.. '

NONLINEAR DYNAMICS OF A HELICOPTER MODEL IN GROUNTJ RESONANCE

D.M.Tang * and E.H.Dowell** Department of Mechanical Engineering

and Material Science Duke University, Durham , NC 27706

Abstract

An approximate theoretical method is presented which determines the limit cycle behavior of a hel icopter model which has one or two nonlinear dampers. The relationship during unstable ground resonance oscillations between lagging motion of the blades and fus elage motion is discussed. An experiment has been carried out on using a helicopter scale model. The experimental results agree with those of the theoret ical analysis.

H

I

J sx,Jsz

t (A-model) nO Po fa

(ll-model) n 2--na P2--Pa fb

Notation

geomet rical length of the mode, see Fig.l mass moment of inertia of blade relative to drag hinge mass moment of inertia of fuse­lage about X and Z axis through A mass moment of inertia of shaft about X and Z axis through A coupling spring coefficient be­tween fuselage and shaft distance from the axis of rota­tion center to drag hinge center hydrauli c damping moment coef­ficient nondimensional hydraulic damping coefficient s dry friction moment coefficient nondimensional dry friction damping coefficients mass of the blade number of blades damping coefficient of blade radius of the rotor blade static mass moment relative to drag hinge time

fuselage damping coefficient natural frequency of fuselage hinged mass ratio 2 nS2H2/2I\Jsz+Jfz+nmbH )

damping coefficients na tural frequencies hinged mass ratio nS 2H2/2I (Jsz+nmbH 2)

* Vi ~ itin<J Schola r, I anjing Aeronautical Institute , China .

** Dean , School of Enginee ring , D u ~e Uni v e r si ty , !Jur . LlT.1 , ll C.

(e-model) nx,nz Px,Pz

fuselage damping coefficients natural frequencies of fuselage hi~ge d mass ratio t c nS H2/2I(Jo+nmbH2)

8 fx ,8fz angular motion coordinate of fuselage about X and Z\ axis angular motion coor d ' r. ate of shaft about X and Z axls angular amplitude of fuselage about X and Z axis

8 fxO ,SfzO

SsxO'S s zO angular amplitude of shaft about X and Z axis

45

angular orientation of different rotor blades angular deflection of k-th blade natural frequency of rotating blade relative to drag hinge di stance of center of gravity of the blade from drag hinge coordinate of the center of gra­vity of the blade system in fixed reference frame nondimensional blade parameter

JLv . hS/I r otor speed

Introduction

Helicopter ground resonance is a self-excited vibration phenomenon rather t 'lan a forced vibration. The safety standards for helicopter strength require that a helicopter with an articulated rotor must have enough stable margin when it is in contact with the ground, especially, in case of rough landing. Such standards are based on linear theory, but i n fact the most common design of the l anding gear has nonlinear damping characteristics. How much of an effect on ground resonance will be induced? Does the nonlinear damping characteristi~ tend to i ncrease stability or not? These questions are waiting for investigators to solve. The present aut hors [1,2) and Tongue (3) have engaged in such research. The purpose of this paper is to further study such problems.

The nonlinear dampi ng in the landing gear of a helicopter brings into action both roll and pitch motion of the fuselag~. In a series of pUblications [3-l3J about helicopter ground resonance instability, this aspect was not taken into account by many authors. Also in the past, many investi9Ators have used Coleman' classical multi-blade coor oinate theory to simplify

. I

..

the st udy of the rotor blade dynamics. Most ~revious authors have considered linear dynamic models.

In the present paper, an approximate method has been used to calaulate the limit cycle behavior of a given helicopter model which has a nonlinearity in the landing gear such as a hydraulic resistance which is approximately proportional to the square of velocity or dry friction resistance. An experimental investigation has been carried out as well. The test apparatus is an improved version of Bielawa's rotor model [4]. The experimental results are in good agreement with those of the theoretical analysis.

In the present paper, we study the effect of combined roll and pitch fuselage motions, and consider the relationship between the lagging motion of the blades and the fuselage motion. Also discussed are the differences in the unstable boundary between the linear and the nonlinear model. We believe that these results will be helpful in further understanding the physical essence of helicopte r ground resonance instability, and in protecting against such an i nstabil ity •

Approximate method of DOnline!! .!. analysis

The analysis is based on a helicopter model with an articulated rotor. A sketch of this modd is shown in Fig.l. We will assume that tlle blades in the plane of rotation are rigid but articulated; the spring and damping coefficients of the fuselage and the shaft are isotropic, but their mass moments of inertia are anisotr opic. Nonlinear damping includes hydraulic resistance which is proportional to velocity squared and also dry friction resi~t~nce; these can,be represented by Mh 1919 and Md signe, respectively. The

equat10ns of motion for the model are six nonlinear differential equations with unknown functions 1 , s, efx' Sfz, Sax and 8s ~' These are as follows (see Ref.[l]):

. • 2 ~ •• "'\+2 nb'l+c.>s~-2Q() r.bt. )=nSH9 sz/2I

~ +2nb~ +W:~+20(~+nb'l) =-nSH9 sx/2I ., , 2 2 ' esz+2 n2 9S?+(P2+P4)9sz-2 n49fz _p 2 e =sHiVJ 4 fz sz •• • 2 2 ' 8s~+2 n19sx+(PI+P3)8sx-2n38fX (1) -P-9 =-SH! / J 3 fx sx .. , 2 2 ' 2 8fx+2ns9fx+(PS+P7)9fx-2n79sx-P78sx

+mhxef~efxl+mdxsign9fx=0 .. ).. 2 2 ).. 2 efz+2n6~fz+(P6+P8)efz-2n8~sz-P88sz

+mhZefZ~fzl +mdZsi9n~fz=0

In order to simplify the problem, we will discuss two special cases:

1. A model with a single nonlinear damper.

We suppose that the motion in the X direction is constrained, i.e, es~ -Sfx -0, and the subscript z in SSZ and Ofz is thus eliminated, and define that:

Ss=mhz9s ' 9f =mhz 9f

~=~mhZSH/J , ~=~mh SH/ J sz z xz

'i'ill:::;' (1) be cor.1 Gs : ;.: '2' .. 1'1. +2nb1[ +Wsif-22(f+nb~) =~9s

46

~ +2nbt +uJ~~+2 n(~+nb1f) =0

ij +2n2~ + (P22+P4

2HJ -2n ~ _p2~ = ~ 5 5 5 4 f 4 f '( (2)

.. , 2 2~'" ). 9f+2n6~f+(P6+P8)9f+~f9(+.si9n8f

:.. 2 -2n89 -P8~s=0

where

For the nonl inear terms Of I 8t I and sign9t , we apply an approximate procedure which is called quasi-linearization.

Let: it '" ,af I cosAt

Representing the above nonlinear terms by a Fourier series and retaining the first harmonic produces(see Ref. [3],[14])

~fl~fl=- 8/3n*A219 f d

2 sin~t signQ f=-4 / ]I' *sin"t

Let:

7i =7) e iAt o

to' - i >. t ;:) =SOe

~ .... ""9 i>l.t "5 sOe

Once again let: ,. ,. i"t 9 f -9 fO e

Subst i t ute these and require coeffi cients to solutions. The equations are:

real part:

expressions into Eqs. (2), the determinant of

be zero for nontrivial final characteristic

aQ fO'" 8 -b~;O >.7 -c~fO" 6 + (d9 f

20+,b)" 5+e \; A 4

2 3 2 fO -(f~fO+,d)A -9~fOA +'fA+h~fO=O (3)

i h1 ugi lla r y ?a r t :

92 ,, 8 ~ 7 ~2 6... 5 a l fo +b l fO" -(cI~fO+bOa)A -d19fOA

( -2 ) 4 ... 3 2 2 e 19fO +,c l " +f 19fO" -(919fO·,e1)"

-h1~fO"- 91=0 (4)

...

. ..

"

The coefficients a,b , c ••• etc. of the above Egs. are given in hppendix B of Ref. [1).

- It' -For the case of Ksf ~(X), 9~ ~ \jf '" 9, i.e th:? shaft is connected together with fuselage , the differential equations become a simpler nonlinear mathematical model for analysing helicopter gr ou,l d resonance •

(5)

where:

nb=~/PO' Ws =U;/Po , nO=nO/PO' ~=Q/PO

For this case, the final churacte risti c equation is the same as in Ref[3).

Eg. (32. or (4) is e ighth order in ". Also , the 9fO a ngul a r amplit(;oe novI appears explici tly . This means that the l i mit cycle amplitude of the nonlinear syst em is directly related to ", the fre~uency of oscill ation , and both must be solved for simul t a neo usly .

The Broyden method [15J was used to solve the above siro ul taneous equi\tions. Let functions fl(gfo,A) and f2(9"fO,A) be equal to the left hand sides of Eg.(3) and Eq. (4), respectively .

fl (lifO')' ) .. 0

f2 (9fC ,A) = 0

2. A model with damper s .

He suppose:

two

Ksf=OO , 9sx·9fx=9x ' 9sz=9fz=9z

and define:

nonlinear

J O=(Jx+Jz )/2, i x=9x ' ~z=9z' ~=~SH/JO

l=rSH/Jo ' Jx=Jfx+Jsx ' Jz=Jfz+Jsz '

~=Jo/Jx E:3 =J o/J z

The general equations of motion for th e model are given by Eqa . (6).

- --_________ "T ,

Eqs. (6) is a set of nonlinea r differ tial equations for determining four un y.no\'l n func tions 't , c, ex and 9z .

Using the retaini ng we take

method only the

of harmonic balance and fundamental ha rmonic,

(7)

Her e '/Z, 'I" , 'I, are respectivel y the phase diff erence of q , 1i and $' with r espect to e

z x tlo\~ He consider the t e r m sign~ anu ~ I ~ I • As . was mentione d p reviously [2), \'Ie can obta in the fol lO\-l ing appro ximate r esult:

For", ~zO-::' O

,:.. - 4 - -singe =-r- (e cos.\t-e sin"t ) z JI~ZO ZS zc

,.:. 8 A2 ,e 19 .- ~ e (9 sinAt-9 cosAt) Z z 3JI zO ZC zs

Ins erting Eq . ( 7) into Eq . (6) ~I e obtain a sys telll of ei gh t siu-, ul t a neous nonl incar al gebraic eq uations for dete r mini ng t ',e c uantities , 'ie , ~s '?c ' ~s ' 9x O' 9zc , 6zs and A •

By elimina tion of the first four unknowns , the ec;uat ions can be rewritten morE: compactly as a matrix eCj uat ion wit l fou r unknow ns , QXO;i3 zc ,ezs ' ".

Let:

(8) where

CV1· [-0.-: =; -: c -b

(p2/ A2_1)9 /c

3+2n 9 / AE 3+ z zc Z zs

- - - 2-8mhz9z09zs/31rE3+4mdz9zs/nA 9 z0t 3

lu)= (p

2/ ... 2_1)6 / £3- 2n 9 / At3-z zs z ze - - - 2-8mh 9 09 /3~E3-4md 9 / KA 8 OE3 z z ze z ze z

(1_P~/A2) 6xo/ C2

- ;. 2-2 2nx8xo "E2+4mdx,1r~" +8mhx8xo/3 1rE2

a=(w!1,,2_ l )/f l

b=2l),ltl" 2

c=2Ql),/f l'~

d=20/f l "

In order IF) must Broyden's equation.

to solve Sq.(8), the elements of be zero. Here we also apply

method to solve the above

Experimental Inyestigation

1. Experimental equipment

An overall view of the test model and recording equipment used in this study is shown in Figs.2 and 3. The details of the various components of the model appear in Ref. [1) and [2). The A-model has only one degree of freedom for fuse l age motion. The B-model has two dearees of freedom for shaft and fuselage motion. The C-model has two degrees of freedom for the [011 and the pitch motion of the fuselage. The characteristic curves of the hydraulic and dry friction damping are shown in Fig.4(a) and (b). The method of determining these parameters is described in Ref.[l).

2. Measurement system

The X and Z\ angular displacements and velocities of the fuselage and the shaft are obtained by RVDT, R30D, velocity transducers located near the dry friction dampers, and a LVDT located near the gimbal support assembly. The output voltage of the transformers and velocity transducers is proportional to angular displacement and velocity of the roll and the pitch of the fuselage and the shaft. The output from the transformers are amplified and recorded on a multiple channel tape recorder, HP3968A.

In order to obtain a one per revolution signal and the lagging motion of the individual blades during rotation of the blades, we mount a l3-ch brush and slip ring assembly on the shaft between the electric motor and hub, see Fig.5. One of these slip rings is not a closed ring. It gives an impulsive signal once per revolution, so a very accurate measurement of rotor speed can be provided. Three angular transducers, R30A, are mounted on the drag hinges. The signals for lagging moti on of each blade were amplified ~nd recorded through brush and slip rlng assemblies. The 8 channel signals were recorded simultaneously on a tape recorder, and further analysed by a Frequency Spectral Analyzer, HP3582A. Finally, the phase plane plots and frequency spectral plots were plotted by a X-Y recorder.

48

The stability test needs an initial disturbance, so two electro-magnets were mounted on the test equipment. Those are used to generate a disturbance in the direction of roll or pitch of the fuselage.

The device for angular amplitude calibration i s shown in Fig.6. An electric motor with stepless variable speed provides a vibrat1~n source, and drives a cam which can generate a sine wave, such that t he relationship betweer output voltage from the transducers and the actual angle can be determined.

3. Frequency response test

The purpose of the frequency response test is to determ i ne the natural frequencies of the system, and to provide an independent cheCk on the system parameters. Before the test, the drag hinges are fixed, and a known block matJs (3 oz) is put on a de ag hinge. A centrifugal force caused by the bias mass will excite the system, and the frequency response versus rotor eryeed can be found.

4. Stability test

Testing the model f or aelf-excited instability regions was accomplished by slowly varying the rotor speed until instability was observed in response to an initial disturbance.

Results and Discussion

Fig.7 shows the limit cycle behavior when only the hydraulic nonlinearity is present in the A-model . The figure shows that the maximum limit cycle amplitude occurs near the critical rotor speed of the linear system, and that the dominant response occurs in the region, (1.7-2.0) *Q. Furthermore, there is an abrupt change in response from (1.9-1.95)*2 in Fig.7. This means that the amplitude response is very sensitive to small changes in r otor speed in this range of rotor speed.

Fi g.8 shows the limit cycle behavior with both hydraulic and dry friction nonlinearities. The curve has a shape similar to that of Fig.7, but the response amplitude is smaller than that in Fig.7. Physically the action of dry friction is to increase the equivalent viscous damping in the landing gear. The instability region becom~s narrower, and the response amplitude reduces.

Fig.9 shows the limit cycle behavior of the B-model. There are two instability regions in the figure. The first one is dominated by the shaft motion. Because the effect of non.linear damping of the fuselage on the shaft is weak, the shaft has a large limit cycle amplitude (for

> • \

•

aome range of Q beyond the measurable angle). In contrast with the first region, in the second instability region there is not a large limit cycle amplitude because of the direct effects of nonlinear damping of the fuselage on the fuselage motion.

As far as the C-model is concerned, Fig.10 shows the results of frequency spec~ral analysis of the fuselage motion at a r otor speed of l.2(HZ). The mode shape 0f the unstable motion is dominated by rJll of the fuselage, and the os:: illation frequency is equal to the r~~l natural frequenc1 of the ~u~elage. SLnul t~neousl~, there is a small llm1t cyclr. ampl1tude 1n the pitch motion~ its oscillation frequen cy is not equal to the pitch natural frequency, but is equal to the roll natural frequency. The phase angle between roll and pitch is equal to Sl~ corresponding to a frequency of O.84(HZ). The phase curve in the vicinity of O.84(HZ) is almost parallel to the frequency axis, and the phase angle is quite ineensitive to frequency.

Fig.11-12 show theoretical and experimental results of the limit cycle amplitude and phase angle behavior. A reasona ble experimental verification of the theory is achieved. Because of the exper imental complexity and calculation approximation, the quantitative accuracy of the verification is not very high. However, the general features of the experimental results are well predicted by the theory. For reference, in Fig.13 the stability boundary results from linear theory are shown and compa~ed to experimental data obtained w1th the nonl inear dampers removed. The agreement between theory and experiment in this case is good.

Fi g.14 shows a phase plane plot which is directly plotted by a X-Y recorder. The measurement point of displacement and velocity is located at the same position. Their phase difference approximates to 9 if, the pattern displays an ellipse approximately, and the higher order harmonic components are included in the ellipse curve . From this pattern, we also can see the transient process leading to the steady state limit cycle amplitude. Different disturbances have different unstable processes. In this case the roll oscillation of the fuselage tends to a limit after about 7 pseudo cycles, and the transient process includes more higher order harmonic components.

Fig.lS shows the results of a frequency spectral analysis of the lagging motion of the individual blades at a rotor speed of 2. 04(HZ). It is found that in the instability region the individual blades have an identical oscillation frequency, p, which is equal to the difference between the rotor speed and the oscillation frequency, ", of the fuselage. This result is in agreement with

49

conclusion which is derived by claasical linear theory. However, the individual blades have different amplitudes and phase angles due to the actual small different blade damping characteristics.

The equation of motion of the k-th blade can be written as follows:

f. +2n;' +w2f = ~ (x sin~k -zcos;k)

k ~k 5 k L v •h

The periodic term can be removed by co-ordinates.

in the above equation using the COleman

n '7 = L: 3"k s in}v,k

k=1

~ = t ~cos~ kozl k

If it is represented oscillation defined as:

expected that ~ and t are well by a fundamental harmonic with

f req uency, A, then t k may be

j k = ~cos (Pt+211 k i n

i.e the follOWing conditions satisfied:

must be

*p=Q-A * the amplitudes of

blades are identical. * the phase angl e of the

equal to 2k"/n.

In the present experiment, condition is satisfied. violations of the latter are the main reasons for the theoretical and experimental

the individual

k-th blade is

only the first The modest

two conditions error between resul ts.

In order to provide a better phYSical understanding of the unstable motion, we discuss the relationship between motion of the hub center and the center of gravity (C.G) of the blades. In the fixed coordinate system XOZ, the motion of the common center of gravity of the blade system is expressed by

X =X- Pc t.. fksin'f,k c ~=I

Z =Z+ Pc trkcosl!,k c nk=l

X= ate c osA. t i=1 xO i 1

Z= ate O . cos(.A . t-~ . ) i-I Z 1 1 Zl

Fig.l~ shows the time history of ~ and S • The test curves include components with frequent;}' (2S2- .... ), and A, in addition, a cos .... t component. This also can be clearly found from Fig.17. The motion locus of the C.G of the blades is not exactly an ellipse, but a somewhat more complicated curve. The starting point at taO does not

~ .

coincide with the end point at t-2(sec.), however over a much longer time interval, the motion does appear periodic. The phase angle between hub center and C.G of blades is equal to 46- from calculation, and l5~ from test. The phase angle depends upon the dampings of the fuse lage and blades, as well as rotor speed.

Fig.18 shows the theoretical and exper imental results for a rotor speed of 1.2(HZ). The test curves are appr oximately cosine waves. The motion locus of the C.G of the blades approximates to an ellipse, and excellent agreement of calculation with test is achieved for the motion locus of the hub center. As compared with Fig.17 and 18, we find that the difference between theory and experiment for the hub center motion mainly comes from the variation of blade motion from one blade to the next.

Fig.19 shows the test results for a rotor speed of 1.96(HZJ. All higher order harmonic components are included in the motion of the fuselage and blades. Their motion loci have guite complex shapes, but they always tend to a limit amplitude.

Conel usions

A numbe r of conclusions may be reached.

1. Unstable motion of a helicopter in ground resonance can occur only when the rotor speed is within a certain region, and at the Eame time a initial disturbance acts on it. This conclusion is suitable for both linear and nonlinear systems .

2. During unstable motion the ollcillation frequency is equal to the natural frequency of the linear system, or approximately the natural frequency for a nonlinear system.

3. The nonlinear dynamic rotor system with multiple nonlinear dampers has more ca.plex motions than the system with a single nonlinear dampe~. The coupled Dotion of roll and pitch of the fuselage is due to blade lagging motion. This coupling motion is important in understanding ground resonance of an actual helicopter.

4. The agreement of theory with the experimental results verifies that the analytical method which is described i ii th1a paper is useful and reliable.

5. The different charact er istics of individual lag blade dampers will lead to different limit cycle behavior.

6. The results show that a gear with hydraul ic nonlinear characteristics and dry frict i on .ay be advantageous for protecting ground resonance instability.

landing damping damping against It is

50

unlikely, perhaps, that a helicopter resigner woul d rely on a nonlinear damper for pri~ary protection against ground resonance. However, such a damper may be useful as a failsafe device to ensure a benign failure mode should a primary lineal damper become inoperable or prove inadequate under some operating conditions.

7. It is expected that the design and operation of blade dampers will also be very important for avoiding ground resonance. This is a logical topic for future study.

We would like to acknowledge to Mr.Wi l liam Clayton of the Duke Mechanical Engineering Lab. and the staff of the Structural Engineering Lab. for their contributions to the experimental work. The experimental rotor system used in the present study is a substantially modified version of that originally designed by professor Richard Bielawa of RPI for his work on ground resonance (Ref.4). It was made available to the authors through the genarous assistance of professor H.C.Curtiss,Jr., of Princeton University.

This work was supported by the National Science Foundation under Grant No. MEA-83l5l93. Dr. Elbert Marsh is the technical monitor.

References

1 . Tang D.M and Dowell E.H, "Nonlinear Dynami cs of a H~licopter Model in Ground Resonance. Part 1: Analysis and Experiment on a Helicopter Model with Single Nonlinear Damper and Fuselage Motion", Duke University, Engineering School Report, March 1984.

2. Tang D.H and Dowell E.H, "Nonlinear Dynamics of a Helicopter Model in Ground Resonance. Part 2: Analysis and Experiment on a Helicopter Model with Multiple Nonlinear Dampers and Roll and Pitch Fuselage Motion", Duke University, Engineeri ng School Report, May 1984.

3. Tongue,Benson H.,"Limit Cycle Oscil­lations of a Nonlinear Rotorcraft Model", AIAA Journal, to be published.

4. Bielawa,Richard L. ,"An Experimental and Analyti cal Study of the Mechanica l Instability of Rotor on Multiple-Degree-of Freedom Supports", Princeton University, ~.S.E. Thesis, June 1962.

5. Coleman, R.P. and Feingold, A.M., "Theory of Self Excited Mechanical Oscil­lations of Helicopter Rotors with Hinged

Blades", NACA Report 1351, Feb.1957.

6. Brook,G.W.,-The Mechanical I nstabi­lity and Forced Response of Rotor s on Mul­tiple Degree o~ Freedom Supports , Ph.D The.is, Princeton University, Department of Aeronautical Engineering, Oct.l96l.

7. Hohenemser,K.H. and Yin,S.K.,-Some Applications of the Method of Multi-Blade Coordinates-, Journal of the American He­licopter Society, Vol.7,No.3 July 1972.

8. Price,H.L.,-The Avoidance of Ground Resonance-, Aircraft Engineering, Vol.17, No.376, June 1960.

9. Done,G.T.S.-A Simplified Approach to Helicopter Ground Resonance-, ARC 33254, 1971.

10. Dowell,Earl H.,et al., A Modern Course in AeroelasticHy, Sij ithoff and Noor dhof f, 1980.

11. Mil,M.L. et al.,-Helicopter-- Cal­culation and Design, Vol. II, Vibration and Dynamic Stability-, NASA TT F-519,May 196 8.

12. Howarth,R.M., and Jones,C.H.,-Grou­nd Resonance of Helicopter-, The Journal of Helicopter Association of G.B, Vol.7, No.4, April 1954.

13. Johnson,W;, Helicopter Theory, Pri­nceton University Press, ·Princeton,N.J., 1980.

14. Dowell, Earl H.,-The Behavior of A Linear Damped Modal System with A Nonlinear Spring-MAss-Dry Friction Damper Attached-, Journal of Sound and Vibration, 89(1),1983.

15. Broyden, C.G.,-A New Method of Sol­ving Nonlinear Simultaneous Equations-, CoIIlputer Journal, Vol.12, No 1, 1969.

51

.. .co

ORIGINAL ' PAGE IS OF POOR QUALITY

L A

', an' fng g~ar

Fig.1

Fig. 2. te s t model

F ig.3. test equ ipment

1 n

:::

, , \

. 5

. 4

.5 . ) , ~

.J

" c • § .2

c

'" . 1

= .,

. 0

0

.18

C 7 . 16

~ -. 10

" c ; • 12

i . 1

'" c

l08 '0

. 06 •

. 04

. 02

n

0

..

Fi • • d .xcit inq frequen c y 1 (HZ)

FiKed exc Lting frequln c y 1.1 (HZ)

.. ; ' .

.1 2

. /

. .:/: ./. ~ .

. 18 .24 • J

II ngu llir apeed (rlid/ •• e )

r 19 . 4 (.) . char.elect.ti c cue VI of d&llper

Fued ex c iting frequency I(HZ)

Fixed exclting frequlncy I. 7(HZ)

.. , . .. , • • ..

.1 . 2 • J . 4

angu lar spe.d (rlld/ •• c)

r .g. 4 (b). c haree teciati c cu rVe of dllaper

Fi g.S ..... ur ... nt .y.t .. ot r otor bl.d.

. 42

. ~

52

" 4

"

ORIGINAL PAGE IS OF POOR QUWt'Y

Fig.6. c.libr.tion device ot .ngul.r amplitude

." A-MOdel

wit hout d ry

. ) to

~ . 2 ".

" 0. • 4

. 1

0 1. 2 I.H 2.1 2. 25 2. 4

r o t o r Ipeed

r 1'il - 7 . lutllt c y c le b' ,. Vl o r

.4[

-: . J

'0 4

.!: • ., 3 . 2

Q. I

. 1

0 1.2 2 . 1 2 . 25 2 . 4

r ig.8 . HIOtt c ycle behavior

~ ,

.9

• 7~

~ .6 .. • '" , !:! . 4 S

} . )

.15

a -model

ca l culation

maximum lIIe4sural:lc anqle

Q

.4~B-----9~6~4~~~----~.~~~~--~~-9"---. 1.92 2. 4 2 .88 ).)6

.;. ~

~ ! '" c~ .'" · · • .c 0.

--I'

... '" . · ~ • '" 3 0. e · ... c:

0

.5

.4

~ • J

~ • '" ~ .2

j . 1

.96 1.08

roto r speed (lIZ)

Fig.9. limit cycle behavio r

C-mode l phase anq 1. between 9

A and 8 z

r~

amplitude of 9. and 9 •

frequency (HZ)

;:- ~ q. lO . frequency spec tra l analysi!

C-model

caLculatio n

teet

8

9

1.68 1.8

rotor speed (HZ)

Fig.l1 . Li_it c ycle behav ior

2.16

53

56

49

ORIGINAL PAGE IS OF POOR QUALITY

C-Rlodel

c alculation

----0- teat

Z'lo '"

.. c ! .!:! .. .. • 0

" '" c

"i '" ~

'" ! n

o~~~ __ ~ __ ~ ____ ~~ __ ~ ____ ~~~ __ ~~ .96 1. 68 1.9

. 27

.24

.21

.18

• J 5

.12

.09

.06

. 0)

.8

rotor speed (HZ)

r1.9 . 12 . Ii_i t cycle behavior

C-model (w ithout nonlinear

1.2 1.4 1.6

r o t o r speed

1. 6

(HZ)

Fig.13. instability region

2.2 2.4

--~~~~I+-------------1~------------fA~--~ift- ·.

Fi g. 14 . pha •• plane plot

'f

•

phase anqle between ' 1 and r3

Fiq . 1S . frequency spectral analys i s

. 6

Flg. l. ime huto ry

54

2.5

1. 5

. 5

!:!. 0

• ." -.5 ~

.!:

j - I

-1. 5

-2

-2.5 -6

4.8

3. 6

§: 2.4

~

a. o

=-l. 2

- 2.4

- 3. 6

- 4.8

-~

ORIGINAL PAGE fro Of POOR QlJAUTY

---calc u lation

--o- t •• t

-4. 8 -3.6 1.2 2.4 o a.,.plitude (X)

Fiq.17. motion locua

3.6 4.8 6

--- c al culat ... on

.• Q

Q ) ' 0 d

-- O~o--O u '

-1.2 - . 9 -. 6 -.3 o .3 .6 .9 1.2 loS

• 0 ."

~ t - 1 4

-2

-3 -6

amp1 Ltudll! (Xl

F ig. 18. mo l ion l ocu.

--- hub c enter

- C. r:: . o f blades

- 4 . 8 - 3 .6 -2.4 -1.2 1.2

ampl i tude (X)

Pi g . 19. mo tion l ocu s

2 . 4 3.6 4 . 8

.. . l,

' f

I (!;

DISCUSSION Paper No . 4

NONLINEAR DYNAMICS OF A HELICOPTER MODEL IN GROUND RESONANCE D. M. Tang

and E. H. Dowell

Jerry Hiao, Si korsky Aircraft: I would l i ke to make a comment. Ear l ier Jing Yen sa id tha~ we may not be able to analyze ground resonance, but apparently from t his paper we can see with a nonlinear damper we can even calculate the amplitude of the oscillat ion. Therefore, if you o~J y know what numbers to put into the analysis you will be able to calculate it. Thank you.

55

't . l,