New aspects on helicopter rotor dynamics

EIGHT EUROPEAN ROTORCRAFT FORUM

Paper No 3.4

NEW ASPECTS ON HELICOPTER ROTOR DYNAMICS

K. HEIER, R. RISCHER

TECHNICAL UNIVERSITY MUNICH, GERMANY

August 31 through September 3, 1982

AIX-EN-PROVENCE, FRANCE

NEW ASPECTS ON HELICOPTER ROTOR DYNAMICS K.HEIER, R.RISCHER

TECHNICAL UNIVERSITY, MUNICH

ABSTRACT

Mathematical modeling of the elastic rotor blade in flight me-chanical investigations is based on the known stubstitute model of rigid beam with phantom flapping - that is respective lagging hinge in the vicinity of blade clamping place. The elasticity of blade is respesented equivalently by installation of a spring on the hinge. The blade model serves sufficiently for statements on the first harmonic oscillations.

In case of dynamic investigations i t is however necessary to represent higher harmonic oscillation forms of blade. The ne-cessary local deviations for this on the blade supplies the solution of the partial differential equations of blade de-flections.

These coupled diff~rential equations for flapping, lagging and torsion are derived by J. C. Huboldt and G. W. Brooks.

For the solution of equations a variation formulation according to Ritz with Hermite-polynomials as formulation functions is drawn up. Based on this solution formulation, a calculation program is set up on which blade oscillation forms and bending procedures for various flight cases can be determined and dis-cussed.

The study was carried out at the Institute for Flight Mecha-nics and Flight Control at the Technical University of Munich by order of the Federal Ministery for Research and Technology.

LIST OF SYMBOLS A c. ~ EI L m* q t t

~1

T u v v -g

"

res system matrix parameter for Ritz-formulation

l i f t coefficient vector of boundary deriviations

bending resistance according to main axis bending resistance in the simplified flap differential equation lifting force

torsional resistance of the blade

total blade length mass per unit length line load res'ul ting

from l i f t distribution time

dimensionless time blade chord

centrifugal force on the rotor blade

matrices for transformation of coordinates

freestream velocity kinetic energy

total velocity in rotor fixed coordinate system velocity vector, geodetical coordinate system

velocity vector of translation

velocity vector of rotation

total velocity vector, helicopter fixed coordinate system

resulting flow velocity of the rotor blade

3.4 - 2

w

"·

~ "eff

e

_u w potential energy exterior work

deflection of the blade radial coordinate of the blade

dimensionless deflection in flap plane

preset angle of incidence

induced angle of incidence effective angle of

incidence

twisting deformation of the blade

torsion angle

angle of roll, pitch, yaw azimuth blade angle

air density

rotational frequency

rotational vector of the helicopter

INTRODUCTION

The derived and coupled differential equations in [1] are of the following form:

Torsion:

- ( (GI + TiA2 + EB e '2 ) 8 I - EB 8 I (y "case + z "sinS ) } I

1 u E 2 u E u E u Te (z "case - y "sinS )

- A E u E u

+ w 2m* e sin8 + w 2m • [ ( i 2 - i 2 ) cos2S + ee cosS ] S

Ro u YE Ro m~ mn ; u o u E

+ m*i 2

8

+ m*e

{Z

case -

Y

sinS )

=

m E E u E u

M

*

+ ( Ti 2 8 I } I

L A u w

2_{m• [ (i 2 - i 2 ) sinS cosS + ee sinS ]}

Ro mt;: mn u u o u

Flapping:

- Te case e - EB 8 'sinS 0 I}" - (Tz I) I - (w 2m *ex case 8 ) I

A u E 2 u u E E Ro u E

+ m

*

(ZE + e case e ) u E

Lagging:

z "

E + (EI l sin2s u + EI 2 cos 2S ) u y .. _E

+ Te sinS s - EB

2S 'cosS B ')

A u E u u E

..

-

(Ty ' ) ' E + (w Ro 2m*ex sinS S ) ' u E

+ w 2m*e sine eE + m•

<Y -

e sinS 8E) - w 2m*y

Ro u E u Ro E

- L

*

+ (Te cos8 )11 _{+ (w}

2m*ex cos8 )' + w 2m* (e + e cos8)

y A u Ro u Ro o u

The solution of this equation falls under the problems of ela-sticity theory.

Mechanical problems of this kind however, frequently do not permit any proper solution. This applies especially for the so called boundary value problems, that is problems as such where a differential equation or a system of differential equations with specified boundary conditions is to be solved. In such cases one frequently refers to approximation solutions adapted to the boundary conditions. The proper methods to find such approximation solutions, especially in boundary value problems in elasticity theory, are based on variation calcu-lations where the Ritz - or the Galerkin procedure are the most significant.

2 SIMPLIFICATION OF DIFFERENTIAL EQUATIONS

The now following method of solution is based on the variation method by Ritz.

First of all to make the procedure clear the simplified un-coupled flap- equation is derived.

For this the following neglections are made:

a) Not only the stiffness of blade EI, but also the mass per unit m* are assum~d to be constant along the blade.

b) Only displacements in the z-direction are considered.

-

y

=

e

=

e

=

o

E E u

c) The center of gravities-, tension - and elastic axes coincide

-

e = 0

d) Time inteJration ensues in small time intervals, so that the process can be regarded as quasistationary .

•

- L _z

=

q(x)

The differential eq~ation thus receives the form: Eiz 1111_{- (Tz ' ) ' +m*Z :::q(x)}

E E E

( 2)

in which for centrifugal force holds' T=m*w 2fxdx Ro (3) ·rhus i t follows Eiz "''- m*w 2xz ' E Ro E q (xl (4)

3 THE RITZ-PROCEDURE

In the elasticity theory the elastic bending line w = w (x)

is according to the Ritz-procedure generally approximated by

c. w. (x)

~

'

(5)

The formulation function·s w. (x) are arbitrary chosen functions which must be sufficient fa~ the boundary conditions of system. The parameters c. must be determined, that is with the assist-ance of the alre~dy mentioned variation, which can be deduced from the energy formulation due to Hamilton [2]

J[o(u-

V)

+owl

dt

=

o

<>+

(6)

From the kinetic energy u, the potential energy V and the exteriour work W follows for the virtual derivations

'

ou

=

Jm*zoz

dx

ov

(7)

ow

f(Ozq(x) L + m*w 2xz0z' + m*w 2xOzz') dx

, · Ro Ro

rn this existing dynamic problem for the approximation function according to eqn. (5) holds

'

2 (x, t) = E c. ( t) z. (x)

::, l l

From this formulation with the derivations the energy equation

L c

m* !IC.z. l:Oc.z.dx + EI [f.c.z." IOc.z. "dx

') ~l l l 0 l l l l L I 10

" "

z' z 1

*

2 ; ...,

+ -₂ m wRo Jx·-Ic z _i I6c ~ 'd

i i i X 2m*w _{R o o} 2 /x'Lc.z. I6c.z. 'dx _{l l l l} L - fq(x) ~oc.z.dx

= 0

•J l l (8) follows (9)

4 THE HERMITE- POLYNOMIALS AS fORMULATION fUNCTIONS

Now formulation functions must be found sufficing the boundary conditions. The Hermite- polynomials are deduced according

to ref. [3] from a boundary value consideration; they f u l f i l l

thus this requisition~ According to the degree of the based derivations of the boundary values, one distinguishes

Her-mite- 'Polynomials of the 4th, 6th and 8th order.

In the following tables and figures the various polynomials are illustrated. 4th _order 6th _order

a,

= 0

-

3 2

a,

_• 0 0 -10 15 - 6

Hz

•

0 - 2 i< Hz

•

0 0

-

6

a

3 ;;Z 0 l ~ ~ - l

H3

• 0 0 3

- z

H3

•

0 2

z

2 2 84 • 0 0 - 1 x3 _H4 = 0 0 0 10 -15 6

I

;;

.z

X ;(3 I

·t

l:

0 0 - 4 7

-;j

l

_::j

H6

•

0 0 l - 1 2 gth _order Hll• 0 0 0 -35 84 -70 20 Hz

•

0 0 0

-zo

45 -36 10 i( il3

•

0 0

_z

1

a.

- 5 10 -~ 2

_.z

2 H4

•

0 0 0 1 ₆

-

~ _6'

-

~ 1 ;(3 6 6 ~5 = 0 0 0 0 35 -84 70 -20 ;;4 116

=

0 0 0 0 -15 39 -34 10 -5

"'

~ 0 0 0 0

2

_{- 7}

.!1

- 2

j

.6

•

~ 2 2 88

•

0 0 0 0 - l l - l l '1.7 6 2 2 6

Table 1: Hermite- polynomials

gth order

Variable ~ of the polynomial is hereby without any dimension. Thus the later integrations result across the blade radius in the interval [0,1) which will be of an advantage.

From the figures i t can be seen that the approximation func-tion ~ (x) from eqn. (8) originates from a superposition of several polynomials.Every polynomial H. is hereby emphasized with the r~spective parameter ci corre~ponding to the specific boundary derivation.

For f i r s t investigations we choose Hermite- polynomials of the 4th order. Since, as is well known, polynomials are dimension-less quantities, i t is necessary to get energy equation (9) into a normalized form.

Normalizing with total blade radius L follows in vectorial representation.

T ';-·zd-

1

*

z

4 T

r·-z-

2

-£

o ~ X + 2EI m WRa L ~ o X ~' dx EI 2 m*w Ro 2L4 ~ T ;"---, -o x~~ dx

( 10)

in which for the vector c of boundary derivations i t is valid that

"

Vectors

!,

!

1 Z

vations.

are the Hermite- polynomials and their

deri-Numerical integration of the with each other multiplied veccors ~of the Hermite- polynomials, lead to constant matrices, the

so called Hermite- integral matrices. Hereby the following definitions apply:

'

iZ''

2

_dX

₌

!!22

,_

0 ;i(O~ dx

J;cz:;, zdx

₌

a

•

-

=Xll

f~l~

dl<

Jzzdx

₌

_~00 o -~

---!xzz dx

_llxot

•

ix .. -n-~ dx 3.4 - 8

!!a

!:!.1

h -n ( 1 t )

The insertion of the from eqn. (11) derived integral matrices is equivalent with the local integration. Thus the time va-riant differential equation system of the 2nd order remains:

E*H C + (H + ..!:.

J!

H

=Oo- =22 2 =X11 2 B* H =X01 ) c - G' '" E q i-i h 0 (12)

One recognizes now the great advantage in the application of Hermite- polynomials integrated numerically only once accor-ding to eqn. (11). The matrices and vectors derived from this integration can be used for further relevant problems at once. For the solution of differential equation system ( 12) i t is significant to transform on a system 1st order of general form: [ref. 5]

x_=Ay+b _{( 13)}

With the introduction of a member c the system (12) is extended

in the following manner:

c

0 E c 0 = = . . . . . _-

-

₊ G*

E*

---

(14)

c

K 0

c

!_q.

k, =m = ,. 1 - l

whereby i t is valid that

-1 '211xo1 1 1 ~22) K H _~X11

-

_E;T

=m =00 2

!>i

H -1 h. =00 - l

Vector c contains, as is well known, the degrees of freedom Z , Z',

z

1,

z-{

of system. On behalf of both boundary

con-d~tiogs

~

z

= 0

0 0

the corresponding equations of system (12) drop out, which is

equivalent ·to eliminate both first rows and columns of the integral matrices. We receive thus differential equation system in component notation

~ z.l 0 0 1 0 . zl 0

Z.'

0 0 0 1

-

'

0 1 21 G* ( 15) = +

--

~ _E* z1 K11 K12 0 0 z1 Eqi ki 1

z '

₁ K21 K22 0 0 ~ z1

'

Eq. k ~ c 2

5 DETERMINATION OF EXTERIOUR FORCE OF AIR DISTRIBUTION The inhomogenous part q(x) in eqn. (2) is equivalent to the

force of air distribution prevailing on blade. Because of the different oncoming flow along the blade radius this force of air distribution is a variable load distributed across distance.

Generally for the uplift generated by a profile

F =

!.

pc v 2

s

A 2 A res

holds.

Since the force of air is variable on the blade radius, i t must be calculated in segments.

(16)

Calculation of dFA thus divides with P and t_{81 known in the}

determination of vres and cA. Both parameters are dependent on blade radius x; that is why a simple integration of dFA is not

possible. The velocity determining the uplift Vres will be cal-culated from the rnomentdry state of flight, in which

v

res vres (uoo,tPbl'x)

For the calculation of vres one needs the velocity components vx, vy, v₂ in the co - revolving coordinate system with re-ference to rotor. For this first of all from the given velo-city components vXg' Vyg' v 2 g in the geodetic coordinate system across the transformat1ons matrix

T =GH

cosBcos~ + sin~sin8sin~

cos~sinlji

cos8sin~sintlJ - sinBcos~

sin~sin8cOsW - cos8sin~ sinecos$

cos$coslji -sin$ sinBsintP + cosBsin~cosW cosBcos$ the velocity components from the translational motion of co-ordinate system vx~' vyT, v₂T with reference to helicopter, are calculated to

v

=T T _{=GH ~} v (17)

In this system the rate of revolution of total helicopter (pitching, rolling, yawing) are superimposed on the transla-tional velocity ~T· I_f one describes this motion by an angular velocity vector ~, the thus resulting velocity ~R is calculated

from

where

Ep

is the vector gated blade point.

w x r -,p

from center of rotation to the

(18)

investi-For the velocity vector ~H resulting from the superposition of translational and rotational velocity i t then holds that

v (r )

-H-p (19)

Velocity components vx, vy, v

2 of coordinate system with

re-ference to rotor under consideration of mast installing angle K are calculated from the components.of vector~ with reference m

-cosljlb

1cosKm sinljlbl -cos!JJ bl sinK m

:f:HR

= -sinljlblCOSKm· -cosljlbl -sin!JJ

1sinK b rn sinK 0 COSK m m to v

=

₍₂₀₎

In the coordinate system with reference to blade the part

V = W • X

Ro Ro ( 2 1 )

supplied by rotation of rotor is added scalarly to the y-com-ponent of ~·

Flapping velocity v

₈

supplies a contribution to the component

vz of ~· It reckons out stepwise from two temporal succeeding

biade derivations.'Since for every time interval for deterJi-ning the local blade derivations an i n i t i a l flapping velocity

v ₆₀ is presupposed, the flapping velocity v

₈

must be

determi-ned iteratively.

Generally for the flapping velocity holds

- ljl.

'

( 2 2)

A share to the uplift give only both components Vy and Vz. For the resulting velocity vres i t thus follows that

v

res + v z 2 ( 2 3)

The uplift coefficient cA depends on the effective angle of

incidence aeff which again depends on radius x and also an

the Mach-number.

The effective angle of incidence is made up of a set angle

of inCidence aA and a variably induced angle of inciden~e ai

For the induced angle of incidence ai holds:

a.

'

arctan v z v y ( 2 4)

For the profile NACA 23012 exists a data sheet from "MBB" [~ which contains the coefficients cA depending on angle of in-cidence a for several Mach-numbers.

If both the actual effective angle of incidence aeff and the actual Mach-number are not comprehended in the data sheet, the c -coefficient is determined _{A .} by a linear interpolation. The values for vres and cA received thus result, according to eqn. ( 16), the uplift force dFA/dx relevant to blade ra-dius. The execution of this calculation on plurality of blade supporting points leads to the searched for line segment load q(x) that is after normalizing to q(x) respectively.

Now the integration of function q(X) requires with the appli-cation of Hermite- polynomials according to eqn. (11) ratio-nal functions f(X 0 ) . Because of this, according to eqn. (16)

the line segment load

q(xJ dFA 1 pc v 2 t

2 A res bl = :=

dx

is approximated by·a Newton-interpolation-polynomial of the 4th order to

q(xJ

The constants q₀ . . . q

4 are hereby determined from the lation procedure. from integration ace rding to eqn. term L_qihi in eqn. (12) results.

6 NUMERICAL SOLUTION OF SYSTEM

( 2 5)

interpo-( 11) the

The setting u~ of the differential equation system (15) and its computational solution is earned out with the computer program ''EBLAMO''. The determination of the system matrix~ is done with elenentary matrix operations requiring short computation times. The time integration following these after is based on the ap-proximation procedure due to ''Runge-Kutta''.

For this a library-routine "DVERK" exists [ 7].

I

By insertion of components z1, !1 in the Ritz formulation from eqn. with various blade support points line reckons to

and the Hermite- polynominals

{8), for every time interval

the approximated blade bending

z ex

J = (3x 2 - 2x 3Jz +

1 ( -x2 • -'Jz I X 1 (26)

For small time intervals ( f . e.,.ljJbJ. = 1°) oscillations can be re-presented for various blade support points (f. e. blade tip).

The determined program ''EBLAMO'' is now applied to th helicopter

BO 105 from MBB, West Germany. The input data hereby are as follows: EI

₌

6800 Nm 2

"

5.54 kg/m m

₌

L

=

4.912 m tbl

=

0,27 m

=

44,5 m/s 2 WRo

The calculations are executed for both flight cases hove ing flight and horizontal forward flight by 200 km/h.

The working of the program ''EBLAMO'' is shown in the following

flow chart.

derivatlm of

7 RESULTS AND DISCUSSION

Fig. 2 presents the oscillation of the blade tip for the first six rotations, whereby the normalized deviation ~ of the blad·' tip is plotted versus azimuth angle 1/Jbl· Figure 2a) shows that in hovering flight the system earlier gets into a stationary status than in forward flight. After the first rotation, the blade tip in both cases oscillates exactly with the frequency of excitation.

The flapping velocity

v8

versus the normalized blade radius X is presented in fig. 3 for an interval of 45 degrees. A com-parison with fig. 2 shows the correlation with the oscillation of the blade tip: Flapping velocity v₆ is positive for in-creasing and negative for dein-creasing deviation.

In fig. 4 bending lines of the blade are presented, which show the elastic behaviour of the blade in radial direction - approxi-mated by Hermite- polynomials.

The effective angle of incidence versus blade radius is shown in fig. 5. In case of the advancing blade radial variation of the angle of incidence is very small. Only at the retreating blade great variations are recognized near the clamping place of the blade (forward flight). The behaviour of uplift coef-ficient cA, shown ~n fig. 6 is logiacally similar. In case of seperated flow the angle of incidence a and the uplift coef-ficient cA are set to zero for plotting.

8 CONCLUDING REMhRKS

The results in the previous chapter 7 make evident, that the approximation solution based on the Ritz-procedure with Hermite-polynomials as formulation functions is thoroughly applicable

for such problems. The setting up of the system matrix ~ only bases on elementary matrix operations and so i t requires-short calculation times. Only determination of the flapping veloc1ty v~ necessitates mare computation time. The outlined procedure is the beginning of a series of continous investigation pos-sibilities. The application of this procedure for the other degrees of freedom of blade motion {lagging, torsion) is already in work.

9 REFERENCE

1. J.C. Huboldt, G.W. Brooks, Differential Equations of Motion for Combined Flapwise Bending, Chordwise Bending and Tor-sion of Twisted Nonuniform Rotor Blades , NACA Rep. 1346, 1958

2. I. Szabo, HOhere Technische Mechanik, Springer Verlag, 1963 3. S. Falk, Mathematische Methoden der Mechanik II, Technische

Hochschule Braunschweig

4. S. Falk, Zeitschrift fUr angewandte Mathematik und Mechanik, Band 43, April/Mai 1963, Heft 4/5

5. P.C. MUller, H. Schiehlen, Lineare Schwingungen, Akad. Ver-lagsgesellschaft Wiesbaden 1976

6. Profile data sheet for NACA 23012 from MBB, West Germany 7. Routine Library from ''Leibniz Rechenzentrum'' of

Tech-nical University Munich

8. H. Schlichting, E. Truckenbrodt, Aerodynamik des Flugzeugs, Springer Verlag", 1962

9. D. Ludwig~ Modal Characteristics of Rotor Blades, 7th Euro-pean Rotorcraft and Poward Lift Aircraft Forum, Garmisch-Partenkirchen 1981

N 0

'"

co

..

0 0 2a) N 0

-'"

co

....

0 0

'

2c) N 0

-"'

a:; 0 0 ....

'

.... ....

....

.... rot .I forward flight

I

_'

~ /

---'\

....

'-hovering flight 110 220 _IJi (OJ ""--l) rot.J .... ~ / / /

'

/

-

--

/ 11 0 :' :l 0 _ljJ [OJ 310 rot.S

-

'

....

...

__

... / : 10

no

'"

[ 0) 310 N 0 IN

.,

0 0 2b) N 0 IN

...

0 ~ 2d)

"'

.: I

J.

"'

0

'-'

_'

_....

1 10

'

'-'

....

--1 --1 0

'

....

....

_....

....

: 10

Fig.2: Deviations of blade tip z vs. azimuth angle iJJbl

3. 4 18 rot.2

- - '

/ / / -220 ljJ [OJ "iJO rot.4 /

...

/ / / / / 220 _ljJ [ 0

l

3 30 rot.6

....

~ / / / / .120 _Vi r o 1 '

.

'10

C< 0 forward flight

I

[

//~

/ / hovering flight 0 ~~~---~ 0. 0 . "l . 6 x/L . g 3a) ·'' ₀

/~~

0

+-~~-~-~-~---~--~---~ 0.0 ."l .G x/L . g Jc) >N

0~~~---~~---"

0 N 0.0 Je) 0 'i=3!5 ."l .G x/L . g

.]

/

. .

~----0 0.0 ) .6 x/L _g \g) N · 0 IN a> 0+-~~~---~ 00 0.0 3b) 0 IN a> .3 .6 x/L . g

----0~~~~---_0.0 N 0 . N IN

"'

0 N 3d) 0 ~=270 0 ·i=360 . 3 .6 X/L ,q

-·

.'1 .G x/L q 0~-e~~---~---~~---0.0 .3 .6 xL_g 3h)

"'

_'

hovering 0

·o.o

4a) 0 0.0 4c)

,,

.'l

"'"'

W=225° >'-8 ~

"'

--"'

0

c.o

,,

4e)

-~-"""'

>'- _~=315° a

"'

---"'

' 0 0.0 .3 4g) Fig.4: Flapping forward flight

l--flight -.G ~/L. g .G x/L,q .6 x/L_g

- - - -

-::::

.6 x/L, g velocity Vg vs. blade

"'"'

> ' ₆ <D 0 0

'

0.0 lb) 0.0 4d)

"""'

·J!:<!70° >

'

8 ~

"'

'"

0 0 ' t' ·1 f I

"'"'

0 >'- _It• .= 360 6 ":;'

"j

~L

_'

_~ 0.0 -I h) r"adius X .'l .G x/L , g .'l .G x/L ,g

--

-

-

---.u x/L . ~~ / .'i lj x/L.O

0 '

'"I

_{~ ~}

I

0 ~::::45

"'J

forward fllqht

-. l ___

f_/ _ _ _

_

0 0 <1!=90 L - - - \ _ ·:> 1 -

~~~li~g-

f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- f-ll~h-t-- \ _ _ ·~~ <---~---~-~-~--~--0 20 S.:t) .GS

~

]1-r

-~---~~-==--.sa · x/L "2 Sc) .:r~

'"

I

-

- - -

--

- - - -

-~1

_{;,;: H .G9 xfL} 0₂ Se)

ol~

j _{o= 315°} <Dj ~, ·n ~ 0 J ) -~ 0

'

0 ~

,_,

~ '20 5b) . 20 Sd) -

-20 Sf)

•••

.«

- -

-, H •/!=270 0

~

-n

_.,

-'~

~

-

-

--

-

--

_-

--

_-

-

-

-

-

-0

"

I'

'"

..----·-.---.

'

20 , u .G3 x/L '12 20 u Sg) ohl

Fig.5: Effective angle of incidence a vs. blade radius x

.68

.63

-.GS x/L _q2

forward flight hovering flight

---

_---

I

0 0 +---~--.--L--.20 6a) 0 i/J=l35

-.68 x/L 92

-0 o+--~---~.,....L---.20 •• .68 x/L.92 6c)

----:c

j

0 0 ~--~--~---~----~---.20 . H .69 x/L · g~ 6e)

"'

---cu

0 _{. 20 .••}

---~---.

_.G8 6gl

Fig.6: Uplift coefficient cA vs.

cr-.

r---c o+---~---~---20 6b)

'"

0 +-'---~---0 .20 hd) 0

'i'

270 20 6f)

•••

.GS x/L .. 92 .GS .G8

"'

~---~~~

-- -- -- ---- -- -i

20 6h) .<4 blade radius X

'I

~L

x/L q2 3.4 - 22