WEAR 349

NOTES ON THE PROBLEM OF SELF-EXCITED VIBRATIONS AND NON- LINEAR RESONANCES OF ROTORS SUPPORTED IN SEVERAL JOURNAL BEARINGS

A. TONDL

National Research Institute of Heat Engineering, Prague (Czechoslovakia)

(Received November zz, 1964)

SUMMARY

Some of the results obtained in an experimental investigation of self-excited vibrations and non-linear vibrations of rotors supported in a statically indeterminate manner in journal bearings are described. The investigation concerned the effect of bearing design, rotor unbalance and distribution of critical speeds on the onset and intensity of self-excited vibrations and non-linear resonances. Sub-harmonic and ultra-harmonic resonances are not the only non-linear phenomena that might arise in addition to the main resonances, i.e. the critical speeds.

NOTATION

n

121,122, . . . ns

qs = %,Z

40 = nO/nk

f

Vl, VP

rotor speed, r.p.m. critical rotor speed, r.p.m. rotor speed at which self-excited vibrations are initiated, r.p.m. rotor speed at which self-excited vibrations disappear, r.p.m. ratio of the speed at which self-excited vibrations are initiated and the critical speed corresponding to self-excited vibrations as shown by the form of the rotor vibrations ratio of the speed at which self-excited vibrations disappear and the critical speed corresponding to self-excited vibrations as shown by the form of the rotor vibrations vibration frequency, c/min twice the amplitude of the vertical vibration of the first and second discs respectively, mm

INTRODUCTION

The self-excited vibrations of the rotor (often called the “oil whip”) induced by the action of lubricating oil film in journal bearings have been studied both theoret- ically and experimentally by many authors (Newkirk, Taylor, Stodola, Hummel, Hagg, Poritsky, Stemlicht, Cameron, Pinkus, Capriz, and others). They are not the

Weav, 8 (I@4 349-357

3.50 A\. TONl~I.

only vibration phenomena (if the critical speed of the machines is not considered) that endanger the smooth running of machinery, particularly of that mounted in several bearings. The non-linear character of lubricating oil films in journal bearings together with rotor unbalance might give rise to secondary resonances peculiar to non-linear systems, i.e. to resonances occurring outside the range of critical speeds. Most notable of these are the sub-harmonic and ultra-harmonic resonances well known from the general theory of non-linear vibrations. Sub-harmonic resonances have often been confused or associated with self-excited vibrations. But these two resonances are not the only possible non-linear phenomena.

The paper presents some of the results of an experimental investigation conducted with models of rotors supported in several bearings, to study the self- excited vibrations and non-linear resonances.

Omitting special cases of tuning, the results of the model tests of self-excited vibrations of rotors supported in several bearings are, in principle, identical with those obtained for rotors mounted in two bearings and are in agreement with experience gained with actual turbine rotors’. For this reason these results are merely reviewed while greater attention is accorded to cases when non-linear resonances were also observed in addition to the self-excited vibrations.

Fig. ~(a).

1100 260 so0 mm

43.4 11.25 36.6 in.

Fig. 1(b).

Wear. 8 (1965) 349-357

SELF-EXCITED VIBRATIONS AND NON-LINEAR RESONANCES OF ROTORS 3.51

EXPERIMENTAL EQUIPMENT AND MEASURING TECHNIQUE

Figure r(a) is a photograph of the experimental equipment used in the tests. The equipment consists of a rotor having two shafts connected by a fixed coupling, four bearing pedestals and a frame. Each shaft carries one disc. Figure r(b) gives a schematic layout of the rotor and bearing arrangement. Details of the apparatus are summarized in Table I.

TABLE I

DETAILS OF TEST APPARATUS

kg lb. mm in.

Weight of heavy disc 40 88.3 Diameter of all journals 40 I.55 Weight of light disc 10.6 23.4 Length of all bearings 40 I.55 Weight of long shaft 16.9 37.2 Long shaft, diameter between bearings 35 1.38 Weight of short shaft II.3 24.9 Short shaft, diameter between bearings 30 1.18

In the tests the rotor was connected through a rubber coupling to a Ward-Leo- nard set operating at speeds from o-6,000 r.p.m. A TESLA capacitive indicator was used as the measuring instrument, particularly for the vertical components of the disc motion (or of the shaft motion when the disc was removed).

TABLE II

TYPES AND PRINCIPAL DIMENSIONS OF BEARINGS TESTED

he No. of bearinq

I

- Cylindrical

2

3

- E///ptico/

4

5

With flexrble

elements

6

7 Loose bushing

_ with flexible

8 elements

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352 A. ‘roNI)I.

The doubled amplitudes of the vertical components of the disc motion art denoted by 1;~ (short rotor) and Vz (long rotor, close to the motor) in the diagrams. The records of the vibrations are marked with these letters and numbered in the same way as the corresponding points in the diagrams. In addition, they are provided with speed marks so that the frequency of vibration can be followed relative to the speed. The diagrams also plot the relation between the frequency of vibrationfc/min and the rotor speed n. To facilitate rapid orientation, each diagram includes a schematic layout of the respective rotor and bearings (see sketches in Table II), discs (the light one having half the thickness of the heavy one) and an indication of the location of additional unbalancing masses. Table II gives the types and principal dimensions of bearings subjected to the tests. Turbine oil with a viscosity of 22.4 CP at 50°C was used in the investigation.

EXPERIMENTAL RESULTS

The value of the rotor speed, as, at which the self-excited vibrations are initiat- ed, the amplitude and width of the interval of self-excited vibrations, or the limiting speed no, at which the self-excited vibrations disappear as the rotor speed is decreased can be used as criteria for assessing the resistance of a given type of bearing to the onset of self-excited vibrations, providing the “inertia effect”* is operative.

The limit of initiation or disappearance of the self-excited vibrations can be expressed relative to the critical speed Sk (k = I, 2, . . .), i.e. relative to those speeds which correspond to the self-excited vibrations as shown by the form of rotor vibra- lions. In almost all cases it is the first critical speed ni. Thus:

The results may be summarized as follows: (I) Supporting a rotor in several journal bearings results in virtually no change

in the resistance of the bearing towards the initiation of self-excited vibrations. Bearings which gave a good performance in two bearing mountings performed equally well in tests with rotors supported in several bearings.

(2) The frequency of self-excited vibrations is almost always slightly lower than the frequency corresponding to the first critical speed of the rotor and increases slightly with increasing rotor speed. The sole exception, the frequency of self-excited vibrations corresponding to the second critial speed, was a rotor supported in three cylindrical bearings (Fig. 2). The form of the vibration is similar to that at the second critical speed, i.e. the discs vibrate in phase (compare records 2 and 5 in Fig. 2).

(3) The rotor unbalance, which has virtually no effect on the limit of initiation of self-excited vibrations in the case of a two-bearing mounting, affects the limit to some degree when rotors are supported in several bearings, particularly when multi- pressure wedge bearings are employed.

(4) Results of tests in which one of the four cylindrical bearings was replaced

* The inertia effect denotes the well-known property of self-excited vibrations persisting down to speeds lower than those at which they were self-initiated.

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SELF-EXCITED VIBRATIONS AND NON-LINEAR RESONANCES OF ROTORS 353

Fig. 2.

by a bearing highly resistive to the onset of self-excited vibrations (a loose-bushing bearing, with flexible elements) gave conclusive evidence that thereplacement of only one bearing brought virtually no improvement in performance.

(5) Far greater effect and more significant results were obtained when two cylindrical bearings were replaced by loose-bushing bearings. The region in which self-excited vibrations occurred continuously disappeared and the intensity of the vibrations decreased, at times being entirely suppressed, especially in the region of higher rotor speeds. Non-linear resonances appeared, however, particularly in im- properly tuned systems,

(6) Rotors supported in several loose-bushing bearings, where neither self- excited vibrations nor non-linear resonances were observed over the range of test speeds (except for several minute disturbances within a narrow speed interval), exhibited the highest resistance against the onset of self-excited vibrations.

(7) The possibility of the onset of non-linear resonances, at which the vibrations apparently possess a character similar to that of the self-excited vibrations, is far greater for rotors supported in several bearings, especially if the critical speeds are unsuitably distributed and the rotor is tuned to the so-called “internal resonance” (some of the critical speeds are close to a ratio of small integers).

Since the non-linear resonances and the self-excited vibrations are basically different phenomena, it appears expedient to compare the characteristics of the two vibrations.

In contrast to the self-excited vibrations which are initiated primarily by instability of the equilibrium state, the non-linear resonances are a phenomenon conditioned by a sufficient degree of unbalance and non-linearity of some of the ele- ments of the system, notably of the lubricating oil film in journal bearings.

Wear, 8 (1965) 349-357

354 A. TONI)I,

The self-excited vibrations usually occur over a wide range of speeds, their frequency being practically constant over the entire range and very close to thus natural frequency (especially the lower one) of the rotor. The non-linear resonances, on the other hand, are observed within a relatively narrow speed interval. The fre- quency of vibration is proportional to the speed frequency, the proportionality constant being equal to a ratio of small integers (most frequently I/Z, r/3, I/,+, z/3). The vibrations need not contain just one dominant component of this type, but might include two or more of them.

The non-linear resonances can generally occur in the speed range given by a simple linear combination of critical speeds %i, 122, . . .

n = ;- $ Nrnr (No, Nk = 0, I, 2, . . .) Ok1

Of major significance are the resonances for which NO, Nr are small integers. It follows from the above that, in addition to the known sub-harmonic

(~2 = 2 nk, n = 3 nk) and ultra-harmonic (n = 1/z a&, n = 1/3 r&k, . . .) resonances which have already been observed in rotor@, other resonances might also arise. Theoretical analyses394 indicate that the most significant of these are the combination resonances of the type

B = nr + n2 or ‘pt = 2m + 922 , $8 = r21 + 2%~

A characteristic feature of such resonances is that, as well as the speed fre- quency component, they have substantial components with frequencies which are close to the respective natural frequencies of the rotor (nr and na in this case). The vibra- tions are generally not periodic but quasi-periodic. In contrast to the self-excited vibrations when the vibration frequency changes only slightly with a change in speed, the frequency of the components at such resonances is proportional to the angular velocity of the rotor. A typical property of such vibrations is that the intensity of the two components is not manifested uniformly along the whole rotor. Thus a case might arise when one component predominates in one field, while the reverse is true in another field.

The occurrence of such resonances is greatly influenced by improper tuning of the system to internal resonance (e.g. n&z = I/Z) when several types of resonance (e.g. a combination resonance and a sub-harmonic one) might merge together. In such a case the vibrations become virtually periodic.

Another feature of systems tuned to internal resonance is that the vibrations of the main resonances-critical speeds--contain substantial components with a differ- ent frequency than the speed frequency. Thus, for example, at nrlnz = 112 the vibra- tions contain, in addition to the component with frequency it, a component with frequency 2n for n N nr, or with frequency I/Z vz for n N no. Such tuning of the system can always be considered as unfavourable relative to the cyclic loading of the rotor shaft; moreover, it renders the system most susceptible to the onset of non-linear resonances outside the interval of critical speeds.

Results obtained in a test of a system tuned to internal resonance (~~~n~ = r/2) are shown in Figs. 3 and 4, The only difference between the cases of Figs. 3(a) and 3(b) is greater unbalance of the rotor in the latter case. The first diagram clearly shows

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SELF-EXCITED VIBRATIONS AND NON-LINEAR RESONANCES OF ROTORS 355

Fig. 3(a).

2b W 40 W W 2& 35 35

Fig. 3(b).

that, next to the main resonances (points I and 2, Fig. 3(a)), there exists a continuous region of self-excited vibrations above the speed of 5,000 r.p.m. and some disturbances within the narrow interval between the second critical speed and the continuous region of self-excited vibrations (points 3, 4 and 5, Fig. 3(a)). By increasing the rotor un- balance the continuous region of self-excited vibrations is suppressed except for sepa- rate disturbances (points 4 and 5, Fig. 3(b)) b u a non-linear-combination resonance t

Wear. 8 (r965) 349-357

n (r.p.mJ

Fig. 4(a).

0 Km m 3oco 4m m

n hp.ml

Fig. 4(b).

is initiated in the vicinity of the speed given by the sum of the first two critical speeds (points 6 and 7, Fig. 3(b)), the combination resonance being of the type n = n1 + nz. Another effect of greater rotor unbalance is the vibration component of the order I/Z

(point 3, Fig. 3(b)) appearing at the second critical speed. Similar results were obtained for the opposite direction of rotation (Figs. 4(a) and 4(b)) when, again, the rotor unbalance was increased in the second case. The self-excited vibrations are again

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SELF-EXCITED VIBRATIONS AND NON-LINEAR RESONANCES OF ROTORS 357

substantially suppressed and the vibration component with a frequency half the speed frequency becomes more pronounced still at the second critical speed. It completely

predominates in the field containing the thinner shaft with disc I (points 4 and 5, Fig. 4(b)) which was of primary importance for the first critical speed. The rotor with least unbalance, Fig. 4(a), exhibited two more or less continuous regions of self- excited vibrations, one in the interval between the first and second critical speeds, the other, with very intensive vibrations, beginning at ?z = 5,430 r.p.m. It is interesting to note that the lower limit actually begins after passing through the first critical speed. No such reduction in the limiting ratio was observed in systems tuned other- wise (n&k2 not close to r/z).

The results confirm the characteristic features of non-linear resonances demonstrated above.

It was established experimentally that greater rotor unbalance exerts an unfavourable effect on the possible vitiation of self-excited vibrations (by raising the limit of its initiation, reducing its intensity or suppressing it entirely) but has a favourable effect on the onset of non-linearWresona&s. distinct physical character of non-linear resonances.

CONCLUSIONS

Th& is corroboration of the

The most important results may be summarized as follows: (I) Supporting a rotor in several journal bearings causes virtually no change in

the resistance of the individual bearing designs to the.initiation of self-excited vibra- tions. Similarly, the basic characteristic features of self-excited vibrations are the same as in the case of a two-bearing mounting.

(2) The rotor unbalance, which exerts virtually no effect on the limit of initi- ation of self-excited vibrations in the case of a two-bearing mounting, affects the limit to some degree when rotors are supported in several bearings, particularly when multi- pressure wedge bearings are used. However, an increase in the unbalance increases the possibility of the onset of non-linear resonances.

(3) The possibility of the onset of non-linear resonances, when the vibrations apparently possess a character similar to that of self-excited vibrations, is far greater for rotors supported in several bearings especially if critical speeds are unsuitably distributed (i.e. the critical speeds are in the ratio of small integers). In addition to the sub-harmonic and ultra-harmonic, other non-linear resonances, e.g. the combination resonances, might also arise. The indispensable condition for the onset of resonance vibration, particularly of secondary non-linear resonances, is the existence of a period- ic exciting force which in this case is the rotor unbalance.

REFERENCES

I J. R. SCHNITGER, Development of a smooth running, double-spool gas turbine rotor system, ASME Paper No. 58-A-r97, x9$.

2 A. C. HAGG, Some vibration aspects of lubrication, L&~&z&n Enp., 4 l1~~81 166. 3 A. TONDL, On the combination-r~on~ce of a nonilinear system u&h’& &&es of freedom,

Rev. de Mecan. Appt., Acad. Rep. Popufaire Roumaine, 8 (4) (1963) 573. 4 A. TONDL, An analysis of resonance vibrations of a non-linear system with two degrees of free-

dom, (in Czech), Roz+avyCesk. Akad. Ved., Serie TV, 74 (8) (1964) 1-82.

Wear, 8 (1965) 349-357