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Full text of "Rotor internal friction instability"

N86-30187 



ROTOR INTERNAL FRICTION INSTABILITY 

Donald E. Bently and Agnes Huszynska 

Bently Rotor Dynamics Research Corporation 

Minden, Nevada 89423 



Two aspects of Internal friction affecting stability of rotating machines are dis- 
cussed in this paper. The first role of internal friction consists of decreasing 
the level of effective damping during rotor subsynchronous and backward precessional 
vibrations caused by some other instability mechanisms. The second role of internal 
friction consists of creating rotor instability, i.e., causing self-excited subsyn- 
chronous vibrations. Experimental test results document both of these aspects. 

1. INTRODUCTION 

In rotating machines, damping effects are conventionally split into two categories: 
external and internal damping. The term "external" refers to the stationary ele- 
ments and rotor environment, as they are "external" to the rotor. External damping 
is related to energy dissipation due to friction occurring between stationary and 
rotating elements, and/or fluid dynamic resistance in the rotor environment (mostly 
in fluid- lubricated bearings). External damping is also supplied by material damp- 
ing of supports. External damping forces depend on the rotor absolute velocity of 
vibration, and their effect on the rotor vibration is usually welcome — they pro- 
vide stabilizing factors. 

The term "internal" refers to the rotating elements, including the rotor itself. 
The same physical phenomena characterize internal and external damping. Internal 
damping forces are due to material damping in rotating and vibrating elements (most- 
ly shafts) and friction between rotating parts (such as joint couplings and shrink- 
fitted disks on shafts). As internal damping occurs in the elements involved in ro- 
tating motion, the internal damping forces depend on relative velocity, i.e., on the 
difference between the absolute vibration velocity and the rotative speed. Thus the 
relative velocity may be "positive" (following absolute velocity when rotative speed 
is low) or "negative" (opposing absolute velocity when rotative speed is high). The 
corresponding internal damping forces can, therefore, act as a stabilizing (adding 
to the external damping to increase the total effective damping in the system) or a 
destabilizing factor (subtracting from the external damping to decrease the total 
effective damping of the system). 

The terra "damping" is traditionally related to the stabilizing effects created by 
the irreversible conversion of kinetic energy to heat. Being physically related to 
the same mechanism, rotor internal damping plays an additional role in rotating ma- 
chines ~ it opposes the external damping and actually transfers the rotational 
energy into vibrations. That is why the terra "damping" does not fit well in this 
situation; by contrast, the term "internal friction" does not introduce ambiguity 
concerning stabilization. 

Internal friction has been recognized as a cause of unstable rotor motion for more 
than 50 years [1-7]. Since the first description of internal friction-related in- 

337 



stability, many other rotor destabilizing factors have been identified, such as rub 
or fluid dynamic effects in fluid- flow machines and/or bearings and seals. The lat- 
ter effects are much stronger than the internal friction effects and very often ob- 
served in the performance of rotating machines. "They result in subsynchronous vi- 
brations (rotative speed higher than vibrational frequency). Internal friction is 
now very seldom identified as a main cause of rotor unstable motion. However, in- 
ternal friction plays a negative role by reducing the system-effective damping for 
forward subsynchronous and backward vibrations caused by other destabilizing 
factors . 

In this paper two aspects of rotor internal friction are discussed: the first, a 
damping-reducing effect and the second, a cause of instability and self-excited vi- 
brations. The experimental test results document both aspects. 

2. ROTOR MODEL WITH INTERNAL FRICTION 

To simplify the considerations, a symmetric rotor will be discussed. Dynamic behav- 
ior of a rotor in its lateral mode of vibrations is usually represented by a set of 
linear ordinary differential equations. For each mode, the set of equations reduces 
to two, which for a symmetric rotor can be presented in the form of one complex 
variable equation [8]: 

Mz + Di + Kz + ^%^]^ = 0, z(t) = x(t) + jy(t), OAa, j = -fl (1) 

where x, y are rotor horizontal and vertical deflections correspondingly, they de- 
scribe the rotor precessional motion. M is the rotor mass; D is the external vis- 
cous damping coefficient; K is the rotor stiffness, including the shaft and pedestal 
stiffness. The rotor parameters M, D, K are generalized parameters corresponding to 
each separate mode; u) is rotative speed. The equation (1) may, in particular, de- 
scribe rotor at its first mode. Eq.(l) contains frequency Q of the resulting pre- 
cessional motion, unknown a priori; |Q-u>| is the value of the shaft actual bending 
frequency. It has been introduced to the rotor model (1) following the way by which 
hysteretic damping is usually included in models of mechanical systems: a viscous 
damping coefficient is replaced by a product Kri/Q*, where K is stiffness coeffi- 
cient, n is loss factor, and Q* represents the frequency of elastic element deforma- 
tion. In the case of rotating shaft, the frequency of deformation is equal to a 
difference between rotative speed and frequency of precession. Note that for for- 
ward low frequency precessions, the frequency of shaft deformation is lower than 
rotative frequency. For backward precessions, the frequency of shaft deformation is 
a sum of rotative speed and frequency of precession. For circular synchronous pre- 
cession (Q=u)), the shaft is "frozen" into a fixed bow shape, so that internal fric- 
tion does not act. 

In Eq.(l) F (assumed positive) is internal friction function. For shaft material 
hysteretic damping F is constant and equal to Kr|. For the synchronous precession F=0. 
Generally, however, F can be a nonlinear function of z and z [4, 5, 9]. 



338 



For F = const and Q supposed constant, substituting z = Zf-e^*, the 
eigenvalue problem of (1) yields four eigenvalues (satisfying the degrees of 
freedom of eq. (11 ))*: 



s = -6 ± (1/V2)[V62-K/M + VA ± jJk/M-62+Va] (2) 

where 

6 = (D + F/|fi-u)|)/(2M), A = (62 - K/M)2 + F2u,2/[M2(fi-u,)2] 

The real parts of (2) are non-positive. I.e., the system (1) 1s stable when 

F2m2M/K < (D|Q-u)| + F)2 (3) 

which for u» > yields the following conditions: 

For u)2 < K/M the rotor pure rotational motion is s tabl e. 

For m2 > K/M it is stable only if |F| < D|Q-u)|/Cuj/Vt7M-l] (4) 

At a threshold of stability, i.e., when |F| = D|Q-u)|/[m/i/K/M - 1] the eigenvalues 

reduce to 

s = ± jVkTM (5) 

The rotor motion is purely periodic with the natural frequency determined by stiff- 
ness and mass (for the stab le motion below the threshold of stability, the frequency 
is slightly lower than VkTM, due to damping). 

If the stability condition Is not satisfied and F exceeds the limits (4), then rotor 
pure rotational motion is unstable. The linear model (1) is not adequate anymore, as 
for high lateral deflections nonlinear factors become significant. Nonlinear factors 
eventually lead to a limit cycle of self-excited vibrations. The latter usually oc- 
cur with the lowest natural frequency determined by the linear model, as the non- 
linearities have very minor influence on frequency. With high amount of probability, 
the frequency Q can be, therefore, equal to the rotor first natural frequency: 

Q = ± VTCITMI (6) 



*Solving the quadratic in s gives 



M 



Expanding •ya+jb = c + jd and solving for (c,d) gives 
+ ya ± jb = +^ fa +Va^ + b^ +jy-a+Va^ + b 



Substituting 
,2 



* - V 2M / M *"° ° - M 
gives four roots: 



339 



where the index "1" refers to the first bending mode. If the model (1) describes the 
first mode, the stability condition (4) reduces to 

|F| < DViqTFfr = 2KiU ^^^ 

where Ci t^s the damping factor of the first mode. 

For the hysteretic damping, F = KiHi and the inequality (7) yields 

Hi < Ki (8) 

i.e., for stability the shaft loss factor has to be lower than the double of external 
damping factor. 

The modal approach to the rotor modellzation permits evaluation of the stability con- 
ditions for several modes. For example, the inequality (4) for the 1-th mode, (index 

"1") Is: 

|F| < D.|VkI7M7 - u)|/Cu)/Vjq7MT - 1] (9) 

Figure 1 Illustrates the condition in which the same amount of Internal friction may 
cause the first mode to be stable and the third mode unstable. T his co n dition takes 
place when the modal damping ratio is sufficiently high, Di/Ds > VKs/Ms/^Ki/Mi 
and when the rotative speed exceeds a specific value, i.e.: 



U) > 



Di/D-^-l 



DiVmI7k;/D3 - vmTTkT (10) 

3. rotor effective damping reduction due to internal friction 

Assume that the rotor performs steady nonsynchronous precessional, self-excited vi- 
bration with frequency Q. This vibration occurs due to any Instability mechanism 
(for Instance, it may be oil whip). It means that the rotor motion can be presented 
In the form 

z = Ae^*"^ (11) 

where A is an amplitude of the self-excited vibrations. Introducing (11) Into (1) 
gives 

-Mn2 + DjQ + K + F (fi-uj)/|Q-u)| = 

The real part of this expression yields the frequency. The imaginary part relates 
to the system damping. The external damping term, 1X2, is now completed by the term 
expressing Internal friction: 

DQ -> DQ + F (n-uj)/lQ-uj| 
or 

J, ^ f D + F/Q for ukQ (supersynchronous precession) 

^ D - F/Q for ui>Q (subsynchronous and backward precession) (12) 

For supersynchronous precession Internal friction adds to external damping and in- 
creases Its level. For subsynchronous and backward precession, the Internal fric- 
tion reduces the level of "positive" stabil izi ng damping In the rotor system by the 
amount F/Q. Taking into account that Q = V^(7R, for subsynchronous precession the 
rotor effective damping factor decreases by the following amount: 

340 



C -> C - F/(2K) (13) 

It also means that the Amplification Factor Q increases: 

Q -» Q/(1-FQ/K) (14) 

If, for instance, the Synchronous Amplification Factor is 5 and internal friction is 
due to shaft material hysteretic damping with loss factor n = 0.06 (F=Kri), then the 
Subsynchronous Amplification Factor increases to 7.14 (the Supersynchronous Ampli- 
fication Factor decreases to 3.85). 

Note that the decrease of the "positive" external damping for rotor subsynchronous 
vibrations does not depend on the form of the function F (constant or displacement 
dependent) . 

In practical observations of rotating machine dynamic behavior, it has very often 
been noticed that subsynchronous vibrations are characterized by much higher ampli- 
tudes than any super-synchronous vibrations. There are many different causes of 
subharmonic vibrations in rotating machines. In each case, however, the role of 
internal friction opposing and decreasing the level of external, stabilizing damping 
is very important. Although not a primary cause of instability, internal friction 
often promotes subsynchronous vibrations and causes an increase of amplitudes. 
Figures 2, 3, and 4 illustrate dynamic responses of some unstable rotating machines. 
The self-excited, subsynchronous vibration amplitudes are much higher than ampli- 
tudes of synchronous and supersynchronous vibrations. More examples are given in 
[83. 

The rotor model considered in this paper is symmetric; therefore, the synchronous 
precession is expected to be circular. In the case of circular synchronous preces- 
sion at constant rotative speed, the bent shaft precesses "frozen" and is not a sub- 
ject of periodic deformation. The internal friction does not act. The regular cir- 
cular synchronous precession of real rotors very seldom occurs, however; usually 
nonsymmetry in the rotor and/or supporting system results in the elliptical syn- 
chronous precession. In this case, the bent shaft is no t "frozen," but deforms 
with the frequency two times higher than the rotative speed. The internal friction 
then brings a "positive" effect: it adds to the external damping. 

4. SELF-EXCITED VIBRATIONS DUE TO INTERNAL FRICTION 

If in the equation (1), F is given in the form of a nonlinear function of the rotor 
radial displacement |z|, velocity of the radial displacement d|z|/dt and relative 
angular velocity u)-8, [4,6] where 

e(t) = arctan [y(t)/x(t)], |z| = V>?+y^ (15) 

i.e., F = F(|z|, d|z|/dt, lu-e) then the rotor model (1) allows for the following 
particular solution 

z(t) = Be^'^^ (15) 

where B and Q are constant amplitude and frequency of the circular precessional 
self-excited vibrations correspondingly. They can be found from the following 
algebraic relation yielded by (1) and (16): 

-MQ2 + OjQ + K + jF(B,0,w-n)(n-uj)/ifi-u)| = (17) 

The nonlinear differential function F becomes nonlinear algebraic function. 

341 



Bolotin [9] quotes several forms of internal friction function F; for instance, 
for a shrink-fitted disk on the shaft, the internal friction nonlinear function has 
the following form: 

F (|z|. d|2l/dt. uj-e) = ^^'^1 - (18) 

where Ci>0, C2>0, n, m are specific constant numbers. In case of the function (18) 
equation (17) for the first mode yields 

Q = VKITMI . B = {[Ca + (ui-WRI)'"] 0#I7fir / Ci)^/" (19) 

Si nce Ci and Ca are positive, the solution (16) with amplitude (19) exists for 
u)>VKi/Mi only. This means that the self-excited vibrations (16) exist for suffi- 
ciently high rotative speed. 

5. INTERNAL FRICTION EXPERIMENT 

During balancing of the three-disk rotor rig (Fig. 5), an appearance of self-excited 
vibrations at the rotative speed above third balance resonance have been noticed 
(Fig. 6). The frequency of these self-excited vibrations exactly equals first natu- 
ral frequency. The self-excited vibrations disappear for higher rotative speed. 
It was noted that when balancing weights, which affect the balance state for the 
third mode, were removed, causing a significant increase of the amplitude of the 
synchronous vibration at the third mode, the self-excited vibrations nearly disap- 
peared (the amplitude decreased from 1.8 to 0.4 m'lls p/p, compare Figs. 6 and 8). 
It appeared that the energy from self-excited vibrations was transferred to the syn- 
chronous vibrations. Higher rotor deflection due to unbalance evidently caused some 
substantial modifications In the self-excitation mechanism. 

Since there was no other obvious reason for the self-excitation, internal friction 
(of the shaft material and disk/shaft joints) was blamed for the appearance of these 
self-excited vibrations. To prove this supposition, an Increase of the rotor inter- 
nal friction was attempted. Half of the shaft was covered with a 4-mil -thick layer 
of damping material, commonly used for vibration control (acrylic adhesive ISD-112, 
3M Company). Applied to the rotating shaft, the damping material increases the in- 
ternal friction and magnifies the self-excitation effect. The expected result was 
confirmed: the amplitude of the self-excited vibrations Increased from 0.4 to 0.7 
mils p/p (compare Figs. 8 and 9). 

The self-excited vibrations disappeared completely when the disks were eventually 
welded to the shaft, and the damping tape was removed. 

The question of why the self-excited vibrations occur at the rotative speed ~7150 
rpm and disappear In the higher range of speeds has not been answered. Nor was the 
internal damping function identified. The analysis presented In Section 2 gives, 
however, some Indications that a nonlinear Internal damping function may cause rotor 
Instability In a limited range of rotative speeds. Figure 10 presents the stability 
chart for three modes. 



6. CONCLUDING REMARKS 
This paper discusses two important aspects of internal friction in rotating ma- 

342 



chines. Firstly, the internal friction in rotating elements causes a decrease of 
the amount of effective damping in the rotor system. This effective damping reduc- 
tion occurs during rotor subsynchronous and backward precessional motion, which may 
be caused by any instability/self-excitation mechanism (such as rub or fluid flow 
dynamic forces). While usually not a primary cause of instability, internal fric- 
tion promotes unstable motion and affects the value of self-excited vibration 
amplitudes. 

Secondly, the internal friction occasionally is a major cause of rotor self-excited 
vibrations due to incorrect shrink fits, loosening of shrink fits by differential 
thermal growth, or. by mechanical fatigue. Known for more than 50 years as a con- 
tributor to rotor instability, the internal friction analytical model has not, 
however, yet been adequately identified. 

This paper documents experimentally these two aspects of internal friction in rotat- 
ing machines and gives a qualitative description of the dynamic phenomena associated 
with rotor internal friction. 



SYMBOLS 



A,B 

D 
F 

K,M 
Q 



Amplitudes of self-excited 

vibrations 

External damping coefficient 

Internal friction function 

Rotor generalized (modal) 
stiffness and mass coefficients 
Amplification Factor 



s — Eigenvalue 

z=x+jy ~ Rotor radial deflection 

(x-horizontal , y-vertical) 

t, — External damping factor 

n ~ Loss factor 

6 — Angle of precessional motion 

uj — Rotative speed 

Q — Angular speed of precession 



1. 
2. 



4. 
5. 

6. 

7. 

8. 

9. 

10. 



11. 



REFERENCES 

Newkirk, B. L. : Shaft Whipping. Gen. Elect. Rev., 27, 1924. 
Kimball, A. L. : Internal Friction Theory of Shaft Whirling. 



Gen. Elect. Rev. 



27, 1924. 

Kimball, A. L. : Internal Friction as a Cause of Shaft Whirling. Phil., Maq. . 

49, 1925. 

Tondl , A. 

Loewy, R, 



Nonlinear Vibration Problems, 13, 



Some Problems of Rotor Dynamics. Prague, 1965. 

,, Piarulli, V. J.: Dynamics of Rotating Shafts. The Shock and Vi- 
bration Information Center, SVM-4, 1969. 
Muszynska, A.: On Rotor Dynamics (Survey). 
Warsaw 1972. 

Bently, D.E.: The Re-excitation of Balance Resonance Regions by Internal 
Friction: Kimball Revised, Bently Nevada Corp. BNC-19, 1982. 
Muszynska, A.: Rotor Instability. Senior Mechanical Engineering Seminar, 
Bently Nevada Corporation, Carson City, Nevada, June 1984. 
Bolotin, V. v.: The Dynamic Stability of Elastic Systems (translated from 
Russian). Holden-Day Inc., San Francisco, 1964. 

Wachel, J. C: Rotordynamic Instability Field Problems. Rotordynamic Insta- 
bility Problems in HIgh-Performance Turbomachinery ~ 1982, Proceedings of a 
Workshop Held at Texas A&M University, College Station, Texas, NASA CP 2250, 
May 1982. 

Bonciani, L. , Ferrara, P. L. , Timori, A.: Aero-induced Vibrations in Centrifu- 
gal Compressors. Rotordynamic Instability Problems in High-Performance Turboma- 



343 



chinery, Proceedings of a Workshop held at Texas A&M University, College Sta- 
tion, Texas, NASA CP 2133, May 1980. 
12. Muszynska, A.: Instability of the Electric Machine Rotors Caused by Irregularity 
of Electromagnetic Field. Bently Nevada Corporation, 1983. 



FIRST MODE STABLE 



FIRST MODE UNSTABLE 



THIRD MODE STABLE 



THIRD MODE UNSTABLE 



"V 



X 






D ^TTW INTERNAL FRICTION FUNCTION F 
Figure 1. - Regions of stability for rotor first and third modes. 



ORIGWAL PAGE IS 
OF POOR QUALITY 




Figure 2. - Cascade spectrum of steam turbine vibrational response, indicating high subsynchronous 
vibrations. Data courtesy of 3.C. Wachel [10]. 



344 



ORIGINAL PAee n 
OF POOR q5al,7? 




Figure 3. - Time histories of six-stage compressor at 9220 rpm. Subsynchronous vibrations due to 
destabilizing dynanic forces generated on last stage. Data courtesy of P.L. Ferrara [11]. 








FREOLEHCY (EVENTS^MIN »: lOa0> 



g^' ■ ■ ' ' ■ ■ . • < .... I ■ . 



Z 



•3. 



Figure 4. - Cascade plot of electric motor response during shut(kMn. At running speed of 510 rpm 
(below first balance resonance), high half-speed vibrations present due to electromagnetic 
field unbalance [12]. 



345 



1. 


BASE PLATE 


2. 


MOTOR 


3. 


FLEXIBLE COUPLING 


4. 


RIGID BEARING 


5. 


KEYPHASOR DISK 


5. 


X-Y NONCONTACTING 




PROBES 


7. 


STEEL DISKS 


8. 


STEEL SHAFT 




Figure 5. - Three-disk rotor rigidly supported. Disks attached to rotor by radial screws. 
(Dimensions are in inches.) 



1102 RPM FIL 

T 




ix FILTERED 


T 




U) 


■:■•;! i : 




x-v. ■ 


, 


■■ ■■ : '/^- . 


f= 


. ^ .,',., 1 




: ,.^ i i/- i i 


f^ 


. ! 11: 1 y. M 




. . W .i 


L 







AT 7150 RPM 






m^^m-M'^ 



t,M mUM*9W 



■Ca^lv !.•• BtLVVIV 



.^(^ 



5-W- 






»C 



■3x; 



rREOJENCY CCVDITS^IH x IBQOi: 



UNFILTERED AT 7150 RPM 




i=M 




_| ■ Vl"« SC«I« ».M aUK'St* 

£— — 

% 



RUN! 5 



Figure 6. - Startup response of three-disk rotor measured at midspan position. Cascade spectrum and 
orbits indicate existence of subsynchronc^s self-excited vibrations. 



346 



ORieiNAL PM 
OF POOR QUi 



ORIGINAL PAGE IS 
OF POOR QUAIIITY 




I 



"OUTBOftRB 2- 



niSSPRN ! 

SPEED, RPM X leea 



• INBOHRO 2 



Figure 7. - Bode plots of Ix (synchronous) filtered rotor response and rotor mode shapes. Original 
state of unbalance. 



1102 RPM FIL 
T 



ERED 



ix FILTERED AT 7150 RPM 



UNFILTERED AT 7150 RPM 

5; 




RUHt 2 



Figure 8. - Cascade spectrum and orbits of unbalanced shaft: Decrease of subsynchronous self-excited 
vibrations. Data from midspan position. 



347 



WAL PME- IS, 



1102 RPM FILTERED 



OF PC 



t 




■-;■■■ 


lA 




i !^-v ■ 


' 


• . 1 ' , :> . , 


E 


■ I ■> ■ - 


. -^^ 


CM 


:■;.-■!: 




V, 


■ ■ ■:>••• 



X FILTERED AT 7150 RPM 



t^ *' .UN FILTERED AT 7150 RPM 



o 



o 
u 



X 




RUN: 11 



Figure 9. - Cascade spectrum and orbits of unbalanced rotor measured at midspan position. Shaft 
covered with damping material. Increase of subsynchronous self-excited vibrations. 



"-t 






1 1^ 

, 1 


v/ 1 / 


W//. 




INTERNAL FRICTMH FUNCTION/^^ 

! _ . ,_ . 

' STABLE 1 

1" ~ , 1 

1 1 


unstable / / 
instability; > 

REGION 





^t 



"Hi ••m. 



ROTATIVE SPEED "> 



Figure 10. - Rotor stability chart for three modes (inequality (9)). 



348