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Produced by the NASA Center for Aerospace Information (CASI)
a

DEPARTMENT OF

MECHANICAL

ENGINEERING

FINAL TECHNICAL REPORT

THE INFLUENCE OF GYROSCOPIC FORCES ON

THE DYNAMIC BEHAVIOR AND FLUTTER OF

ROTATING BLADES

By

F. Sisto and A.T. Chang

December 1983

Prepared for

Lewis Research Center

National Aeronautics &

Space Administration

Cleveland, Ohio 44315

NASA Grant NAG3-47

Report No. ME-RT-82006

OF

' C bEiiAvl"

Final

i I Auy. 198.s

G3/C7

N84-20524

J11Cid

18869
FINAL TECHNICAL REPORT

under

NASA Grant NAG3-47

THE INFLUENCE OF GYROSCOPIC FORCES ON THE

DYNAMIC BEHAVIOR AND FLUTTER OF ROTATING BLADES

F. Sisto and A.T. Chang

Department of Mechanical Engineering

Stevens Institute of Technology

Hoboken, N.J. 07030

December 1983

Prepared for

Lewis Research Center

National Aeronautics and Space Administration

Cleveland, Ohio 44315

Period of the Report: 1 June 1980 to 31 August 1983

Approved for Public Release; Distribution Unlimited
TABLE OF CONTENTS

CHAPTER PAGE

1. Summary 1

2. Introduction 2

3. Blade Idealized as a Cantilever Beam

"rider Rotor Spin and Space Fixed Precession 4

3.1 Introduction 4

3.2 Kinetic and Potential Energies 4

3.3 Uniform Blades 9

36

3.4 Blades with Different Geometry

40

3.5 Time-Varying Precession

3.6 The Effect of Damping 44

4. Arbitrary Blades Under Combined Rotor Spin

and Spaced-Fixed Precession 57

4.1 Introduction 57

4.2 Stiffness Matrix 58

4.3 Kinetic Energy and Mass Matrix 72

4.4 Stability Analysis 80

5. Forced Vibration and Flutter 83

5.1 Introduction 83

5.2 Forced Vibration 84

5.3 Flutter 85

6. Conclusions 91

7. Acknowledgement 93

8. References 94

Appendix A A-1

B B -1

C C-1

D D-1

E E-1
I t -1-

c

l.m

P

t

1. Summary

This report summarizes the results of a three year investigatio

into the structural dynamics of a cantilever turbomachine blade

mounted on a spinning and precessing rotor.

A cumulated list of publications issued during the course of t

research appear as references 1,2,3 and 5 of this final technical

report. In addition to these references which are conference

proceedings and Journal publications, reference 17 is an interim

report covering the first eighteen months of research under the

Grant. The present final technical report incorporates that

interim report, with corrections, and augments it with the results

of the final one and one half year of effort.

Both stability and forced vibration are considered with

a blade model that increases in complexity (and verisimilitude)

from a spring-restrained point mass, to a uniform cantilever,

to a twisted uniform cantilever, to a tapered twisted cantilever

of arbitrary cross-section. In every instance the formulation

is from first principles using a finite element based on beam

theory. Both ramp-type and periodic-type precessional angular

displacements are considered. In concluding, forced vibrating

and flutter are studied using the final and most sophisticated

structural model.

The analysis of stability is presented in some detail

and a number of numerical examples are worked out. One example is

given approximating a shroud-type restraint at the 3/4 span point.

One other set of calculations demonstrate the role of structural

r.

f.
damping in the phenomenon. The practical occurence of this type

of instability is discussed.

The forced vibration problem is treated, with forcing

present at one and two times the rotational frequency with

amplitude dependent upon the precessional rate. Finally there

are presented some considerations on the effect of subsonic

aerodynamic damping on the dynamics of twisted cantilever blades.

The conclusion is that when flutter occurs it is of the coalescent

type with the two lowest modes coupling to account for the vanishin

of the aerodynamic damping.
t:

f.

2. Introduction to Precession-Induced Instability of Rotor Blades

An operating turbine-type aeroengine may be considered a

gyroscope with the turbomachine rotor blades representing a large

number of radially disposed beams. These blades are subject to a

wide variety of vibratory forcing mechanisms as well as flutter,

or self-excitation. Hence rotor blade vibrations, as well as those

of stator vanes, have received a great deal of analytical and

experimental attention, forming a very large sub-discipline in the

field of aeroelastic and mechanical vibrations. The excitation and

self-excitation studied in this present research program relate to

the fact that the engine spin axis may be forcibly precessed with

angular rates that are developed by the entire vehicle in which the

aeroengine is mounted.

The potential sources of precession of the rotor shaft axis

are several in number. In highly maneuverable aircraft the pilot

may intentionally institute rapid pull-up, nose-over or yawing

motions with angular rates approaching, or even exceeding one radian

per second. with a maneuver lasting only a few seconds the rotor

may turn through hundreds of revolutions providing ample time for

self-excitation to occur or for forced vibrations to build up.

In addition to this ramp-type precession, the rotor may be

subjected to an harmonic precession of the engine axis due to flying

through a turbulent atmosphere. A wing-mounted engine would be

subjected to the same harmonic angular rate as the wing chord in a

flutter situation, assuming the flutter mode had an appreciable

wing torsion component. In the latter case, however, the fluttering

system would consist of the wing/pylon/engine and the inertial and

aeroelastic characteristics of the engine and its nacelle would have

to be included in the flutter stability determination. 

Another potential source of harmonic precession is the operatio

of turbine engined aircraft on rough runways, or the operation of

land vehicles in rough terrain. In these cases, as well as in the

turbulent atmosphere, the precession might be expected to have a

certain statistical distribution centered about the natural frequency

of the entire complex vibrating structure.

-2-
` Other vehicular applications of gas turbines and the seismic

motion of stationary gas turbine mounts are additional situations 

which gyroscopic influences on blade stability may be important.

For the ultimate treatment of these gyroscopic phenomena in

turboblading the aerodynamic influences should be included in the

anaylsis. However, initial.y it is convenient to ignore the aerodynamic forces and deal with only Plasticity and inertia. If the

assumed operation is far from a fluttPr condition this assumption

is tantamount to ignoring the aerodynamic damping provided by the

working t.uid (air, or comb:stion products in a turbine).

The order of investigation is therefore as follows:

i) Develop a finite element for a tapered, twisted beam

ii) Qualify the element by static loadings and rotating natu

frequency determination E

t•

iii) Conduct stability analyses and forced vibration analyses

under both precession histories, steady (or ramp variatio

and harmonic

iv) Include the aerodynamic forces in the forced vibration

analysis, and, if possible, in the self-excitation analy

v) Include the shaft restraint in the analysis (i.e., allow

for a "coning" ofthe shaft as additional degrees of freed

In the present report items i) to iii) are included, represen

the results of the first 18 months of work under the present contr

have F (These results been reported in 3 published papers 1, 2, 3).

ti

i9

-3-

r 
l

^g

I F

i"

i

3=

3. Blade Idealized as a Cantilever Beam Under Rotor Spin and

Space-Fixed Precession

3.1 Introduction

This section deals with the study of vibratory behavior of turbo

machine blades under combined rotor spin and angular precession.

The turbine blade is idealized as a cantilever beam. Pretwist

is not considered. The effects of blade taper, variable precession

rate, damping and Coriolis forces are studied.

The purpose of this study is to gain insight into the complicated problem of blade vibrations under combined rotor spin and

precession by first studying a simplified model. The results of

this study can also be used as references for the numerical solution

obtained from the more complete model employed in Section 4.

The equations of motion are derived from Lagrange's equations.

The kinetic and potential energy expressions required are developed

in the sequel.

3.2 Kinetic and Potential Energies

It is assumed that the precessional velocity vector

is space fixed and the origin of the rotor fixed xyz coordinate

system (see Fig. 1) is a fixed inertial point.

The angular velocity of the rotor is

W we 4 1'L (C's ^ S^n B k (1)

i r:

I I

r

I -4-
i"

i

OGNG9

OF POOR QUALITY

where 8`-(vf and W is the magnitude of the rotor spin

velocity. The position vector of a displaced point on the beam can

be given as

where 9'x, gy, Ss are the centroidal displacement components

of an arbitrary beam cross section. For the purposes of this sectio

St can be considered to includc only the fore-shortening effect

(4) and is given by

1SV 

2-1

a

Taking the time derivative of equation (2) the velocity

of a point on the beam is

+Sys +irk+ w x (4)

Using equations (1) and (2) the velocity can be written more

explicitly

' (5)

(2)

i

r

r

-5-
(6)

ORIGINAL PAGE 19'

OF POOR QUALITY

The kinetic energy is obtained from

T's^•^Pdv V

We neglect all the terms which include x and y in the above

integral because these terms are due to shear deformations and/or

due to rotary inertia which are usually neglected in beam type

analysis. Equation (6) can be simplified

R+ L

T = ; S V e Af}) ^} (7)

R

where A(z) is the variable cross sectional area. In practice

blades usually have a very nearby linear thickness variation

where A is the cross sectional area at the root of the blade,

and t is the ratio of tip thickness to that of the base.

In order to reduce the beam problem with many degrees of

freedom to a single degree of freedom only bending vibration

perpendicular to the major axis of cross section is considered.

This leads to

^y = 5 (^,,^) cosh

-6-
I

MAL PAGE (9

)OOR QUALITY.

Y. 69

x ^ ;

Y

Fig.l Space-fixed Precession in the Rotor-fixed Coordinate

System xyz

Fig..2 Setting Angle for Straight Blades

i -7-
ORIGIti

AL 7-1_

rk OF POOR ti v LII Y

where 0 is the setting angle measured from the axis of spin

to the major axis of the local cross-section,(see Fig. Z) and

is the displacement in the direction of the minor axis of cross

^,. section. Furthermore the fore-shortening effect can be written ass

S (,O

IHence the kinetic eneigy integral (7) can be completely expressed

in terms of the displacement, S (z,t) in the flexible direction

of the blade. For completeness we retain the terms that contain

cross sectional coordinates and rZ _ —

v - v + ^a Ca^ + C S \ a ^^^^^ d^

R

`1 YL a^

^ s aa S c^ as ^

>z

c ^s z

2 01 1

^- R

R

- 

JZ Costf9^ + WZ 4 - a'SIA ^ ^i Cos

♦ A%? ^ sS,n^R -4- Y'C_ID ^3 ♦ n Sin ? -^ 0 Sin

l S

ql -} 1j d 51 /1 Sf X 1 2 1^

— X 12 51^ — a^ — (

/ v) ^ 'oSB

R

2 Cos 43 C — ,^ ? ( (,s `1 1 ^ `^ ♦ Sin (3 4

- a^

R

at O' J

' rl CO s 1 . rL S, A 8 1

r

I

-g- (continued)

(lo)
C

-9-

N

1

t

t'

ORIGINAL PAGE '9

OF POOR QUALITY

°^ €^ (^s ) ^ ) ( ;^)(?^)a^ ^ €w^ ^snp —^c-0s^ +Cos )

7a R a^ a: ^

P

4 It COS a (^ S ^i1 YI coS R+ S S, n ^3 j

: z (11)

3 _- S

ca 'S ^ - r^ s,n ^ ^ S cis -0.''s;O26

R

-L

.1L

Y?

CzSap1w-aw3e

Neglecting shear deformations the potential energy is given as:

U a 5 El:

With an known displacement function this integral can be

evaluated.

3.3 Uniform Blades

3,3.1 Equation of Motion

To study the behavior of uniform cantilever b.ades for which

t 1, we may assume a reasonable distribution of displacements

along the blade as follows

a (^,t) = U (t) (I - Cos 3 (,t- R) ,

ZL

r

r

r

r

r

(12"

(13)
ORIGINAL

OF POOR QU/A*_h­i,

being the tip displacement of the blade. This assumed form is

inserted into the kinetic energy T and potential energy U expressions

and the Rayleigh-Ritz procedure is applied to both integrals.

From Lagrange's equation

DL

Zo IA aU

where L T U, the equation of the beam tip motion is obtained

C1 (A

+ 4 e,cl. -4- C_ cos E"L % C-015 Z

d9

C _X —3

Q LA 4- 2. C dy

(15)

CE F S-I

where the quantities 00 as R 344

and F are 0

all dependent on blade parameters as follows:

(dimensionless time)

-R I

LO e (typically about 0.0001)

co5 4- -4- (,1'73

L m L LA.)'

IF 9 -4- 0, C 0.809 (16)

4 . L

E. 

L

-10- (continued)
1 ^^,.z ORIGINAL

OF POOR QUALITY

O 6 -3 1 + o,5- cc.5 0.809 L

Q2

Q3 

= — 1 , 094 5 !n ^L

4

0.60% /L^

L C-s (3 ( p,S^9 1 + 0, % 1 R )

^y

The blade pares sters are setting angle (stagger) (3 , blade

length L, rotor radius R, mass of blade m, section modulus EI. w

is the -tor spin velocity whereas -a is the magnitude of the space

fixed processional velocity.

The parameters given in equation (16) are for uniform blades.

If the blade is tapered these parameters have different expressions

and they reflect taper effects. However, the equation of motion

(15) retains the same form and the characteristics of the solutions

presented in this section remain general.

It should be pointed out here that xyz coordinates used in

equation (15) are rotating with the rotor angular velocity (U

Therefore, the tip deflection u is located in a moving coordinate

system. This in turn produces the Coriolis term ( E c-b sin 6 (

which is not explicitly dependent on velocity.

t

Comparing the parameter q 4 to the other parameters in the

t= differential equation (15), q 4 can be neglected for large blade length

-11-
r

r

L. Since gyroscopic instability is important for relatively long

turbine or fan blades this nonlinear term will be omitted from

further consideration in this section. For the time being it is

worthwhile to mention that this term is due to higher order effects

of the foreshortened axial displacement.

The two terms on the right hand side of equation (15) represent

forcing functions. They are generated by the centrifugal acceleration

associated with the rotor rotation. The first term is a time function

with twice the frequency of the rotor spin i.e. the force becomes

zero four times in every rotation. This force vector is in the y

direction of the moving coordinate system. The second term is generated

by a force vector that is in the x direction of the moving coordinate

system. Both inhomogeneous terms on the right hand side of equation

(15) represent components of these force vectors that cause blade

bending in the u direction.

3.3.2 Stability of the Linear Equation of Motion

Equation of motion (15) is nonlinear even though the higher

order effects of foreshortened axial displacements are ignored by

dropping terms that are dependent on q 4 . In this section we shall

omit the other nonlinear term E 1 

3 

sin Q ,2' Since the stability

of the motion is determined by the homogeneous part of the equation

of motion the linearized equation (15) is reduced to

dZ z + CQo # C- 0 l^q^ C-0 S0 -+ E ZQ^ co 3Z87 C.1 = 0 (17)

C1 9

The stability characteristics of this equation was discussed

-12-
in reference (S). The equation is identical to the Mathieu equation

if the CO' terms are ignored. The stability of Mathieu ' s equation

is studied by means of the so-called Strutt diagram, ( 6). The terms

that are of order 0( C-% generate additional unstable regions and

augment the standard stability regions in the Strutt diagram. The

derivation of these stability curves is based on Floquet theory.

The period of the time -dependent coefficient in equation (17)

is 2 7T The perturbation method of strained parameters ( 7) can be

used to determine the periodic solutions of u. This requires the

expansion of the solution u(8) in terms of

z

Z = Q -+ Cr (4 + G (4 z

(18)

Also it is necessary to expand the dimensionless squared

frequency a in terms of E- .

40 = )0a + F6 , 

-+ Eib 2 (19)

Substituting equation (18) and (19) into (17) a set of differential

equations are obtained

j u o

60 Q. = o des

o1 2u, + 6

o u, — (b, q:cose) uo

d el

z

d ul

d ez

b n LA = — tb z -+ CI 4^ c.5 2e)ue

— ( b, + q,co so) u,

etc.

-13-

(20)
ORIGINAL PAGE E$

OF POOR QUALITY

The differential equations (20) are solved successively and

secular terms are suppressed by imposing conditions on the expansion

parameters bit b2 , etc. This is based on Floquet theory for the

transition behavior between stable and unstable solutions. The

solution of the first differential equation (20) is a harmonic function

CAo -, A cos V eo a -,. (3 0 5 in V "-, a (21)

with constants A and B o . Substituting this expression into the

second differential equation (20)

2

d u,

do 2.

b ° u, _ - C6, +Q,cos19) L4

(22)

and solving this differential equation yields

Ao b, Sin Jbo6 B 6, a c°s,^b 8

,2 J-6. 2

Ao q, r cos ( — 470" a Cos ( 1 + Jbo) 8

2 V bo ZVbo -} 1

2 L b - I ZVbo ^-

(23)

The first two terms on the right hand side of (23) are secular

terms. Additional secular terms arise when r6. = I/2 If this

analysis is carried out further by inserting the above solution (23)

into the third differential equation ( 20) it is determined that

1

-14-
(bs -+ C4 , + I q % 2- = o

and

(h-x+ Q, -4- ^

 4,'")A. = o

9

if

-15-

ORIGINAL PAGE 19

OF POOR QUALITY

besides f b• ' 1/4 also gives rise to more secular

terms. Analysis in the higher powers of Cr generates even more secula

terms but for most practical purposes O (Fz ) is accurate enough.

Starting with ^, = 1/2 equation (22) becomes

c^ cd^Z

u^ 6^ — I' ) AO CO5 8 ^- — hr `- Q' )BV Sjn i 9

d8

QilQo 30 Q ► QD

36

^ Gos a a S ^ n 2

(24)

Eliminating the secular terms in (24) requires that either

-4.

61 a

130 O

(25)

or

b = 4^i A =o (26)

The two values of b 1 in equations (25) and (26), when inserted

into equation (19), correspond to two different stability transition

curves in the ( Qo,F) plane. These curves separate stable

from unstable regions.

Using these results, the second order approximation of the

transition curves can be obtained by eliminating secular terms in the

solution of the third differential equation in (20). This requires

that
ORIGINAL FAGS E.

Equations (27) and (28) indicate that

OF POOR QUALITY

is the sufficient condition for the elimination of secular terms

in u2.

Equations (25), (26), and (29) are inserted into (19) to

obtain the equations for the two transition curves that emanate from

ao = 1/4, F =0

Q

z

")

The above process is repeated at the other singular point:

ao = 1 and C- =0 . The transition curves are described by

s

2- [ TZ C12. a

and

+ ET Q%

Q ° L is P.

"" CIO] (32)

Emanating from each singular point on the a o axis of (ao , F)

plane there are two transition curves. The loci of transition values

separate the (ao , F ) plane into regions of stability and regions of

instability. Additional unstable regions can be derived for higher

values of ao . However, these regions of instability are narrow

(order of E 3 ) and do not have much practical significance.

-16-

(29)
In the stable regions equation of motion (15) can be approximated

by the following differential equation.

d u t 4o a

Es F ^^^ a _ C- F f`an p s,^ 9

d8 :

(33)

This approximation is not valid in the unstable regions of the

(ao , E► ) plane.

3 `

l

3.3.3 Numerical Example

F

g The equations (30), (31) and (32) for stability transition

curves derived by the perturbation method contain blade parameters

P , R/L etc. In this section using these equations for some

typical blade we may construct a stability chart. The stability

chart of Fig. 12 is given for a particular set of blade parameters,

R/L = 1.384, L = 10 in., p = -450 which yields the following parameters a1 = 1, q1 = 1, and q 2 = 2.

For every pair of curves that emanates from the discrete

ao values, the region between the transition curves correspond to

(ao , C- values which give unbounded solutions of the equation of

motion. Therefore these are regions of instability. It should be

noted that the unstable region at ao = 1 is considerably narrower

than the region at a o = 1/4.

We now check the validity of the stability transition curves

that are given in Fig. 12 by an independent numerical procedure.

- According to reference (8) the solutions of the homogeneous differenti

equation (17) are stable if

(34)

-17-
3

ci

0 0.0

9

0

w

w

a

uN

0.0

ORIGINAL PAGE 19

OF POOR QUALITY

V. A. 's 1/Y V. do V.1V 1 I.VI

DIMENSIONLESS FREQUENCY a0

Fig. 3 Stability Chart for the Linear Equation of Motion

-18-
A;V7

f

ss

Ft

where vl ( 6 ) and v2 ( B ) are numerical solutions of (17) with

the following initial conditions

V, (o) '(0) -o

F ' (35)

r.

V-X (o) =o V } (o) = I

The predictor-corrector numerical scheme of Adams-Bashforth

was used to obtain v I ( 6 ) and v 2 ( 9 ) for given (ao , E ) values

when al = 1, ql = 1, and q2 = 2. The stable and unstable points

are marked on the stability chart of Fig. 3. Excellent agreement

is observed with the results of perturbation methods.

3.3.4 Effect of the Nonlinear Coriolis Term

If the nonlinear Coriolis term, E q3 Sin6 VIZ

is retained in the equation of motion ( 15) a general stability

analysis becomes impossible. However certain predictions can be

made about systems which have the form

dlu 

-^ C?. 1A F 4(-0 a a d6

C1 G'L

where E is small and co8, u , dd is periodic with

a period 27f in the product variable 'd19 In these systems

"resonance" occurs only if

ro P

where p and q are small and mutually prime integers (9). If

is irrational and cannot be obtained by the division-of two integers

the solutions to (36) are not periodic but usually bounded. Therefore

-19-

(36)

(37)
3

(

i^

it suffices to make the stability analysis of (15) for only those

^q values which can be represented as a division of two mutually

prime integers

^o = (38)

P

Neglecting terms that depend on q4 , the equation of motion

(15) can be written in the form of equation (36). The coefficient

E in this equation is the ratio of the precessional angular

velocity A to the spin angular velocity w . In general this

ratio is very small compared to ao . Therefore it can be expected

that the solution. of (15) is nearly harmonic. An approximate linearized differential equation can be obtained by assuming that the

solution consists of two parts:

u = y j. + EiA (391

The first part u satisfies the equation:

diu ` + q o y L. = 0

dei (ao)

which has the solution:

UL= AL C'5 VGe e -4- BL Sin fqo a (41)

Substituting equations ( 39) and ( 40) into equation ( 15) and linearizing

by neglecting the ( E u) 2 term, results in

-20-
x

dsu -.1, ( Qc ,^. Elu ^ -+ E Q ^ coS 6 + E Q s cc 16

J 8' 4

tc sin B u ` N

OF POOR QUALITY

z E F Sin. 20 - ^2E F(-qn P S^i+8

4 43 SIA B uL2

(EQ - -^ 4i Coa9 4 GC.COt:a6)UL

(42)

The stability of the above differential equation can be

studied by considering the homogeneous part only.

d 4 -. [00 + EL0 1 -^ E C) Cos 8 -4-61Tz Cas ;2

del L

— E 2q-j Sin 6 u,. ] CA z 0 (43)

Under the condition of equation (38) the period of the coefficient

of u in the above differential equation is 2F 7r Furthermore

the stability of this differential equation can be studied by means

of Floquet theory.

Here we apply the perturbation method of strained parameters,

again.

LA = U + E (A- + E 2u, (44)

40 = 6. "+ F6, -+ E x b2 (45)

Substituting the above equations into (43) a set of differential

equations are obtained

-21-
ORIGINAL PAGr' 19

OF POOR QUALITY

d21W 

-+ Lo u o - O

do"

z

d ^^ -4 b o ut c 09 1 2 ^(4o

8

zC A)uo

de

— (b, -"ci, COSS -

,2q, 51^ 8 4a ) CAI

(46)

etc.

The differential equations (46) are solved successivelv

and secular terms are suppressed by imposing conditions on the

expansion parameters b it b2 , etc. The solution of the first

differential equation in (46) is an harmonic function

uo = A. "5 Zo e t 8o Silo \Tbc 9 (47)

with constants Ao and Bo . Substituting this equation into the

second differential equation in (46)

C1 u,

4 6ou, = — (6, Co30 — zq, s^^6 (A..) uo (46)

do IL

and solving this differential equation it is determined that secular

terms arise when be =1, J b

. = 1/2, and V 6. = 1/3.

-22-
As before, the differential equation (48) has to be solved for

these specific values of b and the secular terms that arise must

be eliminated by imposing conditions on b1.

Starting with bo

1/9, equation (48) is

i

Ali ^' = e p

bl A. Cos 3 — 6 1 pC s^' 3

de 9

' At AL S ► n 

 A3 f ^ .61. 1; cos

.

B. A L Q3 Cos 36113L 73 Sin 3

# (49)

Eliminating the secular terms in ( 49) requires that

6. At ^ ;. 6L q3 Ao " a A'- q3 6p = 0

-6, r3oVL 13 AL 1 3 A* _ o (50)

Equations (50) can be written in matrix form

-6, ♦ '^ 6 ^ -L'A L A4 0

qa _ b _ a3

A^ 2a^ Qo 0

(51)

OF POOR Q: s+L9 B Y

-23-
ORIGINAL P,-^

OF POOR QUALITY

To obtain nontrivial solutions the determinant of the coefficient

matrix must be zero. This yields the following equation

ti

61 1 - ^; ( A L '` 4 62) = o

or

6, _ ± I- A, z ,

 BLS

a

The two values of b in equation ( 52), when inserted into

equation ( 45), correspond to two different stability transition

curves that emanate from ao = 1/9, f = 0.

For each b that is obtained from equation ( 52) the corresponding

relation between Ao and Bo is determined from equation (51).

Ao

Bo AL (53)

When this analysis is extended to the second power of f we face

an inconsistency that presents us from eliminating the secular terms.

For each transition curve that corresponds to a particular b value

we obtain two conditions on b 2 instead of one.

9 z 9 z

z

— 9 A^^

' 33 gL \ 13

3 ;L

-24-

(52)

(54)
i

e

t-

)

r

x

r

I

y

I

-25-

9 Z 9 Z z

61 z

` CIO` ToQ^ t g ^3 6L AL 4 BL

L 1 7510 ^- 3 AL. T3

32

e,

(55)

Obviously conditions (54) and (55) cannot be satisfied simultaneously for arbitrary AL and BL . Examining the two expressions for

b2 in these equations, the difference is in the last term of the

equations. If these terms are neglected the perturbation solution

(44) will not be exactly periodic due to u 2 . The prerequisites for

applying Floquet theory to the transition curve will be violated.

The method of strained parameters fails to determine the stability

transition curves to the second and also to higher powers of E .

Therefore the solution so obtained is not uniformly valid as t--*-Co .

Other perturbation methods have been tried without success in resolving

this inconsistency. This remains a mathematical problem that may

be tackled rigorously at a future time.

Fortunately, in most practical cases the ratio, E , of

precessional speed to rotor spin is small. The numerical error resulting from omitting these inconsistent terms is negligible. In

order to verify these approximated stability curves the independent

numerical method of previous section is applied to the linear equation

(43) directly. This is presented in the next section.

The approximate equations (54) and (55) for b 2 and equation

(52) for b are inserted into (45) to obtain the equations for the

two transition curves that emanate from ao = 1/9, E = 0.
ORI '.j"^AL

Q' ao = 9 ± E z A^^+ Blz

E Q^ + 9 z^ 9 2 2 z C 1o 1, $ ^.3 6, A g

(56, 

The above process is repeated at other singular points: E = 0

and ao = 1/4, 1 etc. The approximate transition curves at ao = 1/4 are

a L u is (58)

C 

^

C 2 -

$ 15 1 (59)

and at q = 1 are

r

q o = 1 -^ E a L 2 

±

^^ AL a

--+B Lz

z (60, 

- c1 [a 33a C A^^ ^ Q ^^ ) 1

^^ J

It should be mentioned that these are not the only stability

curves. Additional unstable regions can be derived for higher values

of ao . However, these additional regions are exceedingly narrow and ther

fore they are less important.

-26-
All of these transition curves contain AL and B L which are

the initial condition constants of the basic solution u L* The

dependence of stability on the initial conditions is typical

for nonlinear systems. But these transition curves do not depend

on the constants A 0 and B 0 in equation (47) of the perturbed solution.

This indicates that the stability curves are valld for any arbitrary

initially perturbed motion.

3.3.5 Numerical Example

The equations for stability transition curves (56), (57), (58),

(59), (60), and (61) contain the initial condition constants

A L and B L as well as the blade parameters P , R/L, L etc. For

each of these parameters one can construct a family of stability

curves while the other parameters are held constant. We choose to

examine the effect of initial conditions on the stability curves

for a given set of blade parameters.

Stability charts of Figs. 4 , 5, 6 are constructed for the

same blade data that were used in section 4.3.3 which yield a 1,

q, = 1, q 2 = 2, and q 3 = 0.1547.

For this purpose two dimensionless initial value parameters, Y?)

and Y^j. are defined as follows:

k

L

(62)

I

I

-27-
^f

t -28-

r

i

Note that AL is the initial displacement, UL (0)and BL is proportional

to the initial velocity.

Fig. 4, Fig. 5, and Fig. 6 correspond to different initial

displacement values, Yj, = 0, rj, = 0.05, and Yj, = 0.1 respectively

In each figure four pairs of transition curves are drawn for four

different values of 0, 0.05, 0.1, and 0.5. Note that Y?1 =0.5

corresponds to a significant initial speed. For example, if the spin

velocity is 4000 rpm the initial speed for the transition curves

emanating from a o = 1/4 is 87 ft/sec.

In the unstable regions, between a pair of curves emanating

from discrete a0 values, the perturbed motion u grows and consequently,

according to equation (39), the blade tip motion u is unstable.

In Fig. 4 the transition curves for Y?, _ Y?-j = O are of

particular interest. These are stability transition curves when the

nonlinear Coriolis term is not taken into consideration as given

in section 4.3.3. From the expressions for stability curves it

can be seen that if the Coriolis term is neglected the stability

characteristics of the blade tip motion will not depend on initial

conditions (AL , BL ). This is expected since the stability of linear

systems should be independent of :;r:itial conditions. Also when

there is no initial motion (AL = 0, BL = 0) relative to the spinning

system the Coriolis acceleration does not effect the stability regions.

An interesting effect of Coriolis acceleration is the creation

of additional region of instability at ao = 1/9. This region is

not predicted by the linear analysis of section 3.3.2.
^r

o

.04

o

W

a

0.02

s.

t _

3 3

H0.04

Ca

w

• w

a

X0.02

1/9 0.24 1/4 0.26 1

DIMENSIONLESS FREQUENCY a0

Fig.5 Stability Chart for Initial Displacement V0.05

and for Different Initial Speed

z

-29-

.05 Numerical results

rti •^ o Stable

• Unstable

.05

. 05

1/9 0.24 1/4 0.26 1

DIMENSIONLESS FREQUENCY a0

Fig. 4 Stability Chart for Zero Initial Displacement 'h,=0

and for Different Initial Speed h2

.05

Numerical results •O A

05 o Stable

.1C`^ .1 • Unstable , 1 1^_ /'.5

.5

0

.5

5

.5,.1)
3

0

H

H

9 

0.04

0

w

w

a

0.02

OF POOR QUALf re

Numerical Results

o Stable

.05 405 • Unstable

DIMENSIONLESS FREQUENCY a0

Fig.G Stability Chart for Initial Displacement =0.1

andd for Different Initial Speed h 

2

-30-
r

t`

Unstable regions at a = 1/9 and a o = 1 are quite narrow

for practical initial values (Y?, < 0. 1 , Y7 a < 0.0 S) and widen for

1. increased values of initial condition parameters V?, and t?Z

On the other hand, the unstable region at a = 1/4 does not depend

on the initial conditions. This region was predicted by linear

analysis and is unaffected from the nonlinear Coriolis term.

Studying the three stability charts it might be concluded

that fcr moderate initial values and low prec ssion rates

the effect of Coriolis force on motion stability can be neglected for

blades undergoing gyroscopic motion.

So far the stability of tip

the perturbation method. Becaus

of this method it was decided to

in Fig. 4 , Fig. 5 , and Fig. 6

described in section 3.3.3. The

(43) are stable if

motion has been analyzed by

of the uncertainty in the accuracy

verify the stability charts presented

by the numerical procedure that was

solutions to the linearized equation

I v% ( 2P7r) + vz ' ( 2 P-n) I < *;?, (63)

where v1 ( 0 ) and v2 ( 6 ) are numerical solutions cif equation (43)

with the following initial conditions

V, (0) = I V,, (0) O

V1(0) = 0 Va"( 0) -

o

(64)

t Using this numerical method the transition curves that correspond

to q2 = 0.1 are checked. The stable and unstable (Q.)6)

points that are predicted b the numerical method are marked on the P P Y

L.

-31-
stability charts of Fig. 4, Fig. 5, and Fig. 6. They indicate

that the stability curves obtained from the perturbation method

are quite reliable.

3.3.6 Forced Vibrations

The equation of motion (15) indicates that the gyroscopic beam

motion is a forced vibratory motion. From the stability analysis

it has been determined that the motion in the unstable regions of

the (ao , E ) plane is unbounded. This is due to the fact that

the forcing function initiates vibrations which lead to parametric

resonance. Hence it is only necessary to give the forced response

away from the resonance regions.

In the stable regions the solution to the equation of motion

(15) can be assumed to be

Zl = U, + E u (65)

where u now includes the forced response

2

C1 uL 

-,Q1 u` 

= Sin 28 +

de

where F is given in equation (16) and is inthe order of 0(L),

and u is the perturbed solution due to the remaining terms in (15)

expecting the q 4 term. As indicated before the terms with q 4 has not b

included in our analysis for the reasons noted earlier.

In the stable regions the perturbed motion is bounded.

-32-
-1

E

z

W

aU

a

a

N

A

a

N

H

OF Poop t: /

DIMENSIONLESS TIME 6

Fig.7 Comparison of Numerical and Approximate Solutions in the

Stable Region with Zero Initial Conditions
4

-z

-4

a

E 2

2

W

U

I

a

N

D

a

N

N

ORIGINAL PAX oIj'

OF POOR QUAL !TY

DIMENSIONLESS TIME

Fig. 8 Comparison of Numerical and Approximate Solutions in the

Stable Region When Initial Conditions are u(0)=2.54 cm

and a8= 2.54 cm

s4- . 
)F POOR

From equation (65) the error in taking u equal to u is in the order

of E . Therefore in the stable region the forced response of the

equation of motion (15) with error of order E can be obtained

from equation (66) directly

(,( a uV = Aces VQo 9 '+ Bsin 40 8

1

+ E F sin 20 + IC- cof P s ►A 9

40 -'I{ a, -I

where

A = u (o)

g = i olu _ 2E F _ 2EF cot

IC119

6•o Qo —^ L7o 1

The peak value of (67) can be determined by summing the

amplitudes of individual harmonics:

z MAX +

+ 2EF Cof p

^ u I = A -^ B C10—o — H qo — i

To check the validity of the solution (67) the equation of

motion (15), disregarding the q 4 term, is integrated directly. The

predictor-corrector numerical scheme of Adams-Bashforth is used again.

Both the numerical solution and the approximate solution are plotted

in Figs. 7 and 8. Fig. 7 is constructed with zero initial conditions

whereas in Fig. 17 the initial values are u(Q) = 1 in., d u I= ( 'n.

d8 9 -,,

(67)

(68)

(69)

-35-
r,

J 

which corresponds to dF l«.o - W in/sec. The solutions

in Fig. 7 and Fig. 8 correspond to a stable point in the (a 

o

f F )

plane which is ao - 2 and E - 0.01. The particular blade data

used in these figures is the same as those used in the stability

chart.

The approximate solution of (15) in the stable region which

are obtained from equation (67) are in very good agreement with direct

numerical solutions of (15) as can be seen in Figs. 7 and 8. For

smaller and more realistic E values the accuracy of the approximate

solution will be even better.

The previous discussion leads us to conclude that for practical

purposes the peak value of the vibratory motion which is governed by

equation (15) can be predicted from equation (69) provided the particular blakie and rotary date correspond to points inside the stable

region of the (ao , F ) plane. It is worthwhile to note that bending

moments associated with peak tip displacements in this case ( F _ 0.01)

are acceptable. Therefore the forced motion of a blade under gyroscopic

disturbance is generally less important than the corresponding parametric stability problem.

3.4 Blades with Different Geometry

In this part we show how to modify the analysis of previous

section to take into account some common blade geometry, namely

tapered cross sections and shrouding of blades. It can be shown

that these properties do not change the general form of equation of

motion (15). The parameters in equation ( 16) are changed however.

-36-
{

1

ORIGINAL PAGE 6^ 

OF POOR QUALITY 

They include blade geometry efects. Since the differential

equation of motion (15) preserves its original form the stability

analysis of previous section remains valid.

3.4.1 Tapered Cantilever Blades

In section 3.3.1 we have given e ;-.ration of motion for uniform

blades that have a taper parameter, t = 1. If t # 1 the parameters

in equation (16) take the following form

e = - Wt

E = ._fL /W

N = O. 2'249 ♦ 0.1$1, (ir

E A' (0.2537 ( 1, -0. 1) (try -^ I^

q o N 6a L y WZ L

ti 7r

+ 0.2 166a -4- 0.36b% R _ 0 , 26 ,9 cosy

+(fr- 1 0.2341 -+0- 2 762 L — 0 ' l o C'OS2^

i Q 1 _ - 0.0$37 -^ 0, 1-234 L + 0. 1134 C652^ N

z

t (ir ' 1^ - 0. 01,9 t 0. 1 A431 L t 0.0° 2. Cos P

i

(70)

Ci - S1^ 2 (3 ( unchanged)

-37- 
L

r

r

ORIGINAL PAGE 6i

OF POOR QUALITY,

1 R ^

N 10.1431 -+ 0.11 34 L -} o.1134 Co5

(fir 1) (0.11 -71 -+ 0.143 1 1 + 0.093 C o5Z (3

__ I S;A

N

1 0.Q 482 + (ir — 1) O. '^ ma ^

— L cos

^0.13y 3 + 0.1 17 R ..+- (f — N 1 L L

( 0.1064 + 0, 134 l}2) ]

(70 cont.)

t

t

t

The new blade parameters which appear in the above equation (70)

are material density, e , the width of the blade, b which is assumed

constant, the root cross sectional area, A and tr , the area ratio of th

tip to the root. N is some shape factor which is defined above.

3.4.2 Tapered Blades with a Shroud at z = R + 3L/4

Except the assumed displacement distribution (13) the general

procedure described in Section 4.2 and Section 4.3 is exactly

repeated. The assumed displacement shape is now

(a, W 4 ^ ^ - R 1 - 3 - 2 ^z

L / L

which satisfies the geometric boundary conditions since

-38-

(71)
'' 3

^T

3

(

i

x

f ^?

ORIGINAL

OF POOR QUAL17-i

a^

a^ (R A 0

(72)

u(t) is again the tip displacement at z = R + L.

For this case the parameters in equation (16) of cantilever

blades take the following form

6. — Wf

E = sZ /w

N = 0.0 57 -- (^, — 1) 0.0714

EA' a 2

qoNP6

z

L 

4 3.933 (-( C -^^ 4 13.8 (fr — I^ -+ 16.5 (f^-1) 4-

w

+

N

^0.057 ► 9 + o.o$S7 L N cos ^- -f^ (o.4s71

(73)

-1. O S 1 x-13 L

( 2 Q _ t^ ^ ao9719 — aN + N cos g +o. 0065-7 L a 

-y- 4r (0, '4 5- 71 -4- 0. 51'43 L

^2 — aN t 0 05719 4- O.o^6 s7 R + N Cos z(6

+ Tr 0.4571 + o.5)y3 9 ,

(continued)

-39-
ORIGINAL PAGE, 

OF POOR QUALITY

F _ L cos 

L 0, 01667 + O, OS L — {r 0, 066 ^ 7

aN

(73 cont)

3.5 Time-Varying Precession

In Section 4.2 we derived the kinetic energy expression for

space-fixed, constant magnitude precessional angular motion of the

rotor. In practice, however, time dependent precessional velocities

are very possible. In this section we w i ll examine the case when

the magnitude of the precessional angular velocity vector is an

harmonic time function

1 _,a I = _ i C-os af. (74)

Under this condition the angular velocity vector is expressed as

(75)

where ^ is the frequency of the precession and 9 = — Wt

as before.

3.5.1 Liner Equation of Motion and the Its Stability

The equation of motion is obtained by the same procedure that

-^

W = W ( -F ^ co5 ^ (f cosh + s^^ e k) J

-40-
OF POOR Ql ALI t v

was discussed in Section 4.2 and -c .:ct-ion 4.3. Disregarding the

nonlinear terms the equation of motion is given

I _ d ,F S 40 + Y1 - I E ^ E 4^ CCO50 4 V)@ + Cos(I-y^e^

de Z 2 2

+ E2

cos + ( Q, - Cos2(I +y) 8

+ a^ Cos2(I-^i^ ] U (76)

= - E F fan g [ Ss'n ( 1 + v) 8 + Sin' ( % -v) 67

E L F : ^r ^8 + 4 :, a ( I + v) @ +

Si^^(1-^)9

where a indicates the tip displacement again. The quantities

a0 , a l , q l , 

q2, and F are related to the blade parameters as follows

8 = - Wt

F = W << I

(77)

w

_ 13,ti2 E7

WO —

PIA L 

3

(continued)

-41-
10 - 

wo

^J 1 - 

g

Cos 1^-+ I . ^ b i y + ► . 6 1 1

L

Q, = —s ►„2^

q'= 0. 631 + 0,5CO5

°•8a9 I- 2

L cos? L

(77 cont)

The equation of motion (76) represents a forced oscillation

whose stability characteristics are determined by its homogeneous

solutions. The homogeneous equation can be rewritten in the form

d z -^ Q.0 = U (78) de

Assuming the solution of the above equation to be in the

form

u - A cos (^ e -+ c^ C = cvnstgnt

(79)

equation (78) may be solved by first substituting equation (79)

into equation ( 78). The amplitude in equation (79) is than determined by the equation

dA

A de ?qo

-+ 1 -v)8 -+- ;^ c] -4. Sin ^(a.I 0 - 1 +?^)6

-42- (continued) 
z

ORIGINAL VA" ^ : `A

0

OF POOR Qil1j%'%l.1'7 Y

2

} £ ', ( q1 - 1 ) ( 5-I n [(2^o `^2`^ )e ^' 2 C] -^ SIn C(2fAo — ?t^^e '^2 C^

$qo 

+ qa S ' n (2 `1 4 0 2^ 8 ?c7 + Sin ^(^^ — 2) e -^ zc)] )

?Aov1

s1

-} G1(sinC(:t^o+2E}+ 2CJ -+ sin L(2.j Qo16Ao^i L

- a-D) 8 + ;?Cl

+ Sin ^(2^40 + 2 -- 2-x)8 + 2 C] 4 Sin E +2 )9 + 2 C^^ .

_r (80)

a

If any term on the right hand side of this equation is not harmonic,

the solution amplitude in (79) will be unbounded. This leads to

p unstable solutions at certain a 0 values. The unstable values are

indicated schematically in Fig. 9.

^. V `v`

`T

1

• 2

p 1)1 tiv0

9-0

(1-U) (t-}vl^

(I-v) I (1+U)1

ti qo

Fig. 9 Schematic Indication of Unstable Values

-43-
r.

The general stability of equation (78) can be studied

conveniently again by means of the parameter plane (a o , E ). The

plane can be divided into regions of stability and instability by the

so-called boundary curves or transition curves, separating these

regions. The perturbation method is used to derive these stability

+ boundary curves at each of the unstable a values given above.

In order to apply the perturbation technique for the determination

of s}ability it is required that the function f( 8 , E ) in equation

(78) be periodic.

The sufficient condition for f( to be periodic is

that 1) is a rational number, i.e., obtained by division of two

mutually prime integers m l and m2.

rn (81)

rn

Under this condition the period of f( 9 ) E ) is 'Tf ml if ml and

m2 are both odd. Otherwise the period is 2')Y m l . Later in this

section we will restrict m1 and m2 to odd numbers only.

- By limiting f( e,E ) to periodic functions Floquet theory

for linear differential equations with periodic coefficients is now

applicable. Here we use the method of strained parameters again.

The solution of (78) is assumed in the form

s` 2

U = U 4- Eu -+ E Uz + (82)

with the stipulation that the solution u is periodic with the period

of f( O, C- ), whereas the fundamental frequency a is given by

.

-44-

p-
OF POOR QUALl V

40 = 6e + E6, (83)

Substituting equations ( 82) and ( 83) into equation (78) a system

of equations is obtained

z

d uo

CI l G -+ I, o uo = o

CIe 11 + 

bd u, = b, uo - a (A E c,)S( 1 Co s (I-v)e]

Ct U 2-

d8 z + bo u, = b,u, + b,uo - u,1, [coS CI +v)e -^ c.05 C - v)e]

+ (Ao -1) C,052 -68 -+ Q2 COS 2e + ^ zco52(1 -4-709

COS

(84)

Equations (84) are solved recursively. Following the same

pattern that was described in Section 3.3 secular terms are eliminated

at each step and the equations for stability transition curves are

obtained. We give the equations for these curves at 40 =-U , Qo =

Qe- I—v

ao =

1 +V

y

and 

At q , = ?z^

A G 2

11' ^ ♦ ^ (32? —I)

—

Q^^ tZ I z

1 z 1

+ , z q q1- I 1

(85)
ORIGINAL PAGE 14

At g, = I OF POOR QUALITY

a s

a q; 2

C49 I

r q r 72 z

a 4 I (1- v)(9 -^

(86)

I-v

At 4, - y

x

y -t 4 (87)

Iry At q.

4

Oo z 1 +v 

y t(88)

— E—%

Other transition curves are obtained accordingly.

3.6 The Effect of Damping

So far we have not considered damping in the analysis. In

reality blade motion is damped by external and internal dissipative

forces. Air .resistance is a typical form of external damping and

internal damping may be due to dissipative stresses in the blade

which is called material damping.

The dependence of dissipative forces on blade motion is

quite complicated. In this section we shall study a simple model

in which damping force is proportional to the velocity. Introducing

-46-
ij

3

4

-47-

I

s.

t -

C o) U

a damping force w d 6

into the linear equation of motion

(17) the equation becomes

d + c ^ 2 + 4' cos^ 6 u=O de' ^ 

du

^'`^ dE' + CQ° E q ^ Eq, cvsE^ E Q' ^ (89)

The constant c in the above equation represents the viscous damping

factor.

In order to study the stability of the motion governed by

equation (89) the equation is reduced to a Hill's type of equation

through the transformation

c

-W6

U (e^ a CA (90)

under which

d u r ci X 2

de 

+ l — L4.;71 # E Ck + Eck, cosh + E c, cos 6^ U = O (91)

From the results of section 4.3.2 it can be inferred that the

solution to equation (91) is sirgular at 0e - W1 = 1

/4 and

Qn - w' = I In other wcrds, the singular points on the ao

axis of (a 00, E ) plane for the u:damped case are shifted to the

left. In general c l is a very small quantity so that the singular

points may be assumed approximately coincident with those of the

undamped system. Since the other parameters in the differential

equation (91) are the same as those of the undamped system of

equation (17) the stability transition curves will have the same form.
ORIGINAL V `

` ' ,

OF POOR QUALITY

In the unstable regions the solution to equation (91) has the

following form

x'16

u(e) = e f(e) (92)

where k(6 4 -27f)_ ^D (e) i.e. is a periodic function

of 9 . The coefficient I governs the growth rate of the solution

u in the unstable regions and therefore sometimes it is termed the

"negative damping coefficient". In the following steps we will

obtain in terms of the parameters in the differential equation

(91) .

Inserting the assumed solution (92) into the differential

equation ( 91) we obtain a differential equation in terms of the

periodic function

y

d^ t

C11 i

4 11 -1- CQ o- C 1 4 M 4 E G I ♦ E Q^ cos 9 4 E Q, cos :Z

d e de w l (93)

=0

Following Whittaker's perturbation method that is given

in reference (7) we expand the solution and the parameters of

equation (93) in terms of E as follows

^=0.+Cz0, +E20a

a^ - w^ = bd E

z

(94)

2

= E^, E.Iz

t-

^t -48-
ORIGINAL PAGES

OF POOR QUALITY

Substituting the above expansion forms into equation (93)

and rearranging in powers of E we obtain

96o ..+.. ^o ^o = O

^„ + bo 0,

0o, 

Q, ca a ^o

0 2 4 bo s^z = —2.j 1 ^do — 2A, 36 1 — b, y^,

— (62 +aO) J,eos6^ti, — Q*e03'2e 00

(95)

Equations ( 95) should be solved in the neighborhood of the

singular points mentioned before, name:_y Ap -s!.. .. '/tI and

z

0 0 - w, _ The resulting secular terms are eliminated

in each power of F so that periodicity of '40 is maintained at

each step. At Qo - ^^^'Wa ^/y we obtain the following

conditions for eliminating secular terms in 01

b, + jl^'L-A,^ (96)

AoT.-I

_

_

z

^. z

Bo A l 2 + c{ -"41 j (97)

where Ao , Bo are the solution constants of 4. . Equation (96)

represents a relation between b and .1, which can be used to

expr;,ss Al in terms of the other parameters. A better approximation

is obtained by eliminating secular terms in 0 & . This requires that

-49-
URIG{T..L PA,"E , 0'1*1

JF POOR QUAL. T Y

`1Z _ ^*a

y 1 2

^^ q2

Q

= 61+'Ail + (98)

^= 1

61 2

QIz

from which it can be shown that

-14z = O

(99)

110)

and

-.w, 2 - 4, - if

(101)

In equations ( 100), (101), and (96) the frequency expansion

parameters b are related to the negative damping parameters -411

in the neighborhood of ao = 1 / 4. We will give the expression for

a constant .Al curve in the unstable region near ao = 1/4. From

equation (94)

s

1

E

and

z

i z

Ao — 

W

., -^ E6, E ba

(102)

(103)

-s0-
Inserting equation (102) into eq. (101) and eq. (96) and then

by substitution into equation (103) we get the expression fora M =

constant curve in the (a 

o

f E

) plane. i

q o - . ^ t

+ 4^: EZ_^z _ E2(A ► "^ Q t ) -- J

(104)

i'''1 y ti St

By finding the extreme of the above expression it can easily

be shown that the lowest value of E is

1I^ (105)

Thin, equation can be used to determine the lowest E value

that will causes instability when damping is present in the system.

The analysis is repeated in the neighborhood of ao = 1 to

obtain the negative damping curves that correspond to a 41 = constant

value. In the first power of E it is determined that

bt =a

,106)

and in the second power of E

z - n 2

1

61 6 — q t t 2 \ l2 ^) - ^bJ^a (107)

Equation (96) takes the following form

2

Qo —-1 = I -^ E2 b . (108)

w

-51-
OF POOR QUALIr

^^ = f (109)

Inserting equation (109) into (107) and in turn substituting

into equation (108) we obtain the negative damping curve that

corresponds to I = constant curve in (a o , E ) plane in the

neighborhood of ao = 1

2 ^

40 ' Cx = I + E qb ' A^- Ey^ — ^^2 ^— ^b/^^ (110)

w j 72

The minimum value of E on this curve is

^M

2 (111)

It should be noted that the negative damping coefficient in

the neighborhood of a = 1 is in the order of E whereas that of

ao = 1/4 is in the order of E Hence the growth rate is slower in

the unstable region near a o = 1. It can be expected that the growth

rate at other singular points will be successively smaller.

So far we have analyzed the growth rate of u that was defined

by equation (90). The next step is to link the growth rate of u to

that of u which is the solution to the equation of motion (89) for

-52-
r

s

•

a -

blade vibrations whin damping is present. It must be noted that

this dissipative damping has no relation to the negative damping that

was mentioned before.

Equations ( 92) and (90) can be combined to give

_ GI- 5,)9

U ( e^ Q C9^ (112)

where is periodic. From this equation we clearly see the

criteria for stability of damped blade vibrations. The vibrations

with damping will grow in time if the negative damping coefficient

associated with parametric excitation is larger than the dissipative

damping coefficient of the system.

C

>,^ 4^ svck6 I, ty

W (113)

The stability transition curves of equation of motion (89)

are obtained by setting

C

W - (114)

Inserting equation (114) into the negative damping curves

(110) and (104) the stability transition curves for the differential

equation of (89) are obtained.

Q Z

2

1, Z

1

t

C ^' 1

Lt wl

1 -; + J

2 2 t

C+° - I + w^ '^ E ( ^i — U i ) .} C-^'i l2 — ^'^\ 2

 — ^^, ^^, ( 116

t ^ J W )

vRit:1

-53-

OF POOR QUALITY
ORICINAL Pf.-01: 7P,40

OF POOR QUALITY

0.0

0.0

E-4

0.0

0.24 1/4 0.26 1

DIMENSIONLESS FREQUENCY a0

Fig.,10 Stability Chart for Damped Vibrations

-54-
These equations for transition curves could be obtained by

directly applying Whittaker's method to equation (89) and searching for periodic solutions.

Equations (115) and (116) are plotted in Fig. 10 for a

particular blade data that yields a l = 1, ql = 1, and q 2 = 2.

The stability transition curves that correspond to a particular

damping factor, c/W = 10-3 , are compared to the undamped case.

If W is 5000 RPM CAJ = 10 '' corresponds to a viscous damping

factor, c = 0.5. As c becomes smaller the transition curves in Fig.

10 approacn the curves for the undamped case.

From Fig. 10 it can be seen that a finite damping coefficient

causes blade instability to occur at lower precessional rates. For a

particular damping factor the range of safe precessional speeds can be

determined from equations (111) and (105) when negative damping

coeffieicnt Al is set equal to the damping factor, C /W

3 Hence we obtain

= aC

_Q = (117) s a1' q o 4

and

CW 1Z S = /

afi Go = I (118)

- where 1Z5 represents the safe precessional speed. From these

equations and Fig. 10 it can be inferred that higher precessional

^R

-55- 
rates are required at ao = 1 for instability as compared to

ao = 1/4. For the particular example (c = 0.5) the safe range

of precession in terms of E = -R /W

are Cs = 0.002

at ao = 1/4 and Es = 0.051 at ao = 1.

i

if"

ai-

-56-
f

4. Arbitrary Blades Under Combined Rotor Spin and Space-Fixed Precessi

r 4.1 Introduction

The stability analysis performed in section 3 are based upon

an assumed blade deformation shape (see equation 3.13) which might

be different from the actual shape that would occur. Another

deficiency of the approach used in the last section is that pretwisted

blades can not be analyzed in the same fashion. In this section a

general numerical formulation to analyze the stability of a pretwisted

blade of arbitrary cross-section is presented. Results obtained

by this method are compared with those computed in section 3. In

addition the behavior of a turbomachinery blade with realistic

._ dimensions is studied. The same approach may be used to analyze

forced vibrations of a pretwisted blade under gyroscopic excitation.

The first step is to derive a stiffness matrix for the

twisted blade. In order to handle blades wiLn arbitrary geometry,

finite element approach is selected. For a twisted blade usually

the shell element is the most appropriate finite element. However,

the numerical method for dealing with the stability problem is

quite complicated. It is desirable not to employ too many degrees

of freedom for the modeling of the blade. For this reason the beam

element is chosen. Since the problem interested in this work happens

almost exclusively to long and slender blades this simplification

is not considered serious. A 12 x 12 beam element stiffness matrix

for twisted blades is formulated based on a theory proposed by Downs

(10). Static deflection solutions are used to study the accuracy

t -57-
-58-

of this stiffness matrix.

The derivations for mass matrix and the changes in stiffness

matrix due to rotational motions are based on Lagrange's formulation

of equations of motion. Before the equations of motion with time

dependent coefficients are analyzed the natural frequencies in both

rotating and ,on-rotating force fields are calculated. The computed

solutions are compared with known analytical and experimental results.

The subsequent stability analysis of the pretwisted blade under

both rotor spin and space-fixed precession is studied via numerical

methods based on Floquet's theory.

4.2 Stiffness Matrix

The elementary beam theory cannot be directly applied to

the study of twisted blade. The modified beam theory assumes that

the undeformed blade is pretwisted along its length about a straight

longitudinal axis through shear centers of cross sections. These cross

sections are parallel to the rotor fixed xy plane (see Fig. 3.1).

The blade surface is supposed to be composed of helical fibers. If

the position of an arbitrary point on he cross section is indicated

by its principal coordinates E and the distance of this point

from the shear center is

d - (r: + q ) ( 1)

By designating V as the initial twist of the blade in radians

per unit length and considering a portion of the blade with unit

length, any helical fiber of length QN which is at a distance d
OF POOR QUAD 

f

Unit

Length

N

b)

Fiq, I- Twisted Fiber Segment

of

Iz

i Q4

a)

Fig. 2 Cregcent Cross Section

-59- 
from the shear center forms a small spiral angle 't (see Fig. 1; with

the longitudinal axis. The length of this helical fiber can be

approximated as

i

Knowing this length, the axial strain of a helical fiber due

to bending, axial load and torsion can be derived. The axial filar ment strain caused by an extension of "a" per unit length is

Z Z %Z : 1 2

( [(i4Ct)2 +dd] -[1+mo d

Neglecting higher order terms in oC this equation may be reduced to

bi

The axial strain due to an elastic twist per unit length can

be given in a similar manner

c 

^

^ 

^2

(5)

2 2 3 +0/

Again ignoring higher order terms in 0 and v4

C- - ^oC^ I 2o<d2 ^ (6)

-60-
E

dW T

.lot

L

^oCc^2^1X

al

[P] ^

-61- 

(9)

JW

dsof

C11%

dd

d^

aa2

oC d ^

►

0R

OF POOR QUN1 6"I

The total axial strain form axial extension, bending and torsion

can be derived

^_?^d

=^ aW _ *dui _ *dui 7

where ^* _

- exple = fi t- e (ex ,ey are positions of the centroid)and

where and [.(,j are the deformations in the principal directions.

The stiffness matrix is obtained from the potential energy

of a beam element

lT - ^y Fdv (8)

where E is the deformation strain and 6' its corresponding stress.

Substituting the axial strain given by equation (7) into the strain

energy expression (8) the stiffness matrix is determined after

performing the integration. Since torsional strain was not included

in the strain equation (7), torsional rigidity has to be added to

the final stiffness matrix. It is worth noting that other effects such

as shear deformation, warping and rotary initial terms can be included

in the strain expression (7) if they are considered important. The

integrand in equation (8) may be written in matrix form
S

z c

ORIGINAL PAGE C4;

OF POOR QUALI i t'

Node 1 Node 2

x

z

Fig.3 Principal Nodal Displacements in Bending

-62-

_ 
For the finite element formulation the deformations within

an element j DUj have to be expressed in terms of nodal displacements.

This is accomplished by assuming reasonable distributions for the

element deformations between end nodes and relating these deformations

to nodal displacements by shape functions. Element bending deformations Uj and Uq in equation (9) are in the principal directions

of the cross sections and hence they rotate with the pretwist angle

along the longitudinal axis. This will lead to complicated finite

element shape functions. In order to keep the shape functions

simple the principal displacements 4 and V at the middle

of the blade element are selected to represent the element bending

deformations. They are related to bending displacements of an

arbitrary cross section by (see Fig. 3)

U,f = U COS aS 4- V S iI1 dS

S I 'lo(S + VCOSdS

(10)

The coordinate s originates front the middle of an element. It

coincides with the longitudinal z axis and they have the following

relationship

h-,1^; -^ ISM <^ (11)

where n is the number of the element and Qi represents the length

of each element. The length of the nth element under consideration

is J. R is the rotor radius.

I

-63-
All the displacements are approximated by polynomials

W = C 'S + C2

N' CS + C,4S"{CSS'-^CbS3

'v = C.-7 a- C%S 

+CQs2 + CIO s3

(k=C„S+C12

(12)

The constants C; are determined by the nodal displacement variables

at the two end nodes (1 and 2)

22 ^ (13)

The expressions of C are listed in Appendix A.

The deformation variableL ID.11 used in equation (9) can now

be expressed in terms of nodal displacement variables € Nij . The

bending curvatures in ^ %,U j are related to curvatures in the

middle of the element.

+ d s 8'S2 Cc 5 as, S,ndS (14)

A ov RZ

- Z s^ n oC S +

d t Cos It

s dS ds

Differentiating equation (12) with respect to s the following

equations are obtained

N POOR
aw

ORiC)INAL^; f,,

d5 C,

OF POOR QUALITY

n

1

ds ` '

c

(15)

d A = aC5 + 6 C^ S

4 - CS + a C, s + a C,o S=

as

The constants C i are known in terms of nodal displacement variables

fail . They are substituted into equation (15), and with equation

(14) the transformation matrix between deformation variables

and nodal displacement variables €'yl is obtained

W,

d0%I as

dW 71 • TI,,z

U1

ds dun

^i I

offd.s f

NSa '^^) . . s

^ VI

d uh Tv

(16)

d5s

''' •

A0

ult

T`'' 1 dufi ds ds

ua I

or btu _ [ T.1 I oil ds ,,

Vz
s

The elements of transformation matrix L711 are given in

Appendix B.

Inserting equation (16) into equation (9) and after performing

the matrix algebra the potential energy is

if =

 

S T

V -11 (A I SIA ^ ,A ii (17)

V

where the matrix [S] is

2 Z T

[S3= E(I - ad)[Tj [Pj[T,1 (18)

From the potential energy equation (17) the stiffness matrix is

derived

^K! = rS^aV (19)

To extend this integration for the stiffness matrix_ to a

relatively complicated cross section of the type which is liable

'_c occur in real turbomachinery blades requires the introduction

of a method known as the isoparametric transformation in the finite

element analysis (11). With this approach the evaluation of the

stiffness matrix represented by equation (19) can include cross

sections of any arbitrary form. This involves the distortion of

a simple basic square form into areas with arbitrary boundary shapes.

The accuracy of the 'mapping' depends on the number of nodes

introduced around the circumference. In this report all results

are based on eight boundary nodes.

-66-
In the standard isoparametric formulation of three dimensional

distortion, mapping takes place in all three directions. In the

present approach, the integral (19) of the stiffness matrix already

contain the blade pretwist effect. Therefore, it suffices to perform

distortion in two dimensions of the blade cross-sectional area only.Y

This reduces the amount of required numerical computation in evaluating the stiffness matrix. It also assumes that the axis of pretwist

i

is a straight line.

" The isoparametric formulation begins with the establishment

of a one-to-one correspondence between the Cartesian and curvilinear

' coordinates. The variables F and YJ in matrix [P] of equation

(18) are related to the curvilinear coordinates x and y

G X N

- ( 20)

Once such coordinate relationships are known, shape functions can

be specified in local undistorted coordinates and by suitable

transformations the stiffness matrix evaluated.

The most convenient way of establishing the coordinates

relationship in (20) is to use the shape functions N i associated

with the eight boundary nodes. This may be written as

(21)

is

3

-67-
in which N i are shape functions given in terms of the local

coordinates (x,y). Based on the basic square form the shape function

expressions are

Corner nodes

t

N; = y l ♦ xx, >(1+ yy,^ Cxx,+ yy; - ^) (22)

Mid-side nodes

Y

s ^

X=O , ^; a ^^—X^^1ty2

z (23)

y; =o , Ni xx,>C^-y }

where i is the number of the shape function, and x i , y i are either

+1 or -1 or zero depending on the location of the boundary node.

To evaluate the stiffness matrix in equation (19) two

transformations are necessazy. In the first place f and Y? are

defined in terms of local (curvilinear) coordinates x, y by equation

(21). In the second place the volume over which the 4_ntegration

has to be carried out needs to be expressed in terms of the local

coordinates. A standard process will be used which involves the

determinant of the Jacobian matrix [J]. Thus

d dads - Jet ^^]dxdyds (24)

1Y

The Jacobian matrix [J] is found explicitly in terms of the

local coordinates by

^f

-68-
i

-69-

dig, d N, d nk ^, n^

0 & d §, ns

^71 s AK;, dNz dt^^

-&J- . .. ay j (25)

E't

After substituting equations (21) and (24) into the integral

express'on for the stiffness matrix (19) the stiffness matrix of an

arbitrary cross-section is

[K7 ° ^^ [S'7 a et [iij,J 1 4, (26)

The above integration must be evaluated numerically. A flow-chart

showing how to compute this 12 x 12 matrix using Gaussian integration

is given in Appendix C.

In order to establish the accuracy of this numerical method

the static deflections of cantilever beams with arbitrary cross-

: section were investigated. For different elliptical and triangular

cross-sections the errors in computed free and deflections using

this approach with six elements remained below 3% of the theoretical

values. This seems adequate for most engineering applications.

In particular for the stability analysis of interest in this work

the accuracy of the approximated stiffness matrix is sufficient.

Another interesting example selected to evaluate the numerical

accuracy in computing the stiffness matrix is the cross - section

defined by a convex and a concave edge and shaped like a crescent

(see Fig. 2). Two nodal distribution schemes were tested. They

)
are shown in Fig. 2a and 2b. For a 6 inch long cantilever beam

with 3000 lb. load at the free end the vertical and horizontal

free end deflections are compared with those obtained by the

numerical method (six elements):

d

recfion recfion

Sheme A

SheMe B 5M .O%qS

QeaM Solat; on .5608 , 0'377

This indicates that the eight nodes in case b are better placed

3

than the ones in case a. The numerical interpolation is quite

sensitive. If three nodal points are employed along the circular

edge only, the results can be as much as 100% off the correct values.

This is due to the fact that the three node parabolic approximation

cannot represent the circular edge correctly.

For twisted blade with rectangular cross section some

experimental results were presented by W. Carnegie (12). In

these experiments the blades were uniformly pretwisted over a 6 inch

(15.2 cm) length and had cross sectional dimensions of 1 in x 1/16 in.

(2.54 cm x 0.16 cm). The blade material was mild steel with density

of e = .000735 lbf-sect /in 4 and with Young's modulus of E = 30 x 106

psi. Using the same blade data and six finite elements the static

deflection under unit load at the tip was computed. The results

are compared to Carnegie's experimental results in Fig. 4. As can

be seen deflections in the horizontal and vertical directions become
OF POOR QU -t'_,-1'f

a

r

e

ti

VoErtica

Y

•

x

x

x

Hori zonta

x

•

0.030

0.025

0.14

0.12

0.10

0.08

0.04

( 20

0.015

0.010

0.02 0.005

0

0 0 20 40 60 80 —0.002 5

T y Tip Section

m

ril-

' aL

Root 0.035

Section

F Angle of Pre-twist 0 20 40 60 80

a` degrees Angle of Pre-twist

a ` degrees

x = Microscopic 0 = Dial gauges -- = Present Solution

a. Load Applied Vertically b. Load Applied Horizontally

Figure 4: Static Deflections of Rectangular Cross-Section Blades

Subjected to Concentrated Loading, ( 6 Elements )

-71-
due to pretwist. Satisfactory agreement is obtained in these

comparisons.

4.3 Kinetic Energy and Mass Matrix

The mass matrix is der^ 1­' "IVm the kinetic energy which

also determines the char;e s in she stiffness matrix due to rotational

motions. For the case .. space fixed precession these changes in

stiffness matrix are time dependent and might lead to instability

of the system. Without precessional angular velocity usually the

primary interest is in the calculation of natural frequencies.

- After deriving the kinetic energy and the associated matrices, the

correctness of these matrices will be tested by comparing the

computed natural frequencies in rotating and non-rotating force

fields w--*.th known experimental and analytical solutions.

Beginning with the displacement vector r of an arbitrary

point (x,y,z) of the beam

= ( ,' +u- 1 0 +(I +V +XO)^

g Z4

2

R

i

The x,y,z coordinates employed in this displacement equation are

the rotor-fixed coordinates shown in Fig. 3.1. The corresponding

velocity is

I

-72-
.1

+(w +Oek(x♦ Y2,— C ap• ^^ a.1•av

R ^^

11 z ' 01B ac) + w x t

(28)

R

where w is the angular velocity of the rotor and was defined in

equation (3.1) of section 3. The velocity V is to be inserted V'

into the kinetic energy

T= z Q v v av ( 29)

For the finite element analysis, the deformation variables

T au '.) Y ')

in the velocity V equation (28) have to be expressed in terms

of nodal displacement variables C Uij employed in the stiffness

matrix (19). Due to the angle between the major principal bending

axis of a cross section and the rotor spin axis the deformation

variables luF^ can be related to deformation variables i u A ^ at

an arbitrary cross section

Q Cos$ - Sin 0

0 O O 0 U^

V sino CcSk3 O O 0 O O Un

W o O 1 0 O O OW

0 - O o O 1 O 0 O

4; 0 0 O 0 Cos O -5n? O au;

a^

Q C7 O O Sin Cosh O

a^ ad 0 O v

0 O 0 1 ?^

P1

a

(30)

-73-
k,

i

X

ORIQNVAL PAGE I$

OF POOR QUALITY(

Y

Fig.5 Setting

 Angle and Global Displacements

-74-
or I Ur-1 = [-T.21 1 (441

where the angle ^ is between the rotor spin axis and the major

principal bending axis of an arbitrary cross section. If the pretwist

angle per unit length eC is constant the angle P is (see Fig. 5)

_ (3 0 -a- of -JEL ( 31)

with ^, as the blade setting angle at the root.

To be consistent with the shape functions employed in the

a

derivation of the stiffness matrix the principal displacements u

and V at the middle of an element are selected to represent the

element bending deformations. The angle of twist of the nth element

middle cross section is

n-1

Q nQ i

+ 2 di Xi "+' DenQn

where oC t and are the pretwist rate and length of elements

before the nth element. With in equation (30) the

deformation variables ^(O are functions of

'rJ

au E Ls^ ^

v , w > > I:? ait ,

Repeating the procedure used to derive equation (16) for th'--

stiffness matrix the above variables can be determined in terms

of nodal variables f Uii

(32)

(33)

-75- 
OF pOOR QUA' I .

[T=1 u rn (34)

The elements of matrix L Ts] are listed in Appendix D. With

SS f UAJ z ) Umj in equation (30) and Z um^ from equation (34)

4

^ J

the variables in the velocity v equation (28) are expressed as

functions of the nodal variables ^uil Substituting this velocity

into the kinetic energy (29), this leads to the kinetic energy of

an element

F

`C = aP ^^u,[M^^^u^^ + ^u, [M2^^u,^^u; [M3^ cc;

-+ ^ M417 4CI + i MS T Fmll A v (35)

where l u;l are the nodal displacement variables and 4;7 their

time derivatives. The matrices [ M il are

[ Mll _ [ T31 ", IT "[A,] [Tal [T31

[Mal T-5 1 "T T,,I'r A I [T21 [ T31

M3^ Y [7,]' [ T-J' A-,I [Tal [ l 3-1

My ` C Ti T 

[T21"

A4 1T21 [T31 (36)

Msj _ [T3] 

T 

[Tal 

T f 

A3j [ 72 1 [7-31

The matrices [A-.] are listed in Appendix E.

-76- 
(r With both kinetic and potential energies the equations of motion

1=

can be derived via Lagrange ' s formulation. The mass matrix is obtained

after integrating matrix C

M

I J . 

This mass matrix is not exactly

the same as the standard mass matrix because of the pretwist effect.

The differences are, howe,Cr, very minor. The elements of matrix

[Mal are time depend--7--nt. They reflect the effects of both

rotational and precessional motions on the beam stiffness. This matrix

is combined with the stiffness matrix of equation (26) to form a

time dependent matrix representing the spring stiffness coefficient.

matrix of the equations of motion. Matrix [ A31 does not appear

in the equations of motion. The forcing terms of the equations of

motion are obtained from matrices J My1 and ^ M;j . The integrations

in equation (35) are performed numerically. For arbitrary cross-section

the same numerical distortion procedure used in the evaluation of

stiffness matrix has to be applied.

Without the precessional angular velocity the integrated

matrix C M11 represents the change of blade stiffness due to

rotor spin. To demonstrate the accuracy of the present approach

the following numerical example was calculated: length, ,t = 7 in.,

width, w = 3 in., thickness, t = 0.09 in., Young's modulus,

E = 1 x 10 7 psi, Poisson's ratio, 't) 0.3, and mass density,

Q = 2.587 x 10-4 lbm/in3 . The results are presented in Figs.

5 and 6 and are compared with the experimental and numerical results

obtained by MacBain (13) using NASTRAN program and 230 plate elements.

The present numerical results, based upon the beam theory, are

obtained using 6 beam elements. In Fig. 6, the first, second and

i

-77-
a

]Lila to

n

So

U

al 4o

Cf'

41

14

E14

ro So

0

a

n^

2.0

0

z

10

j

OF POOR QUALITY

O

O

Ie

1%

NASThAN 2T

-- — — present solution

i

/ 38

OV

000—idr

pe

r ^ ^

IB

100

200 Soo pd•

Total Blade Pretwist Angle

Fig.6 Natural Frequencies vs Pretwist Angle

-78-
( IT

I oOc

N Boo

v

U

d ^D OO

a

b'

v

14 w

^4m

T^lo

1.

OR10is AL

OF POOR QUAL V

n^ 'r

L

^+ r _ I r

I 

i r '

goo

O 4000 8000 /1000 1f(oeo

Rotational speed

Fig-7 Natural frequency vs rotational speed

(30 deg pretwist angle)

0
third bending frequencies and the first and second torsional

frequencies at zero spin velocity are plotted vs. the total pretwist

angle of the blade. in Fig. 7, the same frequencies with 30 0 deg

pretwist angle are plotted vs. the rotational speed. In this case

t

the blade setting angle 0 and !ub radius R = 7 in.

The same examples were used by Chen and Dugundji (14) to show

i

the accuracy of a twisted beam finite element derived from the

i

governing differential equations of motion provided by Houbolt and

Brooks (15). Since the theoretical assumptions made by Houbolt

and Brooks are essentially the same as those made in this report

there is practica.1 y no difference in the numerical solutions.

Comparing the derivation of stiffness and mass matrices for the

finite element analysis the present approach is superior because

of is straight-forward simplicity. Nonlinear material r geometrical

effects can be easily included in the derivation. The evaluation

of cross sectional properties are based on the shape of the area and not

its moment of inertia which is often difficult to determine.

4.4 Stability Analysis

The homogenous equa- .s of motion for the study of stability

due to precessional motion can be formulated as a system of linear

differential equations with periodic coefficients. The system

has the following form

M K*l^^ = o ( 37)

E

-s0--
[Ml] is the mass matrix defined by equation (36). [K*] is the

effective stiffness matrix which is the combination of the structura -l

stiffness matrix [K] defined in equation (26) and the [M 2 ] matrix

given by equation (36). The periodicity enters equation (37)

a

through the matrix, [M 2 ], whose time dependent terms are Sue to

the precessional motion.

A vast array of literature exists on the subject of ordinary

differential equations with periodic coefficients. Some of them

are procedures to study the stability problem. After reviewing these

procedures, it was determined that most of them are not suitable

for systems with many degrees of freedom such as equation (37).

,. For large systems a general method of solution proposed by

Friedman (16) was found to be most appropriate. The details of

the method are not presented in this report because the procedure

in reference (16) was adapted here without any major revision.

Following the suggested procedure and employing the improved

numerical integration scheme based on the Runge-Kutta method a

computer program was developed to study the stability regions of

the equations of motion governed by the matrix equation (37).

The first problem studied was the straight beam with constant

cross section. This case was already analyzed using another method

in section 3.3.2. The resul^s were plotted in a stability chart

in Fig. 3 of section 3. The linearized finite element solutions

based on equation (37) with one and two elements were able to

suplicato the same stability chart. This demo-.3t.rates that the

basic_ approach employed i:: the finite element solution is correct.

i

-31-
Furthermore, since the stability chart is now obtained by two

different solution methods, there is much more confidence in the

validity of this chart.

9

d

a

1

4-

{

The next case studied was the pretwisted beam with constant

cross section. The stability chart for this case is basically very

similar to the stability chart of the beam without pretwist. The

effect of beam pretwist is to increase the structural stiffness

in the effective stiffness matrix [K*] of equation ( 37). This

leads to an increase of the parameter a of the stability chart

in Fig. 3 of section 3 depending on the amount of pretwist. Since

the regions of instability decrease with increasing values of ao,

the pretwisted beam is less prone to unstable dynamic motion under

precessional rotations.

The computation for each point in the stability chart using

the finite element solution i s very expensive and time ccnsuming.

Not enough stable or unstable points were calculated to chart

a complete stability diagram.

-82- 
5. Forced Vibration and Flutter

4

5.1 Introduction

I The complete time dependent problem under precessional

T' rotation solved by the finite element method can be reduced to a

'r

system of ordinary differential equations of the characteristic form

in which NO and (K*j are assembled mass and effective stiffness

matrices. The 7 orce matrix { F}is obtained from matrices { M4 } and

{M5 } defined in equation (36). In general, the above equations

are non-linear; only linear cases will be studied. The forcing

matrix {F} contains functions of first and second order in spin

velocity; hence it is periodic. In the following the forced vibration problem will be solved by the determination of periodic responses

to equation ;38).

The availability of the twisted beam finite element

program and a subroutine for unsteady subsonic aerodynamics (17)

made possible a short study of "fan flutter" under realistic

t - conditions of structural coupling due to twist. The effect of

i

centrifugal forces on the flutter mode is taken into account as

3

well. The terms associated with precessional rotation were removed.

a

The equations of motion to study flutter have the following form

r I ^ 6 

c ^ i

i

LM43 L^ - ^Cj + ^K*l

(39)

(38)

-83-
Matrix [C] represents aerodynamic damping and contains complex

terms. Substituting

eOct

(40)

into equation (39) the characteristic equation is obtained

I * "1) ^—^ (41)

where d and {u} are complex. The real part of the solution

represents a decaying vibration. The complex eigenvalue problem

involved in Eq. (41) is solved by a numerical procedure given in

reference (18).

5.2 Forced Vibration

In section 3.3.6 the forced vibration of the simplified

single degree of freedom system was studied. It was shown that

dynamic responses are not sensitive to time dependent quantities

in the stiffness term. It seems reasonable to remove time dependent

quantities from the effective stiffness matrix [K*] and reduce

the coefficients of the differential equations (38) to constants.

The forcing term in Eq. (38) is periodic. It can be

expressed as

F 

e 

act

42

where 0( is complex. A general solution can be given as

k I u 

e 

dt

(43)

-84-
Substituting (43) in Eq. (38) gives

(0z ^Mll + I K* I) ^ IA) ^ F^ (44)

By inverting the matrix

(45)

and pre-multiplying with the force amplitude {F} the dynamic response

{u} can be determined.

The first computational example selected for comparison is

the straight beam without pretwist. Using a model with one finite

element the same tip displacement responses were reproduced as those

given in Fig. 7 and 8 of section 3. A few pretwisted beams were

studied as well. In general, the peak displacements are smaller for

pretwisted beam.

5.3 Flutter

The computational model chosen for examination is a steel

cantilever rotor blade uniformly twisted from root to tip such

that the stagger angle varies from 30 degrees at the attachment

to 60 degrees at the free end. The blade is untapered and the

cross section is taken to be a thin rectangle of 5.08 cm chord

and 4% thickness ratio. Pitch/chord ratio at midradius is 0.878.

The hub/tip radius ratio is 0.5. Axial inflow is assumed, uniform

radially, such that the relative flow angle at the blade tip is

60 degrees. This specification determines the air velocity triangle

at every radius; the aerodynamics are computed at the mean radius

-85-
) element using the formulation outlined in [17).

The method of solution of the equations of motion was

essentially the same as that described in reference [18], and known

as the "p-k method". For a given Mach number, a value of reduced

frequency, k, is assumed, the unsteady aerodynamic forces are computed

i using the aforementioned subroutine. The equations are then solved

for the complex eigenvalues, w = w + i w I . The motion having

been assumed to have a time dependence described by exp(iw-t), the

computed value of w R provideri an improved estimate of the reduced

frequency, k = w RC/(2V) and the procedure can be iterated until w 

does not change. The converged value of w gives an estimate of

the flutter frequency; the associated value of w at convergence

is an indication of the nonaerodynamic forms of damping (positive

or negative) that must be supplied to maintain the constant amplitude

"flutter" condition. In the absence of such external damping,

the flutter condition occurs with w I = 0; thus w < 0 indicates

instability, or potentially divergent motion.

In the previous series of calculations the interblade phase

angle is a parameter. In practice this is limited to a finite

number of possible values 2Tr n/N where N is the number of blades

in the row and n is any integer, 0< n < N. For the extraction

of the greatest amount of information it is convenient to let the

interblade phase angle range continuously over 0, 2Tr. Thus, at

each Mach number, a closed contour is obtained in w R , w I coordinates

with interblade phase angle as a parameter.

The true flutter condition occurs at that Mach number where

a closed loop is obtained, lying entirely in the positive w I region

-86-
expect fo g one point at which the contour ; tangent to the w axis.

Figures 4-1, 4-2, and 4-3 show the contours obtained for

the model blade at absolute approach Mach numbers of 0.75, 0.85 and

0.95. It appears from these figures that flutter will occur at a

Mach number just below 0.85 with an interblade phase angle in

the neighborhood of 1000.

a

In each of the figures a cusp-like tangency or near tangency

of the contour occurs in the neighborhood of w = 1023 to 1253 rad/sec,

depending on the relative Mach number, and hence rotor speed.

These points, and the irregular nature of the curves near these

tangencies, are att-ibutable to the "aerodynamic resonance"

phenomenon. At the combination of parameters which produce

aerodynamic resonance the unsteady aerodynamic loading on the

blades vanish [18] and the blades vibrate as if in a vacuum. Thus,

the computed eigenvalue is real and corresponds to the rotating

natural frequency in vacuo. Although the aerodynamic damping does

not go negative, operation at this point could result in large

response to forcing and may be shown in some instances to lead

to the accumulation of fatigue damage.

True flutter of the coalescent type is illustrated in Fig. 4-3.

i

Here a slight amount of mechanical damping would produce a flutter

point at w 2330 and 100°. The mode shape represents a

combination of the modes associated with the two lowest eigenfrequencies at w R =1138 and w = 3296.

One concludes from this brief study, and other supplementary

data, that the structural coupling introduced by blade twist results,

at least in these fairly repzesenative cases, in a classical

3

"coalescence-type" flutter.

87

low
ORIGINAL PAGE IS

OF POOR QUALITY

O0N

i OONN00

' oN00

ao

r.00

0

rn r, -

11y

oN ^ N

°

o

0

° o°

0

0

0

c

OG

^l

.-r OGN00000

00

$4O

.Oazu

c^VO

it

C3. a

Lnn0Ow

a^a

caGO

GO WOOxhOOa

co u

-88- 
N

O

N

0)

O

z

N

0)

3

O

O

3

F

W

O

00^ N

u .O

N

-• z Cd

O .c

u u

e^

0

W

u

N f0

d O

a it

-^ a c a

>a

c

O un

Oo co

wo

N

Q

00

.,4

LL.

ORIGINAL PAGE

OF POOR QU ALITY

0 •

^

o

M

.4

u ^

O O O O O O

H N

O O O O

3 O W ^D d N

—89—
ORIGINAL PAGE iii

OF POOR QUALITY

cv tt

-90-

N

a^

.c

z

c:

u

co0

k

a

a

Ln

o^

c

ti0

w

v,

a

co

ya

CD

0o

lilt

w

N

3

V

M

QO

Ls.
1

Y 3

t

s

2

i

i

6. Conclusions

The 3 year program of research supported by NASA and reported

herein has resulted in four papers listed as references 1,2,3 and

S. Additionally, an interim technical report listed as reference

16, was issued after completion of the first half of the contract

period and has been incorporated into the present report.

Analytical representations coupled with computer-based

numerical solutions have been completed for a number of ajgnificant

blade dynamics problems. The earliest of these are point mass

and uniform cantilever models. A beam-type finite element, displaying the important foreshortening property, has been developed

and forms the structural basis for all of the later programs.

Blade twist, taper and cross-sectional shape are accounted for.

• Stability of a blade on a spinning and precessing rotor has

been studied using this beam element. Both steady (ramp function)

precessional displacement and harmonic precession have been

studied and instability has been shown to be possible in both

operational modes. The critical parameters discriminating unstable

motion are found to be dependent primarily on the ratios of

rotating natural frequency to spin frequency and of constant*

precession rate to spin frequency.. The blade setting angle and ratio

of blade length to attachment radius are also of significance.

With harmonic precession the critical values of spin rate are

found to he dependent upon the frequency of the harmonic variation

of the precessional motion.

* In the case of harmonic precession the maximum rate (or amplitude

replaces the constant of the steady precession case.

-91-
• Forced vibrations, outside the regions of instability and

^T

excited at three basic frequencies (a subharmonic, the spin

frequency and twice the spin frequency) are attributable to

the interaction of precession and spin and may be calculated

Y in a straightforward manner.

d.

• Blades ( beams) modeled as pretwiat, - d beam-type finite elements

were subjected to subsonic cascade flutter analysis. It was

demonstrated that it is important to take into account the

structural coupling due to pretwist and to properly model the

effect of the rotational field on the rotor blade mode shape.

The general conclusions of greatest importance are:

1) the rotor blade model for dynamic analysis in complex

' rotational fields and in air aerodynamic environment

should properly reflect foreshortening and pretwist

for reliable predictive capability, and

3

2) thin blades, on the order of 5% tip th4ckness, should

properly be analyzed using a plate-type finite element

incorporating foreshortening (as yet unformulated) and

with arbitrary orientation in the force field due to

arbitrary rotational motion (spin plus precession).

r

} 3) Twisted cantilever rotor blades at subsonic speeds tend

to flutter in the coalescent mode in which the two gravest

mode couple to produce the negative aeiod.ynamic damping.

This result is interesting since it is at odds with some of

the theoretical predictions for untwisted blades.

1

-92- 
7. Acknowledgement

The authors are grateful to L. James Kiraly and Robert Kielb

of NASA-Lewis Laboratory who acted as Technical Officers during

the course of this work.

i

f

I

-93-
P

References

1. Sisto, F., Chang, A.T., "The Influence of Gyroscopic Forces

on the Dynamic Behavior of Rotating Blades", The Fifth International Symposium on Airbreathing Engines, Feb. 16-21, 1981,

Bangalore, India.

2. Sisto, F., Chang, A.T., and Sutcu, M., "Blade Excitation due

to Gyroscopic Forces", IFTOMM Conference on Rotordynamic Problems

in Power Plants, Sept. 27th-Oct. 3rd, 1982, Rome, Italy.

3. Sisto, F., Chang, A.T., and Sutcu, M., "The Influence of Coriolis

Forces on Gyroscopic Motion of Spinning Blades", 27th Internationa

Gas Turbine Conference and Exhibit, London, England, April 18 -22,

1982.

4. Vigneron, F.R., "Comment on Mathematical Modeling of Spinning

Elastic Bodies for Modal Analysis", AIAA Journal, Vol. 13, No. 1,

January 1975, pp. 126-127.

5. Sisto, F. and Chang, A.T., "A Refined Finite Element for Vibratio

Analysis of Twisted Blades Based on Beam Theory," Paper No.

AIAA-83-0917-CP, 24th Structures, Structural Dy-amics and

Materials Conference, L. Tahoe, Nevada, May 1983.

6. Stoker, J.J., Nonlinear Vibrations, Interscience Publishers,

New York, 1950, pp. 202-70-8-.

7. Nayfeh, A.H., "The Mathieu Equation", Introduction to Perturbatio

Techniques, First Edition, John Wiley & Sons, 1981, pp. 235-249.

S. Magnus, W., Winkler, S., "Basic Concepts", Hill's Equation,

Dover Edition, Dover, 1979, pp. 7-8.

9. Bogoliubov, N.N., Mitropolsky, Y.A., "Influence of External

Periodic Forces", Asymptotic Methods in the Theory of Nonlinear

Oscillations, Gordon and Breach Science Publishers, New York,

1963, pp. 196-198.

10. Downs, B., "The Effect of Substantial Pretwist on the Stiffness

Properties of Thin Beams of Cambered Section", Journal of Applied

Mechanics, Vol. 46, June 1979, pp. 341-344.

11. Zienkiewicz, O.C., "The Finite Element Method",Third Edition,

McGraw-Hill Book Company (UK) Limited, London, 1977, pp. 178-209.

12. Carnegie, W., "Static Bending of Pretwisted Cantilever Blading",

Proc. Inst. Mech. Engrs., Vol. 171, No. 32, 1957.

13. MacBain, J.C., "Vibratory Behavior of Twisted Cantilevered

Plates", Journal of Aircraft, Vol. 12, No. 4, April 1975.

i

{ `

-94-
14. Chen, Lee-Tzong, Dugundji, J., "Investigation of the Vibration

Characteristics of Shrouded Bladed Disk Rotor Stages", Journal

of Aircraft, Vol. 17, No. 7, July 1980.

f

15. Houbolt, J.C., Brooks, G.W., "Differential Equations of Motion

for Combined Flapwise Bending, Chordwise Bending, and Torsion

of Twisted Nonuniform Blades" NACA TN3905, 1958.

16. Sisto, F. and Chang, A.T., "The Influence of Gyroscopic Forces

on the Dynamic Behavior and Flutter of Rotating Blades",

Report No. ME-RT-82006, Stevens Institute of Technology,

Hoboken, NJ, August 1982.

17. Smith, S.N., "Discrete Frequency Sound Generation in Axial Flow

Turbomachines", British ARC, R & M 3709, London, 1973.

18. Dowell, E.H., et. al, A Modern Course in Aeroelasticity,

Sijthoff S Noordhoff, The Netherlands, 1978, p. 109.

it -95-
[APPENDIX A ORIGNAL V: AE

--- OF POOR QUALI T

---- - dy ces as + = d

s -

ds ci s

s nca

CIUq CIF

'a —' sen of s + =-v C.os oC S cis ds ds

d

OR = u,

d^

dS

(A

at s = IAL

w = w1

^t

y

ds ^

aq = v2.

A-1
C,

Q W1 + W^

C2. V41 4 \AJ2.

C t, t

clx a z

ORIGINAL PAGE 18

OF POOR QUALITY

Cs QmCO3 Z

( ^XI 01C.2.)

.. '^ ^ Sin Z

C jell

Cos

a

OCR

(u.+(A,) n ( V, —V2.)

__ C®5 

^Z 3 Sid (v, 4 v^,.)

_ COS

--14—

A-2
C _ Cos ?Q

-^- Sin ed

7.R

C6 = CoS dZ 

OKI

S iA iQ

_ y^

ORIGINAL PACE Fg

OF POOR QUALITY

i

2casd set

Q

C?- —sin2

os od

Z ^

Cos°`

8

CS = 3 Sin

0

OeR

ZQ LP l I

a^

-^- S' z

--- ( S^x>I " ^xz)

Lf

Cd

4

A-3
4

C : Sin°-^ 1 2

.29 

q(XI T Xt, ORIGINAL PAGE 13

OF POOR QUALITY

-^- Cos

Z \o ,-^s/ ^Q y y

C to _ ! _ 2 S ^ n °{Z +"Vx) 2CoS `c

^C

Q

Sin

Z ^xt)

IL

` Co3 a

A'a C ^j 1 4 9(1z1

A-4

r+-- 
r APPENUX e

^ These are

^ 1-ran sforM afi or

oR,NL^ E

OF POOR QUALITY

non -zero elements

m oitri X E T

of the

TV I _ '/fit

cos z COs°=^ Cos(04s)

zQ

Sin «Q Sid .'

(%<s)

29

T:,3 — ( 2 03x3 C ;L S C.S(o(S)

. •C Q S' n 3Sin (d S)

23

Tx,'t ^' Si0°=a_ Q z -}- (^ S'—=- S Co s (a< S)

-}- CO S

- Co S d Z _ .-^. Z S Sin (,<S)

zQ

6 ,,

KQ

2^6 2Sin z 6 --- s Cos (°^s) Q3

+2Ces S sin Cis) 6 ,

B-1
T[.,(.. C os,e COS Z

ZIS —+ 6 s c os ( ot s)

at si^=Q + s;n + c s srn (,qs)

4^

_ cos Z S 

Co'

^

,

^ s^

T2.19 — s ORIGINAL PAGE 14

OF POOR QUALITY

+ 6 2 s S^n(gs)

_, to - 2

OIL

L)

'^ 6 s= S Co s eCS)

z^ Q:

J ,e

T -F G C4s .^ in C s^

TZ,^^ b 2s-

'—^ s cos ^s)

Q' J

C

+ _6 ^cas Z Sin ^o<s) s

T44J4

TI , ,

vf

T3 z

c-Cos

Q

sin

+ a -^- ` S Co s (as)

2.Q Q2

B-2
T31 %

aCOS

= ; 2 S SI n (d S^ ORIGINAL PAGE 14

Q

OF POOR QUALITY # r_

e 1 1' ^ ^ •f

;,L sin 2 `

+ 6 s cosecs)

sine—”

^ To zQ ^

.}. Cos _ cos zP

+` - $ cos(tgs^

z. Q

^- 6 . .^ aCoS 2^ s co s C ^ s)

.. - cos ?P (LolIX 1

OCR

.c. Q

^. a sin z _ s n i s co s

j Qe1

x CO3 

.^Q

T31 i - - 6 _____ S s n (d=) ^a

+

_ 

2 S Cos^^s^ L ^ a

T31

m

10 — Sin? -}- 6 -Sin

S Sin 1

z

Cos

Z^

29.

j
,tlsin4

T,, 6 7- )s S-I'n cgs)

31nd

s I co sacs

Fg

B-4
fe r C-1-ft-us — Lelen.tr+e ^^^9r h.^

wily kt f"rs

WF(11 = .23692(pt4So

W F (al :. LI'M2-U 70 4

WPCs) s ,witnttt9

WF(4) = UJF(2)

wr-(s) WF ( 1)

DQfiv,e

Five ps;'t frroomiq

ED ot,s

XW z – -90G1-79t,1+S-4

'C(2) a ;5^44G9SIo1

- tCzl "L O.

x(y) 'm —Y(2)

XM) 'a — 1( (1)

APPENDM

c

s

r

c

Sub rat►hne STIFF (aw)

Gavo — Lv4r* l4tru'"on

A - Pmtwitt angle Per unit leMA

fzr one tlema%V

START

i

i

i

i

I

i

i

i

3

'r.,: 44 ! 1*1 a of Cl `

modNluS of elackcity — AMATO

P.iss o,ns es to — AMAT (a)

Ler,9A, W i J*h +Thickness

— A L ,W ,T

of +ke it' elementr

Co ocJt4^t'ts of the n o 1es on fit .r-,i+c mmfer*Ace

of the ubss rer-Ron (on4i; at Stitsr' c*Aftr')

rn x ^^i-cc^'on — SgP C i) .

C = i ^ s$

:^ y d^nc ton — ETAP (t)

C-1
A URIGINAL PAGE IS

OF POOR QUALITY

InAW Variaklet

Vo loo

'

Int"Nifen 1; aicrefie"

DO 1(l = t, tWe f *14% '6rnw► 14

De Lo VO KZ = t,S'

^o K3 =t^5'

r0sica4ien In X CIAJ en

five pol;t: ^nhyl^

Inttinnatbo ^n y d^rK^ien

J^Ve P0shth ^► rnutq

Call Subroutine SHAP'(X(K7)8Y(K3))

- Evaluwii.a of 4k "I f%w io Ni

qo-.J Ox deviv&N%j^ ; ; t 4) N_

Nju SHOO)

^81 CAI;

Wa * ,^ aK

1)SNAFE (I,?)

f - ST Y? — ET

N^ C Na 0 N1 O )sipm

O N

ETAP'

ST P (t)

ETA P(tl

1 IM )N: _ _ _ )NI SLKI^ ETAP(

C -

1 FTAceP] 'JL '

['^Ms 'aNa

 - --

'4NI

^^ ^^ 3'q j SIP(t) ETAP(1

I?acebionl a

C F -TAt.oe,

6

C-2
. 1

•

t

F

t

i

_

3

F

a d

° UkIGINAL PAGE E9

OF POOR QUALITY

Evalw4+t6i n e4 dd"wt o* 7cm*6im 1 —VETTAG

CAL, Swbrboint KSTIF(A)

EVAIMwilOA of 4+*, sliff a rno 1mx

[FKj; 1n One Poi.mt et I;tt rutYvn

EV6l^tA4t'IIn ^F T ^^ ^ M4^11k

CSM^

 ^^ toto! 6wuts ' Lt9.^,^re ^;,tr^o►Ko/^

n -no. of 6-^ ^^tr^r^tioA P.^^ts n:k1lK^fK3=

[SMl z : [FK]i'WF(K2) • WF(KS) •T•W•AL-DErJAC•.125

ST VENANT TORS ON TERMS

G3 it AMAT(t) /6,f(;.+AMAT(2))

SM(4,k) = SM4,C) +G3 - W•TI AL

SM (4,13) S $M (4, j) --63- w.-r"/AL

S M ( 1 2,L, - SM (m,0 —G3 • W1 • T'/AL

3M(12,

ra) -SM(tz,Q)- &I-%0•TVAL

RETURN

END

C-3
S,,bra tint KST1 F (A)

Eval..ji.A e^ i^► c S*%Jess mck ix CFK]

.6r every Poet of Cr- L I' lelcu"On

START

gib=rty"3 

FA,,-C- is

Q.'

or pp0R QUN'-ITY

INITIAL DATA

A - Prctwist aan9le Per ulnit 1449+6 itic ilk eJw#At

ST, ET, ,l , lntt jrO*i6n p003 rn 9;1,?- CIIMC"onS r"Pe lel

AL.— Len94h of i 1h eleme4t

AMATM — MoUms of Elcaiic;ilj

InstA CDl n►a+rix

fom Fq. ( +7)

Insert [Tj Matrix

{6m Ltq. (:ts)

[ F K7 ' CT,T [ DJ [T]

CF K^ - [TIT[ E7

[Ftc^ = AMAT(i)* (i. — A1•(si +E))

. [FK]

R ETON

1 5 TOP I

k

1

7

C-4
Su6r^un^e SHAP(%)Y)

EvallAc 'i on of At shaft funs, +*n Ni

QA) i4-3 dQlivafive S * a^^

^ a^

START

ORIGINAL PAGE M

OF POOR QUALITY

,jNIT1AL iDATA

x 4 am ibots Vann

Gauss — LegvJm Zntgrkt'ion; In

f q dictAons re.sp&divtly

Ni it SHAPEM

N; _ 10SNAPE (I )T) a^

s ^Sr1APE(^,r)

a^

in$ert g (A trl) eo,fttl of 2

luoArotti c e6ment

s

1h Matrix fx" ERLI

((RL(I,3') ,r`1,t ),T=112)

/

-110 1 I t —I t l^_I^p^ 1 r t o 1"

I Vo I=1,'%

SX - R(I)

r) * x -'1 n,

^ J J 0

p P Eval^ Sft^¢ ^vn- trvt true

C410A Ni its derma "

-For mIC141e pou► tt ^n

Evaluate Shaft w,-

ch.n N; ih deri

vo&i-b -fur me d1k

po; h A x jimNon

E-ja►tWate shye

tvfkfibn Ni f ift der;

vatives for corner

PO4AT-s

lZEYVkN

EN D

c-s
APP5ND M D

F1 = + s -o,as s -°=-S + s $ C z^ ^ L

_ ,S ^ ^ ors F 2- L

a S - 3 -- -^- 

— (I)-j- L

_os_ - $ s 01 2S + s( L ,. L

Fl+

p,s S z

3 L L ^

S s

^S = 0.257 + L

_ L

6

0. s S z

L L T)

7 = o, 257 _S C^ + 3 CS

w here oC S e p

sfi and le re tw?

Q n 't A element"

Per un

L 4s fhe I^^^+h of Ae. A elemen

S is the coord inate a(o n3 the jA

demenfi , its origin is qt the mA- le

of the ele e-At

D-1
O O O

H +^

f

jc u?, ^(^ ? O

SIN ^ ^h .o

;n o LL

• O .= !.. } y o ti O

V

c

IL O o

'^ u U_it ^ h ^ N

y o

.^ U. p p ti LL. c t^ O

V N V H

rt

IM LL ^1^ 

,,_ o O

^I^'

E

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