National Advisory Committee for Aeronautics

ARRNO. 3~29

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

ORIGINALLY ISSUED

Advance -:port 3G29

m0RY OF SEW-EXCITED MEC"?ICAL OSCmTIONS

OF RINGED ROTOR BLADES Robert P. Coleman

Langley Memorial Aeronautical Laboratory Langley Field, Va.

WASHINGTON

NACA WARTIME REPORTS are reprints of papersoriginally issued to provide rapid -stribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

THE OR? OF SELF-EXCITED NL3CHANI CAL OSBILLAT I ONS OF ZIBGED SOTOR BLADES By '&%ert P. Coleman

Vibrations of E~tnry-wfng aircraft mar derive their enc?rgy frorn the rot-nct9on 0% thd rotor rather than froin the air forces. B theoretical analysis of these vibrations is described anC methods for its application 3re explained herein.

?he presant paper also supersedes and. extends tho C scope of the Advanae Xdotricteh Bepart entitled "Theory of Self-Excited Xech##ical Oscflhations of Hinge6 Rotor i Elartes," parts of which are in error. The theory has ic bsen exte-lde?, to include the sffects of unequal stiffness of t-:a pylofi foi deflections in different directions and the effect of darqing in the hinges and in the -pylon. 30th the dsrivation OP the characteristic aquation aiid the aetkorls of application of tlie t5eory are given. In particular, the theory predicts tlie so-called "odd- frequency" self-excited speed range as well as the shaft- critical speed. Charts are presented from which the shaft-critical and the solf-excited instabilities can be predicted for a great variety of cases. The influence of each physical Dayameter upon the instabilities has been obtainr-d. The comprehensive treatment applies to a rotor that has any number of blades greater than two. Only a Srief discussion and the formula for shaft-critical s3ced are given for the one- or two-blade rotor.

Ths use of complex varirtbles in conjunction with Lagrange's eauations ha,s been found very convenient fgr the treatment of vibrations of rotating systems.

IKTEODTJCT IO5

A rotary-wing aircraft that, has hinged blades will, under certain conditions, be subject to vibrations which 2 derive their energy 4 fron the i;otation of the rotor instead of fron the ais. forces. The term "ground resonance" usually refers to vibyations of this type. 81- though such vibrat ion3 have apparently caused accidents in some rotary-ving aircraft and have impaired the flying qualities of othi;rsI very little actention has been .;iven this proSlcn in the literature. A theoretical analysis has therefore been undertaken, and the purpose of thc present paper is to present the t3eory and to describe the application of the t'ngory to rotary-wing aircraft.

General vi'oration theory n:id itG application to al- lied rrobl.erns as well as to the particular problem of rotor vibration are 3iscussed In references 1 to 4. good genez-al backgrcxnd for the present problem is provid- ed in the chapters 02 rgtating !xichir,ery and on self- excited vibrations in reference 1. EZeferences 2 and 3 treat i.;i iflore abstrcct faskicn the topics of rotztion and $-amping, B discassion of the variety of nodes of vibration that oxi.st iiz rotors and a number 03 frequency forxu- las obtained by considering sey~ra.telyeach degree of 1.. freedon! are givec in reference 4. This discussion does not, however, lead. to n prediction of self-excitad mo6os of vibrakioa,

Experiezce hcs. shcwn that two ty~esof Eeckanicctl vibration ixq: 0ccu.r in rotors. The vibration froquercy of the ~ylonis equal to the rotational spaed in one type, unequal in the other. The first type is sometirnas called the even-freguoncp vibration or tho one-to-one frequeacy, znd the second tgpe, the odd frequency'. The one-to-one freq.uency vibration resenibles the phenomenon occurring at a criticel speed of the shaft of rotating machinery and will consequently be referred to in this na?er as P, " shnf t - cr iti cal vibrat i cn e I' The 0 c? 6- fre qu B i: cg vibr at i on is properly called a self-excitod vibratior,.

derivation of tke equations of notion for vibrnt,ions 4 of a rotor for the case in which the pylon stiffnaas is equal in all directious or" deflection 1.s contained in reference 5. The equation for the shaft-critical speed is obtained and checked by tests of simple Kiociels.. Reference 5, however, contains incorrect statemsnts rogarding tila existence of self-excited vibrations. The error was due to a confusion in the use of conjugate complex quantit.ias which has now been cleared xp. The present !)Etper thera- fore super6od.e~reforance 5 t;n,d, in order to nake the present paF.sr in6ependent of reference? 5, the cospleto 3

dsrivations are included herein without reference to tho earlier repart.

An alternative derivation of the characteristic equation for the whirling speeds of a three-blade rotor has been given b;~>Jagnei* of the X2llstt Autogiro Corporation. By considerlng 0111~the CaEe of a pylon having eqaal stiffness in all directioos of deflection, t'jagaer has shortened the analysis by considering directly the equtlibrium of forces and nomcnts under cocditions of steady circular whirling, Some exaaples of the dependence of whirling speed upon ratattonal speed are given, and the formala for the shaft-critical soeed is obtained.

In the grosent repart,, the theory is extended to include the effects of danning In tks hinges and in tile kab and the effects of clifferent stiffnesses of pylon deflection in different directiocs. The Z2ethod of analysis, particularly the usa of coqlex varicibles in the eqxations ..c iaotloz, is exp1a.iszd iz saxe ?..etail and all the previous results ai'e snown to bo a special case of the more general problem here treated.

SYI.:BOiJS

a reiial position of vertical hinge

- A12 = A12 elements of the cnaracteristic deter- - i' ninant (sao equation (31)) c i 8.3, = A,, I

b distance fron vertical hingo to center of inass of blade I

B damping force per unit velocity of pylon disTlacement F Ex +2 ... (af ".y> 4

R, - By La = -- 2

31 coefficient defined in equation (35)

BR ccefficient defined in equation (34) c,CI, ... C, arbitrary constants

CI ccefficient defined in equation (36)

' 'R coefficient deyined in equation (34)

D t ime-deri vat ?sa operat or (a/at)

F dissipation fu.nct ion

I ~icmentof inertiF- of %lade sbaut hinge

cce""'' (37) I,, . . . I, J.L~cientsdefined in equation .. j;k indices 8.nd subscripts ussd with hinge. coordinates (equation (141)

IC spring constant (Kf = --- 2

M total effective mass of blades and pylon (m + nab) AW mass added at hub for vibration test n total nunber of blades 5

r radius of gyration of blade about its center of mass R,, ... R, ccefficisnts defined. in oquation (37) Q 3 N) 8 stiffness ratio (KY/&)

;- .i t time

17 kinetic energy

Tr kinetic energy of rotation of blade about its center of Eass

Tk kinatic energy of translational motion of kth blade kinetic energy of p;ylon TS v potential energy

x,y displacesent * s

xo 9 Ya values of x and y when t = 0

z complex displacement (x + iy) - 2 co3p1.ex conjugate of z (x - iy)

a angl e b e t v e e 11 b 1ad e s Bo, B,, , .. Bk angular displacements of blades

value of Pk when t = 0 @kO

ilaP' ck variables representing hinge deflections tJhen equations are expressed in fixed coordinate systeri b c0, 6,, *.. Gk 'variables rspresenting hinge deflections when equations are expressed in rotatfng ' t coor ?.inat e syst cm

A, = ?.x -BY in applications My :M7w, ) 1 6

hg. = in applications

hl = - a

EE. in applications A, = I (2x )

A3 = CL 2 (1 +-r2\ \ b2 /!

v1,v2 eqressions define6 in aqaation (3)

w angulcir velacity oI" rotor (the dimensionless ratio w/ar is callcd w in applicatioiis)

wa acguiar whfrl4.ng velocity aaasured in rotatlng coordinate syrsea (used in nondimensional fora in applicat ior,~)

Wf angular whj.~l-i.r~g'velocity nc.esared in fixed coordinate sgr,.:,~;!:: :I: si?d in nc:i3-.i-!pcnaional fora in ap- pii cat i e 1; s 1 . .--- wr reference frequency ( JiX/~,) Sub script s :

f fixed coordinate system

a rotating coordinate systen!

J ~ ~~~~ 7

8 hinge deflection

XSY component diroctions in fixed coordinate systen b blade

APPZOACB TO THE VIBBBTIOrJ PROBLEM

Stabf?ity a3d Instability.

If a vibrator vere attached to a rotorcraft, several modes of vibrati-on coult be excited at. any rotor speed. Only the modes that are likely to be excited during oper- ation of the aiTcraft, however, &re inportr-int.

In the present discussion, it is convenient to classi- fy the modes of vibratioa according to ths circucstances required for their excitation. The different types of vibration are identifiod analyticelly by the nature of the roots of the characteristic equation. A hinged rotor nay encouoter three types of vibration which, for convenience, are herein designated ordicary, self excited, and shaft criticsl. At zero or slow rotational speeds, an externzl force is require2 to excite vibratibn. The friction al- ways present in such systems c3uses the vibration to bc dsrmped out when the force is reaoved. Xodes of vibration requiring an external applied force to naintain them will be cslle0 ordinary vibrations. The mathematically ideal- ized case of zero damping will also be considered an ordinary vibration --Then it is understood to bo an ap;?roxima- tion to a systen actually dan?ped. Self-excited nodes of vibraticn are those with nega-tive damping and are recognized analytically by the fact that a root of t3e charec- teristic oqaation is a complex number which has 8 negative imaginary part. A slight disturbance will tend to in- crease with time instead of damping out.

When a rotating system is not perfectly balancsd, tha centrifugal force of the unbalanced mass may excite vibrations that have peak amp1itud.e~at certain rotational speeds. Vibration excited bg unbalance and in resonance with the rotation will be called shaft-critical vibration. This type occurs at the rotational speed at which ths spring stiffness of the pylon is neutralized by the centrifugal force. In the analysis, ths shaft-critical vibration is recognized in rotating coordinates as a zero frequency and in fixed coordinates as a frequency equal to 8 the rotational speed. The critical speeds of a rotating shaft are o common example of this class.

Thz purpose of a theory of rotor vi.bration is nainly to predict the occurrence of and, if possible, to show how to avoid self-excited and shaft-critical vibrations.

Choice of Degrees of Freedom

Of the large number of degrees of freedom of a hinged rotor, the imTortanis ones for +he present problem have been found to be hinge deflection of the blades in the plane of ::otation and horizontal aeflections of the pylon, Other degrees of frearlom, such as the flapging hinge motion of the blades, tte bending or torsion of the blattes, and the torsion of :he drive shaft, tire considered unim- portant iu the yroblen of self-ezcited oscillations. Some rcctions, scch as landing-gear deflection, that produce lateral deflection at the top of %lie pylon may, however, - be impor tznt .

P h y s i c a1 Paran c. te r s

The theoretical resiilts given later provide a ae8.n~ of predicting :he nataral frequencies and, in part;LbuALCrr, -.-'I - the c-itlcal speeds and unstable spencl ranges in terms of certain physical parameters, such as sass, stiffness, and length. The successful apglicnfion of the theory depends upon the determinatlon of the proper values of these physical paramtters for the aircraft or nodel being studied.

The important parameters to be dctorinined are: a radiel position of vertical hinge. b distance from vertical hinge to center of mass of blade. 4 m% mass of blade. Flexibility of the blade structure may have to be ta.ken int.0 account by thc use of an effective valua of ab different from the actual blade mass. The effective blade xass can 3e taken as tha value requfred to make the theory predict the correct pylon natural .frequency when the rotor Cas a zero 3r very slow rotational speed.

I moment of inertia of 'trlade a,bout hinge [mbb' (l+$)] 9

sprlw constant of self-centering springs, which can be determined by a force test or from the hinge frequency with the hub rigidly supported. Q) 0 m +,my effective mas8 of pylon for deflections in x- and y-directiono. I d KXilzy effective atiffness of pylon. The effective mass of the pylon is the value af a concentrated mass that would bavs the same kinetic enoi'gy expressed in teras of the deflectiom at the hub RS tho actual. pylon and hub if it were placed at tha rotor hub in tho plum of rotation. The effcctive stifrhesa of the pyla is tho otii'fnccs of n spring that, if placed tit tho hub in the plmb of rotation, would have the sme Po-Lcntiial eiierm in toms of deflections at the hub ~EIthe actud pylon. Equivalent definitions as0 that, if a sin@o ~pringand mass were irmgined to be substituted at tho hub in tho plane of rotation for tho pylon and hub, the natural frequency md the change of natural frequency with addod mass would be the sane as for the Gctual pjlon.

An experimental method of measuring the effective mass % an0 stiffness KX of tho iglon is to replaco the rotor by M approximatou oqual, rigid, concentrated mass @M at tho hub and to measure the natural frequency for two or more vdlues of this added mass. The qumtitice are then related by tho equation

If maasurod values of l/wf2 axe plotted agdnst added mass LW and Q otraight line is drawn through the points, the intercopt Rnd the clop of the line will dctcrjnine the effectivo values of Kx and %. For the parumoters a adb, the actudL geamstric lengths ahould be used unless the flexibility of tho hinge offset m n is compay.able in nqnitcde ni% the bingo spzfng sLiffnese. in this case, it is rocmend& that an effective vduo of a bo guessed and that b bo dotermined by subtraction from tho correct goomotric value of Q + b. 10

The &ing parameters may be defined %y the form of' a diosipatlon function F. The function

is equd to %he rate of dlssipation of energy by d.auping* The pnrmet,ers B, and By thus measure the damping force per unit velocity referred to linear disphcements of the top of the pylon and BB is the damping torque per unit angular velocity at n blade hinge. If the damping force per unit velocity is not a constant, effective values should be used that will represent the same dissipation of energy per cycle as actually occurs with a reasowble mplitude of vibration . The nmplitude of free vibration in il single degree of freedom is given in tern of B,, By, mld Bp by -9B q=~gea4

The dmping parameters are probably the most difficult ones to measwe accwaiely. In practice, it is advioablo to We calculationa for a given case, first on the basis of no damping and then with the we of tho estimated valuea of the damping pmmters.

MATBEMATICAL LIEvELOPMEllTS

Method of Analysis

Tho derivation of the characteristic equat-on that is used a8 tho basis for predicting the unstaB2.e oscillations of a rotor is presented in this section of the report. Readers interested solely fn applications can omit this secti~nand proceed imnediately to the section entitled ?%&hod of Applying Theory." co Tha method of aualysis treats the equations of notion 3 V) for small displacements Tram the equilibrium condition with steady rotation, A proper choice of coordinates laads to equs5ions with constant coefficients. The solu- tions are exponential or trigonometric functions.

Tha mathenatioal manipalations involved in treating the niotions of a m,%sg in a plsne of rotatjon are faclli- tatod by the use of a complex variable, Xihe typical. disturbed notion obtaiaerl by eclving the equations of m3- tioo is art elliptic whirling motion, vr'nlch 1s roj'esentcd in terzs of a com2lox variable z = x t iy. A5 i;,~y ic- stsnt, z rapresents the displacemant of tho pylon from its equilibrium position. An expression such as iwft e = ce

represents whirling of the pylon i3 a circle of radius c with angular velocit,P wf. The sign of wf determine8 the sense of the rotation. Two rotations in opposite sense with the same radius are equivalent to a vibratory motion in a straight line. iwft

= 2c cos Wft

Two opposite rotations of different radii are equivalent to whirling in an ellipse. complex value of Wf represents whirling in a spiral, which may be either a damped or a self-exci.ted motion depending upon the sign of the imaginary part. A self-excited motion exists when the imaginary part of wf is negative, and the magnitude of z increases with time.

The displacements my be referred to a fixed or to a rotating coordinate system. If ef and za are the displacements with respect to a fixed 2nd to a rotating ref- erencg s;Tstem, respectively ,

ef = z,e i wt

If L

iwat za = ce then i ( ua+w) t zf = ce

A whirling speed tua with respect to the rotating coordinates thus corresponds to a whfrling speed wf = w, + CL' with resFect to the fixed coordinates. A shaft-critical vibration corresponds to w, = 0 in the rotating coordinate system or to (of = w in the fixed coordinate system.

Example of Eotor with Locked Hinges

An example that involves a partial use of complex variables is given oil page 253 of reference 2. The problem given there of a cass pzrticle aoving on the inner surface 02 a rotatfnp; spherical bowl is mathematically equivalent to the disturbed motion of a flywheel and shaft or of a rotor with fockod bingos. The eauations of motion ,- obtained in real forni in rotating coordinates

Bats IC ya + 2wjta - w2ya = - - - - M bf ya

are combined in the single equaticn

where

4 za = xa + iy, ..

is the complex position vector in the rotating coordinate . system. The conplete solution, if small damplag is assumed., is iwt -u, t+iwr t - u, t - i+ t 'a e = C,e + C,e (3) 13

D, = -1 -Ba (1 4. 1 t) 2M t=i The path of the motion is represented by rotations of n compl.ex vsctor in a plane.

and za can be treated as a generalized coordinate in the Lttgrc?nz;ian equatlonw-of motion. The kinetic; aid potential energy oxprassicus can be ixnediately written as

.A dissipation fv-nzticr: for damping thst cie2ends upon motion rolatlfe to tl:B rotating system can be written

The equations- of motion are now obtained by considering za and za as generalized coordinatss in the Lagrangian 4

equotions. Substitution in the equation

thcs yields the equation previously given

?he same method CP.Q be used to obtain the equzttions of ;nolion in the i5.xsd coordinsto syLi=i::n. In th: :: case, 7 i

The oo.uation of ~otionin terms of zf becomes

.. ij E; Zf e --A (if - iWZf) 4- -- Zf = 0 M 1;

end the solutio= COT s~allvalues of $.amping is

t (7)

This sclution ~~CWSthat the moticn cons53ts of two circu-

This exampla illustrates a shaft-critical spesd for W = /xp and a self-excited instability for W > Jic/z.I. 15

A discussion of the physical picture of this instability due to damping is given on page 293 of reference 1. cga The effect of damping in a nonrotating part of the cr) system c?,n be included in tha analysis merely by adding to t the previous dissigation funstion the term GI

The oquation of. Eotion then bacones

The solution for smsll values of damping bccomes

-Bf - B, (1 - [- 2bI 2M zf = C,e

Tho motion is now unst8,ble abovo the speed

(10)

Hinged Rotor

Inclusion of the effect of hinga motion in the plane of rotation increases tho number of degrees of freedom and the number of equations of notion. For axample, three hinged blades and two directions of pylon deflection give five degrees of freedors to be considered. If special linear combinations of the hinge deflections pk are used as generalized coordinates, no more than four degrees of freedom need be considered simultaneously, The use of complex variables reduces these four equations to two equations.

Appropriate variables in the rotating system for a three-blade rotor are .

I .d These variables and their corrplcx con jugntos satisfy the relations - _-_ 6, = - e, e, = - 6, and also

The vpriabies Bt, by virtue 02 their meaning, are rc- ferrod to a rotating coordinste syston. The spocial linear combinations of the denoted by 61~ are also relorred to a rotating c9oriiinate c;rsfern. The appropriato variables to represent the hings defluctions when fixed coordinates &re used are deficed by

Geometrically, 0, or 5, is the coniplax vector representing the displaceinent due to hinge deflection of the center of mass of all the blades, just as z represents the position of the shaft due to pylon deflection. It .rill be shown later that, in the eqzlations of motion, 8, is 4 coupled with z and 6, is an independent principal coordinate. Equations (21) when solved lor Bo, p,, and B, become

80 * 81 + 8, = 5iB0 17

Then, in a mode invoiving 8,

8, = 0

el=-eb iwa.t 2

e, - - 81

8, = sin w,t

B, = sin (wat - a) (13)

En.uations (13) show that, in the 6,-c;ode, the blades are uniiergoing sinusoidal vibrations 120' out of phase with one another in a manner analogous to three-phase electrical . currents General formulas for any number of blades are n- 1 --.

I k- 0 I a=271 n - -- -6n- j i (14) 6, = 6, n- 1 2 n 0 i k= -. Derivation of Equations of Motion

The equations of motion and the characteristic equaticn of whirling spec s are herein derived for the general case of three or more equal blades on a pylon that may have different stiffnoss properties in different dircc- tions of deflection. The effects of damping in tha blade hinges and in the pylon are included. The equatiolis are first formulated in a nonrotating reference system. The 18 required modificatim me then given for the cme of isotropic support Etiffnoss. Tha correspcnding equations referred to the rotating coordinates are then obtained.

Let the position of the center of mass of the Lth blade be represented by the complex quantity Zk in the plane of rotation. (See fig. 1.) Let the bending deflection of :'le pylon be represented by zf in a nonrotating coordinaie system and let B, be the hlnge deflection of the kth blade. Then

+@+beiBkj e i (ak+wt) = Zf (15) :r :r i The corcplex velocity is

Because only small displacements are being considered, the exponential factors cor~taining Bk can be expanded and on13 the terms that lead to quadratic terns for the kineticenergy exyression need be considered.

Some terms can be ignored either because they cancel after sunnation for all the blades or because the ccrre- sponding derivative expressions in the Lagrangian equations vanish. The substitution

leads to an expression for the kinetic enorgy of translational motion of the kth' blade. 1 .. . = - mb Zk Hk (17) 2 where Y

The kinetic energy of rotation about ths center of mass of the blade is T, = P -1 mb r" 8k2. 3 (18) m 2 The cffactive mass of the pylon may be different in the xf- and in tlie yf-directions. Allowance for this possibility .is ns.de by writing the kinetic energy of the pylon as T, --- 1f =a jm,xf -imjiifa) (19)

where . ax + n m= 7 2

mx Am = - my 2

The total kineticenergy is tho sum of the expressions for the seFarate kinotic energies.

The pylon spring constant may differ in tha xf- and in the yf-diracti >ns and, consequently, the potential energy can bo expressed 8.8

1- 1- .@-- u J The effszt of dainping will 50 ck~-"sscdwit.h the &id of fi. Cissi;;tion fu:;ction. If dp-rnpicg exlsts in tf;3 pylon, in the rotating shaft, ani in the hir-gcs, this function be- c 0132 s a n-1 Lf2 + 'L, .. F=&[*'2 BZ~Z~ 4 AB -- 2 - + B,za-i& + d7 3,b,"] (21) k= 0 The eum of tho various energy expressions for all the 20 blac',ce, oxpresscd in terms of tho veriablos zf and lk in the nonrotating coordinates, becones

The Lagrangian eqcations of motion are

aT as

and si~iiarcxpressioos for the other varia5les. The equations of roticn in fixed coordinates then bacome

(rIi+D?Z~~)Cf + EE.vL 4 3a( Zf-iWZc) f KZf + Clll%f + AZ?f $. OEf $. mb = O 21

where Ck refers to the t-variables other than p,. The complex conjugates of these equations are also obtained but gfve no sdditlonal information. Each complex equation 9 is, of course, equivalent to two real equations. It is M noticed that the first two equations contair only the var- I and that the third equation iables zI", zf, 81~6 I4 represents n-2 eciuations, each containing one independent, principal coordinate Ck. The physical meaning of this partial segara'cion of variatles is that a blade motion repyesented by lSinvolves n motion of the common center of mass of the Slades and, thus, a coupling effect with the pylon. Blade notions in which the common center of nass cioes not move are represented by [z, . . . cn. For three blades, the only such moCle is the one corresponding to c0. In this rnoCe, all the blades move in phase; the moticn is alvays damped and does not lead to instability.

The equations of motion of a one- or two-blade rotor are soinairhat different from equations (24). The differ- ence is connected with the circumstance that a rotor of three or nore equal blades has no preferred direction in its plane; vhereas, a one- or two-blade rotor has diffes- ent dynamic: properties in directions along and nornal 30 the blades. Ocly a brief stater;errt+and the final equation for shaft-critical speed vi11 be given for the one- or two-blade. rotor. The equations of motion ir,volving zf and c1 can be written more compactly by use of the notation

D=-d D2 = -dZ at dta

and the substitutions

a -"a _I M = A,- 22

} (251

The Characteristic equation

The general form of solution of equations (26) is an elliptic vhirling motion that can 3e represented by

Special c~isesof this motion include whirling in 8 circle - C, = CtJ = 0, and linear .vL'oi*ation. c, = c,, c, = c,. " Substitution of equations (27) in equation (26) gives 23

In orger for equations (27) to be a solution of equations (26), eq,ua.%ions (28) nust be satisfied for each value of iwf t - ii5f t t. The coefficient of each time factor e or e must therefore separately vanish. Because each bracketed. expression represents a corrplex qaantity that vanishes, its complex conjugate also must vanish. The condition for a solution can therefore be expressed by the vanishing of the flrst bracketed terms a,nd the complex conjugates of the second bracketed terns. Hence,

.J -- where kIi(iwf) is the conplex conjugate of A,,(-iWf) and is o’stained- from B,,(iwf) by changing iu, to -iu witlioGt changing iwf. The characteristic eq’lntion giving the rotational- speeds- is the determinant of the coefficients of CIS C,, C,, and C, equated to zero. With the second and thir& columns interchanged for symmetry, the determinant becomes

0 0 A1 2 822 =o

0 Ai1

. The expanSed form of this determinant is

A- -- 24

Ths roots u'f ~f this equation are tha characteristic whirling speeds of the rotor.

For the case of Isotropic supports, GB,, = 0

the equations of noticn are satisfied. by equations (27) vith C2 = C4 = 0. 'I'k ciini.ac~,e:3.~'iicoq7,mtion is thep sinpkv

In a rotating coordinate system, the complex coordinafes are z, and El,, where iwt zf = zae iwt 51 = 6,e Then 25

If the whirling speed in rotating coordinates is represented by w,, iwet co z, = C,e 0 m hat I e, = C,e 4 The characteristic equation is then obtained by substitutl. ing wa f w for uf.

The chayacteristic equation can thus be stated In terms of a whirling speed in either the fixed or the rotating co-

ordfnst e sy st em s,

MSTHOD OF APPLYING THEOBY

Application Neglecting Damping

In glotting curves for use in applications of the theory, it is convenient to consider one of the pylon bending frequencies wT = +jmas a reference frequency and to refer all othi:r frequencies as well as the rotational speed w to ths reference frequency as unit. The nnmbsr of independent parameters is thus reduced by 1. All quantities in equations (31) to (33) are then expressed nondinensionzlly.

The natural whirling speeds and the three types of vibration - ordinary, self-excited, and shaft-critical - can now be predicted from a study of tha roote of equation (31) in which wf is considered a function of w for ffxed values of the other parameters. The c&se of no damping will be considered first. Be-. cause eqxation (31) with damping ternis omitted is of the fourth degree in wf 2 and of only the second degree in w’, it may be solved conveniently by first choosing values of uf and then solving the equation for w2. Simi- lar indirect msthods can be used with equations (32) End (331, Special methods to be used when damping is included will be discussed later. 26

Tho mtsnlng of eq.~...atlons (31) to (33) will bz $11~3- trated b:; ~,xzrisles. Pns real part, of uf trill ba plol;t,d ag.zi2-st w Ior selected valuPvs of the parameters -4,: Az, A3, and. so The simplest case is that in which the cass of tke blades is so s~allt3at a-r,y force on the pyl?n due to >lade inotinns is nogligibie. The pyloc motions zre then indcpendert of the blade notions. This case is obtained 'bjr putting I?,& E 0. The characteristic equaticn (31), (S?), 3r (33) thon factors into expressions yielding straight lin9s an& hyporbolas.

An e:;cm?lc of a rotor with psrticular values of the paranaier:: is plottcl. as 1ong-iJ.asn lines in figure 2. The ko-iaontal straight lines corresFond to yy3.on bonding and the sltinting h:rprrbolas corrfes2cn6 t c hinge deflection. Each curve re7,resGnts ths trend of one of the rcaZ roots wf. 1s A3 incronscs slightly from zero, the greatest c!=ang,:c in tke curves cccyir in th; vicinity of the intersections of thc straight linss wit5 the hypcroolss. Here cack kranch breaks av.ray froo tnc intorsection a24 rc- n joins t52 other branch, At 2 gap, sxch as v in fiprr3 2, the nambc:r of real roots of the frcqucncp ea-uation is reduced by 2. The missing rootn arc, ccn?lcx conjugata numbLrs; pnd. onc: cf ti-em ml;et have a nep.tivo imaginary part, which im?Zics a sell-axcitod vibration.

Consider the interpretation of figure 2 3s r;! is .gradually incrccctscd from zero. At zcro rotational spccd, the ?ralucs of ui: arc the natural frequencies thet could be excited as orClinsry vibration by applieg. vikrating force. Positivc ar,d cogativz valnes occur in pairs of equal magcitudc. nnd correspond to lincar vibration modcs rcprcscnt-6 in complex notr?.tion as -iqt zf=c e +c iwft As W increases fron zero, the positfve acd negative values of wf no longer are equal in nagnitado. The nor- mal :oodes are therefore itrhirling rcotioos with angular ve- locities eq-ual to the piotted values of wf.

The shaft-critical speed is tho rotatfon'al speed at which ;?if = w and hence is given by the point A whore a'45' line through the origin intersects the wf-curvea This speed corresponds to the psak for vibrations escitsd * 27

by un3zlance in tho rotating system. As U) increases above the shaft-critical speed, the modes of vhirltng are stable until, for the case of no damping, the value of wf r--;r 3 becones complex at the value of w at which a vertical w line is tangent to the plotted curve. This point B is i the begincine of the self-excited raqge. At the yoint D, )=I the moti3n zgaip. becomes stable. The real part of wf is plotted in the region C as a short-dash line, The COD?- plex roo%s in the region G have 'seen calculated ard plotted in figure 3. The point E, at which wf = 0, is of some interest. At t.his qeed, a vlbIation of the blades could be excited by a steady force (cut = 0, w, = -w) such as the force of gravity if the plane of the rotor is not horizontal.

Because the most imTortant iaformation to be obtained fron the frequency ocfuation is the critical value of w for the shaft-critical and self-excfted vibrations, a. set of charts that givas this information for a large varioty of values of the physical parameters has been prepared. These charts are given in figures 4 to 6, which correspond to values of stiffness ratio K~/K, = s ci 1, w, and 0, respectively. The use of the charts is illustrated by a numerical cxaniple. Suppcso the values ol the paraseters for a certain rotor are A, = 0.07, A, = 0.22, ti3 = 0.1,

e = 1, an6 t!lr = 155 cycles per minute. A straight line, such as 853 in figure 4, is first drawn to represent the function w2Al + A,. This line i2tersects coi1tours

A3 = 0-1 at (L)' = 0.77 for the shaft-critical point and W' = 1.6 and 4.85 for the beginning and for the end of the self-excited range, respoctively, With a referanca frequency of 155 cycles -per minute, these values correspond to actual rotational specds of 136, 196, and 342 rpm. All possible values of A,, A,, and A, are thus cov- . ered by suitably changing the straight line AB. Tho general effect of the stiffness ratio s is not large; any case can therefore bo ostimatecl witk a fair degree of ac- curacy by use of figures 4 to 5. 28

, I Possibility of Avoieing Occurrence of Vibration

Figures 4 to 6 can alsq be used for the inverse prcb- lem of fincling the values of tins parameters that a22 required to obbain given vnluas of critical rotational s3!2sd. These figures show that to alfninate entirely the self- excited instabiliby requires that A, be equal to or grcater than 1. The shaft-critical instability can be entirely aliminated only with a vR1ue cf iLl in 6 srnsll range near 4 and with s = m or s = 0. 'Phase valuos of A, differ rnciical1.r;. from presont designs in which a typical value is 0.07.

The satisfsctorg requirzmcnt of kseping the instnbil- ities outside the o:?erabing range of rotational sp2ed is found by first picking a rcaeonnblo valu,: of the pylon freq.uenc;y fixp to fix the scala unit for w and by the11 obs3rving the combinations of AI and h, that cm be used to avoid thc? critical &-contours.

3ffect of Damping

Ths affect of darping has been includcd in cquaticn (31) throcgh tlie paraEeters A,, A,, and hp. k nctho6 cf cumputation similar to that usad in flutter theory apnc3rs prcfarablo to t%ttc?qtingto solve the QOUation dfroctlp for wf0 The beginning arid the end of an unstable range can bo found by the following method: At a limit point between a stable and .?tn unstslblc speed range, thc value of Wf is realo Equation (31) is first separatad into 1.~~311 and inaginzry parts with wf considered real. Each part is considered a functional relaticn between wf and UJ and is plotted for a given set of values of the parameters. The intersections of the real 2nd the imaginary equaticns give the rotbr speeds and frequencies corresponding to the beginning and the erid of the unstable rwiges. In the com- putations, it is preferable to choose values of Wf ~iid to solve the equations for the corresponding values of U.

The explicit form for coinputation in the sibplest case of j.BoT;ropic SqyortB, ~r.i*L danping in the pylon and in the hinges but not in the rotating shzft (ha = O), is obtained from eountion (32) rearranged c.s fol~ows: 29

f t-4 w= whore 3313 =

and

Imaginary equation w2 - 2B1w + CI = 0 (35) where

The most general' case obtained frorn equation (31) is written:

Real equation ,

Coefficient OT w6 w4

1' 30

Itmginary quation

Coefficient of W" I (1 -A.,)" + X$ f3 1

(u2 R I + R,I, + La2 I, (3.7) 13. 1 R112 + R,I - I,

t-- --.

Examples of calculated cases with [email protected] are shown in figures 7 to 9. The Zresonce of small amounts of damping in both the pylon and the hinge detgees of freedcm does not greatly change the predictions that would be made *con the equations with no damping. Tho plot of the real equation is practically the same as the plot obtained when damping is neglected. The intersections of the cui'vos of 31

the iniagrlnary and the real Equations with any reasonable value of hf/hp are near the points that would be con- co sidered the limits of the unstable range if damping were 0(VI neglected. Increasing the amount of damptng decreases

4 the gap between tho limits of stability until the unstable 4 range is finally ellminated. An approximate solution for the amount, of damping required to eliminate the self- excited instability is obtained by requiring that the damping be at least largo enough to make the curve of the real equation pass through the point where wf = 1 and w is the value given by the equation

The vnlucs required in the case of s = c)r, have been com- puted and plotted in figure 10. The elimination of self- excited vibration by damping thus looks promising and merits further stuCy with reference to specific application. LIMITATIONS AND FURTHER DEVELOPMENTS OF THX THEOiiY

lolar Symmetry

An important idea in the rotor vibration theory is the concept of polar symmetry, This concept implies the absonco of a preferred direction in the plane of the rotor. A rotor of throe or'more equal blades has polar symmetry. A rotor of two blades or one with unequal centering springs does not have polar symmetry. A pylon 3, By, and has polar for which K, = ICY = m, = my symmetry. The possibility of solving tho rotor vibration problem in terms of exponential or trigonometric functions depends upon the existence of polar symnctry in the rotating parts or in the nonrotating parts or in both. The general case of no polar symrnotry would lead to Mathieu functions or something similar.

Two Blades

A brief comperison between tho two-blatde and tho gen- oral case is presented heroin. Polar symmetry of the pylon is assumed. The shaft-critical spoed is obtained by substituting W, = 0 in the characteristic equation as 32

expressod in 2, rotzting coordinato system. For one or two blades, the equation obtained is

The first bracketed factor gives tho beginning of a self- excited range and the second factor gives the end of the range.

Equation (38) can be compared with the equation for the shaft-critical speed of three or more equal blades and for polar srmmetry

A useful chart based on equation (39) is given in figure 11; some experimental results of tests of a simple mcdel in figure 12. These tests demcnstrate the essential differ- ence between the two-blade and the general case.

Langley Yemorisl Aeronautical Laboratory, Kational Advisory Connittee for Aeronautics I Langley Field, Va.,

REFEREWCE S

1. Den Bart og, J. P. : Mechanical Vibrat i sns. XcGraw- Hill Bock Go., Inc., 1934, pp. 237-328-

2. Lamb, Horace: Higher Elechnnics. The Univarsity Press, Cambridge, 2d ed., 1929, pp. 200-201, 241-255.

3. Lamb, Horace: Bydrodynamics. The University Press, Cambridge, 6th ed., 1932, pp. 307-319, 562-571.

4. Preltritt, R: E., and Wagner, R. k.: Frequency and vibration Problems of Rotors. Jour. Aero. Sci., vol. 7, no. 10, Aug. 1940, pp. 444-450.

5. Coleman, -5obert P.: Theory of Self-Excited Mechanical Oscillations of Hinged Rotor Blades. NACA h.R.X., July 1942. NACA

\ \

Fig-ure 1.- SimTlifiod machanical system raprGscnting rotor. XACA Figs. 2,3 4 ,’

3igur.3 2.- Tine effect of coupling between pylon and hinqe motiom.

, -.2

-.4

Figure 3.- The complex frequency iii the umtebie rsnge for J$. = .07; A2 = .22; A3 = .L; 5 = 1. c ITACA 5.0 -- . I

3.9

. ,2

2.0

Fipre 4.- Stability chart for s = 1. NACA Fig. 5 5.: _I- I Io

4.( -___ t----- 4 I

3.1

,2 ,2 c

2.1

Zigure 5.- Stability chart for s = co. Fig. 6

u)%1 4- h2

7igire 5.- Stabilitjr cliart for s = 0. NACA . Figs. 7,s

1.0 2.3 3.0 u

3’igxe 7.- Plot. of rm1 and imginary cqcatiore for a. tJTical C~SB. s = 1; bl = a075 A2 = ,233 A3 -198.

c) 1.3 2.0 3.G UJ

FigAre 8.- Plot cf real ad imqimry eqwtiohs for case of s = co ; = .O7; -!= .22; f~ = .198. NACA Yigs. 9,lO 2.0

L3 3 M

c!J f

1.G

1.0 2 .!I 3'.0 u

2.Q ---

1.5 I___-_ Figura 16.-

1.0 self-

oscillation for 6 = m. .5

0 .2 .3 .a ,5 .s NACA Fig. 11

1.;

1 .(

-. ,E

d -a Pb

.I *; NACA 12

& h'

0 .2 .4 .8 '. 1.0

Fi,gure 12.- Eqerixental critical speed.!s os small models.