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Produced by the NASA Center for Aerospace Information (CASI)
NONLINEAR ROTORDYNAMICS

ANALYSIS

1

i^

L'

(NASA-CH-171425) NONLINFAR ROTORDYNAHICS

ANALYSIS Final Report (Auburn Univ.) 38 p

RC A03/MP A01 CSCL 01A

Final Report

on

Contract No. NAS6-35992

Prepared for:

George C. Marshall Space Flight Center

Marshall Space Flight Center, AL 35812

Prepared by:

William B. Day

Department of Computer Science and Engineering

Auburn University, AL 36849

15 March 1985

N85-22364

Uncles

G3/02 14681

P

Yi

12
`Y

^I

1

i

is

y!

A

a

ii

i

l

t

J 9

ABSTRACT

This report is an e v )imination of special nonlinearities of

the Jeffcott equatione in rotordynamics. The immediate

application of this analys i s is directed toward understanding the

excessive vibrations recorded in the LOX pump of the SSME during

hot firing ground testing.

Deadband, side force and rubbing are three possible sources

of inducing nonlinearity in'the Jeffcott equations. The present

analysis initially reduces these problems to the same

mathematical description. A special frequency, named the

nonlinear natural frequency, is defined and used to develop the

solutions of the nonlinear Jeffcott equations as singular

asymptotic expansions. This nonlinear natural frequency, which

is the ratio of the cross — stiffness and the damping, plays a

major role in determining response frequencies.

Numerical solutions are included for comparison with the

analysis. Also nonlinear Frequency — reponse tables are made i'or a

typical range of values.

Finally the regions of stability of these nonlinear problems

are bounded using inequalities of the system parameters.

i

.y W_ - . - _._____...
Ill

`^

^ a

M

s

r

k

s

r

R

CONTENTS

SECTION

1. Introduction

2. Nondimensionalization and Heuristics

3. General Theory

3.1 Nonlinear Natural Frequency

3.2 Equivalent Problems

3.3 Method of Multiple Scales

3.4 Resonance

4. Examples

5. Stability

6. Conclusions

7. Bibliography

11

PAGE

1

7

8

9

10

12

14

17

19
LIST OF FIGURES

01 FIGURE PAGE

y vs. z - Heuristic example 20

2. r vs. t - Heuristic example 21

s

3. PSD of y - Heuristic example 22

4. Y vs. Z - Example 1 23

24 a 5. Y vs. Z - Example 1

i 6. PSD of Y - Example 1

25

?. PSD of Y - Example 1 26

9

8. PSD of Y - Example 1 27

9. Y vs. Z - Example 2

28

29 10. Stability regions

i

^ r

l^

111
LIST OF TABLES

TABLE PAGE

1. Frequency response, E-0.5 30

2. Frequency response, E=1.0 31

3. Frequency response, Em1.5 32

i^jrs

;>ss

^i

^t

i^

^r9

t

i

^

L't E

i

'v

iv
I

1. Introduction

Vibrations are inherent in rotating machinery. Mathematical

e explanations of vibrations began with Jeffcott's description of

i the shaft's natural frequency of lateral vibration [SJ.

Unfortunately, Jeffcott!s linear model cannot account for all

frequencies that have been observed experimentally. In

particular, destructive vibrations have occurred in hot firing

ground testing of the LOX pump of the SSME with no clue to these

vibration's origins being offered by the linear model.

Specifically, examination of the power spectral density (PSD)

plots reveals unaccountable frequencies. Consequently, numerous

investigations have been undertaken to study such rotors and to

provide descriptions of the solutions of the two, coupled,

second-order differential equations which describe the motion of

the rotor's center of mass. Following the early work in

rotordynamics by Yomamoto (7), one introduces a nonlinearity to

r7 the Jeffcott equations by including the effect of bearing

clearance or deadband. In the pump, this deadband refers to the

7 load carriers (ball bearings) and physically describes the

{ clearance between the outer race of the bearing and the support

housing. The work of Yomamoto did not include cross-stiffness,

but a straight-forward derivation with this modification is

easily obtained. A more limiting gap in his work is the

j assumption that the response is simply a perturbation of the

forcing function. This is tantamount to assuming that one always

i

has the graph of a circle as the solution. It is shown in this

report that this generally is not the case. Both empirical

results by Childs [1,2] and Gupta et al. [4) and numerical

solutions

Dynamics

using the

Company

fourth order Runge-Kutta algorithm by Control

[3] have been helpful in understanding the

' rotor's motion for the nonlinear problem. This report extends

the earlier work by using analytic expressions obtained from

singular asymptotic expansions (method of multiple scales) to

quantize the solution.

The primary objective of this report is to describe

analytically solutions of the nonlinear Jeffcott equations with

deadband (or an equivalent variation such as rubbing). To this

end emphasis is placed on determining which frequencies one

expects from the nonlinear problem, how these frequencies are

related to the parameters of the differential equations, and

where the solutions are stable.

( ul

fi Section II is a heuristic development of responses to a

nonlinear system. This discussion is based on a numerical

j solution and is intended to help develop some intuition about the

solution of the nonlinear problem.

Section III is

analytic solutions.

the

This

formal

section

mathematical

begins with

development

the discovery

of

of

the nonlinear natural frequency and proceeds to incorporate it in

an asymptotic expansion of the solution. This leads to

considering the two cases of resonant and nonresonant driving

tt

9

1
I

frequencies. Herein also lies an explanation of why one expects

i

the rotor's motion to be either a circle or an annulus.

r Section IV contains numerical examples which verify the

a

theoretical expansions. Typical frequency-response descriptions

are also included in this part.

Section V discusses bounds for stability regions. This is

i done with comparison lemmas for ma,jorieing the nonlinear Jeffcott

equations.

Section VI concludes the report with a summary of the

report's applications and directions for future studies.

I

I

r

z

0

2
I

2. Nondimensionslizstion and Ileuristics

{ The linear Jeffcott equations which describe the displacement of the rotor

center from its equilibriumposition in the inertial, Cartesian coordinate

system ( y,z) (each in meters) are these:

(1.) my -Cay - Kay-Qsz+muwzcovwt

(2.) mz•-Cg z+Qs y-K.z+muw z ainwt

where the shaft of the rotor lies along the x-axis and

,a

m n mass (kg.)

C

s - 

seal damping ( kg./s.)

K . seal stiffness ( kg./s.2)

s

Q s• cross-coupling stiffness of seal ( kg./s.2)

u . displacement of the shaft center of mass from the

a geometric center (m.)

W - angular velocity of the shaft ( rad./s.)

r

For the model to include bearing forces which hold the rotor in

position, one adds the terms

- K B(Y-Y d/ Y2 +z 2 )+u KB (z-z6 //y +zz )

and

r

-yRB(Y -Y6/ }^ +z2 ) - KB ( z - z 6//-Tii )-

respectively, to right-hand sides of equations (1.) and (2.)

Here

KB = bearing stiffness (kg./s. z )

6 = clearance or deadband between housing and bearing (m.)

P = coefficient of friction between housing and bearing (none).

These bearing forces occur only when 67_+_Z">6; otherwise, they are zero.

Since V is nondimensional and typically small, one may regard it as zero

4 without affecting the qualitative results.

Equations (1.) - (2.) then become

(3.) y+(Cs/m)Y+(1/m) [ Its+KB(1-6 /r)]y +(Qs/m)z•uwzcoswt

(4.) z+(Ca /m)z-(Q.1m)y+(1/m)[Ks +K B (1-6/r)]z•uW2 sinwt

[ when r w hy—+"z-T)d; otherwise, Ka •0. Equations ( 3.) - (4.) can

put in nondimensional form using a displacement g and a frequei

c. One pair of candidates for g and o would be g . 6 , t

deadband size, and o2 .wo •K siKB,. the natural frequency of i

^ 3
^rQE POOR QUALITY,

corresponding linear problem (6-U). Thus, using Y-y/g, Z-z/g,

ands -at, the dimensionless equations are these:

(5.) Y '' +CY'+(A+k(1-4/RJY+BZ.E02cosOT

(6.) Z " +CZ'-BY+(A+k(1-4/R)IZ-EO2sinOT

where prime denotes differentiation with respect to T and

t C-Cs /m /a , A-Ks /m / 02 , k ` KB / m/ o . B-Qs/m/a=, 4 . 6 / g , R- r / g , E-u/g,

and 0 -u/o.

is Equations ( 5.) - (6.) can be reduced to the following single

equation by defining W-Y+iZ:

t(7.) W''+CW'+(A+k(1 -4/I WI)-iB)W-E0 2 exp(iOT )

Furthermore, the polar form of equations (5.) - (6.) is

(8.) R''+CR'+(A+k(1-4/R)-(0')21R-E 02 COS( OT-0)

(9.) R0''+(2R'+CR)0'-RIB-uk(1A/R)]+F.4 2 sin (OT -0)

3

where R . (Y 2 +Z 2 ) 1/2 and 0 -Arctan (Z/Y).

The nondimensional Jeffcott equations with deadband and mass

imbalance are easily solved numerically using a fourth-order

Runge-Kutta algorithm. If these solutions are then plotted and

analyzed in a power spectral density (FSD) investigation, the

resulting graphs provide a direction for initial, analytic

descriptions. To this end, consider the following special case:

M-1 . u-1.56

C-240. Q-200,000.

K .0. S 6=.0000285

K B -1,305,000. w-50UHz

I The units used are those given after equations (1.) - (2.) and

are only of interest in a relative sense. Therefore, with

wo -(1,305,000) 1/2 and 6 •.0000285 as the nondimensionalizing

parameters, the Jeffcott equations become

Y''+(C / wo) Y'+(1-I/R)Y+(Q /w o ) Z-1.5(.J/wo ) 2 cos(w/ wo)T S s

Z"-(Cs / W O ) Z'-(Qs / wa ) Y+(1-1/R) Z. 1.5(w/wo ) 2 sin ( w/wo)T

where T- wo t and R-(Y 2 +Z 2 )

1/2

=(y 2 +Z 2 )

1/2

/6. Figure 1 shows the

motion for this case. The motion has become periodic for the

Ell time interval (.5<-t<-l.s.) shown. The graph of t vs. r - 6 R,

given in Figure 2, suggests that r has the form

r-.71+.39cos720n t-ao+a2 COS YT.

It is this form that leads one to consider approximating the

i Jeffcott equations with

(

l

k

^. *mss r
Y"+CY'+(L-1/(a 0 +a,cosy T ))Y+BZ.1.50 2 cos 01

Ottt6fNiU: PAGE R

Z"+C7.'-BY+(1-Mao +a l cos yT ))2 . 1.50 7 sinoT . OF POOR QUALIV

Examination of the PSD of R shows that R is nctuolly an infinite

t sum:

R. I a cos k YT

k=0 k

with a and a the dominant terms. The other coefficients

decrease exponentially to zero as k- m Therefore, the

replacement of R with a +a l cos yT is a reasonable

approximation. Furthermore, ao can be considered the average

value of R. Since this value will be nonnegative, one may write:

i' 1/R-1/(a0 +a 1

cos YT )

-0/ao ) f (-a1/ao)It (cosyr )k.

k=0

If the original infinite summation of R had been used, the above

series for 1/R would still be a summation of cos YT terms, but

with different coefficients.

I

Now consider a t /a 0as an asymptotic expansion parameter c

Thus:

: 1/R-( 1/a 0) ( 1- eeos YT J+0( e' ).

This last approximation may ultimately lead to poor

approximations for the magnitudes (but not the frequencies) of 

the Jeffcott solutions since higher (01) powers of cos yT

actually contribute to the coefficient of cos 0 T and cos yT

Therefore, thu following derivation for a straightforward 

asymptotic expansion may be used primarily for predicting

response frequencies, but will generally be poor for describing 

the corresponding magnitudes.

With this prelude, one replaces the nonlinearity 1/R of the

Jeffcott equations with 1/a o- Ecos YT and expands the solution as

" Y-Y + cY1 +0 (c' ) and Z-Z0 +c 0 Z 1 +0 ( 0 ). The resulting low-order

pro^lems are these:

0(1):

Yo "+CYO

'+(1-1/a 0 )Y +BZ -1.5 1) 7 cosmT 

0 0

Z0 "+C; ' -B 

c 

+(1 -1/a0 )Z 0 -1.5 02sinOT

^LI Y1 ''+Ci '+(1-1/ a 0 )Y +BZI --(1/a 0 )(cos YT )Y0

71 ' '

+CZ 1 '-BY +(1-1/a 0 )ZI .-(1/a 0 )(sin yT )Z0

w

0 5
problem. The initial conditions for all higher-order problems are

all zero.

" Thera is a circular argument involved here. One is using

the answer, which was found numerically, to solve the problem.

T

Although this is unacceptable mathematics, the results obtained

in this heuristic discussion are not derived solutions but only

suggestions of solutions. With this in mind, one proceeds to

solve the zero - and first-order problems. The zero-order

} problem has solutions of the form:

Y a -C ( T ) Sin BT +C 2 (T)COS BT+C5 cosmT -C6sin,T

Z o -C3 ( T )sinBT +C 4 (T )COS BT +C S SinmT +C6cosmT

where C 1 , C 2 , Co and C,, are all functions of time and all involve

decaying exponential functions. Thus, in steady-state

y COS (r-r +mo)

r 5

,ss ^

-C sin( OT + ,p )

Z o ' SS 7 o

In the first-order problem, one is presented with the same

left-hand side as the zero-order problem. Thus, in steady-state

it is only the forcing function of the right-hand side that

provides non-trivial solutions. Hence, one finds combinations of

' the form cos YT ^os OT and sin •/T sin OT

d

Thus, the steady-state solutions of the first-order problem

+ should look like these:

Y 1 -C b tos( Ot Y ) T

Z 1 -C e sin(ms Y) T

Since higher order problems contain right-hand sides of the form

(cos ky t) f(Yo, Z a , Y 1 , Z 1 , . . . , Yk, -1 Zk-1),

" it follows by induction on k that the kth-order steady-state

solutions contain terms of the form

Y 

- cos( o`- ky ) T

Z - sin( 0`- kY) T

Finally,y, ne anticipates p that the PSD of Y ( or equivalently 7,)

will, show frequencies at 0! ky This is verified in Figure 3,

^w

which is the PSD of the (numerical) solution of the test problem.

( ORIGINAL' PAGE M

L' 6 OF POOR QUALITY,

L^
e(

'

r^ 

3. General Theory ORIGINAL PAGE M

OF POOR QUALITY 3.1 Nonlinear Natural Frequency

Consider the homogeneous (E-0) equation corresponding to

equation ( 7 .). If this equation were also linear (A -0), then

4 r exponentially growing or decaying solutions would generally

result for a given set of system parameters. In the special case

that (9/C)'-A+k, a sinusoidal solution is obtained with frequency

B -B/C. To see this, consider the characteristic equation for

W - eX P( m t):

J

m2+Cm+(A+k-iBJ-O

m--C/2s (C'/4-A-k+iB}1/2--C/2'(C'/4-(B/C)'+iB)1/2

m--C/2ti(iC/2-B/C) 1/2 .-C-iB/C, iB/C.

In the nonlinear, homogeneous problem, k is replaced by k(1-A/R);

hence, if R is a constant, then there is a wide spectrum of R for

which (B/C)' may be A+k(1-A/R); i.e., if 

(10.) A<(B/C)'<A+k,

then there is a constant value of R (with R>p ) for which

(B/C)' -A+k( 1-A/R). This value of R is denoted by a and the

corresponding frequency by Bo -B/C. This frequency is labeled 

the nonlinear natural frequency. Thus, whenever 

Inequality (10.) is satisfied, equations (5.) - (6.) with F-0

j have steady-state solutions Y-a cos(3oT ) and Z-a sin (B0T)-

The same results, can be found from the polar equations

- (9.) by assuming R and 0' are constants. Then with EwO those

equations become 

(A+k(1-A/R)-(0')' } R -0 . C 0'-B•

Thus, 3 0 -0'-B/C and a-R-kp/(A+k- 301).

Notice that 00 -B/C<(A+k)1/2-wo, the dimensionless natural

frequency of the linear system. Thus, in considering the general

nonhomogeneous problem, it is necessary to be aware of these three

dimensionless frequencies:

the nonlinear natural frequency,

j ao - the natural frequency,

p - the driving frequency.

Either U 0or wo is an appropriate choice for a the

nondimensionalizing frequency. Correspondingly, one would select

V , either a (with 3 0 } or u (with wo ) as the base displacement g.

7
Ed

emphasis the nonlinear natural frequency:

(11.) W '+CW'+KW-f(W)+E0 = exp(iC; )

v ORIGINAL PAGE MA

where K- A+k(1-A/a)-IB and f(W)-kv (1 /!W1- 1/aJW. OF POOR QUALITY

iy 3.2 Equivalent Problems.

In

the form

this subsection the forcing function is assumed to have

F(w) exp(iwt), where F(w ) >-0 (or equivalently F(w)

and F is single-valued for w >-0. The following are special

cases of physical interest:

11s a. Mass imbalance. As discussed above, F(w) n uw2

for a

mass u.

b. Side Force. This force may be introduced into the

Jeffcott equations (1.) - (2.) as constant replacements

for the mass imbalance. In such cases; equation (7.)

becomes:

:

W''+CW'+(A+k(i- a/ IWI )-iB) W- uW/ 69' n constant.

is Thus, a side force is the special case F(w)-constant and

W -0.

T

C. Rubbing. Rubbing contact between a rotor and its

r di, housing produces a Coulomb damping force. This force

Iwould modify the original Jeffcott equations by the

addition of the terms:

K at (1- 6/ r ) y - u K at (1- 6/r)z+K st(1 -6 /r )G

/ r )Y/r)z+ K at ( 1 - /r)C

Kat ( and 1 - u

6 + R at ( 1 - 6 u6

respectively, to the right-hand sides of equations (1.)

- (2.). Here, G n constant-stator offset in the ydirection, R at n stator stiffness and u -coefficient of

friction, which may not be small. As be ore these

forces 2 would be included only when r-(y2 +z2)' > 6. On

t replacing y-G by y, equation (7.) (and corre5, pond ingly

its equivalent forms) again occurs but vith these

modifications:

1. i(-B) is replaced by i[-B +u

(K st/g= )(1- A /R) J

2. the forcing function Ed e

exp(IOT ) is replaced

by E 0' exp i OT )+(- wo) / g2 ( G 6 ).

If the mass imbalance term is omitted, then one may

regard the deadband with side force problem and the

rubbing problem as equivalent. Notice that the range of

values of the parameters for these two problem may not

be the dame since the rubbing problem includes the nonnegligible term u (K /g' )( 1- / r ).

at

i?

8
3.3 Method of Multiple Scales. OF POOR QUALITY

" Not consider a more formal development of analytic

Solutions. Straight-forward asymptotic expansions are not

general enough for this g g problem since they always lead to a zeroorder approximation of the form:

W-M ex 1 BOT

C

The heuristic example considered in section 2 suggests that

Bot must be replaced by B (T ) which is close to but not exactly

+i BoT. There are several methods of attacking such singular

g

robbl ms these are d scribed i Nu ffeh rr 66 o0 e method the

et,o^ of averaging , Is appropr^ete y For t6^'Jof^Cott equations

since it begins with the assupmtion that W = M (T ) exp (iB (T)) +

N exp ( i OT). Another method, multiple scales, is also appropriate

I because one can envision the action of the rotor being based on

two different time scales. The following development uses the

method of multiple scales. The results are identical for the

method of averaging. In this subsection the forcing frequency

M m is restricted to differ from the nonlinear frequency B ; i.e.,

the problem is not resonant. Resonance, is dealt with in

the following subsection.

Instead of one time scale T, assume the problem depends on

many time scales:

a

T o . T, T I . ET, TZ =E 2 T , ..

w

6 Henceforth, only T and T are used. Let W (T ).W(T ,T ) . Wo(T0tTI)

+EW1 ( T o , T1)+... 

o 

Equat1on ( 11.) becomes a partial Id ifferentiai

equation since

d/dT= ( a/3T o ) ( d% /dT )+( a/aTl)(dT, / d T) -Do+eDI

and ( d 2 /d r )=(D +c D )_ .

o I

Thus, one finds

(12.) (D +ED (W +EWI+...)+C(D0+EDI)(W0 +E W1+...)

0 1 0

+K(Wo+EWI+...)=Ef(Wo+EWI+...)+E©Z exp( ii T0).

k' Equating like powers of c yields

I

(13.) CO :Do 2 Wo+CDoWo+KWo . E0 2 exp(ioTo )•

1 

1= This is a linear problem with this steady - state solution

Wo =M exp(iB, T o )+N exp(i^To )

where N= EO 21 (- O 2 +iCO+K ) and M-M(TI ). To determine M one must

i rII examine the c -order problem and choose M to eliminate secular

C4! terms; see Nayfeh [6J:

11

9

L
5 cl:D W 1 +CW I +KW 2D o D 1 Wo -CD1 Wo +f(W 0 ). OF POOR QUALIT.b

With L•kA/c the right-hand side of the last equation becomes

-21poM'exp(i(ioTO )-CM'exp(iRoTo )

+L(1/ IW O 1-1) (M exp(iQoTo )+N exp(i^T0)J

where IW o 1 I M1 2 +INJ Z +MN exp(i(0-'uo )T,] +MN exp(i(a, -m)T o])1/2

To avoid secular terms one requires, that the collective

coefficient of exp ( i8o To) be zero. Although an analytic solution

of the differential equation for M ( T 1 ) has not been found, one

K

can qualitatively aasess M based on a similar problem ( van der

Pol'9 equation) and specific numerical results ( presented in the

next section).

Since M ( T 1

) is complex, it may be written as

q^ M(T, )° P( T 1 ) exp [ i B( Ts ) ] . Thus,

t 's ^

W'o-P(T 1 ) exp(i6oTo+iR(Ts)]+ N exp (iQTo) or, assuming R (T1 ) is

analytic near t=0, 1b .p(T 1 ) cxp[i ( bo 4 ER1 ) T +... ]+N exp (i.^T)• Thus

the fundamental frequency of the nonlinear problem is not g

°

but.

d Ro+ G 1 +...; however, p must reduce to Eo when U 2 =0. This

frequency shift can account for the phenomenon of "tracking" that

has been observed experimentally ( 2). Similarly, the frequency

Y =' - p o that appears in - th expression for IW o I should

considered as Y 4- G. Then 1/ TWAl shows all fre q uencies nY

be

and

11

Wo/I W shows all frequencies nY±:s, for n=0,1,... This suggests

 that has a complex Fourier series of the form:

n• s exp(inYTO.

1

Another factor of M must also be included since numerical

examples show that M30 if E0 2 is greater than some fixed value.

This is sir : ilar to the behavior of the van der Pol oscillator;

see (6]. ?'.sus, one may speculate that M has a factor of the form

F - 1/(1+exp(- nT )] where n =n( E0 2 ). This would imply that F - 1

as T-'O0 when n >b and F - 0 as T- m wher. n <0. Thus, M looks like:

1/(l+exp(-nTI )]E sn exp(inYTI).

PSD plots of R show only frequencies of nY while plots of Y show

frequencies of p and nYzp.

3.4 Resonance.

q

The resonance problem 3=w can be handled in a manner quite

similar to the nonresonance case. The approximation 0 =1+a E is

made where E is the nundimensional mass causing the imbalance and

Qo is the nondimensionalizing frequency. With the same multiple

scale expansion as before, one finds the following two low-order

problem:

[^! 10
j 0(1): a 2 w e / a T 1 2 +C( 3W p/ 3To )+[1-1 / a-iBIWo- 0

ORIGINAL PAGE 15

3 with the solution W o -m(T )(axp(iTo), OF POOR QUALM

1

0(E): 3 2 W I / a T^ 2 + C( 3W aTo )+( 1-1/a-iBJWI

+23 1

W O / 3 T o 3T +

I/ 

C(3 W / ,TI )- ( 1/E)(1/ ^W,1 - 1/a)W o + exp(iT a ) exp(io T1).

Elimination of the secular terms of the latter equation requires

0--21 M'exp(iT,)-CM' exp(iTO )+(1/E)(1/M-1/a)M exp(iTo ) +

9 ((('''I exp(ioT d exp(iT o)

t

or with M n m(T I ) exp(i a ( TI ))

(21+C)(m'+i a'm)-(m/E)(1/m-1/a)+ exp(ia T l ) exp(-in ).

, 

Now let V - o T I - a separate real and imaginary parts, square

both sides of the two resulting equations and add to find the

steady-state solution (m'-0- a '): 

m 2 (1/a 2 -4 a /a+a 2 (c2 +4)) + m(-2/a+4 a )-0.

This magnitude result, a quadratic equation, is identical to that

of YomPmoto (7) although the expansion there is with the base

exp(1 0 To ). 

W-m exp(i(T,+ a))

where m is found above and the frequency is

(d/dt) ( To+a )-(d / dt)(t+ aEt- y ).

But d y,/dt-0. Thus, the frequency is l+ cE- 0. This phenomenon

is often referred to as entrainment; i.e., the nonlinear

frequency, 4, is entrained by the forcing frequency, w

I

$.

11

d

n
URIGHVAL PAGN R

4. Examples. OF, POOR ,QV LrM K

This section contains three numerical examples of the theory

i

presented in Section 3. The first example shows how the solution

changes for a fixed forcing frequency but increasing forcing

magnitudes. Example 2 is similar to the first example but with a

different set of parameters. Example 3 is a frequency-response

s ' illustration of variations in the forcing frequency and

corresponding responses.

i

a Example 1. In this example the system constants used are

these: u -0, m-1 lb.- s. 2

/in., C -240 lb.-s./in., R -0.,

R b -1,305,000 lb ./in., -200,000 lb.^in., 6 =.0000285 in.s and

w -500 Hertz-1000 n re ./s. Thus, 8 0

-833.33 rad./s. and

a-,000060915 in. The system is made nondimensional using a for

the g-displacement and 6• for the o-frequency. With these

choices, the constants of this equation:

W, ,+CW'+[k(1- A / I W I )-iB]W°EOZ e xp ( i mx)

have these values: C-.2886 k-1.8792. A -.467865, B-.288, and

0-6 n/ 5 .

Figures 4 and 5 show changes in the solution Y vs. Z as E assumes

the values 100n /( 10001) a for n-0,1,...,7. The graphs are

plotted for .2 < t(, .5s. The initial circle ( for E-0) opens into an

annular region, vhi.ch becomes larger and thicker as E increases

until a ( transition) value of E occurs and the coefficient of

exp(iBT) becomes zero. Thus, W-N exp ( iOT), a circle of radius

I N I .

As E increased beyond this transition value, the solution

remains a circle ( figure 2.d) with radius I N I Em2/(- @Z+iC

+k(1- A/I N I)- 1B ) I.

The nonlinear natural frequency 8 0 is the angular

frequency of the circle when E-0, but this frequency increases as

E increases. This tracking phenomenon is displayed in the PSD

plots of figures 6 and 7 using the dimensional frequency s.. In

- these figures only the 120 - 180 Hertz range is shown. At

E-7/10 , 000tr z a, there is no frequency in this range; instead, the

circle is tranversed at the forcing frequency, w.

Figure 8, a typical full PSD plot, is the case

1 E-4/10 , 000?T a. As shown earlier, one expects f e uencies of

t` wand g to appear, as well as harmonica of nytB w^ia a y-w -B

and n-1, 2,... Thus, with 8 -150 Hertz, and w-500 Hertz, one

predicts that the PSD plot will exhibit peaks at 150, 200, 500,

F

550, 850, 900 .... Hertz. Figure 8 confirms these predictions.

rr

Example 2. In this example, the system constants from

CDC[ 31 are these: v -0, m-.20422 lb.-s? /in., C -200 lb.-s./in.,

Ks -200,000 lb./in., Rb -1,000,000 lb./in., w -500 Hertz,

Q -C w/2 lb./in., and 6 -.0005 in. Thus, B -250 Hertz =

5s00ns rad./s. and a-.0007183 in. Figure 9 summarizes changes in

Y vs. Z as E varies by 250n/(1000n) 2 a, n-0, 1,...,4. These

12
j

graphs are plotted for .45<t<_.5s. URIGINAL PAGE m

POOR QURLM.Yj

aj In general, if the time interval is sufficiently large, the

plots of annular regions will be completely filled (visually if

^ not mathematically). However, when the ratio of 8 toy is p/q

dt for small, positive integers p and q, then the curve actually

falls on itself as time evolves and some attractive patterns

appear. Figure 9.d is a case where 6/y°5/4, and the picture

tr would look essentially the same whether shown for .45<t<.5s.

(actual) or for .45<t<50s. On the other hand, figure 6.c is more

typical and would show a black annulus for .45<,t<50s.

Example 3.

As described above, one expects either a circle or an

annulus at each fixed forcing frequency m. Furthermore, near

resonance one always finds a circle since the characteristic

frequency is entrained by the forcing frequency. Considering;

that the forcing function is a continuous function of this

frequency 0 , it follows that the solution is a circle or an

annulus over an interval of the ^ -axis and that the number of

i

transitions between the two shapes is finite over a bounded

portion of the positive 0 -axis.

TABLES I, II, and III illustrate these results for the

a problem with these parameter values:

I

C °240

s

./ wo 6 °.0000285

a

Q 

s

°200,000 / wa F =0.5, 1.0, or 1.5

K °0. K =1,305,000. = wz

n

s s o

Tables I, II, and III are for the three cases E=0.5, 1.0, 1.5,

respectively. In all three cases, wois used as the

nondimensionalizing frequency. Each of the three tables

describes the response curve by listing the radius of circles or

the inner and outer radii for annuli. Furthermore, the frequency

Y JS-wl is given for annuli. The values listed for 0 are

every 0.1 except when

1. more refinement is required to bound better the

transition point

2. less refinement is needed because of small variation in

the response at different frequencies.

Particular attention should be paid to Table III since this table

i shows three different regions in which the circle appears as the

solution. The first two.tables contain only one such region.

1 ^` 13

tl
ORItifNAL' PAGE 15

5. Stability. OF POOR QUAUM

To examine questions of stability for the nonlinear Jeffcott

equations, it is helpful to use the ;:ollowing complex form:

w+81 +[KB +K a (1- 6/r)-i% ]w-F( u,w)

where w-y+iz. Since the solution's stability is determined by

the corresponding homogeneous equation, one has

id+Ca w+[%+ Ka (1- 6/r)-iQa]w -0.

Furthermore, if a solution becomes unbounded then r > r o for all

constants r o . Thus, as a comparison one may consider

w+8w+[KB +K s (1— 6/ro ) — iQs]w-0.

;i Solutions of this equation are of the form w-exp(Mt). Hence:

m 2 +C am+[% +Ks (1— 6/ro)— iQa] -0.

The stability question is reduced to determining whether Re(m)<,0.

But

m s /2 ± [(Cs /2) z — KB - Ka (1- 6/ro)+IQa]1/2

and

Re(m)--Ca /2 ±Re[C a I /4-KB -K a (1-6 /r )+I%]1/2 0

--Ca

z

/2 tRe[C +iC ]1/2

i 

where

3 C1- a z/4-K B Ks (1- 6 /r a )

C =Q_ .

z

In this modified form, the stability question becomes this:

Solutions are stable if and only if Re(C 1 +iq )1/2<Ca/2.

The following lemmas are useful for bounding the value of Qs/C s

in order to find stable solutions.

(. Lemma 1 . Re(C 1 + iC , ) 1/2 e(^C1 /2)1/2 [

(1+(C z

/C1 )

; ) 1/2 y

1 1 /

2 where the

minus sign is used for C 1 <

D Proof: Case C < 0. C 

1 

< 0 implies IC

1 ' 

=-C Then:

7

' 

1

(] Re(C +i.0 

z ) 

1/2

^; 1 

H

=(CL +C2 ) 1 /4 cos[1/2 Atan(Cz/- IC1^)]

LJ

l14
-(C 2 +C 2

)1/4 cos[ ( n -e) /2] 2 ORIGINAL PAGE 15

where e - Atan(C 2 /I C 1 I) OF POOR QUALITY

-( C 2 +C2 )1/4 sin e/2)

" -(C2 +C2 )1/4 [(1-cose)/21, 1/2

vi

-(C2

1

+C2 )1/4 (1/2)1/2[1_10 I/(C 2

+C 2 )1/2]1/2

-1/21/2

2 1 1 2

( 

(C 2 +C 2 ) 1/2 _ 

IC Z I ] 1/2

-( 1011 /2)1/2 ((1+(C2/CZ)2)1/2 _ 1] 1/2

Case c 1 >0. C >0 implies C -1 C I Then

1 1 1

Re(C +iC )/2

1

-(C 1

2

+Cz 1/4cos(1/2 Atan(C2 /IC1I)]

)

_ =(C2 +C 2

) 1/4 [(1+cos Atan C /IC 1)/2]1/2

-(C1

1/4[(1

+1C11 +CZ /(C 2 +C 2 ) 1/2)/2)1/2

-1/2

)

1/2[ (C 2 +C 2 ) 1/2 + C 1] 1/2

t 1 2 1 1I

-( IC11

/2) 1/2 [

(1+(C 2 /C 1 ) 2 )+l ]1/2

gI Lemma 2. If ( Q s / Cs ) 1 0K , then for all 6 <r o<

+^+ Re(C1+iC2)1 /2<C /2.

s

z Proof: Case C 1 < 0. If C Z CU, then ICZ 1 = —C1'

G

(Qs/ s ) 2 <KB

-KB<-(Qs/C s)2

-(Ks +K B0- 6 /r0 )) < -K B < -(Qs /Cs)2

"

C1=(Cs /2)2-(Ks+KB(1- 6/r o )) < (Cs/2)2-(Qs/5)2

3. - 1011 < (C g -4Q s 2 ) /4C 2

-4C 2 I <C° -4Q2 s 1 s s

.. 4Q < Cs"+4Cs2 I C

s2 1 1

j 4(Qs2 + C 1 2 ) < Cs ° +4Cs 2 I C 1 1 +4 IC112

4 (Qs 2 + C 1 2 ) < ( s1 +21C11)2

2 1 C 1 I [ 1 +(Q /C1 )2 ] 1/2 < Cs2 +21 C1I

[1+(Q /C1) 2]1/2 < C s /2101 1 +1

15
I

E

6x

I

I

I

I

I

it

.W

I

ORIGINAL PAGE 15

OF POOR QUALUX

r 1 1 /2([1 +(Qs /C 1 )x ]1/2 _1) < Cax A ( IC11 /2) 1/2 ([1+(Qa /C 1 ) x ) 1/2 -1) < Cs /2.

But by Lemma 1, the last left-hand side is Re ( C 1 +iC 2 ) 1/2 Thus,

Re(C1+iC2)1/2 < C

s

 /2.

Case C1 >0. This case is identical to the previous argument

except that one has + 1 C1j instead of -IC1 l on the fifth line of

the inequalities and thereafter the corresponding term switches

signs.

Lemma 3. If ( Q a /C s ) 2 > K B +K a , then for all 6< . r 0 < m

Re( Cl+iC2)1/2 > C /2.

Proof: After noting

(Qs /Cs ) x > KB +K s > P +K,( 1 - 6/r a)

one has

-KBs( 1- 6 /r o ) > -(Q s /C d2

and

C 1 --K$ Ks (1- 6/r0) + (C2/2) 2 > ( Cs /2) 2 -(Q s/C s)2

The remaining step-ps in the derivation are identical to Lemma 2

for both cases 

f12 

C 1 except that the inequalities are reversed.

Thus, Re(C 1 +iC 2 ) > Cs/2.

Figure 10 summarizes the stable and unstable regions as

functions of (Q /C ) 2 for the linear and nonlinear homogeneous

problems. In %hi g figure, there is only one point, on the

(Q /Cs ) 2 - axis at which oscillations may occur for the linear

prosblem. For the nonlinear case, oscillation, and hence

transition between stable and unstable still occurs at only one

point on the ( %/C,)'- axis, but that point may be anywhere

between K B and KB+

s

[.

l^

F,

16

tJ
El`

6. Conclusions

This report has shown how vibrations at frequencies which

are unexplainable

Jeffcott models which

by linear theory can be expected in nonlinear

consider deadband, side forces or rubbing.

These frequencies and their regions of stability are bounded by

parameters of the differential equations. Although the

asymptotic analysis is weak in quantizing exactly the frequencies

and the corresponding magnitudes, there exists simple numerical

methods which may be employed for the desired precision. The

analysis, then, serves as a guide in locating nonlinear

vibrations, which the numerical techniques then find accurately.

This will apply to systems of higher degrees of freedom as well.

In studying the Jeffcott rotor with deadband or rubbing and

sinusoidal forcing (including constant side force), one must

consider these three frequencies: ( a) the forcing frequency w

(b) the natural frequency, wo , of the associated linear problem

m (deadband- 6 -0); and (c) the nonlinear natural frequency Bo

The frequency w depends only on the forcing function; wo depends

only on the system parameters; B , with its base value Bo

depends on both the forcing function and the system parameters.

For a specific set of equation parameters, one can find this

°

nonlinear

damping

natural frequency Bo as the ratio

and then use it with the (linear)

of cross - stiffness to

natural frequency to

bound the frequencies B

FGr a given system and a nonzero, external, sinusoidal

force, the y-z response is either a circle at the forcing

frequency or an annulus composed of the (major) frequencies w and

B as well as the (minor) harmonic f.requencies n(w - 9) ± B, for

positive integers n.

There are many unanswered questions that remain. First,

this report has failed to cast this problem as one in bifurcation

theory. If such a characterization is achieved in the future,

one may expect a more fundamental understanding of the mechanics

of the nonlinear solution as well as the fringe benefits that

accompany all advanced, well-developed theories.

It would also be nice to know, for a given set of

parameters, the exact frequency values at which the response

switches between circles and annuli. Rased on similar results

for the van der Pol oscillator, these transition points should

exist as analytic expressions thereby avoiding numerical

iterations.

The asymmetric stiffness problem, which many reports have

hypothesized as being the culprit of instability, is vastly more

complex than the symmetric case. Preliminary Runge-Kutta

solutions show not only that the circle/annulus plots become

elliptic and occur with their axes rotated with respect to the y,

axes, but that there may be other shapes and more than one

ii

z

i.^

17

^1
transition point to consider. These problems, however, greatly

extend the model's mimcry of an observed rotor's behavior.

This report is also limited to single forcing functions.

Realistically, one must consider multiple forcing functions.

-^ Here again, superposition will fail for the nonlinear problem

although it may be a first approximation. Certainly, harmonics

L1' (sums and differences of driving frequencies) may appear and

could possible dominate. These problems appear, to introduce no

new theory, but do increase the computational complications.

Stability for all these problems remains the central focus.

Even in the symmetric nonlinear problem with a single driver, it

is still an open question of whether the response may move from

an annulus to a circle ( or vice versa) when it is perturbed.

11,

I

I

I

I

T

U

18
I

7. References:

1. Childs, D. W., "The Space Shuttle Main Engine HighPressure Fuel Turbopump Rotordynamics Instability

Problem", Trans. ASME, Journal of Engineerinm for

Power, Jan. 1978, pp. 48-57:--^'

2. Childs, D. W., "Rotord ,ynamic Characteristics of the

HPOTP (High Pressure Oxygen Turbopump) of the SSME

(Space Shuttle Main Engine)", NASA MSFC Contract

NAS8-345U5, Turbomachinery Laboratories Report

RD-1-84, 30 January 1984.

3. Control Dynamics Company, "Effects of Bearing Deadbands

i

on Bearing Loads and Rotor Instability", NASA MSFC

Contract NAS8-35050, 20 January 1984.

4. Gupta, P. K., Winn, L. W., and Wilcock, D. F.,

"Vibrational Characteristics of Ball Bearings",

z+ Journal of LubricationTechnolo ASME Trans.,

pp. 2 -2

5. Jeffcott, H. H., "The Lateral Vibration of Loaded Shafts

in the Neighborhood of a Whirling Speed - The Effect

= of Want of Balance", Philoso hical, Magazine, Series

6, Vol. 37, 1919, p. 3

6. Nayfeh, A. H., Perturbation Methods, J. Wiley & Sons,

1973.

7. Yamamoto, T. T., "On Critical Speeds of a Shaft",

Memories of the Facultyof Engineerin , Nagoya

Japan University, Vol., No. 2, 1 54.

i

f

1

10
1x

^i

ORIGINAL PAGO M

OF POOR QUALM,

U)

m

m

m

Ln

m

m

(n M 

m I

I m

w -+

m x }

Ln

m

m

m

m

U)

m

.1

c*r In m In w In m LD

I .1 m m m

m

m m w Is; m m

X M 1 I I

C

N

20
OMQ WAC PAGE 15

OF, POOR QUALITY.

P

m

m

(D

m

m

m W

r+ 6

m m

f*7 N

m ^

X

i

5

Y

In+

^ Q1

7

m

^w

^T

N

m

m

00

Of

m

w

rn

C;

V,

C)

m

N

0I

m

CD

C)

C; N

m

m

c

21
m

m

C

CDLA

m

m

v

N

^ vS

m ^ M UZW

O

W

um

N

m

9

.r

9

ORIGINAL PAGE IS

OF POOR QUALITY

'Ia

s

f

F

@l

^tx

^1n

I

m m

r•i x

L^

v N m

m L m ^

N

w. ^ o 0

LL Es4-- O •^

O V--

N C

C -'

22
E=2/(100Cn)Za

I .'P ..v Pte`

•%^'^Y'o -fie/I

•sus fi.i^/ •K'.:Y

a

z

•a

d

ORIGINAL PAGE IS

OF POOR QUALITY

far

, •

E=0.

^A d

I

1

r^a

1Z Z

^a a I a

r -•

-t a e , -• -a •

y y

(c.) (d.)

T

J..

t

^^^ YY

MMM

M' 9 '

,

Y

t

y y

Figure 4

1 ^=

1 1 _!

I

E=1/(100 )ir)Za

I

I

-

I

=3/('000 )'a
(a.)

1.

°z .

!w

^M

E

i'z .

p

s

II

Y

a

Z

ORIGINAL PAGE 15

OF POOR QUALITY.

(b.)

E=5/000010 a

%' r

r r ^^

,Y

(d.)

.

z'

_.

Y

Y

Figure 5

24
144 166

ORIGINAL PAGE 11a

OF POOR QUALM

1.6

ly F I

too Itt 1-0 166

1.0 Ice

it

bt

ll

I

1.0 166 

(d.)

I.2{

ly 

F

1.00

0.76

0.64

4.9s

•.69

IF4` it@

Figure 6

25
6.6

In ISO

OA

1. 6

ORIGINAL PAGE IS

OF POOR QUALITY

8.0

ly

I.S

(d.)

1.0

Y 

F

0.1

0.6

0.2

8.4

Ito ISO

Figure 7

t.5

IF

I

ISO 164

F

1.

1-4 ISO 

"a 140 

26

I 
I

N

H

m .gym.

10

U

d

c

r°. m W m

G

N

+i

W

r

^r

1

i

^rf

^I

i

I

t

i

i

i

I

ORIGINAL PAGE 15

OF POOR QUALITY

1

i

i

I

w r p y

mmm

mmm

m

mN

m

27
j 1Z.

I

g

Z,

°i

1.

i

Z^

(d.)

,

Z,

Y

y (e.)

Z.

,

y

ORIQNVAL PAGE M

OF POOR QUALIT

,

Y

y Figure 9

28
IwIhomog

a

++x

W^

^K

^w

is

Ll

z

w0 `a• -a)

=KB+Ka

'W lha 

..B o a /Cg) 2 __a

ORIGINAL PAGE IS

OF POOR QUALITY

Linear

V

Figure 10

29

I
a ORIGINAL PAGE tS

TABLE I OE POOR QUALITY,

E - 0.5

Shape Radiu s (radii) y(Hz.) E

(Circle / Annulus)

0,0 C 2.137 -

0.1 A 2.130 - 2.145 115

! 0.2 A 2.105 - 2.169 97

0.4 A 1. 959 - 2.305 61

0.5 A 1.747 - 2.470 41

E 0,53 A 1.618 - 2.549 34

0.54

0.6 C 1.833

0,7 C 2.441

0.8 C 3.656 -

0.9 C 6.954

0.92 C 8.249 -

0,93 A 1.212 - 4.018 13

1,0 A 1.369 - 3.230 40

1.2 A 1.551 - 2.858 81 _- 

" 1.4 A 1.643 - 2.822 1T8

t 1.6 A 1.648 - 2.748 154 -

1.8 A 1.694 - 2.633 190

2.0 A 1.804 - 2.679 222

3 2,2 A 1.939 - 2.830 263 

2.4 A 1.993 - 3.162 294

2.6 A 1.632 - 3.125 333

?

a

2.8 A 1.533 - 2.864 370

3.0 A 1.528 - 2.769 410

M 

9da

4w

ii

^Yp 

1

I

r3

t

30

L^
ORIGINAL PAGE IS

TABLE II OF POOR QUALITY,

E - 1,0

Shape Radius (radii) y(Hz.)

(Circle/Annulus)

0 .0 C 2.137 -

0.2 A 2.073 - 2.201 98

0.4 A 1.767 - 2.463 60

0.41 A 1.737 - 2.486 58

0.42 C 1.398 -

0.5 C 1.650 -

0.6 C 2.118 -

0.8 C 4.548 -

0.9 C 9.174 -

0.97 C 18.524 -

0.99 C 10.436 -

1.0 A .872 - 2.033 12

1.2 A 1,003 - 4.194 68

1.4 A 1.145 - 3.972 96

1.6 A 1.431 - 3.839 145

1.8 A 1.151 - 3.283 185

2.0 A 1.341 - 3.296 222

2.2 A 1.598 - 3.499 256

2.4 A 1.671 - 3.932 286

2.6 A 1.732 - 4,456 322

2.8 A 1.062 - 3.745 357

3.0 A .981 - 3.390 400

31
'

ORIGINAL PAGE fb

TABLE6 III

L

OF. POOR QUALITY.

R 1, 5

ii 0 Shape Radius (radii) y (Hz .)

(Circle/Annulus)

0.0 C 2.137 -

4, 0.2 A 2.040 - 2.230 96

0.3 A 1.885 - 2.371 78

0.35 A 1.755 - 2.476 68

0.36 C 1.364 -

' 0.4 C 1.449

0.6 C 2.398 -

0.8 C 5.427 -

i

t^ 0.9 C 11.269 -

0.98 C 27.247 -

0.99 C 25.102 -

r w 1.0 C 19.448 -

'_ 1.2 C 2.585 -

1.38 C 2.028

1.39 A .357 - 4.628 99

1.4 A .529 - 4.832 100

i 1.6 A .837 - 4.660 139

€j 1.77 A .359 - 3.832 172

{ p 1.78 C 1.717 -

1.8 C 1.709 -

1.91 C 1.676 -

1.92 A .100 - 3.121 195

2.0 A .496 - 3.282 217

Gf 2.2 A 1.062 - 4.076 250

2.4 A 1.496 - 4.564 286

2.6 A 1.692 - 5.341 312

2.8 A .222 - 3.987 359

2.81 C 1.565 -

2.9 C 1.560 -

k, 3.0 C 1.555 -

1]

11

F

U

32

PTO ,