blade

vector,

vector,

Eq.

Eq.

(4.85)

(4.86)

function base vectors,

base vectors,

G J, G_A, G_A, G¢¢A, G_A, GCrA, G,tAr/b, _¢A_b, _3_Ar/c, G_A_c, G_J, G_J

modulus weighted Eqs. (4.13k-v)

shear

airfoil

plunging

h_

offset

of beam

h x, hy, h z

components

HR

total

Eq. (3.2a-c) Eq. (3.12a-c)

modulus cross

section

integrals

of the beam,

velocity element

in-board

node

from

blade

root

of he in the (ex, ey, ez) system

longitudinal

hub

force

A

ib, Jb, kb

unit vectors blade-fixed

unit

vectors

coordinate A

space,

Eq. (4.4)

longitudinal

A

shear

force

GET

A

Eq. (6.57)

Eq.

force

det[g i- gj],

undeformed

in modal

of i-th clement,

=

g_

vector,

in Eq. (6.5)

vector

g

gx, g,,

force

A

associated coordinate

associated

with the system

preconed,

pitched,

with

the

preconed,

blade-fixed

with

the

nonrotating,

system

A

inr, Jnr, knr

unit

vectors

associated

XX

hub-fixed

coordinate A

lr,

A

A

.Jr,

kr

unit vectors associated coordinate system blade

|b

I m_n,

lm_,

system

lm_¢

flapping

with

moment

mass weighted moments section, Eq. (4.56d-f) function,

Eq. (7.3)

J1

objective

function,

Eq. (7.8)

J2

objective

function,

Eq. (7.9)

objective

function

hub-fixed

about

of inertia

objective

km

rotating,

of inertia

J

kA

the

blade

of the beam

of approximate of blade

cross-section,

mass

of blade

cross-section,

of gyration

cross-

problem

polar radius of gyration k2A = (Eln0 + EI¢_)/EA radius

root

k_m= k_ml+ k_2 kml,

kin2

principal

mass

radii

of gyration

[K]

system

fK]

stiffness

matrix

of linearized

[K3

stiffness

matrix

of i-th element,

[K_F]

element

centrifugal

[K']

element

applied

[KL]

element

linear

[K _]

element

nonlinear

I

length

of the elastic

length

of-beam

stiffness

L

aerodynamic

m

mass

per unit

matrix

of blade

in modal

stiffness

Eq. (4.87) matrix,

stiffening matrix,

stiffness

Eq.

matrix,

portion

of the

element lift per unit length

xxi

span

of the

Eq. (6.11)

system

stiffening moment

space,

cross-section

blade

Eq. (4.85) matrix,

Eq. (4.86)

(4.83) Eq. (4.83) blade

reDo-

mass weighted warping section, Eq. (4.56g-j)

mD3

mr/_,

constants

mass weighted first moments section, Eq. (4.56b-c)

m_m

m A

distributed

aerodynamic

m!

distributed

inertial

M

aerodynamic

of the

moment

moment

unit

moment resultants Eqs. (4.12a-j)

of the

beam

MIt

vector

of total

hub

moment

MRk

vector

of root

moment

[M]

element system

[M c]

element

[Mi]

mass

n

number

M z, M',,

g,, __p,,T,, S,,

M_,

moment

beam

beam

vector,

vector,

per

My,

of the

cross-

cross-

Eq.

(6.62)

Eq. (6.59)

span cross-section,

S'_

nx',

n_',

n( !

Nm

for k-th

blade

mass matrix, Eq. (4.85); also mass matrix in modal space, Eq. (6.13) Coriolis matrix

damping

of i-th

element,

of elements

components

matrix,

in the

Eq. (4.85)

Eq. (4.87) finite

element

model

of 8_) in the (_,, %, ^' ^' e_) system

number

of modes

used

number

of harmonics

in modal retained

transformation in Fourier

series

expansion Px, Py, Pz I

components

of P in the

(ex, ^ êy, Cz) system

P,7 , PC'

components of aerodynamic forces _ and _ directions, respectively

P

distributed

q_, q_',

qy, q_',

q_

q,/

force

vector

of the beam,

components

of Q in the (i x, _y, ez)

components

of Q in the

xxii

per unit span

Eq. (4.60)

system

(_,, _,_, _) system

in the

q

vector

of finite

clement

nodal

qi

vector

of nodal

degrees

of freedom

Q

number

Q

distributed

[Q]

reduced

EQi]

modal

Qij

coefficients

r

radial position of a point on the to center of rotation, Eq. (5.38)

degrccs

of freedom for i-th

clement

of constraints moment beam

vector

material

transformation

of the

stiffness matrix

beam,

Eq. (4.79)

matrix, for i-th

Eq. clement

of [Q] blade

with

respect

position vector of a point on the undeformed with respect to fixed point in inertial reference I'o

position vector of a point undeformed beam

R

rotor

R

position

vector

Ro

position

vector

RB

position rotation,

vector of blade Eq. (4.21)

R C

position respect

Rex, Roy, Rcz

components

t

time

T

kinetic

TR

total

[T j,

[Tb,],

[Tj,

[Tj,

[Tec],

[Tpb ]

['Tde ]

(3.50)

axis

beam frame

on the

elastic

of the

of a point

on the

deformed

beam

of a point

on the

deformed

elastic

respect

to axis

deformed

beam

radius

root

with

vector of a point on the to blade root, Eq. (4.22) of Rc in the

axis of

with

(_,,, _y, ez) system

energy thrust

generated

by the

rotor

transformation matrices between coordinate Eqs. (2.2), (2.4), (2.7), (2.9), (2.13), (2.17) transformation

matrix

xxiii

defined

in Eqs.

(2.15),

systems,

(4.40)

[T c]

transformation

matrix

defined

in Eq. (4.105)

[TK]

transformation

matrix

defined

in Eq. (4.104)

[T ,_t]

transformation

matrix

defined

in Eq. (4.106)

U,

V,

components

W

of u in the ($_, Sy, cz) system

U

displacement vector of a point the blade, Eq. (3.25)

OR

resultant Eq. (5.3)

U

strain

U

velocity vector of a point on the elastic blade relative to air, Eq. (5.12)

airfoil

velocity

on the elastic

relative

to air,

energy

A{,

components

I%.#

Ap

velocity

of airfoil

V

velocity vector of a point on the the inertial reference frame

{vi,{wi,14,i,{u},

vectors

of clement

nodal

values

st, p,_ and p_¢, respectively, V A

velocity inflow

vector

of air due

V B

velocity

vector

of the blade

Vby,

Vbz

axis of the

of U in the (ex, %, e¢) system

free-stream

Vbx,

axis of

components

beam

with

respect

for v, w, _b, u,

Eq. (4.79) to forward

root,

flight

and

Eq. (4.24)

of Vs in the (_x, _y, ez) system

Vc

velocity relative

vector of a point on the deformed beam to the velocity of blade root, Eq. (4.28)

VEA

velocity

vector

VF

flight

VF

magnitude

velocity

components

of a point vector

on the blade

elastic

of helicopter

of V_ of V A in the (ex, ey, ez) system

xxtv

axis

to

components

of

stress resultants Eq. (4.11 a-c)

_T X

in the (ex, ey, ez) system

VEA

of the

beam

cross-section,

total axial inertial force due to the portion blade outboard of the element, Eq. (6.3) weight

of

of helicopter

work of external nonconservative curvilinear

loads loads

including

the

effects

of the

coordinates

aerodynamic center offset from elastic axis, positive for aerodynamic center ahead of elastic axis XI,

X2,

X3

indicial offset

XFA

notations

for x, n and

of fuselage

center

of drag

(, respectively from

hub

center

A

XFC

in the

inr direction

offset

of helicopter

center

of gravity

from

hub

A

center Yl,

Y2, Y3

local

in the

-

cartesian

in_ direction coordinates

Y

vector

Y0

blade

Yb

vector

of generalized

Yt

vector

of trim

Yu, Y_,

Y,,, Yw, Y_,, Y_, Y¢

of generalized static

equilibrium

coordinates position blade

in hover

degrees

of freedom

variables

notation used for writing the beam strain expressions in concise form, Eqs. (4.17a-g)

energy

zo,z,, _, , Z_,_, Z;, Z., -_ Zw, __Z_, __Z_, Z_, Z¢,

notation used for writing the beam kinetic expressions in concise form, Eqs. (4.51a-i) Eqs. (4.55a-i)

energy and

ZFA

offset

center

Z_,

Z_,

of fuselage

XXV

center

of drag

from

hub

A

ZFC

in the

-

knr direction

offset

of helicopter

center

of gravity

from

hub

A

center

in the

-

Greek amplitude

effective

Gt R

angle

fl(x)

pretwist blade

PJ

Symbols

of warping

multiplication the amount O{A

k., direction

factor of critical damping, of structural damping added

local

angle

of attack angle precone

of attack

of the

rotor

at spanwise

for the

Lock

7

blade

disc coordinate

x

angle

blade pretwist angle at junction portion and swept tip pretwist measured

indicating to a mode

between

angle of swept tip portion relative to flj

straight

of the

blade

of the

blade

number

7n;,

,Vx¢, )'x_

engineering

_x_,

_x¢

transverse

shear shears

strain

components

at the

elastic

axis

c_u

virtual

displacement

(_We),,

virtual

work

due

to distributed

forces,

(awo)Q

virtual

work

due

to distributed

moments,

virtual

rotation

vector

vector,

of engineering

xxvl

(4.64) Eq.

(4.66)

Eq. (4.63)

non-dimensional parameter magnitude of typical elastic sub-arrays

Eq.

representing the order blade bending slope strain

components

of

_ij

strain

_'XX

axial

strain

blade

pitch

0

tensor

in the

local

cartesian

at the elastic angle

with

coordinate

axis respect

to free stream

OG

total

Op

blade pitch angle due to pitch control 0p = 00 + 0,c cos 4' + 01_ sin 4'

geometric

rotation Euler

due

pitch

angle,

to bending

angles

used

Eqs.

(5.24),

(5.27) setting,

in Timoshenko

to describe

the

beam

transformation

the undeformed curvilinear coordinate system deformed curvilinear coordinate system 00, 01o

Kn,

K_

E,,]

Ols

collective

and

cyclic

deformed

curvature

system

pitch

control

of the

beam

from to the

inputs

transformation matrix between and its derivatives, Eq. (3.34)

the triad

(f:;,, %,^'e¢"')

['Ko]

transformation matrix between and its derivatives, Eq. (3.33)

the

(ix, _, _z¢)

2

inflow

A_

blade

Ah

ply angle

A_

blade

A_

ply angle

[A]

local-to-global

transformation

matrix,

Eq. (4.92)

[AC]

local-to-global

transformation

matrix,

Eq. (4.111)

[A K]

local-to-global

transformation

matrix,

Eq. (4.1 I0)

[A"]

local-to-global

transformation

matrix,

Eq. (4.110)

[Av ]

local-to-global

transformation

matrix,

Eq.' (4.1 ! 2)

ratio,/l

=

VF sin _R + v _R

tip anhedral

angle,

in horizontal

tip sweep

triad

angle,

in vertical

xxvii

positive

walls positive walls

upward

of box beams for sweep

of box

back

beams

V F COS 0t R

advance

ratio,/_

v

rotor induced

VLT

longitudinal

=

velocity Poisson's

non-dimensional p

mass density

p^

density

O"

rotor solidity

O'xx ,

O'qC, O'x__ O'x_

{oJ

ratio

coordinate

of beam

element

( -- x__) le

of the beam

of air

components engineering

0"_._/, O'CC,

glR

of stress tensor; stress components

sub-arrays

of engineering

T

twist of the deformed

Tc °

notation

To

initial

defined

also

stress components

beam

in Eq. (4.41)

twist of the beam ( = _-_--Px )

torsional

elastic deformation

second order elastic Eq. (3.41)

of the blade

twist effect of the beam,

u

{_l_c},

{_c'

{(De"

¢,

},

{¢_q'

{_q"

Eq. (4.109)

arrays of Hermite cubic and quadratic interpolation polynomials, respectively, for the beam element

{(_q}

},

fl + _,

first derivatives of {q_¢} and {_q} with respect to x

}

}

second derivatives with respect to x blade azimuth; non-dimensional out-of-plane

, respectively,

of {Oc} and {_q}

, respectively,

also time (_ = fZt) warping

xxviii

function

for the cross-section

rotating

flap,

lead-lag

fundamental frequencies,

_5

angular system

and

torsional

rotating flap, respectively

lead-lag

velocity of the undeformed (_,, _y, cz), Eq. (4.29)

speed

of rotation

components

of the

of _ in the

Special

quantity associated specifically defined

( )o, ( )o

quantity in the global specifically defined

( )l., ( )L

quantity in the local specifically defined

).,.(

),_

derivative

of (

d( )

differential

a( )

variation

of(

a( )

a_( )

(),()

at

'

{ }

vector

[]

matrix

[ ]i-

transpose

with

) with

)

of ( )

at 2

of [

and

torsional

clement

coordinate

rotor (ex, ey, ez) system,

Eq. (4.32)

Symbols

( ),

( ).,.(

frequencies

]

xxvix

the

tip element,

coordinate

coordinate

respect

system,

system,

unless

unless

unless

to x, r/, _, respectively

SUMMARY This report describes the development of an aeroelastic analysis capability for composite helicopter rotor blades with straight and swept tips, and its application to the simulation of helicopter vibration reduction through structural optimization. A new aeroelastic model is developed in this study which is suitable for composite rotor blades with swept tips in hover and in forward flight. The hingeless blade is modeled by beam type finite elements. A single finitc element is used to model the swept tip. Arbitrary cross-sectional shape, generally anisotropic material behavior, transverse shears and out-of-plane warping are included in the blade model. The nonlinear equations of motion, derived using Hamilton's principle, are based on a moderate deflection theory. Composite blade cross-sectional properties are calculated by a separate linear, two-dimensional cross section analysis. The aerodynamic loads are obtained from quasi-steady, incompressible aerodynamics, based on an implicit formulation. The trim and steady state blade aeroelastic response are solved in a fully coupled manner. In forward flight, where the blade equations of motion are periodic, the coupled trim-aeroelastic response solution is obtained from the harmonic balance method. Subsequently, the periodic system is linearized about the steady state response, and its stability is determined from Floquet theory. Numerical results illustrating the influence of composite ply orientation, tip sweep and anhedral on trim, vibratory hub loads, blade response and stability, are presented, it is found that composite ply orientation has a significant influence on blade stability. The flap-torsion coupling associated with tip sweep can induce aeroelastic instability due to frequency coalescence. This instability can be removed by appropriate ply orientation in the composite construction. The structural optimization study is conducted by combining the aeroelastic analysis developed in this study with an optimization package (DOT) to minimize thc vibratory hub loads in forward flight; subject to frequency and aeroelastic stability constraints. The design variables, during optimization, consist of the composite ply orientations, of the primary blade structure, and tip swecp and anhedral. A parametric study showing the effects of tip sweep, anhcdral and composite ply orientation on blade aeroelastic behavior is used as a valuable precursor in selecting the initial design for the optimization studics. However, the most appropriate combination of the design variables, for vibration reduction, can only be selected by the optimizer. Optimization results show that remarkable reductions in vibration levels, at the hub, can be achieved the most

by a judicious combination dominant design variable

of design variables; and for the cases considered.

m

that

tip sweep

is

Chapter INTRODUCTION

1.1

AND

I

LITERATURE

REVIEW

INTRODUCTION Structural

optimization

flight

has

been

as an

important

and

their

sign

process.

typical ingly

recognized area

reduction

bration

interact

at reducing

each

vibration

into the fuselage.

amount

of

research

in

rotor

other. flight

levels

as the

The

area

design,

i.e.,

been

the

become

because main

de-

levels

at

increasinter-

numerous

optimization

the

rotor

helicopter

where

it is not surprising has

by

by the highly

effective

source,

Therefore this

have

use of structural

at the

academia

for vibration

complicated

is particularly

and

in the

seat,

blade

in forward

generated

criteria pilot

is further

helicopter

organizations

of concern

design

such

reduction

vibrations area

decade

problem

in forward

the

propagates

of

research

because

fuselage,

with

reduction

last

The

nature

for vibration

a principal

the

in the

stringent.

disciplinary disciplines

of endeavor

During

blades

by industry,

represent

locations more

of rotor

for vi-

it is aimed

rotor,

before

it

that a considerable

performed

during

the

last

decade[28]. The

majority

to straight

of the structural

isotropic

of composite

blades.

materials

tolerance

than

comparable

processes

for composite

optimization Modern

because

such

metal blades

studies[28]

helicopter blades

blades.

facilitate

have

have

been

restricted

rotor

blades

have

better

fatigue

life and

Furthermore, the incorporation

current

been

built

damage

manufacturing

of refined

planforms

and airfoil geometriesin the blade design process. Blade manufacturing costs are alsolower becausethere are fewer machining operations. Composite rotor blades also offer the potential for aeroelastic tailoring using structural optimization, which can produce remarkable payoffs in the multidisciplinary design of rotorcraft. Rotor blades with swept tips, shown schematically in Fig. 1.1, experience bending-torsion and bending-axial coupling effects due to sweepand anhedral. Swept tips influence blade dynamics becausethey are located at the regions of high dynamic pressureand relatively large elastic displacements. Thus, tip sweep and tip anhedral provide an alternative for the aeroelastictailoring of rotor blades. Swept tips are also effective for reducing aerodynamic noise and blade vibrations. The general objectives of this research are to develop a new aeroelastic analysis capability for composite helicopter rotor blades with swept tips and to conduct a structural optimization study combining this new analysis capability with a structural optimization package. In the next section, a review of the state of the art is given in the areas pertinent to theseobjectives. The specific objectivesof this dissertation are then described in the last section of the chapter.

2

1.2

LITERATURE

!.2.1

Structural

During stability

jority

Modeling

the

twenty

of helicopter

rotor

blade

of studies

use

a beam

strains

and

with

dealing

finite

limit

of the blade

rotor on

first blades

a linear

neglected.

established

is an inherently

the structural

or large)

rotations,

between

axial, forces.

strain

bending,

due

The

overwhelming

ma-

made.

be designed

This

are being

stating

is due to the

life considerations.

of the

rotor

blade

properties.

models

Several

to account

well

that below

available

typical

for asso-

are small

requirement levels

small

strain-displacement

strains

to fatigue

number

blades

deformations

of the

at strain

due

used

torsional

that

rotor

incorporating

to operate

material

material

of helicopter

In the derivation

assumption

aeroelastic

phenomenon

kinematics,

and

that

nonlinear

modeling

(moderate

to isotropic

are discussed

been

beam

must

A substantial

The

with

is usually

blades

it has

Nonlinear

a small

rotor

stricted

blades

the centrifugal

to unity

years,

Blades

model.

effects

relationships, pared

five

Rotor

deflections[23,24,25,26,32,70,71-].

type

coupling

ciated

of lsotropic

last

to moderate

the

REVIEW

as comhelicopter

the elastic

have

isotropic

been

blade

re-

models

below.

analytical was

developed

theory As

model

and

a result,

for the flap-lag-torsion by H_)ubolt

nonlinear the

and

displacement

bending-torsion

of pretwisted

Brooks[-51]. terms coupling

This in the effects

nonuniform model

is based

derivation due

to the

were ge-

ometrical

nonlinearities,

absent

in this

In order of small

to incorporate

and

and

the

formation configuration

were

plane

derived

nonlinear

validated

by

flection

and

and

isolated

the

triad

blade.

blade

analysis,

were

to the

operators,

developed

in

Refs.

and

elastic

these

were and

used

in the

A moderate 40

and

76,

with

beam

the vec-

theories

in the

aeroelastic

de-

which

pro-

stability beam

was

derived

were

moderate

appropriate

deflection

and

in the derivation

beam

the

that

plane

theories[40,76]

also

theories

of unit

remain

used

These

with

the

assumption

performed

combined

trans-

undeformed

triad

axis

were

beam

the

de-

Friedmann[76],

Euler-Bernoulli

tests

the

the

associated

undeformed

deformation,

Subsequently

blades[41,81].

the

the

to static

with

deflection

relationships.

them

operators

derive

and

vectors

undeformed

axis after

blade,

Rosen

and

assumption

between

moderate

and

to the

distinguish

of unit

with

due

associated

In the

deformed

together

elastic

rotor

of the

Dowell[40], the

comparing

structural

should

vectors

strain-displacement

aerodynamic

those

of the

regime[21,77]. the

and

which,

to the

nonlinearities

one

of unit

perpendicular

perpendicular

vided

triad

between

sections

of the

for the rotor

configurations

blade

Hodges

transformations tors

the

configuration by

rotations,

undeformed

of the

developed

important

the geometrical

finite

between

deformed

are

model.

strains

formed

which

theory, by

inertial analyses

of

similar

to

Kaza

and

a large

num-

Kvaternik[54]. During ber

the derivations

of nonlinear

to the

assumption

terms

of a moderate are generated.

of small

strains

and

deflection Many moderate

beam

of them

are

rotations.

theory, relatively

Therefore,

small

due

ordering

schemes

can

terms

be useful

in a consistent

on assigning

orders

governing

the

are assumed of order

allows

rium

blade.

for the

approximation i.

perience

where

with

Hodges[43]

the

of moderate magnitudes

was

small

of the

(and

were

less than

used

instead nonlinear

beam,

which

element served terms

used

was in

equations

removed.

used

such

are

terms an ap-

of equiliba third

compared

schemes

which

that

equations

therefore

kinematics

The

only

order

to terms based

a certain

of

on

ex-

degree

of

larger This

of motion

than

for examining

of motion.

5

on

the

extensional

strain

and

that

the

orientation

angles

theory

was

used

parameters in the

pretwisted

effects

basis This

of higher

were

development

rotating

theoretical

GRASP[44-1. the

assump-

introduced

90 °, Rodrigues

as the

program

assumptions

the

that

of a straight

employed

in which

were

to unity,

angles.

computer tool

and

parameters

rotations

subsequently the

neglected

slopes,

using

also

based

parameters

implies

dynamic

beam

compared

equations

as a valuable in the

was

kinematical

For

bending

1; and

ordering

are

implementation.

of orientation

of the

such

configurations,

negligible) 90 °.

of order

e a were

a nonlinear

rotation

blade

studies[15,16,78,79]

that

nonlinear

physical

approximation

conveniently,

blade

developed

order

to derive,

to note

in their

of the

to terms

of order

order

schemes[40,41,76,81]

compared

terms

actual

is used

e. A second

higher

non-dimensional

in terms

A few latter

It is important

flexibility

tion

one

neglecting

ordering

to the

problem

neglected

and

Most

of magnitude

to be of order

e 2 are

identifying

manner.

aeroelastic

proximation

order

for

isotropic

for the beam

order

beam

element nonlinear

1.2.2

Structural

Modern

helicopter

therefore have

Modeling

during

been

aimed and

attributes

of such

shear

tion

by

assumed used

earlier,

in the cross

section

for

the

nonuniform can,

which cross are

beam. therefore,

lead

into

one-dimensional

functions, theory.

and

from

it needs

The

Anisotropic account beam

the

nonlinear

two

materials

kinematics

once

stiffness and

category. suitable

coupling,

an

the

transin addi-

A review rotor

out

were

Hodges[32]. In a beam

of the

plane,

approach

analysis

of the

blades

and

in and

is to determine

and

cross

sectional

analysis

one-dimensional

global

for rotor

each

cross

blade (l)

properties composite

(2) Structural for composite

are

commonly

two-dimensional

categories:

of the

in this

linear

important

to represent

location,

of composite

into

The

as beams.

blade

center

to be done

discussion

determination

rotor

The

blades.

modeled

Therefore,

shear

suitable

Friedmann

both

for composite

for the

composite

typically

cross-section,

be divided

to the

sections. taken

models

is decoupled beam

are

are

nonlinearities.

and

or neglected.

on a linear

cross-section

ysis

of the

warping

based

blades

studies

and elastic

for modeling

materials;

of analytical

the capability

warping

Hodges[45]

small

structural

require

which rotor

of geometric

suitable

rotor

to be either

of models

model

of composite

number

of composite

representation models

built

a substantial

cross-sectional

deformations

properties

ing

a structural

Blades

are frequently

analysis

Friedmann[26],

As mentioned the

blades

Rotor

development

aeroelastic

structural

theory,

the

deformation,

presented

the

at

to an adequate

existing

the

rotor

the past few years,

structural

verse

of Composite

models rotor

of a model-

approaches

of arbitrary nature

anal-

section

structural

Modeling

for

blade

of the which blade

blade use

analysis.

an

A typical structural model of this category should include geometric nonlinearities, pretwist, transverse shear deformation and cross section warping. Many of the existing composite rotor blade models in category (r) were discussedin detail by Hodges[45]. Mansfield and Sobey[63] made the first attemp to the study of this subject by developing the stiffness properties of a fiber composite tube subjected to coupled bending, torsion and extension. Transverseshearand warping of the crossscetion were not included in the model. This model wastoo primitive for composite rotor blade aeroelasticanalysis. Rehfield[75] used a similar approach but included out-of-plane warping and transverseshear deformation. This was a static theory for a single cell, thin-walled, closed cross-section composite, with arbitrary layup, undergoing small displacements. This relatively simple theory was subsequently correlated by Nixonl'69] with experimental data. Hodges, Nixon and Rehfield[47] also conducted a comparison study of this model[75] with a NASTRAN finite elementanalysisfor a beam having a single closedcell. W6rndle[101] developeda linear, two-dimensional finite elementmodel to calculate the cross sectional warping functions of a composite beam under transverseand torsional shear. With thesewarping functions, the shearcenter locations and the stiffness properties of the cross section could be calculated. In this theory arbitrary crosssectional shapescould be modeledbut the material properties were restricted to'monoclinic. A more general model for calculating the shear center and the stiffness properties of an arbitrarily shaped composite cross section was developedby

Kosmatka[56]. He useda two-dimensional finite element model to obtain St.

Venent

solution

composite

of the

cantilever

sumed

beam

to be prismatic

consistcd

of generally

analysis

was combined

structural

stiffness

et a1.[37]

equations

central

solutions

had

end

by

stiffness

Borri

associated

Bauchau[3] that

plane.

out-of-plane

so-called cellcd was

the

extended

orthotropic

with by material

Bauchau,

and

theory

section

This

theory

the

location

that

the

The

to applied

this

work

geometric

re-

loads to the was

ex-

section

formulations.

beam

material and

materials

does

for

8

center

anal-

correspond

so-called

is valid

properties.

for the

solutions.

due

Subsequently,

was

Coffenberry

sectional

element

was

solutions

warping

isotropic

shear

central

for anisotropic

of the

blade

suitable

finite

displacements

the extremity

cross

transversely

theory

as-

blades[56,57].

solutions

effects.

was

The

of this formulation

displacement

section

beam

loaded

beam

this cross

functions,

to include

a beam

cross

eigenwarpings. beams

to end

large

dcveioped

aspect

while

Merlinil-10]

assumption The

effects,

with

warping

to the warping

due

and

Subsequently,

a two-dimensional

extremity

correspond

The

tip

nonhomogcneous.

propeller

sectional

of a

section.

deflection

formulated

both

displacements

tcnded

materials.

A special

functions

cross and

of advanced

the cross

considering

warping

uniform)

a moderate

also

properties.

sulting

without

with

warping

an arbitrary

(axially

analysis

ysis for determining and

with

anisotropic

dynamic

Giavotto,

cross-section

the

not

based

deform

on the

in its own

expressed

in terms

of the

thin-walled,

closed,

multi-

properties.

Rehfield[5]

Subsequently to allow

for general

it

The

studies

based

on

on composite

a separate

warping

once

for

each

Kim[58] can

and

taper

and

This

such

cross

section.

of the cross and

warping

extension.

and

linear

Stemple

and

Lee[88]

flections

of beams

beams.

This analysis

it was

never

The

is coupled

and

as well

as the

is much

is decoupled used

structural

in the

from

aeroelastic

models

for

modeling

approach

associated

termining

the

center,

shear

posite

cross

section.

is the

one-dimensional

rotor

blades,

two

For

types

in the free

the

more the

with

analysis

category

of theories

of rotor

blade

category

(I),

are

available

9

bending,

by

slatic

de-

composite

whose

cross

and

sec-

therefore

blades.

where

so far the

cross-sectional

suitable

treatment

extended

of large

analysis,

section

out-of-plane

of rotating

discussed

(2) structural

kinematics

beam

the

the

those

which

cross

partially

study

and

warping.

of the beam

analysis

Lee

section the

only

than

by

out

spanwise

Thus

treatment

preliminary

nonlinear

and

element.

fi)r

general

over

it was

expensive

earlier

fl)rmulation

cross

considers

vibration

composite

warping

beam

with

were

to be carried

sections,

nodes

finite

Subsequently

used

has

arbitrary

formulation

problems.

approach

type

mentioned

developed

cross

allows

above,

the cross-sectional

element

warping

beam

This

a finite

arbitrary

and

as

approach,

uses

with

of regular

warping

tional

Lee[87],

by distributing

node

and analysis

alternate

distributions

section

properties,

An

described

to determine

a two-dimensional

beams

accomplished

modeling,

analysis

stiffness

and

thin-walled

at the

torsion

the

beams,

planform

was

situated

and

Stemple

represent

structural

two-dimensional

functions

non-uniform

blade

modeling, for the depending

emphasize

emphasis

properties where analysis on the

the

is on deof the com-

the emphasis of composite level of ge-

ometric The

nonlinearity

first

type

is capable use

is based

thus

between blade

the

displacement

the

axial

an

ordering

terms

For is that

For

quantities

While

scheme

to

magnitude

such

theories

the

models

based

are

rotor

blade

on large

consistent

than

ration

of such

models

could

be complicated. deflection

theories

large

deflection

theories.

sented

first

aeroelastic

those

blade

was

trary

lay-up

deflections

treated

taken

theories

using

are

of

respect

to

do not utilize and

to neglect

aeroelastic

also

model

be more

by Hong

higher

plies.

The

Hodges

10

scheme

scheme;

roorder

more

the

involving

those

blade

Chopral-48]. box

beam

Dowell[40-1,

with

associated

in hover

was

In this model, composed relations

which

and

incorpo-

forward

modest

rotor

Blade

elegant

however,

than

theories

is used.

associated

strain-displacement and

deflection

requirements

and

laminated

ordering

analyses

for a composite

study

moderate

mathematically

an ordering

computational

may

from

ro-

in terms

displacements

used

analysis,

a consistent

as a single-cell,

of composite were

that

into general

in a comprehensive

assumption

aeroelastic

deflection

The

erate

only

and

with

theories

blade

usually

transformation

derivatives

deflection of

the

typc

small.

provided

more

The

the

strains

adequate

the

theories

bc exprcssed

their

large

second

displacements and

coordinates

kinematics.

the

deflection

of blade

(u, v, w, Oh, and

limit

while

relations

x) explicitly.

helicopter

usually

magnitude

undeformed

beam

theory

Moderate

strain-displacement and

coordinate,

tations.

are

deformed

the

one-dimensional

deflection

deflections.

to limit

the

in the

a moderate

large

scheme enable

rctaincd

on

of modeling

an ordering

tations,

being

flight modwith

prethe

of an arbifor moderate

do not

include

the

effect of transversesheardeformations. Each lamina sumed

to have

orthotropic

obtained

using

discretize

the

lag-torsional fects

Ref.

aeroelastic

structural

model

the

forward single-cell,

suitable was

a strong

rotor also

flight.

models

propeller

geometry

was

location

were

blades,

general

Panda

that

Smith

in Ref.

48 to include

a more

refined

stability

cross

and

loads

in Refs.

48,

modeling model

pre-

study

the

to

blades

in for-

modified

shear

analysis[-85],

and

the

of transverse

of composite 72

84

ef-

to the

Chopra[84]

section

flap-

stability

rotor

the effect

to

on blade

Chopra[72]

and

used

coupling

structural

hingeless

aswere

coupled

the

extended The

_;as

the

influence

and

of composite study,

for the structural which,

modeling

with

of curved,

Kosmatka['56,57].

general.

which

showed

in hover[-49]. by

used

for

was

of motion

model

results

was

to inves-

rotor were

blades

in

restricted

to

box beams. analysis

by

used

response,

rectangular

for the

blades

recent

with

The

this analysis

response

presented

aeroelastic

developed

model[56],

have

and

A comprehensive composite

construction

laminate

equations

element

blades

was

together

finite

rotor

In a more

deformation, tigate

48

A

Numerical

Subsequently

stability

flight.

principle.

The

of hingeless

bearingless

in

properties.

of motion.

in hover.

of composite

ward

equations

to composite

boundaries

sented

Hamilton's

behavior

due

material

of the

The

obtained has

been

some

a

discussed

model

stiffness

linear briefly

11

modeling

of advanced

modifications,

pretwisted

In this

cross'sectional from

dynamic

composite the

blade

properties

two-dimensional earlier

could

be

rotor cross

and

in this section.

blades, sectional

shear

finite

also

center element

Bauchau and Hong[4,6,7] developeda seriesof beam

models

acroclastic noted was

which

analysis.

by

Hong

suitable

large

such

as small

warpings.

were

made

on

the

fore,

the

were

retained

small

strain

ial and

neglected

when

relative

magnitude

this

assumption,

in beam

having

for

free

models and

large

results

are

of the with

obtained

strains,

deformations were with

both

in the

axial

same

isotropic

of anisotropy,

by

comparing

as well

12

as studies

strain

and

shear was

strains.

Thereexpression

was

be

on

the

often

axused

materials.

inadequate

analytical

beam.

used

that

anisotropic

it might

studies[50],

small

strain

of magnitude,

kevlar

that

A frequently

that

a thin-walled

out-of-plane

restriction

assumption

or slightly

effects

no assumption

assumption.

order

The

of the com-

axial

shear

an additional

theory[7]

strains.

a revised

showed

for

the

and

terms

were

undergoing

and

the

axial

and

to incorporate

however

strain

includes

Hong[7]

amounts

vibration

small

small

to unity,

coupling

revised

of their

on an extension

assumption,

between

dynamic

beams

only

combined

strain

composite

models[4,6]

twisted

kinematics

plane

two

version

and

shear

compared

strain

final

Green

in the

small

which

strains

Bauchau

perimental

shear

under

shearing

However, beams

revised

order

of

transverse

in its own

first

based

deflection

structural

undergoing

were

definition

this

second

successfully

in the

while

assumptions

blade

curved

theory

curvature,

is rigid In

this

the

basic

cross-section

rotations,

with

initial

The

strains

rotor

The

naturally

and

using

assumption.

for

shortcomings

modeling

associated

approach,

used

Some

displacements

mon

intended

in his dissertation[-50].

for

kinematics

the

were

large

This beams

for

and model

exwas

undergoing

large static deflections. However, an aeroelasticanalysisof rotor bladesbased on this model is not available to date. Minguet and Dugundji[66,67] also developeda large deflection composite blade model for static[66] and free vibration[67] analyses. Large deflections were accounted for by using the Euler angles to describe the transformation between a global and local coordinate system after deformation.

However,

transversesheardeformation and crosssection warping were not incorporated in this model. Thus this model is more suitable for the study of flat composite strips than actual rotor blades. Hodges[46] presenteda general beamtheory basedon a nonlinear intrinsic formulation for the dynamics of initially curved and twisted beamsin a moving frame. This beam model is valid for both isotropic and compositebeams. The nonlinear beam kinematics was basedon a theory developedby Danielson and Hodges[17,18]. The final set of equations of motion were derived using a mixed variational principle, which provided the basis for finite element formulation.

Subsequently Fulton and Hodges[22] developed a finite element

basedstability analysis for a hingelesscomposite isolated rotor in hover.

1.2.3

Structural

Only

a limited

modeling by

Tarzanin

hub

loads

of rotor and

Modeling number blades

of Swept-tip

Blades

of analytical

studies

with

swept

Vlaminck[90]

of an articulated

tips.

An

to investigate rotor

system.

13

have

addressed

analytical the

In this

effect model,

study of tip

the

aeroelastic

was

conducted

sweep

tip sweep

was

on

the

simu-

latcd approximately by manipulating the relative positions of the shear center, aerodynamic center and masscenter of the cross sectionsof a straight blade. The mathematical model consistedof coupled flap-torsion and uncoupled lag equations of motion. The numericalresultsobtained led them to conclude that tip sweep influencesboth bladevibrations and stability. Celi and Friedmann[12] developeda comprehensiveand consistent model which was capable of simulating the aeroelasticbehavior of a hingelessrotor blade

with

a swept

presented tural,

tip.

in Ref.

inertia

presenting

[81].

and

while

of Galerkin

and

systematic

and

forward

briefly

next.

of hingeless

of

design

not

axially

coalescence

its comprehensive

limitations

because rigid

important

element,

blade

However,

addition does

it approximated it also

of small not the

occur, model the

employed

14

was

instabilities

used

swept

detailed

in both in Ref. the on

a

hover 12 are

dynamic a number of blade

by tip sweep and

are can-

damping.

is usually

stabilizing.

in Ref.

linear

using

first

on

re-

of structural

sweep

tip a

the

are strong,

amounts tip

modeled

combination

induced

struc-

element

depends

the

instabilities Such

was

influence

and

the

stability

its effect

as precone

of motion

finite

obtained

a powerful

aeroelastic

nature,

and

of the

beam

conclusions

has

coalescence.

the

a special

on blade

such

by

for

equations

by developing

of tip sweep

parameters,

frequency

modeled

portion

sweep

The

the

This

blades.

frequency

Despite

Tip

on

elements[27,89].

rotor

be eliminated

When

most

based

tip was

operators

finite

frequencies. with

was

the straight

The

behavior

associated

swept

of the effect

flight.

fundamental

The

type

study

described

blade

analysis

aerodynamic

the tip,

number

The

12 had

portion

of

a number the

transformation

of

blade

as

at

the

junction where the swept-tip element was combined with the straight portion of the blade. It was latter shown that such a transformation could be inaccurate for large sweep angles[73].

Furthermore, it should be noted that the

studies presentedin Refs.90 and 12 wererestricted in the sensethat they could only representtip sweep,but not anhedral (seeFig. 2.3). Bcnquet and Chopra response finite

and element

however tip

loads

still

the

Chopra1-55]

each the

using

All isotropic

model

on

a linear

was

of the

given

into

finite

in the on

swept

Ref.

8 to

tip and

constraint

and

anhedral,

the swept

include

oriented

portion

developed

and

elastic

Fuselage

dynamic

mentioned

above

and

nonlinear

formulation pretwist

using

Kim

relations

the

anhedral,

the straight

an aeroelastic

of arbitrarily

elements.

sweep

Subsequently,

in

sweep,

flight

for combining

the swept and

to calculate

in forward tip

blade.

developed

as a series

included studies

the

between

variable

both

transformation

formulation

with

analysis

blade

included

of

Chopra[9]

modeled divided

hingeless

transformation

blades

was

segment blades

the

Bir and

geometry

blade

This

in the assembly

Panda[73].

The

the

an aeroelastic

tip

portion

extended

blade,

vanced

based

straight

transformation of the

of an advanced

method.

it was

with

I-8] developed

by

for adplanform.

segments interaction

with with

formulation. tip

blades

blades.

15

were

restricted

to

1.2.4

Structural

A fairly using

Optimization

recent

survey

structural

straints,

was

grated

optimization, by

multidisciplinary

tional

by careful hardware

mentioned

in this

section.

on helicopter

rotor

blades.

Friedmann

and

methods

helicopter

rotor

the

oscillatory

ratio/_ blade and

and

The

used

hingeless levels, Lim

as rotor

duction contribution

masses,

been

the

indicated

weight

reduction

carried

rotor

blades

in Refs.

61

out

Cross

16

addi-

a few studies

will

optimization isotropic

for

pro-

reduction

of

consisted

of

at an advance placements

sectional

outboard

be

systems.

function

frequency

of the

dimensions

portion

of the

typical

soft-in-plane

reduction

blade,

in vibration

obtained.

aeroelastic

62

can

requiring

moments

a comprehensive

with and

rolling

a 15%-40%

were

for

mathematical

objective

results

that

potential

to vibration

in hover.

Numerical

inte-

to straight

applied

in the

the

which

structural

restricted

included

located

variables.

Chopra[61,62]

made

of

hub

constraints

levels,

or isolation

The

or the

the

without

concepts[80]

constraints

configurations

in helicopter

majority

con-

that

in Ref. 28, only

flight.

shears

stability

design

analysis

Shanthakumaran[29]

hub

tuning

and

have

in forward

as well as a 20% and

The

blades[-28]

behavior

aeroelastic

vibration

reduction

multidisciplinary

offers

in

presented

vibration

shown[28]

rotorcraft

absorbers

approximation

blades

nonstructural

were

and

vertical

= 0.3.

design

the review

studies

gramming

of

vibration

and

It was

particularly

as rotor

duplicating

be

l=riedmann['28].

preliminary

To avoid

on helicopter

aeroelastic

optimization

such

Reduction

research

with

improvements,

achieved

Vibration

describing

presented

substantial

for

consisted

study

constraints. of using

of vibration An

important

a direct

analytical

re-

approach

for the calculation

stability[60], were

with

obtained

finite

approach

able,

respect

of the

difference

only

such

cost as that

explicit

variables,

hub

loads[59]

These

computational

when

of the design

of the

variables.

method,

is applicable

as a function

derivatives

to the design

at a fraction

conventional this

of the

sensitixitv

associated

used

in Ref.

analvtical

in the

and

derivatives with

the more

29.

However

expressions

calculation

blade

are

of the

avail-

sensitivity

derivatives.

Davis

and

analysis

which

natural modal sis

was

frequency

capable

indices.

of

can

Frequency

Icvcl.

shears

lead

bration

indices

bration

reduction.

wind-tunnel

correlation mentioned

This

However

was

was

those

being

measured

aeroelastic

and stability

vibratory the

these

for

hub

most

effective

results

tests the

the

constraints

rotor rotor,

was

indices with

criteria

verified

baseline

optimum

experiments

17

were

obtained.

were

vibration

to achieve

loads,

analy-

levels. minimum

and

vibration

modal

vi-

for

rotor

vi-

extensive levels

reasonably It should

not considered

modal

the

by fairly

and

of

optimization

lower

vibration

of

tailoring

modal-based

of modal

lower

minimization

through

significantly

as blade

an automated

to bc inadequate

minimization

In these

theory

with

the

optimization such

and

characteristics that

shown

Subsequently,

between that

analysis

to substantiallv

tests[99,20]. with

modal-based modal

blade

problems, shears

design

minimization

dynamics of hub

concluded

blade alone

rotor

rotor

minimization

blade They

produce

a modal-based

to various

optimizing

placement

vibration

compared

applied

properties.

analysis

developed

placement,

vibration

structural

hub

Wcller[19]

were good

be also

in Ref.

19.

Young

and the

Tarzaninl-102]

study

on

study

two diffcrent

identical

rotor

bration

rotor

procedure, loads.

and

and

using

objective

tested

in the

procedure

consisted

moment

for the

Ref.

102

for the design

while

The the

ref-

low

vi-

optimization fixed

reductions low vibration

provides

having

tunnel.

of the

substantial

In this

rotor;

structural

showed

Thus,

wind

approach;

analytical

function

results

ratios.

an

design.

a low vibration

a conventional

overturning

advance

were

and

to rotor

system

hub

in the

4/rev

rotor

at both

a validation

of low vibration

of

rotors

the

in for-

flight.

state

and

Mantayl-l'l

of integrated intelligent

various

proach

plan

is one offers

The

studies

of the

areas

and

on straight

objective

was

the the

from

towards

the

this document

an integrated

capability

for

produced

composite

rotor

bladesl35,36].

36 was

of the an

4/rev

extension

18

current included

integration

helicopter

of

vibration design

ap-

gains.

analysis

minimization

the

which

complete that

available

laminated

on

multidisciplinary

for performance

of the

report

of rotorcraft,

stability

ply angles

Reference

optimization

where

potential

modeling

response

were

a comprehensive

development

It is evident

excellent

variables

moments.

for future

improved

aeroelastic

edited

multidisciplinary

disciplines.

reduction

tion

using

test

optimization

Adelman

an

the

rotor

airfoil,

designed

tunnel

shear

high

structural ward

was

wind

hub

and

analytical-experimental

optimization

a reference

designed

in which The

vertical

twist

was

a combined

of structural

rotors:

planform,

erence

low

application

conducted

of the

rotor

a few structural In Ref.

walls hub

composite

of the

box

blade

optimiza-

35, the beam,

loads;

both

hub

study

performed

design and

shears in Ref.

the and 35

by allowing the ply anglesto vary from element to element in the spanwise direction, and performing a multi-objective optimization to minimize the hub

loads Only

on

and

a limited

swept-tip

and

be used

blade

both

with

optimization

has

studies

constraints[14,34].

to isotropic

OF THE study

simultaneously.

aeroelastic

as an important

OBJECTIVES present

moment

of structural

restricted

effectively

The

root

number

blades;

34 are

1.3

the

4/rev

blades, design

they

were

While indicated

variable

that

for vibration

conducted

References

14

tip sweep

can

reduction.

RESEARCH

a number

of important

objectives

which

are

listed

below: 1.

Development ior

of an analysis

of composite

forward

flight.

computational

of

2.

to be critical blades

multi-cell

blade

Conduct

detailed with

rotor

analysis

with

swept

swept

is suitable

since

modeling

of the

(c) ability

tips

behav-

in hover

and

include:

(a)

for the repetitive (b)

capability,

and

aeroelastic

analysis

optimization;

accurate tips;

the

of this

the analysis

structural

for the

with

features

so that for

of modeling blades

important

response

rotor

blades

The

required

trim/aeroelastic found

helicopter

efficiency

calculations

capable

fully

this

coupled

feature

dynamic

to represent

was

behavior arbitrary

cross-sections.

straight

studies and

on

the aeroelastic

swept

tips

19

behavior

to determine

the

of composite combined

rotor effect

of

sweep,anhedral and composite ply orientation acroelastic 3.

4.

Conduct

studies

entation

on

forward

flight.

Combine

the

blades 5.

stability

Conduct posite ply

with

ducing

hub

new

shears

basic

vibration

effects

and

moments

analysis

structural

sweep

levels

of sweep,

optimization

and

tip

in forward

20

the

anhedral flight.

anhedral

of composite

package,

to illustrate

blade

response

and

flight.

capability

optimization

configurations tip

in forward

the

acroclastic

a structural

orientation,

and

illustrating

the

a few blade

in hover

on

rotor

for swept such

and

ply oriblades

in

tip composite

as DOT[106].

studies potential as design

on

two-cell,

benefits variables

com-

of using for

re-

Chapter il MODEL

In this analysis The

chapter,

the

scheme

is described

next.

2.1

used

The

ASSUM hingeless

the axis 2.

used

in the blade

various

in the

development

with

formulation

SYSTEMS

of the

a swept

tip are summarized.

of the moderate

coordinate

derivation

aeroelastic

systems

deflection

theory

related

coordi-

and

of the equations

of motion

of the

The

the

PTIONS blade

of rotation

blade

pretwist

has

is cantilevered (see

distribution

at the hub,

with

a root

offset

e_ from

Fig. 2.2).

a precone

angle

r 0 about

tip (see the

elastic

Fig.

2.2)

axis

(line

and

it has

of shear

a built-in centers)

of

blade.

3.

The

blade

has

4.

The

blade

consists

tation

5.

the

COORDINATE

rotor

in the

Finally,

AND

are defined.

BASIC 1.

helicopter used

transformations,

blade,

assumptions

of a composite

ordering

nate

DESCRIPTION

relative

and

an anhedral

The

blade

no sweep,

droop

or torque

of a straight

to the angle

is modeled

straight

portion

(A_) (see by beam

portion

offset. and

a swept

is described

tip

whose

by a sweep

orien-

angle

(A_)

the elastic

axis

Fig. 2.3). type

of the blade.

21

finite

elements

along

6.

A single

7.

The

blade

center, 8.

The

cross

stiffness are

The

blade

Note

to vary

BO-1051861,

in stiffness

mately

75°/.

relatively in the

with

this

ior of the

blade

which

and

the

blade

length)

The

elastic

portion;

built

and

its chord

of the

blade.

the

line of shear

of a stiff,

blade

the

line

extent

and

centers

pre-

of the

in which

outboard the

blade

large

vari-

(approxi-

properties

blade

of shear

such inboard

portion

of the

line.

helicopter,

nonuniform

length)

deformations

shear

by a straight

for a typical

where

is to a large

of

blade,

a flexible

axis of the whole

is

of mass.

is approximated

thus

distinct

center

consists

occur,

with

and

distribution

usually

segment

elastic

with

25*/0 of the

outboard blade

the span

can

uniform.

rily

The

of

of the

tip.

shape

center

coincides

stiffness

the swept

arbitrary

tension

blade,

(approximately

ations

have

along

axis

blade

to model

properties

of the

the

MBB

portion

mass

feathering

that

can

center,

and

portion

as the

is used

section

allowed

straight

10.

element

aerodynamic

twist, 9.

finite

occur

centers

representative

are prima-

associated

of the behav-

generally

blade. orthotropic

materials,

and

it

is

anisotropic. I I. The

blade

has

completely

coupled

flap,

lead-lag,

torsional

and

axial

dy-

namics. 12. The are 13. The and

effects

of transverse

shear

deformations

moderate

deflections,

and

out-of-plane

which

imply

warping

included. blade moderate

undergoes rotations.

22

small

strains

14. Two-dimensional quasi-steady aerodynamics, based on Greenberg's thcory, is used to obtain the distributed aerodynamic loads; this simple unsteadytheory is justifiable becausethe principal objectivesemphasize the structural modeling of the blade and its optimization for vibration reduction. 15. The induced inflow is assumedto be uniform and steady. 16. Stall and compressibility effectsare neglected. 17. Reverseflow effects are included by setting the lift and moment equal to zero and by changing the sign of the drag force inside the reverseflow region (seeFig. 5.3). 18. The rotor shaft is assumedto be rigid and the speed of rotation (ff_)of the rotor is constant. 19. The helicopter is in trimmed, steady and straight flight. The assumptionslisted above are used in various stagesof the formulation of the aeroelasticmodel. Additional assumptions needed for the structural modeling of the blade, such as the kinematical assumptionsand the assumtions used in the developmentof the constitutive relations, are discussedin Chapter 3.

23

2.2

ORDERING

SCHEME

An ordering nonlinear

scheme

terms,

a beam

generated

element

ordering

blade

nitude

0.10

is based

are

non-dimensional

).

on

of _.

In the

terms

of order

neglected

that

of magnitude

the

are

governing

with

respect

higher

slopes

manner. of the

assigned

the

aeroelastic

to terms

This

a mag-

to the various problem

it is assumed

of order

for

deformed

to have

then

equations,

order

of motion

_ is assumed

governing

of the

delete

in a consistent

e (where

parameters

derivation

_2 are

assumption

and

of the equations

deflections,

of order

Orders

to identify

the derivation

the

and

physical

terms

used

moderate

moderate,

_< _ _< 0.20

and

during

undergoing

scheme

elastic

is defined

in that

1, i.e.,

O(l)+o(_2)_ _ o(1) The

orders

study

are

O( 1 )-

of magnitude listed

for various

non-dimensional

used

below:

x l'

R 1 '

he l '

As,

A_,

O(_,/2) •

Op, fl

O(_)"

'7 !'

_

1'

!___._ _b, _= c3x' c3_b

1 a n 0t'

sinAi,

cos A,,

sinAi,

cos A,

v

v,,w

4_,_1,"

!'

1, l'

w

1'

,x,

0 x, 0_, 0_

O(_2)"

parameters

U 7-'

u x, _:xx, Y_,

W Yxr,, -12,

24

mf_212 EA

W,, t 1

sin

W_ '

1

¢,

e_ '

1'

COS

¢,

tip,

in this

In general, it is assumedthat rotation terms such as v.×,w,×and order

_, while

amplitude

strain

u is assumed

scheme

material

of magnitude. are

tions.

on

matrices,

common

coordinatc

deformation

of the

to note and

A.

ordering

experience

the

are

used

the

systems blade.

are

required

Each

coordinate

unit

vectors.

system

preconed,

to position

motions,

ey, ez) and

uration

as

_

, is used

unique

configura-

both

care

and

describc

system

is symbolically

represented

three

namely,

first

the geometry

systems,

and

,

blade,

to represent

relative

, the

rotating,

the

in Figs.

respectively,

as shown the

blade-fixed

orient

to the

hub-fixed

A

system

blade

relative

the

A

system

A

(ib, Jb, kb),

respcc-

to the

hub

through

next

two

systems,

2.1

and

2.2.

The

are

used

to

position

and

orient

each

A

(ib, Jb, kb) system

in Figs.

orientation

and

^

pitched,

shown

(e^x, _, _)

element

of the

order

to fully

The

(i_,l_,k_)

A

finite

not

blade

requires

of the

same

are

A

, and

and

of thc

actual

scheme

strains

coefficients

schemes

with

This

SYSTEMS

hub-fixed

rigid-body

beam

that

warping

as _._.

(small

^

(it, it, kr) tively,

theory

I, 5, 6), are

of the ordering

^

A

that,

The

of magnitude

deflection

Q,i (i, j =

sense

of orthonormal

nonrotating,

order

d.

of

of flexibility.

COORDINATE

a triad

same

_:x¢ are of order

it is assumed

the application

degree

and

a moderate

stiffness

Therefore,

the

Furthermore,

Several

by

as u,x, _

It is important

based

a certain

2.3

with

rotations).

reduced

such to have

is consistent

moderate

and

terms

q5 are

2.3 and of the

25

in the

2.4.

local

undeformed

A final

blade

system,

geometry

configAj,

¢x¢

(fz',, co, e¢) after

defor-

A

mation. used

An

additional

as explained

2.3.1

A

A

the

in Subsection

Nonrotating,

The

system,

preconed,

blade-fixed

system

A

A

, is

(ip, jp, kp)

2.3.7.

Hub-fluted

Coordinate

System

A

(inr, Jar, knr) system,

shown

in Fig. 2.1,

is an inertial

reference

frame

and

A

has

its origin

at the hub

center.

A

The

vector

i,r points

toward

the

helicopter

tail;

A

in, points A

to starboard;

and

kn, coincides

with

the

rotation

vector

of the

rotor.

A

in, and

j_, are

in the

in this coordinate

2.3.2

^

A

of rotation.

Hub

shears

Coordinate

System

and

moments

Hub-luted

rotates

defined

A

(i,, Jr, k,) system,

shown

in Fig. 2.1,

also

has

its origin

at the hub

A

but

are

system.

Rotating,

The

plane

with

a constant

angular

velocity

center

A

tqk r . The

vector

ir coincides

with

A

the

azimuth

position

of the blade,

while

A

k r is coincident

with

the

the

blade

vector

A

knr ; ir

A

and

j, are

2.3.3

also

in the

Preconed, A

The

A

plane

Pitched,

of rotation

of the

Blade-fixed

rotor.

Coordinate

System

A

(ib, Jb, kb) system,

shown

in Fig.

2.2,

rotates

with

and

has

A

origin

at the blade

incides

with

the

root,

offset

pitch

axis,

of the

blade.

from

which

the

is also A

straight A

A

portion A

The

hub

A

the

center

A

by e_i r . The

undeformed

its

elastic

vector axis

ib coof

the

A

(ib, Jb, kb) system

is oriented

by

rotating

the

A

(i,, jr,kr)system about - jraxis by the preconc angle _p, and subsequently in-

26

A

troducing

a second

angle

In the finite

0p.

global

coordinate

2.3.4

Undeformed

The of the axis;

finite

the

model

rotated

ir axis

of the

Element

element. the

geometric

pitch

the ('ib, lb, kb) system

is the

A

blade,

2.3,

has

its origin

_x, is aligned

_z arc

of the

defined

blade,

with

in the

, and

about

the

-

For A

the swept-tip

components

A

2.3.5

modulus

weighted

respectively, Effects ment

at

of blade

beam.

the same

orien-

the

A

A

A

anhedral for the

applied

Curvilinear

(Cx, e_, e¢) system,

is defined

of the

about

-

k b by the sweep

angle blade

loads

A a .

finite of the

The

model.

finite

As

A

(ex, ey, ez) system

element beam

,'_.

angle

The

element

is also

displace-

are defined

system.

Undeformed

In the

fl(x)

system

elastic

the (e x, ey, ez) system

A

and

in this coordinate

has

element,

node

element

section

the (ex, ey, ez) system

the (ib, Jb, kb) system

Jb by the

coordinate

inboard

beam

cross

A

mcnt

at the

A

A

by rotating

local

A

A

is oricnted

the

the

System

in Fig.

vector

_y and

portion A

shown

The

vectors

A

Coordinate

as the (ib, Jb, kb) system.

then

by

systcm.

the straight

tation

about

clement

(cx, _y, Cz) system,

while

For

rotation

Coordinate the

principal

vectors

axes

System _

of the

and

cross

as the

change

in the

orientation

any

location

aldng

the

pretwist

strain-displacement

are

properly

relations

beam

27

defined

parallel

section;

and

pretwist

of _,

_ with

element,

accounted in the (_,

_¢ are

the

respect

as shown

for by deriving _,

_¢) system,

which

the

to the angle

to _y, _ in Fig.

2.4.

beam

ele-

rotates

with

,

the beam pretwist. The strain components, the material properties, and the cross sectionwarping function are all derived in this coordinate system.

2.3.6

DeJb_wted A t

The

A t

Curl,ilinear

A I

three

rotated

of the

Ap

Euler

angles

to be tangent

2.3.7

Preconed, A

A

A

was

sequences

local

Blade-fixed

in chapter

deformation.

are

deformed

Coordinate

also

elastic

The

A

the (e x, %, e¢) system

0_ about

chosen

A

is identical

_:_, rotated

following

the

possible.

The

i n and

work

of pre,,, ex is

vector

axis.

System

to the

preconed,

pitched,

A

blade-fixed A

the

pitch

angle A

by rotating

the

A

0p is equal

to zero.

A

The

A

canceling

the

A

A

(ib, Jb, ku) system

pitch

system.

Expressing

the blade

system

is convenient

when

similar

results

available

rotation

inherent

response

and

comparing

in the

sys-

(ip, jp, kp) system

about

-

ib by the

pitch

angle A

thereby

3,

A

(ib, Jb, kb) when

is oriented

after

by rotating

of 0_, 0n and

sequence

other

to the

(ip, jp, kp) system A

geometry

is obtained

in the order

but

chosen

blade

detail

A

This

authors[40,76]

The

local

in more

A t

_:_, respectively.

vious

will be discussed

of the (e x, %, e¢) system

through

tern

which

the orientation

orientation

System

A I

(c x, %, e¢) system,

represents

Coordinate

the

literature.

28

in the blade results,

definition

root for

loads these

of the

Op, A

A

(i b, Jb, kb)

in this coordinate quantities,

with

2.4

COORDINATE

The

coordinate

transformations

described

in the

equations

of motion,

2.4. I The the

Rotating

previous

between

section,

are defined

to Nonrotating

transformation

nonrotating,

TRANSFORMATIONS

are

various

needed

for the

coordinate

systems

fl_rmulation

of the

in this section.

Transformation

between

hub-fixed

which

the

the

rotating,

coordinate

system

A

hub-fixed

coordinate

is defined

as:

system

and

A

(2.1)

and

the

transformation

matrix

[Tm

where,

_, is the

blade

azimuth,

] =

[ Tf. ]

[

-

is given

sin 0

_ = f_t.

29

by

cos _, 0

o] 1

(2.2)

2.4.2

Blade-fixed

The

to Hub-fixed

transfi)rmation

system

and the

Transformation

between

rotating,

the

hub-fixed

preconed,

pitched,

coordinate

system

A

blade-fixed is defined

coordinate as:

A

(2.3)

kb and

the

transformation

[ Tbr ]

matrix

pitch

_p is the control

0

cos 0p

blade

setting,

-sin0p

precone

cyclic

0 o is the

sine

2.4.3 The and the

pitch,

Element

=

collective

0

I

0

cOS0p

-sin#p

0

cOSpp

and

0p is the

blade

pitch

angle

(2.4)

due

to

by:

00 +

01c

pitch,

cos0

0_c and

+

0is

0_s are

sin0

the

(2.5)

cyclic

cosine

pitch

and

respectively.

to Blade

transformation preconed,

by

sin 0p

angle,

expressed

0p

in which

is given

I o o?[os, o,,1

=

0

where,

[ Tbr ]

Transformation between

pitched,

the

blade-fixed

undeformed

element

coordinate

A

system

is defined

system as:

A

f'l f't _y

coordinate

= I" T eb "]

^Jb

A

A

ez

kb

30

(2.6)

For

the straight

portion

of the blade

[Tcb]

For the swept-tip

=

0I 0

(2.7a)

1 0 00] 0 !

element

[co, As ,hAs 0][cosAa 0,'nAa]

[ Teb ] =

sin A s 0

cos A s 0

0 !

0 sin A a

-

I 0

0 cos A a

(2.7b) =

where, thc

blade

2.4.4

blade

tip anhedral

Undeformed

The and

A s is the

I

transformation

the undeformed

sin A s cos A a cosA s cosA a - sin A a

tip sweep angle,

cos A s -sinA s 0

angle,

positive

Curvilinear

positive

element

the

and

the

transformation

[Tce

A,

is

coordinate

system

as:

A

= [ Tce ]

ey A

e_

ez

] =

and

Transformation

is defined

A

matrix

sweep,

curvilinear

system

A

en

Element

undeformed

coordinate

,

for backward

1

upward.

to Undeformed

between

sin A s sin A a cos A s sinA aq cos A a

[ T,_ ]

0 1 0

is given

cos # 0 -sinfl

31

(2.8)

by

sin fl 0 ] cos ,8

(2.9)

where, fl axis.

is the

blade

Differentiating

local

pretwist

Eq. (2.8)

angle

with

which

respect

varies

along

the

blade

elastic

to x gives

A

{-ti °t ^

eq,

(2.10)

TOe(

x

=

^

^

_

e_ ,x

Toe q

where

_o =

2.4.5 The and the

Deformed

to Undeformed

transformation undeformed

P,x

(2.11)

Curvilinear

between curvilinear

the

Transformation

deformed

coordinate

system

Ap

[T_]

tra'nsformation

matrix

is defined

system

as:

= [ Tdc ]

[ T_

]

is given

(2.12)

e_

by

=

s [1

coordinate

A

e_

and the

curvilinear

cos00x O0 -sinO,

sin0 1Fc°0 x

007

0

-s

1

cosOxJLSino, I o

-

_OPII[

cosO,lJ

cos0_

sin0_

sin 0_

cos 0_

o

o

(2.13) !]

A

where,

0_,

0.,

and

0x

are

Euler

angles

respectively.

32

about

_,

rotated

%, and

A

rotated

ex,

2.4.6

Deformed

The and

Curvilinear

transformation

the

to Undeformed

between

undeformcd

clement

the

Element

deformed

coordinate

,

where

the transformation

e_

transformation

is defined

= [ Tde ]

e_

ez

is discussed

system

as:

[ Tde ]

(2.14)

ey A

matrix

coordinate

A

Ap

[Tde]

This

curvilinear

system

At

Transformation

is given

by

(2.15)

=[Tdc][Tce2

in greater

detail

in Chapter

4 and

Appendix

m.

2.4.7

Preconed,

The and

transformation

the

section

Blade-fixed

preconed, 2.3.7,

to Preconed,

between pitched,

is defined

the

Pitched,

preconed,

blade-fixed

Blade-fixed blade-fixed

coordinate

system,

Transformation coordinate

system

described

in Sub-

as:

A

jAp

A

= [ TP b ]

(2.16)

^Jb kb

where

the

transformation

matrix

[Tpb]

=

[ Tpb ]

0 I 0

is given

cOS0p 0 sin

33

0p

by

(2.17)

-sin0p 0 cos

1 0p

Chapter STRUCTURAL

The

derivation

model such

MODELING

of

is presented

nonlinear

rods

(Ref.

shear

97 and

Ref.

(Ref.

curvilinear

coordinate for.

cartesian

coordinate

listed

The

warping,

of elasticity

stress-strain

are

of curved in curvilinear derived

in a

are

properly

ac-

transformed

relations

are

included.

first

of pretwist then

beam,

arc

mechanics

components

are

blade

for a composite

on the

effects

rotor

to

assumed

a

local

to be de-

system. in the

derivation

of the

structural

oper-

below:

deformations

2.

The

strain

components

and

shear

strain

components

ever,

the

relative

magnitude

due

order

of the cross

with

warping

section

are small

to material

not be neglected Higher

the

used

The

3.

strain

coordinate

assumptions

features

the theory

BLADE

composite

out-of-plane

components The

ROTOR

the

is based

so that

1.

assumed

and

8), and

4).

strain

cartesian

kinematical

are

100, Chap.

system.

in this local

The ator

These

for

Important

of deformation

system

COMPOSITE

operator

deformations

98, Chap.

counted

fined

structural

kinematics

coordinates

OF THE

in this chapter.

as transverse

The

the

!I I

to unity

neglected

with

between anisotropy,

terms

to axial are

plane

compared

are

respect

in its own

the e.g.,

strains

neglected.

34

axial

are neglected.

such

respect and

squares under

that

axial

to unity.

How-

strains

is not

shear of shear this

both

strains

assumption[7].

can-

The derivation of the strain componentsbasedon theseassumptionsis valid fi_r small strains and large deflections. However, quantities such as displacement components( u, v, w) and elastictwist angle (_b)do not appear explicitly in the resulting expressionsof the strain components. Subsequently,explicit expressionsfor the strain-displacementrelationship are obtained by considering the deformation procedureduring the finite rotation from the undeformed to the deformed configuration and using an ordering schemeto systematically identify and neglect higher order nonlinear terms which are generatedduring the derivation[40,76]. Thus, the final strain-displacement relations are valid for small strains and moderatedeflections.

3.1

KINEMATICS The

the

hub

position center

OF

vector

DEFORMATION

of a point

P on the undeformed

A

e I is the blade

in-board

node

interpretation

of the

=

root

offset

the

represent

the

undeformed

respect

to

vector

from

2.2-2.4.

vector

both

tip portion.

35

_Ten +

center,

is facilitated

Figs.

swept

A

xe x +

the hub

element

position on the

A

h ei b +

from

combinationof

as well as a point

+

finite

of this position by

A

e lir

beam

described

tion

with

is:

_x, 17,_)

where

beam

the

and blade

(3.1)

r_

he is the root.

The

by considering Equation for a point For

(3.1)

of the

physical

the geometry can

on the

a point

offset

be used straight

on the

swept

to

portip

clement, heequals the length of the straight portion sponding

undcformed

based

vectors

at point

r,x =

A

Cx-

The

blade.

corre-

P are

A

gx =

of the

A

_"T0e)1 + r/T0e _

(3.2a)

A

g,1 =

r, rt = e)t

g(

r_

(3.2b)

A

=

=

e_

(3.2c)

A

where

the

derivatives

initial

twist,

ro,

of the

of the

orthonormal

undeformed

triad

beam

A

=

0

ê_,x

elastic

axis,

a unit

vector

evident

from

be obtained and

the

0

from

initial

Eq.

twist

-

0

r0

er/

TO

0

_

(2.10).

Note

TO is nonzero,

that

then

(3.3)

if point

the base

P is not vector

Since position

the

orthogonal

to the

cross-sectional

plane

on the

gx is neither

A

nor

to the

A

%,x

can

related

by:

A

which

A

( e x, e,), e_ ) are

A

of %

and

e_,

as is

Eq. (3.2a). in-plane

vector

of the

deformations point

of the

beam

P in the deformed

cross-section configuration

are neglected, can

be written

A B

R(x,r/,()

=

Ro(x)

+

r/Erl

+

_Etj

+

0c(x)W(r/,_)e

x

the as:

(3.4)

whcre

R0(x ) = R(x, 0, 0)

36

(3.5)

is the correspondingposition vector of a point on the deformed elastic axis; and Ei(x) = R,i(x,0, 0), are

the

first

base

three

the

last

known the

vectors

terms term

of a point

represent

is the

amplitude

out-of-plane

to the The

A I

q'(q,

definitions

orthonormal A

e¢ ).

the

direction

the

orientations

triad

of

be viewed of

E,, the

Eqs.

In Eq.

(3.4),

of the cross-section, cross-section;

out-of-plane

the

while

c_(x) is the

un-

function

of

warping

_l-', _. (0, 0) = 0

(3.5)

and

deformed

Without

(3.7)

(3.6),

respectively.

curvilinear

coordinate

of %

of

and

rotated

curvilinear

loss of generality,

system,

the

unit

to the deformed

of the

coordinate

system,

P_t

vector elastic

version

e X is assumed axis

of the

to be in

beam;

while

A t,

and

e r are

strains[100].

in terms

translated

undeformed

of E X, i.e., tangent

of the

expressed

as a rigidly the

I', t

p.356]

_, tt (0, 0) =

of R 0 and

triad

A

( e x, %,

count

of the

axis.

Af

( e X, e¢, e_ ), can

A

(3.6)

elastic

rotations

r) is the

£

with

orthonormal

A I

and

warping

• (0, 0) =

due

the deformed

translations

of warping;

cross-section,

on

i = x, r/,

The /_'t

Ap

e x,%

nearly

that

deformed and

A I

e_

of E_ and

E_

vectors

of the

base

by

the

following

but

differ

elastic

definition

on

ac-

axis

are

[Ref.

I00,

:

Ex =

A

--

A

t

(1 + _xx) ex

t

(3.8a)

Ap

E_t =

2 _xn ex + (I + _rtn)eq

E_ =

2_x_e

At

A#

x + _(e

37

A

+

t

_n_ e_

_

A

n + (! + _,_()e(

(3.8b)

t

(3.8c)

With the assumption that are

neglected,

in-plane

base

vectors

of the deformed

_£_ =

it will

transverse ply

shear

that

cross

( e.g.,

_-_

( e.g.,

E_-E_

E,7 =

2_x_

E¢ =

2gge

be shown

latter

strains, sections

plane

beam

cross-section

plane

is not

The

deformed

base

Gx =

R x =

Ex +

17E_, x +

_

A t

(1 + gr,x)ex

(3.10a)

e_ =

yxnex

A t

At

--

e¢ =

y_e

x +

At

_xx,

Y_

at the

normal

to E x ) due

vectors

A s

and elastic

to the

(3.lOb)

%

At

x +

(3. l 0c)

e¢

yg

are

axis.

the

axial

Equations

and

the

(3.10)

im-

axis before

deformation

axis after

deformation

to the elastic

at point

presence

of transverse

shear

P are:

A

djE¢, x +

As

+

to the elastic

be normal

normal

(3.9)

become:

At

are

strains.

axis

0

ex +

that

) will no longer

=

A t

respectively, which

_£

elastic

Ex =

where

of the

Le.,

_n_ =

the

deformations

ct,x_e

t

x +

A t

0t_,x As

= [(I + _,=)+ _(2g_, x - K,_)+ ((2g_, x - x¢)+ e,x't'] ex (3.1 la)

+ [2_%_

+ _(2%g_

+ [_(2K¢_ + T) + 2(K_ A

G,_ =

R,_

=

En +

_tW,,le

r) + e_K,t] ^' + _1'_:_] _

t

x (3.1 lb)

A t

=

(2gxn + _tW n) e x +

A t

e.

38

A

G_ =

R,_

=

E_ +

I

cz_'_e× (3.1 lc)

= (2_x_ + c_,

_)e x +

A !

where

the

derivatives

curvatures,

of the

K_ , K_, and

orthonormal

twist,

r,

triad

(ex,

of the deformed

Ap

A t

%,

e¢ ) are related

beam

A_.

to the

by:

At

Kff

A,

--

O

Ap

- K(

Kr

A

t

(3.12)

3.2

STRAIN The

set

vector the

nor

beam

tation

since

the

base

orthogonal with

(x, r/, _') arc, vector

to the

nonzero

initial

(x_, x 2, x3) will be used

3.2.1

Strain

The fined

COMPONENTS

of coordinates

coordinates

Components

components

by (Ref.

Combining

0

Eqs.

Ref.

gx,

base

expressed

vectors

twist

non-orthogonal in Eq.

g_ and g¢ for

z 0 . In the

in place

derivation

of (x, t/, _') whenever

in Curvilinear

of the strain

97 and

in general,

is neither

an arbitrary that

follows,

a unit point

on

the

no-

convenient.

Coordinates

tensor

in the curvilinear

coordinates

are de-

98, p. 113):

! fij = "_-( Gi" Gj ) -

(gi-

(3.2)

Eq. (3.13)gives:

and

(3.2a)

curvilinear

(3.11)

with

39

gj ),

i,j = x,r/,_

(3.13)

(3.14a)

+T

I (_i2+ {2)(2_

_2)

%)]

(3.14b)

+ _-[_',¢: + n(_- %)]

(3.14c)

f_ = f_x = _,_ + _-[_',_-{(r-

f_ = rex = _

(3.14d)

f_-----0

f¢¢ _ 0

(3.14e)

fu¢ = f{n _ 0

(3.140

In the derivation neglected relative

with

vectors

Components

parallel

no assumption

of local cartesian

relations

coordinate

coordinates

system.

(xl, x2,x3)

(Y_, Y2, Y3), consider

40

components

was made order

presented

triad

regarding terms

the

contain-

in this section.

(Yl, Y2, Y3) at point

P with its

(_x, _,_,_¢) of the cross

section,

of the beam

are assumed

To find the transformation and

were

Coordinates

coordinates

to the orthonormal

strain

Higher

in the derivation

in Local Cartesian

The stress-strain

in the local cartesian curvilinear

but

axial and shear

of axial and shear strains[7].

a system

respectively.

the

to unity,

were also neglected

Strain

Define unit

respect

magnitude

ing warping

3.2.2

of Eqs. (3.14), both

the

local

cartesian

to be given between coordinates

^

_r

_r

Oxi

Oyj

Ox i Oyj

(3.15)

ej --

^

Oxi

gk" ei =

(3.16)

( gk" gi ) cyj

Thcrefore,

the

transformation

relation,

, can

be expressed

in matrix

form

as:

Oxi ]

OY i

=

[ gk" gi

=

[l

[ gk"

]-l

_'r 0

l+(2T

-'-o

=

I

êj ]

2

oll

--r/t_rO 2

0!

-_'rO 1

-,7¢¢o _ 1+,Ag . o

(3.17)

o

1

_o o,o

1 -- q_O

0

0 1

where

[ gk" gi ] =

- _z 0 1 1 + (,12+ _2)_o2 _ _o r/ZO

The from

strain (Ref.

tensor

97 and

defined Ref.

in the

local

98, p. 118):

41

cartesian

0

0 '7_o1

(3.18)

1

coordinates,

e,j,

is obtained

( tIR _ O.vi #R Oy i

zii =

3

3

='zx( '" 2

k=l

3 =

Substituting components ponents

30x

Z

Z

k=l

I=1

Eq. (3.17) in the in the

.... Ox k

I=1

k

Oyj

Eq.

_ 0Xk

°Xl

Ox 1

Ox k

Ox I

)

Oy i

-Oy i

(3.19)

fkl

the

transformation

coordinate

coordinate

system,

system,

f,j , can

between

e,j , and be written

2r/r0fx_

the

strain

the strain

com-

as:

e_x

=

f_x +

2_r0f_

e_l

=

%x

=

fxn

(3.20b)

e,Tx =

fx_

(3.20c)

ex_ =

-

Or

(3.19),

local cartesian

curvilinear

r)r

Ox 1

Oy i

into

OR

(3.20a)

erln _ 0

(3.20d)

e¢¢ m 0

(3.20e)

Combining cartesian

Exx

=

Eqs.

(3.14)

coordinates

_xx -

r/K,1 -

with

Eqs.

(3.20),

the

strain

components

in the

local

become:

C.x( +

_,x •

+

_ro((_,n-

r/_,_)

I

+ T (n2 + _2)(T-- Z0)2 + rl(Yx_,x-- rOYn_ + (2x)(hy + v)- (f_x 2 + f22)fn_+ w)- nyVbx + f_xVby+ Vb_ + [(_'_

f2zVby

q- _22) (hx

-- _yVbz

+ x)-

-- _/bx]

(_'_x_'_y -

_:2z)hy

-

W,x } -- (q2 + _2)(_

(_"_x_'_z + _a_y)h z +

+ _x)

+ (f_2 _ f_z) [(r/2 - _2) sin fl cos fl + r/( ( cos2fl - sin2fl)] - _-Qz

[(r/2-

{(f_2 _ f_)

¢.2)( cos2fl _ sin2fl)_

4q( sin fl cos fl] -

[(1/2 _ (2)( cos2fl _ sin2fl) _ 4r/( sin/? cos fl] -

4f_yf_z[(r/2 _ (2) sin fl cos fl + r/C"( cos2fl - sin2_)]}

87

4'

Z_

_'{(r/sinfl+ ( cos fl)[- nxf'/z- ny + (;¢,x-2nz4, + (-Qx-Qy-

nz)_

- (f2_+ _2)w x] + (,/cosfl- ( sin fl)[ - _xt)y + _z + V,x

-

2f2y_

- (f_x_qz + _')y)q, -

(f2_ + f12) V,x] (4.51g)

- i_- _i - n_;._- _< + 2flz;, - 2_y_V +(f_

Z_ 1

+flz)(hx+X+U+_Ot+r/?x,

-- (nxf_y-

Oz)(hy

+ v) -

+ _'2zVby

-- f_yVbz

-- _rbx

7+_¢)

([2x[l z + Oy)(h

}

,1{(q sin p + _ cos p)[ - flxf_ z -

-

(f_

+ f_z) _ ,x] + (r/cos

-

2ny4

-

ii - Wii - r/_xn - _.K

- (flxnz

+ fly)'/'

z + w)

fl -

-

hy +/i,',x-

2f_z4, + (f_x_y

( sin fl) [ - f_xfly

(hi

+ 2f2z9

+ flz:) V,x]

-

(4.51h)

2.Qy,.'v

+ (n_ + nz:)(h x + x + u + V_ + rtTx. + ¢_,Z) -- (f_x_y

-- _z)(hy

+ v) -

+ f_zVby

-

-

_yVbz

Vbx

(f_x.Qz

+ l'_z + V,x

+ _y)(h

}

88

z + w)

z¢ = 4"{(17sin fl + _ cos b')[ - f_z

- _,, + 'a',x - 2f_z_ + (_xf_y

") , ,x] + (_ cos fi - (fl_ + _)_)_ _"sin ,6')[ - f_,_f_y+ fl, + _,x

(4.51i) + (_

+ _1;)(h x + x + u + q'_ + rlYxn + _'_ fi+Dy(h z+w)-f_z(hy+ v)+ Vbx ] W+_"Ix(hy+V)-i')y(h x+x+u)+ VbzJ

121

(5.14)

The

velocity

vector

of air due

to forward

flight

anti

A

inflow,

A

V a, is:

A

V A = _2R(pcoS0ir-pSin_jr-2k

r)

(5.i5) A^ Vx ex +

=

A^ Vy ey

A^ V z ez

+

where

p cos

_V/).x_

=

_)RETcb]

[Tbr]

-psin

lv )

-a (5.16) p COS @ -- tip), COS _b sin f

_R[Tcb]

In Eqs.

(5.14)

-

and

(5.16),

(Vbx, Vby, Voz ) and

[T_b]

respectively, (4.27)

and

The (5.12),

for

the

(2.7b),

velocity (5.13)

the explicit can

portion

respectively,

and

(5.15)

of the

for the U,'

and

U_')

coordinate

transformation system

A !

(4.33),

blade,

swept-tip U;'

can

(4.38), and

(h_, hy, hz) ,

(4.26)

in Eqs.

and

(4.34),

(2.7a), (4.39),

element. be obtained

by combining

Eqs.

as:

U_/_,

the

Op -

for (_x, _y, flz),

in Eqs.

fV EA

Ux'

where

4, cos

sin

expressions

be found

straight

component

Op -/.t

2 sin 0p} cos _, cos Op + p sin ¢, sin Op - _. COS 0p

pflp pflp

matrix, Ap

E dol

tvz

[Ta_]

, between

(5.17)

}

Vz

the

deformed

curvilinear

Ap

(e,, %, e_) and

the

undeformed

122

element

coordinate

system

(_x.Cy,_)'_"

, has

[Td_ ] is givcn Thc (5.17)

bccn

defined

(2.15)

and

the

second

order

expression

for

by Eq. (4.40).

accclcration with

in Eq.

component

respect

U¢'

can

be

obtained

by

differentiating

Eq.

to time:

{'VxEA _

"A

o¢')

Vya {Vy vx } EA

+

(5.18)

tVz - Vz _

I.-v_l:_a Vz j

whcrc

f

iJ + _yW

-- _z _"+ _y (h z + w) -

_z (hy + v) + _/bx

"_

= ,}_,+c,,u -C_x,V' + az(hx + _ + u)- ax(hz + w)+ %y_, (5.19)

v_AO

Lx_,' + _x _' -- f2yU + _x(hy

+ v) -

-

(rA_,

=

QR[Teb

]

_y(h

x + x + u) + '_/bz J

f_# sin

{f_l_ (fit, sin ff sin 0p - cos ff cos 0o) - 0t_ (btflp cos _, cos 0p - _ sin ff sin 0p + _. cos 0p)}

(5.20)

{f_# (tip sin ff cos 0p + cos _, sin 0o) + 0o (#tip cos _, sin 0p + _ sin ff cos 0p + R sin 0p)}

The

matrix

and

(_,

(5.19)

[Td_ ] is given

_y, f_) and

spectively,

(5.20) with

are are respect

in Eq. (4.43);

given

in Eqs.

obtained

by

(4.52)

while and

differentiating

to time.

123

the

expressions

(4.53),

of (_/b_, Vby, Vbz)

respectively.

Eqs.

(5.14)

Equations and

(5.16),

re-

5.3

BLADE

The

wherc

blade

PITCH pitch

0c3 is the

For the straight

ANGLE

angle

total

with

geomctric

portion

WITH

RESPECT

TO FREE

respect

to the

0 =

0G +

4'

angle.

The

O =

/_G +

_

(5.22)

0 =

0G +

_

(5.23)

pitch

free stream

STREAM

is:

(5.21)

time

derivatives

of 0 are:

of the blade

(5.24)

0G = 0p(¢,) + fl(x)

For the swept-tip

is the

tion of the blade tip with

respect

0p

(5.25)

0G =

0p

(5.26)

element

0G =

where/_j

0G =

[Op(_b) + flj] cos A s cos A a +

(5.27)

_T(X)

0G =

0p cos A s cos A a

(5.28)

0G =

0p cos A s cos A a

(5.29)

blade

pretwist

angle

and

the swept

tip,

at the and

junction

between

PT (X) is the pretwist

to the junction.

124

the angle

straight

por-

of the swept

5.4 AERODYNAMIC

FORCES

UNDEFORMED

ELEMENT

The

components

AND

MOMENTS

IN THE

COORDINATE

of the aerodynamic

forces

SYSTEM

and

moments

deformed

dynamic

where

lift and

the

in terms

curvilinear

blade

coordinate

pitching

local

of Uo' and

system

moment

unit

(_z'×,%, e¢) are relatcd

per unit span

span

by (see

Fig.

to the aero-

5.2):

P_t' =

L sin czA -

D cos _A

(5.30)

p_'

LcOS_A

+

Dsincz

(5.31)

qx'

M

angle

=

=

of attack,

U¢' (see

A

(5.32)

_A, and

its sine and

cosine

can

be written

Fig. 5.1) as:

O_A=

_tan-'(U_',

']

(5.33)

\u. j U¢' sin _A

-

--

=

The

aerodynamic

ment

coordinate

(5.32);

and can

forces system be written

and

U¢'

--

(5.34)

UR

COS (xA

in

'_1

At

the

per

N/Ur/,2

U_/'

Ur/

-

UR

\//Ur/2

moments

per

(ex, ey, _z_) are

+ U¢ '2

unit

obtained

as:

125

(5.35)

+ U(_'2

span

in thc

from

Eqs.

undeformcd (5.30),

(5.31)

eleand

{px} Py

(5.36)

Pz

(5.37)

5.5

TREATMENT Reverse

terized

by

flow the

existence

_k < 360°),

section

is from

reverse

flow

flow

known

a-priori.

the

the

to zero.

reverse

FLOW

It should region

flow

the

trailing the

to forward

be noted

air

to the

tangential

A commonly

flow

relative edge

requires

region

due

of a reverse

where

region,

reverse

REVERSE

is a phenomenon

180°
0 (7.12)

Gi =

The

approximate

optimization feasible

7.3

problem

package

directions.

vided

f 1 D0i] Di

lag

1-50-

flap

o.oo O.

I

-0.50

!

I

|

I

I

I

I

6-

torsion 4-

2-

0

.ooo

Figure

8.5:

I

_o

I

._oo

I

l

I

._ .2o0 .25o rrrcx _..c _)

Effect of axial mode on the of isotropic blade. Analysis

258

I

.350

.400

imaginary part of hover with substitution.

eigenvalues

0.20---. exial oxlol mode used not used I

0.10-

x

O_X]

0

Z 0

z_z-_0.10_ VI

....°**o

°°"

•°•*.***

j

O |

...

i-********_

flap

i

I

I

I

|

I

I

I

I

E

0.00 -0.10-0.20-0.10-

torsion ..............

-0.40

,

.ooo

Figure

8.6:

Effect tropic

_o

,

.loo

J

,

. ........

.1so ._o aso rrr_ _=.E (_)

of axial mode on the blade. Analysis with

259

.T_,...°.*.•*****

i

,

_

real part of hover substitution.

,

i

__o

.Loo

eigenvalues

of iso-

Figure

8.7:

Single-cell

composite

rectangular

260

box beam

0.050 -

0.000

eeOo. aa*°ee°°*ol.

_ -0.050

._

_

•

ill...........

•

ver. ply

r_

ver_ ply ....

-0.100.

g

ver. ply == -30

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

deg

•

Fulton & Hodges

•

Hong k

•

E

-0.150 .000

Figure

8.8:

, .020

Chpre ,

a . , • w . .040 .080 THRUST COEFF./SOHDITY

,

.

i , •080

Real part of hover eigenvalues for single-cell a function of (thrust coefficient/solidity).

261

.

.

, .1 O0

composite

blade

as

ISOTROPIC

0.300•_'0.250
-0.10Z

>< . ,.z,-o.2o-

0

10

20

30

40

SWEEPAN(3LE (DE(3)

Figure

9.8:

Effect of tip sweep on the real part of hover tropic blade, baseline configuration.

287

eigenvalues

of iso-

TORSIONAL FREQUENCY=3.263/REV

V

_-,,. ...................

0

'

0

2FI.2LI... 3FI

_ ....................

I

10

'

]K....................

I

'

20

X ....................

I

30

'

X

I

40

SWEEP ANGLE(DEG)

Figure

9.9:

Effect of tip sweep on the imaginary of isotropic blade, modified torsional

288

part of hover eigenvalues frequency.


0.10L_

unstable _.

0.00

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

z_ -0.10-

_.

.

_::_ -0.20 -

/ 0

z

0

,3

-0.30

ta,.I

"

0

-0.40-

I-e_

o.. -0.50-

p

p

r

-..,J

-0.70

'

0

I

10

'

I

20

'

I

30

'

I

40

SWEEPANGLE(DE(;)

Figure

9.10:

Effect tropic

of tip sweep on the real blade, modified torsional

289

part of hover frequency.

eigenvalues

of iso-

I....1Ll-1_i*2Fi_2 _1-1 TB÷3F i _

y

v

v

_6I..==1 __1 ,¢=E >-5Z I.=.i t_9

Illil_liiNiiliililjillll_lll$

m

w 4 • I.=_ 0 I.-tx: 3 .

rt_ 4=:2" Z

0

-20

'

I

-10

'

I

'

0

I

10

'

I

20

ANHEDR/_/. ANGLE(DEG)

Figure

9.1 l:

Effect of tip anhedral on the imaginary part of hover lues of isotropic blade, baseline configuration.

290

eigenva-

I....1,1-1_1-_1 -_-' _l--__t÷_FI 0.10-

unstable _,

0.00

Z 0 Z

Pm.eeoo*oeoe'D°'°*m°*_°JeJ°°*e°_fDm.n

_

: ......................................

.

-':_:'.-.l:

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

...........

**

":'"" .....

_

mJeoaoe,eo_oleee.o,6

staDle )c

U') I.,=J

:_ -0.10 ....._1 ,,,¢E Z LIJ

'" -0.20 I.t,. 0

k ...a -0.30

-0.411 -20

!

-10

'

I

'

0

I

'

10

I

20

ANHEDRAL ANGLE(DEG)

Figure

9.12:

Effect of tip anhedral on the real part of hover isotropic blade, baseline configuration.

291

eigenvalues

of

TORSIONALFREQUENCY=4.340/REV

_

0

_-

0

,C,

_"7. w n,,-

i....1.1-1FI-2 I-.,TI'"2LI÷ FI

0,)6" I,.iJ :=J ,.,,.,J

>,.5" Z LIJ

• .....

•

a_m°°le_°°X °°oo°o,,°°°°°°,_.°o

....

°°°°

....

oo°oo_(o°°

°o.°

o.o°o,,°

°°°°°°X

°°°°°°°°*°

w 4• t,a0 I--

,,v 3 . 13_ >,--

r,,,' 2. Z m

:el m

0

'

-20

I

-10

'

I

'

'"1

0

10

'

I

20

ANHI:'DRALANGLE(DEG)

Figure

9.13:

Effect of tip anhedral on the imaginary part of hover eigenvalues of isotropic blade, modified torsional frequency.

292


0.10-

A

_,

unstable

0.00

.,..........),,z I.z,J

"" 1.0450-

0

z,,

1.0400-

4S

o..

._ 1.03511z

-45

-- 1.03011-

1.0250-

1.0200

-.3340

Figure

9.24:

,

,

-.3300

,

I

,

,

,

I

-.3260 -3220 -3180 REALPART(DAMPING)

,

I

-.3140

,

,

-.31 O0

Root locus of first flap mode eigenvalues as a function of ply angle in vertical wall for two-cell composite blade in hover.

303

3.3703.3603.350Z

30

3.340-

wo3.330N 3.320-

90

20

_ 3.31o¢J

< 3.3003.290-

0

3.2803.270 -.2250

Figure

9.25:

, -.2200

, , -.2150 -.2100 REALPART(DAMPING)

, -.2050

Root locus of first torsion mode eigenvalues as a function of ply angle in vertical wall for two-cell composite blade in hover.

304

I""

As =0 deg. I---

As =20 deg. I

0.770-

0.760-

0

0

_'0.750-15

I.m.t I.m.#

=07_V

"

0.730-

< 0.720(1)

90

0.710-

0.700 -.0300

,

,

-.0250

-.0200

--

90

, -.0150

, -.0100

, -.0050

.0000

REAL PART (DAUPm)

Figure

9.26:

Root angle

locus of first in horizontal

lag mode eigenvalues as a function wall for two-cell composite blade

3O5

of ply in hover.

J--- As=0deg. J--.- As=20deg.J 1.15020 20

tll

1.100k

A

>-

-

z I..=J

,..., 1.050-

_0 •

0

I.i-

"I,

I,'--" n.," O.

,',- 1.000-

"_"_-20

z

0

0.950-

0.900

-.3_o

-20

,

l

-.32o

,

,

,

-.31o

,

-._

,

,

-.29o

REALPART (DAMPING)

Figure

9.27:

Root locus of first flap mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover.

306

I'"

As =0 deg.l-.-.

As =20 deg. I

3.803.70-30

r_

.._..3.60-

-30 T-t

z L,mJ

3.50-

s'-

-15

45;........

_

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

n.,.

'- 3.40-

o

_.i

""'-.....15

....

.

..10 J

0

< 3.30z

'' 90 _E

-- 3.203.10-

e-.

3.00

I

I

I

I

I

I

-.25o -.2oo -.15o -.loo -.o5o .ooo .oso RE_pART (OAMF'iNO)

Figure

9.28:

Root locus of first torsion mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover.

307

I'"

As =0 deg.l.°-

As =20 deg. J

3.40-

3.30-

...... 0

Z I.tJ 0 I,i.I

lS 90

''3.10ILL

"'".......,,-.1..5

IleQm

Ill |

::..:"

-

D_

_ 3.00Z

-10

___2.90-

90 -30

2.80-

2.70 -350

Figure

9.29:

, -.500

, -.450

, , , , -.400 -350 -300 -.250 REALPART(DAMPING)

, -.200

, -.150

Root locus of second flap mode eigenvalues as a function of ply angle in horizontal wall for two-cell composite blade in hover.

308

TWO-CELL COMPOSITE BLADE TIP RESPONSE, LAG MODE (MU=0.3) 0.00

o_-0.50-

z_ - 1.000

z

z

D ¢1.

-2.00

I

0

Figure

10.1"

60

Effect mode

'

'

I

120

of horizontal (/_ = 0.30).

'

'

I

'

'

I

'

180 240 AZIMUTH (DEG)

wall

ply angle

309

'

I

i

.300

.360

on blade

tip response;

lag

TWO-CELL COMPOSITE BLADE TIP RESPONSE, FLAP MODE (MU=0.3)

o_65-

-_..-..- .......

!" ::E

za3 0 Z

J b°n !

%.J

Z

--o2(.1 Ld

hor. ply 15 deg

'"1

hor. ply -15 deg I

13I--

0

'

0

Figure

10.2:

I

60

Effect mode

'

'

!

120

of horizontal Cu = 0.30).

'

'

I

'

'

I

'


wall

ply angle

310

'

I

'

300

on blade

'

I

360

tip response;

flap

TWO-CELL TIP RESPONSE,

COMPOSITE BLADE TORSION MODE (MU=0.3)

°_' 2hor. ply 15 deg i

hot. ply -15

deg

,°a J°eleleelole

J!

.J

baseline

oo oQ o*

to%el'*lela**le.o.o

n***

"aeiI_.eeela

eaooeooe*Q°

_oOOe°

o Z 0 i I.-(J Lid .--I

I_-2

_

_'_D

_

'_',_

a Q.

In

-3

•

0

Figure

10.3:

I

60

Effect mode

'

'

I

120

of horizontal (# = 0.30).

'

'

I

'

'

I

'

'


wall

ply angle

311

on blade

I

'

300

'

I

360

tip response;

torsion

TWO-CELL COMPOSITE BLADE TIP RESPONSE, LAG MODE (MU=0.3)

ver. ply 15 deg

I

ver. ply - 15 deg I

60

Figure

10.4:

120

Effect of vertical (/_ = 0.30).


300

wall

tip response;

ply angle

312

on blade

360

lag mode


65.

7°3. 0 Z Z

o2. I-t.I Ld .J

vet. ply 15 deg

v-'_,_ -_g

I

O.

0

I

0

so

I

120

I

180

I

24o

I

300

'

'

I

3so

AZIMUTH (DEG)

Figure

10.5:

Effect of vertical (U -- 0.30).

wall ply angle

313

on blade

tip response;

flap

mode



1

.L

1

_

boseline

--

ver. ply 15 deg

.... ver. ply -15

deg

0

o

•

I

60

Figure

10.6:

Effect mode

'

'

I

120

of vertical (/a = 0.30).

'

"

I

'

'

I

'

'

I

'

'

I

1BO 240 AZIMUTH(DEG)

300

wall ply angle

tip response;

314

on blade

360

torsion

TWO-CELL COMPOSITE BLADE TIP RESPONSE, LAG MODE (MU=0.3) 0.00 Im

Qi

bJ

%

z_-0.50-

_'_

0

i

:

•

.

.

_

.

.-".............

.

". %

***

I-- " 10 deg tip sweep

"".... ".

I_

"

.-

.

."

I-'" 20 deg tip sweep

"

.." /

/

,'7

,,

_

,e_

Iml

_

•

P

i

-.":;'Z ...-. _

% "-. % "-. % "*°.

_ - 1.00-

_

baseline "

I *%

.."

/

..

, .......

/

..."

/

-y

N -1.5o-

-t i.J o

D.

P--

-2.00

0

Figure

10.7:

•

I

60

'

'

I

120

Effect of tip sweep (/x =0.30).

'

'

I

'

'

I

'


angle

315

on blade

'

I

300

tip response;

'

'

I

360

lag mode


0o,5.

Z

o2. t(.P w .J

10 deg tip sweep I._... boseline 20 deg tip sweep

Lb1 e_ O.

0

I

0

Figure

10.8:

'

60

Effect

'

I

'

120

of tip sweep

'

I

180

"

'

AZtMUTH (D(C)

angle

(# =0.30).

316

I

I

240

300

on blade

tip response;

'

'

I

360

flap

mode


_

? =.1


boseline

--- 10 deg tip sweep .... 20 deg tip sweep

0

o v--

1 S S

."

"....

%

.-

Ililli11111111111tii111t11111

_

i

_

%

e-. illllilttiilltiilitlllltlilllilllltllitl

el

-3 0

Figure

10.9:

I

I

60

120

Effect

of tip sweep

I

'

"

I

'

1BO 240 AZIMUTH (DEG)

angle

(V= 0.30).

317

on blade

'

I

300

tip response;

'

'

I

360

torsion

mode

TWO-CELL COMPOSITE BLADE TIP RESPONSE, LAG MODE (MU=0.3) 0.00 • %

_% ,_ -0.50-

_,

- \',

i

;

I I_

, ,. DOSellne

i--"

I0

I.....

S _" '"

deg tip anhedral

/

10 deg tip onhedrol

I'

_",

,_/_.. .................

F'

_ -1.00-

p--

"" "

S

"'"'-.

%

-"

.."7

i=.eo

ojao

I°

_

.

_ -1.50% a

# %

-2.00 0

s

I

I

I

60

120

180

I

240

I

'

300

'

I

360

AZIMUTH (DE(;)

Figure

10.10:

Effect of tip anhedral (/x =0.30).

angle

318

on blade

tip response;

lag mode


,6-

............. i-.........

o=

Z

i_bos,,,n,

_02w _J

10 deg tip onhedrol

w 1

-10

o.

o

'

0

Figure

....... '_

10.11"

I

60

'

'

I

120

I

'

'

I


Effect of tip anhedral (/_ =0.30).

angle

319

on blade

deg tip onhedrol

'

'

I

'

300

tip response;

'

I

360

flap

mode


COMPOSITE BLADE TORSION MODE (MU:O.3)

---- baseline - - 10 deg tip anhedral .....

10 deg tip anhedral

o o i-r_ Ld

b-2 a (1. i--

-3

I

0

Figure

10.12:

60

Effect mode

'

'

I

120

'

'

I

'

'

I

180 240 AZIMUTH(DEG)

of tip anhedral (/z = 0.30).

angle

320

on blade

I

'

300

tip response;

'

I

360

torsion

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU=O.3) 0.200 -

Collective

z

_o.loo. eL

- cyclic sine

z Z U

Vel

'" 0.050Q

cyclic cosine

m

0.000

I

-go

Figure

10.I 3:

-60

Effect setting

I

'

'

I

'

'

I

'

'

I

I

-30 0 30 60 HORIZlNTALWALL PLY ORIENTATION(DEG)

of horizontal _ =0.30).

wall

321

ply angle

on trim

variables;

90

pitch

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU=0.3) 0.100-

rotor angle (rad) < 0.080I,,.. 0 I.,,I ._1 Z

< 0.060Q Z 0

._0.040-

inflow ratio

0 .--I

la.

z 0.020o¢ 0 I-0

0.000

I

-go

Figure

10.14:

-60

'

'

I

I

I

'

'

I

-30 0 30 60 HORIZONTALWALL PLY ORIENTATION(DEG)

Effect of horizontal wall ply angle on trim variables; and rotor angle of attack (/a = 0.30).

322

I

90

inflow

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU=O.3) 0.200-

collective

_ 0.100"

- cyclic sine

o,.

z__

I_nmmm_mmmm

-l¢Ji

i.a..

'" 0 0.050.

cyclic cosine

m

0.000

' -90

Figure

10.15:

'

w -60

, ' ' i ' ' , I -30 0 30 60 VERTICALWALL PLY ORIENTATION(DEG)

Effect of vertical (,a = 0.30).

'

wall ply angle on trim variables;

323

'

= 90

pitch setting

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU=0.3)

0.100-

rotor angle (rad) 0.080.

i

0.080

0

inflow ratio

__ 0.040 0 .J

z 0.020 0 I-0

0.000

I

-9O

Figure

10.16:

-60

"

'

I

'

'

I

'

"

I

"

'

I

'

-30 0 30 60 VERTICALWALL PLY ORIENTATION(DEG)

Effect of vertical wall ply angle rotor angle of attack (/_ = 0.30).

324

on trim

variables;

'

I

90

inflow

and

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU=0.3)

col|ective _0.150-

o - cyclic sine

Figure

10.17:

Effect of tip sweep (/z =0.30).

angle

325

on trim

variables;

pitch

setting

TWO-CELL COMPOSITE BLADE TRIM VARIABLES (MU:O.3) 0.100-

o_

rotor angle (rad)

_

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