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© N.V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands, 1972.Articles or illustrations reproduced, in whole or in part, must be
accompanied by full acknowledgement of the source:PHILlPS RESEARCH REPORTS SUPPLEMENTS
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ISSUES OF PHILIPS RESEARCH REPO~TSSUPPLEMENTS
1972, No. 1, On the stability of rotor-and-bearing systems and on the cal-culation of sliding bearings, by J. P. Reinhoudt
1972, No. 2, Extraction of particles from a compact isochronous cyclotron,by J. M. van Nieuwland
1972, No. 3, Generation and detection of sound by distributed piezoelectricsources, by R. F. Mitchell
1972, No. 4, Common-bandwidth transmission of data signals and wide-bandpseudonoise synchronization waveforms, by L. E. Zegers
1972, No. 5, Production and perception of vowel duration. A study of dura-tional properties of vowels in Dutch, by S. G. Nooteboom
1972, No. 6, Contribution à l'étude de l'effet tunnel dans les semiconducteurs,by G. Schréder
1972, No. 7, Quantization effects in semiconductor inversion and accumula-tion layers, by J. A. Pals
'"
)ON THE STABILITY OF
ROTOR-AND-BEARING SYSTEMSAND ON THE CALCULATION
OF SLIDING BEARINGS *)
BY
J. P. REINHOUDT
*) Thesis, Technological University Eindhoven, February 1972.Promotor: Prof. Ir W. L. Esmeijer.Copromotor: Prof. Ir H. Blok.
Philips Res. Repts Suppl. 1972, No. 1.
\
AbstractThe stability of a number of "symmetrical" rotor-and-bearing systemsis treated; the symmetry makes it possible to resolve the motion into atranslational and a conical mode. The gyroscopic effects occurring inthe conical mode appear, in most cases, to exert a stabilising effect onthis motion. Forces of unbalance, too, are found toohave a stabilisingeffect in the single case so far examined. With the particular kind ofbearings having a rotationally symmetric response it turns out that, ifmoreover both the rotor and the bearing supports are rigid, an easilyhandled criterion of the stability can be established in respect of eithermode of motion. This criterion makes it possible to say something aboutthe stability also if the excursions from the equilibrium position becomeso great that the bearing response can no longer be assumed to be linear.With rigid rotors having flexible supported bearings it is found that theparameters of the support greatly influence the stability; with a supportconsisting of springs and dampers it is possible in certain cases to indi-cate limits of the support damping, within which the rotor is inherentlystable; with a support consisting of an adjoint bearing, e.g. a floating-bush bearing, the theory provides an indication of the optimum choiceof the parameters in respect of stability, so that new designs of thiskind of bearing suggest themselves. Thereafter a method, based on theprinciple of finite elements, is developed for the calculation of groovedand grooveless sliding bearings lubricated with a Newtonian liquid. Theversatility of this method facilitates the calculation of bearings withgreatly different geometries and the design of bearings with certain de-sirable characteristics, e.g. optimum stability characteristics.
Finally several types of cylindrical and spherical bearings are cal-culated. An idea of the effect of these bearings on the stability of arotor-and-bearing system has been gained by determination of the sta-bility of a symmetric system with a rigid motor and rigid bearing sup-ports.
1.1. High-speed rotors; bearings . . . . .1.2. Stability of rotor-and-bearing systems .1.3. Methods for determining the stability1.4. State of the art . . . . . . . . .l.S. The goal of the present investigations
References . . . . . . . . . . . . . . .
. "
1123467
\
CONTENTS
1. GENERAL INTRODUCTION
2. THE STABILITY OF A SYMMETRIC ROTOR-AND-BEARINGSYSTEM IN WHICH BOTH THE ROTOR AND BEARING sup-PORTS ARE RIGID " 82.1. Introduetion . . . . . . . . . . . . . . . . . . . . . .. 82.2. Stability of a symmetric rotor-and-bearing system with statically
loaded bearings , . . . . . . . . . . . . . . . . . . 92.2.1. Equations of motion and characteristic equation. . . " 92.2.2. Application to a rotor with smoothjournal bearings. .. 15
2.3. Stability of a symmetric rotor-and-bearing system with a constantbearing load rotating synchronously with the shaft 172.3.1. Equations of motion .. . . . . . . . . . . . . 172.3.2. Application to a rotor with smooth journal bearings. 19
References . . . . . . . . . , . . . . . . . . . . . . . . . 21
3. LINEAR AND NONLINEAR ASPECTS OF THE STABILITYOF A SYMMETRIC ROTOR-AND-BEARING SYSTEM WITHRIGID BEARING SUPPORTS AND ROTATIONALLY SYM-METRIC BEARING RESPONSE. . . . . . . . . . . 223.1. Introduction. . . . . . . . . . . . . . . . . . . 223.2. Types of bearings with rotationally symmetric response 223.3. A stability criterion of the case ofrotationally symmetric response 243.4. Physical interpretation ..... . . . . . . 26
3.4.1. Physical meaning of the parameters . . . 263.4.2. Equations of motion in polar coordinates 27
3.5. The effect of large bearing displacements 283.5.1. The possibility of a circular orbit . 283.5.2. The stability of the circular orbit . 30
3.6. Conclusions and supplementary remarks. 31
4. THE EFFECT OF FLEXIBLE BEARING SUPPORTS ON THESTABILITY OF A SYMMETRIC ROTOR-AND-BEARINGSYSTEM. . . . . . . . . . . . . . . . . . . . . . . . .. 334.1. Introduction. . . . . . . . . . . . . . . . . . . . . .. 334.2. The expected behaviour of a flexible bearing support consisting
of springs and dampers . . . . . . . . . 344.3. Supports consisting of springs and dampers . . . . . . . 37
4.3.1. Equations of motion .. . . . . . . . . . . . . 37. 4.3.2. The choice of the dimensionless support parameters . 404.3.3. The characteristic equation if the mass of the support and
gyroscopic effects are negligible . . . . . . . . . . .. 404.3.4. Application to an ALG bearing and supports having
rotationally symmetric response ..... . . . . .. 424.3.5. Application to an ORS bearing and supports having rota-
tionally symmetric response .. . . . . . . . . . 434.3.6. Example of the application of the stability diagrams 494.3.7. Possibilities of designing a flexible support . . . . . 504.3.8. Asymmetrie supports . . . . . . . . . . . . . . 51
4.4. Simplified method of calculation applicable if the bearings as wellas the supports have rotational symmetry 534.4.1. Analysis . . . . . . . . . . . . . . . . . . . . .. 534.4.2. Examples . . . . . . . . . . . . . . . . . . . .. 56
4.5. Bearing supported by an additional bearing (floating-bush bear-. ing). . . . . . . . . . . . . . . . . . . . . . . . . .. 574.5.1. Equations of motion and characteristic equation . . .. 574.5.2. Approximate torque balance for the determination of the
angular velocity of the bush . . . . . . . . . . . 604.5.3. Application to ORS bearings . . . . . . . . . . . 614.5.4. Floating-bush bearing with a freely chosen bush speed 624.5.5. Floating-bush bearing with smooth bearings 63
References . • . . • . . . . . . . . . . . . . . . . . . . . 64
5. THE FINITE-ELEMENT METHOD FOR THE CALCULATIONOF SLIDING BEARINGS . 65
5.1. Introduetion . . . . . . . . . . . . . . . . . . . . . .. 655.2. Principles of the method. . . . . . . . . . . . . . . . .. 665.3. The finite-element method based on the "ordinary" Reynolds
equation. . . . . . . . . . . . . . . . . . . . . . . .. 685.3.1. The Reynolds differential equation . . . . . . . . .. 685.3.2. Transformation of the Reynolds equation into a variation
integral . . . . . . 705.3.3. Boundary conditions . . . . . . . . . . . . . . .. 71
5.3.4. Introduetion of the finite elements5.3.5. Application of the boundary conditions . . . . . . . .
5.4. The finite-element method based on the "generalised" Reynoldsequation .5.4.1. The generalised Reynolds equation . . . . . . . . . .5.4.2. Transformation of the generalised Reynolds equation
into a variation integral . . . . . . . . . . . . . . .5.4.3. The integrals L3' L2' 1+1 and 1+3 for rectangular and
triangular grooves. . . . . • . . . .5.5. Example of an element: the triangular element
References . . . .
6. CAVITATION.6.1. Introduetion .6.2. Cavitation conditions .
6.2.1. Conditions for smooth journal bearings6.2.2. Modification of the Jakobson and Floberg conditions for
grooved bearings .6.3. The dummy-flow method ............•....6.4. Examples . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1. Effect of the parameter Pcav on the load capacity of asmoothjournal bearing with circumferential feeding . .
6.4.2. Effect of the parameter Pcav on a helical-groove bearing6.5. Dynamically loaded bearings.
References . . .
7. ACCURACY OF THE FINITE-ELEMENT METHOD ANDRESULTS OF CALCULATIONS OF LOAD CAPACITY, RE-SPONSE COEFFICIENTS AND STABILITY OF VARIOUSTYPES OF BEARINGS . . . . . . . . . . 1047.1. Introduction. . . .' . . . . . . . . . 1047.2. Short description of the computer program 1057.3. Inaccuracies ofthe finite-element method. . 106
7.3.1. Sources of errors and principles for determining the inac-curacy. . . . . . . . . . . . . . . . . . . . . .. 106
7.3.2. Comparison with the Sommerfeld and the Reynolds solu-tion. . . . . . . . . . . . . . . . . . . . . . .. 107
7.3.3. Comparison with the results of Sassenfeld and Walther . 1087.3.4. The accuracy ofthe pressure build-up in a grooved, centric,
journal bearing. . . . . . . . . . . . . . . . . " 1107.3.5. Comparison with analytical results of a spherical spiral-
groove thrust bearing . . . . . . . . . . . .' . .. 110
7377
7878
85.
878890
91919292
949698
98102103103
7.4., Load capacity, response coefficients and stability diagrams of· some types of bearings . . . . . . . . . . . . . . . . . .7.4.1. Determination of the response coefficients . . . . . . .
· 7.4.2. Smooth journal bearing with.axial lubricant-supply groove(ALG bearing) . . . . . . . . . . .'. . . . . . . .
7.4.3. Smooth journal bearing with circumferential oil supply(CLQ bearing) . . . . . . . . . . . . . . .
7.4.4. Grooved journal bearing with optimum radial stiffness. . . . ,(ORS .bearing) .' '.' . . . . . . . . . . . .· 7.4e,5.. The. spherical spiral-groove bearing . . . . . .7.4.6. Spherical spiral-groove bearing with optimum axial thrust
(SOAT bearing) . . . . . . . . . . . .· 7.4.7.. Spherical spiral-groove bearing with optimum radial stiff-
ness (SORS bearing)References . . . . . . '.' . . . .
8. DISCUSSION, CONCLUSIONS AND RECOMMENDATIONSOF. FURTHER INVESTIGATIONS.8.1. Discussion. . . . . . . . .
8.1.1. Outline of chapters 2-78.1.2. Conclusions .
8.2. Final remarks . . . . .8.2.1. Flexible rotors ..8.2.2. Asymmetrie rotor.8.2.3. Experimental verification.8.2.4. Computer program for transforming the characteristic
determinant into the characteristic equation . .8.3. Possibilities of further investigations and development . . . .
8.3.1. Extension of the above theories. . . . . . . . . . . .8.3.2. New forms of sliding bearings and of rotor-and-bearing
systemsReferences .
Appendix IAppendix nAppendix In .
List of Symbols
111111
112
114
116.120
121
125127
128128128132132132133134
134135135
135136
137140141
147
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1. GENERAL INTRODUCTION
1.1. High-speed rotors; bearings
High-speed rotors have been finding ever wider application in the last fewdecades. There has been a steady expansion of their field of application, andattempts are being made to raise the speed of existing types to ever increasingvalues. Examples of fast-running rotors are to be found, for instance, in navi-gational systems, in .some types of electric motors, in centrifuges and ultra-centrifuges, in turbines and compressors. High-speed rotors are used also inproduction machines, and sometimes a new production technique cannot beimplemented until a rotor-and-bearing system capable of reaching the requiredspeed is available. Examples ofthis aspect are to be found in the textile industry,in the manufacture of filaments for incandescent lamps, and in the enrichmentof uranium in the ultracentrifuge.Depending on the kind of application, there are various reasons for raising
the speed. For example, a certain angular momentum is usually required ofrotors in a navigational system: the mass, overall dimensions, or both, can bereduced if the speed of operation is increased. With centrifuges the attainablecentripetal acceleration, and hence the separating effect, increases rapidly withspeed. With rotary-flow machines, e.g. the so-called expanders, ever smallerunits are demanded: as the circumferential speed must remain approximatelyconstant, this again leads to an increase in angular velocity.If a type of bearing has to be selected for a rotor-and-bearing system, then
the required life of the bearings often plays a major role. The service life of thecommon rolling-element bearings, such as ball bearings, is normally limited byfatigue and wear; the attainable life decreases in proportion as load and speedincrease. A sliding bearing, however, is an element on which the load and speedhave relatively little effect as far as the service life is concerned. This is due tothe characteristic feature of this type of bearing that the bearing surfaces are,under normal operation conditions, fully separated by a film of gas or liquid.Thus mechanical wear is prevented and the life of a sliding bearing, even athigh speed and under heavy load, can be very long. For instance, in applica-tions where the specifications of speed or load, or both, are severe, a slidingbearing may be a good alternative to a rolling-element bearing or even offerthe only appropriate solution.Sliding bearings can be divided into hydrostatic and hydrodynamic (self-
acting) bearings, or into gas-lubricated and liquid-lubricated bearings.The self-acting, liquid-lubricated bearing (for example, the oil-lubricated
bearing) is in many cases better suited for application in high-speed rotor-and-bearing systems than the other types, and while the theories developed in thisreport often have a wide applicability, special attention will be paid to such
-2-
self-acting, liquid-lubricated bearings. Let us examine this statement a littlemore closely.In hydrostatic bearings the lubricant has to be supplied under pressure from
an external pressure source, while in self-acting bearings the pressure, fromwhich the bearing derives its load capacity, is generated in the bearing itself.Thus a system with self-acting bearings may be less complicated.
Gas has a low viscosity as compared with liquids. As a resuit a self-actinggas bearing must have a larger size, for the same load capacity, than a self-acting liquid bearing (cf. "volume rule", VogelpohI1-18)) and as the size hasa greater effect than the viscosity on the energy consumption of the fluid bear-ing, this consumption tends to be higher in the case of a gas bearing. Moreover,the boundary-lubrication properties, which become important when a bearingis overloaded, heavily favour oil in comparison with, for example, gas.A major advantage of gas lubrication is that. a gas-lubricated system can often
work in the lubricating medium itself, thus simplifying the problems of sealingmid lubricant supply of the bearings. This advantage, however, has lately beencompensated in part by the development of simple and effective seals for oil-(or grease-)lubricated self-acting bearings 1-22). This makes it possible to pro-vide such a bearing with a "lubricant supply for life".
Considering the vast literature on gas-lubricated bearings, the liquid-lubri-cated bearing seems to have been neglected in the last twenty years as a bearingfor high-speed rotors. The author is of the opinion, however, that the latterbearing opens up great possibilities for high-speed rotors and deserves moreattention than it has received hitherto.
1.2. Stability of rotor-and-bearing systems
In the foregoing the point was made that, at heavy loads and high speeds, asliding bearing is often superior to a rolling-element bearing as far as the ser-vice life of the system is concerned. Yet in most applications it is not possiblesimply to replace a rolling-element bearing by a sliding bearing, even were theconstruction to allow of this. For one thing, a sliding bearing affects the dynam-ics of the rotor in a far more complex way than does a rolling-element bear-ing: sliding bearings can readily induce self-excited vibrations in a rotor-and-bearing system and these vibrations may give rise to damage of the bearingsand the rotor. In the case of liquid-lubricated bearings the phenomenon is oftenreferred to as "oil whirl", "half-frequency whirl", or "resonant whirl" (see e.g.Newkirk 1-19)). The onset of self-excited vibrations is caused by the fact thatthe rotor-and-bearing system becomes unstable. Because the instabilities occurmore readily as the dynamic effects become stronger, for instance at higherspeeds, one may say that the application of sliding bearings in high-speed rotorsliterally stands or falls with the possibilities of controlling the dynamics of thesystem in such a way as to avoid instabilities.
-3
Let us examine; this in more detail and consider a rotor loaded by a staticforce (cf. fig. 2.1). This rotor may take up a position in its bearings such thatthe forces generated in the bearings are just in equilibrium with the load. Butthis equilibrium position may be unstable, by which we mean that, if we disturbthe equilibrium situation by giving the rotor and initial deflection with respectto its equilibrium position, the rotor does not return to the equilibrium posi-tion but will move further and further away from it.In a stability analysis the motion about the equilibrium position is investi-
gated, to determine whether the rotor, after an initial disturbance, returns tothe equilibrium position (stable) or not (unstable).In chapter 2 the following set of equations, describing the motion of a certain
rotor about the equilibrium position will be derived (cf. eq. (2.13)):
1 A"- x + bxx x + axx x + (bxy +-) y + aXY Y = 0,WW"
(1.1)
Here the dimensionless coordinates x, y determine the deflection of the rotorwith respect to the equilibrium position. The properties of the bearings areexpressed by the response coefficients axx, aXY' bxx, etc. On the assumption thatthe deflections are small, the bearing response may be linearised and then theresponse coefficients are constants. 1j W may be considered as a dimensionlessmass, and A is a measure of the gyroscopic effects. The first and second (dimen-sionless) time derivatives of x are denoted by respectively x (= dxjdt) andx (= d2xjdt2).The goal of the stability analysis is to determine from such equations as
givenby (1.1) whether the rotor, after being subjected to an initial disturbance(x = x, y = y), will return to the equilibrium position (x = 0, y = 0) or not.
1.3. Methods for determining the stability
If the equations of motion are linear, as in (1.1), we may distinguish threemethods for determining the stability.
Method 1. Starting from an initial disturbance (x,y) and with given values ofWand A, we calculate the motion by integration of (1.1) with respect to time (.)and then check whether this motion converges to (0, 0) or not.
Method 2. As the equations (1.1) are linear we may use the substitution
x = x exp "(s r), y = y exp (s r), (1.2)
and deduce the characteristic determinant and characteristic equation of
-4-
(1.1) (cf. eq. (2.15)). The quantities S of (1.2) are the roots of the characteristicequation. By calculating all the roots we are able to check whether all theroots have negative real parts (stable) or not.
Method 3. For the' determination ofthe stability it is not necessary to calculateall the roots, but on applying the Routh-Hurwitz stability criterium (see e.g.Malkin 1-7)) to the characteristic equation we can check whether the roots liein the left half of the complex plane (stable) or not.
Method 1 (cf. e.g. Elrod 1-23)) provides much extra information about themotion and is not restricted to linear equations of motion. However, especiallyat the border of stability, the solving of the problem by this method may requirea great deal of time because of the difficulties of establishing convergence ordivergence.Method 2 gives also much additional information about the motion and may
be equally tedious. A variant of this method, the zero-point method, utilisesthe fact that, at the boundary of stability, a pair of purely imaginary roots, s,should exist and thus the boundary curves of stability may be calculated. Asthe existence of such a pair of roots is a necessary but insufficient condition,it is always necessary to check whether the remaining roots lie in the left halfof the complex plane (cf. appendix I).Method 3 gives less information but works very quickly if only the charac-
teristic equation is known (methods 1 and 2 can be used without deducing thisequation). In this work method 3 will mainly be used.
1.4. State of the art
It is almost impossible and not very meaningful, within the scope of thiswork, to give a historical review of the literature published in the field of rotor-and-bearing dynamics. For this the reader is referred to the existing re-views 1-2/6). Here we shall be concerned particularly with the stability aspectof the rotor-and-bearing dynamics; a survey of this subordinate field will befound in Sternlicht et al. 1-1). Recent progress in development can be outlinedas follows.(A) The feasibility of calculating the performance of a sliding bearing is rapidly
increasing, particularly because of the advent of large, fast computers, sothat the effect of a certain bearing on the stability of a rotor can be deter-mined more accurately than before.
(B) Designs well known for their favourable influence on rotor stability, suchas tilting-pad bearings, flexible supports, etc., have been and are still beinginvestigated, both theoretically and experimentally.
Concerning the greater feasibility of calculation, the Reynolds equation for
-5-
an incompressible flow in the lubricant film has in many recent publicationsbeen solved by means of a finite-difference method. It proved possible to modifythe Reynolds equation in such a way that turbulence in the lubricant film couldbe taken into account 1-20.21). In 1967, Orcutt .and Arwas 1-8) gave the coef-ficients of bearing response of smooth, fully circular and partial-arc bearingswith laminar and turbulent films.. In articles by Hirs 1-10) (1964) and Vohr and Chow 1-11) (1965) journal
bearings with helical grooves were calculated for small values of the eccentricityand it was pointed out that bearings of this kind were stabIer than smoothjournal bearings. In 1969 Chow and Vohr 1-9) published the results of calcula-tions of helical-groove journal bearings working with a laminar or turbulentfilm in respect of both small and large values of the eccentricity.For quite some time use is being made in mechanics of what is known as the
finite-element method, for instance for calculations of strength. Reddi 1-12)
seems to have been the first to apply this method to an incompressible lubri-cant film. It has, in comparison with the finite-difference method, the greatadvantage of being highly versatile, so that bearings with greatly differing geom-etries can be calculated by means of the same program of calculation.As far as favourable constructions are concerned various experimental
investigations and practical experience have shown that the bearing supportplays a significant role in rotor-and-bearing dynamics and that the stabilityof a rotor-and-bearing system can be improved by a design providing a meas-ure of elasticity and damping. Kerr 1-13) gave values of the spring constantand damping obtainable by means of an O-ring support. Powell and Tem-pest 1-24) reported experimental results gained with bearings supported byO-rings. Orcutt and Ng 1-14), in their paper on floating-bush bearings, pointedout that a bearing of this kind had satisfactory stability properties, but theywere unable to explain this.Elasticity of the shaft and bearing support without damping of the flexible
element in general worsens the stability. This can be seen from articles bySternlicht, Poritsky, and Arwas 1-25), Marsh 1-15), and Lund 1-16). Lundconsidered also the effect of damping in the support of a rotor in gas bearings.These authors all used the zero-point method but did not check the positionof the not purely imaginary roots. Then there is always a risk that there arecomplex roots with a positive real part at the moment when a pair of roots ispurely imaginary; it might then be erroneously inferred that one was' dealingwith a boundary between stability and instability.
Gunther 1-17) examined the effect of the support on the stability of rotorswith internal damping. Here the effect of the bearings on the dynamic behav-iour of the rotor was left out of account. The results are of qualitative im-portance for rotors in journal bearings, as the internal damping of the rotoris found to cause the same kind of effect as does a self-acting bearing.
-6-
1.5. The goal of the present investigatlonsNext to providing more insight into the influence of some effects, such as
gyroscopic effects, the purpose of the investigations was to look for solutionsthat could improve the stability of a rotor-and-bearing system equipped withsliding bearings.The stability can in principle be improved in two ways.
- One can try to remedy the real cause of instability, namely the way in whicha sliding bearing reacts to deflections. This leads to looking for bearingswith good stability properties or to trying to affect the bearing geometryin such a way as to yield good stability.
- One can also try to compensate, somewhere in the system, the destabilisingeffects of the sliding bearings, for example by using special bearing supports.
In chapter 2 a symmetrical rotor-and-bearing system with rigid rotor andrigid bearing supports is investigated. Special attention is paid to gyroscopiceffects.In chapter 3 the system of chapter 2 is again considered, but now for the
case that the bearing response possesses rotational symmetry. Such symmetrysimplifies the analysis and leads to a simple and easily usable stability criterion;moreover, an extrapolation to large bearing deflections and nonlinear behav-iour becomes possible.
In chapter 4 the effect of flexible bearing supports on the stability of thesystem is investigated. The floating-bush bearing is considered as a special caseof a flexible support.In chapter 5 a versatile method for the calculation of liquid-lubricated bear-
ings, the finite-element method, is developed. This method permits of the cal-culation of bearings with greatly differing geometries without any significantchange in the computer procedures; it forms a tool that can be used in de-signing bearings with good stability properties, because it creates the possibilityof rapid assessment of the effects of geometric modifications.In chapter 6 consideration is given to the problem of how to include cavita-
tion (which is very important in liquid bearings) in the finite-element methodand, furthermore, examples are given in which the cavitation conditions fre-quently used in the literature lead to incorrect conclusions.In chapter 7 the accuracy of the finite-element method is checked and several
types of bearings are calculated.Chapter 8 contains an evaluation of the work, supplementary remarks, and
recommendations for further investigations.
Note. Although the theories' and results are often of general applicability, sothat they are valid also for gas bearings, etc., the assumption has. beenmade throughout this work that the lubricant can be considered as anincompressible Newtonian fluid.
-7-
REFERENCES1-1) B. Sternlich tand N. F. Rieger, Rotor stability, Conference Lub. and Wear, London
(Sept. 1967), 182, part 3A, paper no. 7. ,1-2) D. D. Fuller, A review ofthe state-of-the-art for the design ofself-acting gas-lubricated
bearings, Trans. ASME, J. Lub. Techn. 91, 1-16, 1969.1-3) T; A. Harris, Lubrication review: A digest of the literature for 1965, Trans ASME,
J. Lub. Techn. 89, 1-37, 1967.1-4) T. A. Harris, Lubrication review: A digest of the Literature for 1966, Trans. ASME,
J. Lub. Techn. 90, 1-34, 1968.1-5) J. H. Rumbarger, Lubrication review: A digest of the literature for 1967, Trans.
ASME, J. Lub. Techn. 91, 225-259, 1969.1-6) J. H. Rumbarger, Lubrication review: A digest of the literature for 1968, Trans.
ASME, J. Lub. Techn. 92, 185-215, 1970.1-7) J. G. Malkin, Theorie der Stabilität einer Bewegung, R. Oldenburg, München, 1959.1-8) F. K. Orcutt and E. B. Arwas, The steady-state and dynamic characteristics of a
full circular bearing and a partial are bearing in the laminar and turbulent flow regimes,Trans. ASME, J. Lub. Techn. 89, 143-153, 1967.
1-9) C. Y. Chow and J. H. Vohr, Helical-grooved journal bearing operated in turbulentregime, ASME-ASLE Lubrication Conference, Houston, Texas (Oct. 1969), paper69-Lub-28.
1-10) G. G. Hirs, The load capacity and stability characteristics of hydrodynamic journalbearings, Trans. ASLE 8, 296-305, 1965.
1-11) J. H. Vohr and C. Y. Chow, Characteristics of herringbone-grooved gas lubricatedjournal bearings, Trans. ASME, J. basic Eng. 87, 568-576, 1965. -
1-12) M. M. Reddi, Finite element solution of the incompressible lubrication problem,Trans. ASME, J. Lub. Techn. 91, 524-533, 1969.
1-13) J. Kerr, The onset and cessation of half-speed whirl in air-lubricated self pressurisedjournal bearings, Proc. Inst. mech. Engrs 180, part 3k, 145-153, 1965/66.
1-14) F. K. Orcutt and C. W. N g, Steady-state and dynamic properties of the floating-ringjournal bearing, Trans. ASME, J. Lub. Techn. 90, 243-252, 1968.
1-15) H. Marsh, The stability of self-acting gas journal bearings with noncircular membersand additional elements of flexibility, Lubrication Symposium, Las Vegas (1968),paper 68-LubS-45.
1-16) J. W. Lu n d, The stability of an elastic rotor in journal bearings with flexible, dampedsupports, Trans. ASME, J. appl. Mech. 32, 911-920, 1965.
1-17) E. J. Gunter, Jr. and P. R. Trumpler, The influence ofinternal friction on the stabil-ity of high speed rotors with anisotropic supports, J. Eng. for Industry 91,1105-1113, 1969.
1-18) G. Vogelpohl, Betriebssichere GleitIager, Springer Verlag, Berlin, 1958.1-19) B. L. N ewkirk, Journal bearing instability, Proc. of the Conference on Lub. and
Wear, London (October 1957).1-20) C. W. N g and C. H. T. Pan, A linearized turbulent lubrication theory, Trans. ASME,
J. basic Eng. 87, 675-682, 1965.1-21) G. G. Hirs, Fundamentals of a bulk-flow theory for turbulent lubricant films, Thesis
Techn. Univ. Delft, 1970.1-22) E. A. M uij derman, Self-contained grease lubricatedjournal bearings (to bepublished).1-23) H. G. Elrod and G. A. Glanfield, Computerproceduresforthedesignofflexibly
mounted, externally pressurised, gas lubricated journal bearings, Gas Bearing Sym-posium Univ. Southampton (1971), paper 22.
1-24) J. W. Powell and M. C. Tempest, A study of high-speed machines with rubberstabilized air bearings, Trans. ASME, J. Lub. Techn. 90, 701-708, 1968.
1-25) B. S ternlich t, H. Pori tsky and E. G. Arwas, Dynamic stability aspects of'cylindricaljournal bearings using compressible and incómpressible fluids, Proc. First Int.Symp. on Gas-Lubricated Bearings, Washington D.C. (1959).
-8-2. THE STABILITY OF A SYMMETRIC ROTOR-AND-BEARING
SYSTEM IN waren BOTH THE ROTOR AND BEARINGSUPPORTS ARE RIGID
2.1. Introduetion
A 'rotor-and-bearing system will here be considered "symmetrical" if it hasa plane of symmetry (plane aa' in fig. 2.1) and if the rotor itself is rigid androtationally symmetrical with respect to the line bb'. It should be understoodthat the external load is part of the system and should be symmetrical withrespect to aa', too.
The motion of this system can be resolved into two modes, viz. the "trans-lational" mode, in which the axis bb' undergoes merely a translation, and the"conical" mode, in which the centre of gravity, G, remains stationary.For the case of a static externalload, stability diagrams of the translational
mode of some types of bearings are known from the literature 2-1.2). Stabilitydiagrams of the conical motion are also known, but here the influence of gyro-scopic effects has generally been ignored 2-2.3). Since in some rotor-and-bear-ing systems gyroscopic effects play a significant part 2-5), they have in thepresent examination of the stability of a rotor been taken into account.
In this chapter the stability of the rotor with two types of external loads isinvestigated. In sec. 2.2 it is assumed that the external load is static. Then, inthe equilibrium position, both bearings have equal static deflections, E. Dataon the stability are found by analysis of the motion about the equilibriumposition. In sec. 2.3 it is assumed that the rotor is loaded by a force of constantmagnitude, rotating in synchronism with the rotor (for example an unbalance
b G D.-.-. ·...;:::.-+-----If=-Z _.
x
a'XL
Fig. 2.1. Symmetric rotor-and-bearing system.
(2.1)
-9-
force). It will further be assumed that the bearings behave rotationally sym-metrically with respect to this force, so that in the equilibrium "position" bothjournals describe circles (E rotates). Again the stability data are found byanalysis of the motion about this equilibrium position.
2.2. The stability of a symmetric rotor-and-bearing system with statically loadedbearings
2.2.1. Equations of motion and the characteristic equation .
Let us consider a symmetric rotor-and-bearing system (fig. 2.1) in which therotor is loaded by a (symmetrical) static force. Let MR denote the mass of therotor, L the distance between the bearing centres, I the moment of inertia withrespect to the centre line (bb') of the rotor, and J the moment of inertia withrespect to a line in plane aa' passing through the centre of gravity, G.In the equilibrium position of the rotor (drawn in fig. 2.1) the static deflec-
tions, E, and the forces of reaction, F, of the two bearings will be equal. Foran analysis of the stability the motion of the rotor about its equilibrium posi-tion must be investigated. Ifwe now define a stationary systemX,Y,Z, then themotion of the rotor may be described by the deflections of both journals withrespect to this system; for the translational mode these deflections are equal,for the conical mode they have equal magnitudes but opposite directions.If X is the dynamical bearing deflection and aF the additional force of
reaction due to this deflection, then for the translational mode we may write(with X,Y,Z an inertial system)
or, if we introduce the components of aF in the X,Y,Z system,
(2.2)
For the conical motion it is useful to define the system U, V, W (cf. fig. 2.2):the coordinate axis W coincides with the centre line of the rotor in the deflectedposition, U and V are defined by the fact that the system X,Y,Z passes into thesystem U, V, W by a rotation, through an angle 8, about a line perpendicularto Zand W. It may be assumed that in U, V, W the rotor rotates solely aboutthe W-axiswith an angular velocity Q.
- 10-
z
Fig. 2.2. Coordinates of the conical motion.
With T, the torque exerted on the rotor, H, the angular momentum, Q" theangular velocity of the U,V,W system with respect to the X,Y,Z system, andeJ' the base vectors of the U, V, W system, the equations of conical motion inthe inertial system are
or(2.3)
With(2.4)
eq. (2.3) becomesdHl
T= --el 4- QrxH.dt
(2.5)
So far all vectors have been defined in the inertial system or, better, in theinertial "space". In a "space" in which the system U, V, W is stationary, theequation of conical motion takes the following form:
dHT=-+QrxH,
dt(2.6)
-11-
(2.7)
and the vectors can be defined with respect to the U, V, W system.With 8 «1, and X representing the deflection of one bearing, the other
one being -X, the components of T, ,Q" and H in the U, V, W system are
TuR::!-LlFJIL,
Tv R::!+LlF"L,
2 dYQ R::!---
ru L dt '
2 dXQ R::!+--
ru L dt '
a.; R::!0;
H." = r û.;
Hu = IQrv,
Hw = I (Qrw+ Q).
From (2.6), (2.7), (2.8) and (2.9) it follows that
2J d2X 2Q dYLlF,,=---+-I-,
L2 dt2 L2 dt
2J d2y 2Q dXLlFJI=-----I-.
L2 dt2 L2 dt
(2.8)
(2.9)
(2.10)
It is possible to combine (2.2) and (2.10) into one set of equations in whichthe parameters have different meanings with respect to the two modes of mo-tion. Meanwhile introducing dimensionless magnitudes, we may write, insteadof (2.2) and (2.10),
in whichLlF"LIl' =-
'J" Fe'
Xx =-
LlR'dx
1LI!" = - (x + Äy),w
1 ..LIJ;,= WCY- Äx),
LlFJILljy= Fe
Yy =-
LlRd2x
dimensionless forces,
dimen~ionless deflections,
(2.11)
-12-
2Fo for the translational stability parameterMRLlRQ2 mode ór reciprocal value
W'=FoL2 of the dimension-
for the conical mode less mass,2J LlRQ2
~0 for the translational
it Imode \ gyroscopic
~for the conical mode
parameter,J
with LlR = radial play of bearing,7: = Ql = dimensionless time,Fo = reference force.
A relation, the Reynolds differential equation, which describes the pressuredistribution in the lubricant film is deduced in chapter 5 (eq. (5.12)).From thatrelation it follows that the pressure, p, in the lubricant depends solely on thedimensionless film thickness, h, and its time derivative, M/b7: (and of courseon the boundary conditions of the differential equation). In a given bearing,for example in a journal bearing, h and M/b. depend on the position and thevelocity of the journal. If the boundary conditions are fixed, or if they are afunction only of the position and velocity of the journal (for example the cavi-tation region under quasi-static conditions, cf. chapter 6) then we may statethat, for a given lubricant viscosity and angular speed, the force of reaction, F,is a function only of the position and velocity of the journal. If we furtherassume that the displacements from the equilibrium position are so small thatlinearisation of the additional reactional forces aF is permissible, and that wemay neglect effects of misalignment (for example, due to the conical motion)we may write
dX dYLlFx = - Axx X - Bxx - - Axy Y - Bxy - ,
dl dl
dY dXLlFy = -Ayy Y - Byy -- AyxX - Byx -.
dl dt
(2.12a)
The response coefficients AXXl Axy, Bxx, Bxy, etc., are only functions of theequilibrium position of the journal bearing under consideration. Introducingdimensionless magnitudes we obtain
Lllx = -aXX x- bxx x- aXYy- bxy y,LIjp = -ayy y - byy Y - ayx x- byx x, (2.12b)
etc.,
-13-
with
.Q LlR .Q LlRbxx = -- Bxx, bxy = -- Bxy, etc.r; s;
From (2.11) and (2.12a) it follows that the two modes of motion can bedescribed by the differential equations
~ x + bxx x + axx x + ( byx + ~) j + aXY Y = 0,
~ji + byyj + ayyy + (byx- ~)x + ayXx = O.
(2.13)
Up till now the reference force, Fo, has still not been chosen. A naturalchoice may be arrived at as follows. From the dimensionless pressure, p, thereactional force, for instance Fx, can be determined with
Fx = J I» Po cos {3 d(/)dP,
in which Po (cf. sec. 5.3.1) is the reference pressure, (/) and P (cf. fig. 6.1) arethe film coordinates, and {3 is the angle of the film element with respect to theX-axis. Introducing dimensionless quantities we may write
ffPPOR2 ,Ix = cos {3 depd7p= J Jp cos {3 depd7p,r;
with
and
(2.14)
(with reference parameters: Ho = LlR, Lo = R). For many boundary con-ditions (which will not be discussed here), p, and thus lx, depends only on thebearing geometry, journal position, and journal velocities; this means that axx,bxx, etc., depend only on the (equilibrium) position of the journal.
Note. The above discussion, based on the "ordinary" Reynölds equation (5.12),will proceed analogously for the "generalised" equation (5.46).
According to the Routh-Hurwitz stability criterion 2-6) it is necessary andsufficient for (2.17) that
ao > 0,
al> 0,
I al ao I= al a2;__ ao a3 > 0,a3 a2
(2.19)
-14-
An alternative to the above choice of Fo is the load capacity, W (W == F = (F.2+ Fy2)1/2). This alternative is often encountered in the literature.An objection to this is the fact that W becomes zero when the bearing isunloaded. In the present chapter, to facilitate a comparison with the literature,we shall use Fo = Wand distinguish the stability parameter (W) based on thischoise by theindex w, i.e. Ww. In subsequent chapters, however, we shall makeuse of (2.14).
Substitution of x __:_Xo exp (s r), y = Yo exp (s r) in (2.13) yields the char-acteristic equation
~ S2+ bxx s+ axx ( bxy +~) s+ aXY
(i;- ~)s+ ayX ~ S2 + s.; s+ ayy1
= - (ao S4 + al S3 + a2 S2 + a3 S+ a4) = 0, (2.15)W2
in which
ao = 1,
al = (bxx+byy) W,a2 = (axx + ayy) W + (bxxbyy- bxybyx) W2 + (bxy- byx) À.W + ,12, (2.16)
a3 = (bxx ayy+ byy axx- aXY byx- ayXbxy) W2 + (aXY- ayx) WÀ.,
a4 = (axx ayy- aXY ayx) W2.
The motion is stable if for all the roots, Sk' of the characteristic equation wecan write:
Re (Sk) < ° (k = 1,2, 3,4).For (2.17)it is necessary (but not sufficient) that
a.:» ° U = 0, 1,2,3,4).
(2.17)
(2.18)
al ao °a3 a2 al = Ct3(al a2 - ao a3) - al2 a4 > 0,° a4 a3a4> 0.
With (2.18) it is possible to reduce (2.19) to
- 15-
(2.20)and
If the coefficients of the bearing response for a certain equilibrium. positionare known, then we are able to examine by means of (2.20) whether a certainvalue of the stability parameter W (or Ww) results in a stable or an unstablebehaviour. Such scanning of the field in which W is of practical value, leadsfor a number of equilibrium positions to a stability diagram in which the stableand unstable areas are partitioned by "stability-transition curves".
2.2.2. Application to a rotor with smooth journal bearings
From the literature the response coefficients of smooth journal bearings areknown (Orcutt and Arwas 2-4)). These coefficients are recorded in figs 2.3 and2.4. The system X,Y,Z is chosen in such a way that the X-axis is parallel tothe direction of the bearing load, W (in the remaining chapters along E).
Q••r
·0 ·5_1e
Fig. 2.3. Dimensionless stiffness coeffi-cients of srnooth journal bearing (BID =1). -- X-- Results of Orcutt and Ar-was; --.-- ALG bearing calculatedwith the FEM.
100....-------------.
Êtw11 y
X
Fig. 2.4. Dimensionless damping coef-ficients of a smooth journal bearing(BID = 1).--x-- Results ofOrcuttand Arwas; --.-- ALG bearingcalculated with the FEM.
16 -
Fig. 2.5. Stability curves of a smooth journal bearing (BID = 1) with a static load W.
WW = MR2~Q2 fortranslation(Ä =O)and Ww = 2J:~~:iforconical motion (Ä = ~).
Because the more accurate data of chapters 6 and 7 did not become availableuntil later, the data of Orcutt and Arwas 2-4) were used to calculate the stabil-ity curves of fig. 2.5. In a subsequent comparison of these data with the resultsof an ALG bearing (cf. chapter 7) the differences (figs 2.3 and 2.4) were smallenough to dispense with a new calculation.In accordance with the above theory, fig. 2.5 applies to both modes of
motion: the curves A = 0·1 to 0·4 relate to a conical motion and the curveA = 0 refers to a conical motion without gyroscopic effects as well as to atranslation. The parameter e is the bearing eccentricity, and e = IEI/LIR.
For the case A = 0 a comparison with data from the literature is pos:sible 2-1.2). Considering the fact that the bearing coefficients there used wereslightly different, we can say that there is an excellent agreement between thosedata and the present curve.It will be seen from fig. 2.5 that in the range 0 < e < ca 0·74 an increase
in the gyroscopic effect (A) improves the stability of the conical motion. Ne-glecting the gyroscopic effect (taking A ~ 0), as many authors do, gives toopessimistic a picture of the rotor stability. The calculation proved that forA > ca 0·5 the stabilizing effect of the gyroscopic action was so strong that,the stability curves came to lie below the e-axis, and hence the conical motionbecame inherently stable. Very remarkable is the range of ca 0·74 < e < 1.
-17-
It is common practice, and justified according to fig. 2.5 for the translationaland conical mode at À = 0, to assum~ that a smooth cylindrical bearing isinherently stable in this range. This, however, is found to be invalid for theconical mode at °< À < ca 0,5, where instabilities can in fact occur. Sinceit is likely that, for example, some turbines operate in this range, instabilitymight unexpectedly occur precisely because of gyroscopic effects. In order togain a better insight into what takes place in this range, a simulation wasperformed with an analog computer. This aspect will not be tIeate~ here.Suffice it to say that the phenomenon could be accounted for by the factthat the gyroscopic effects influenced the mode of motion of the bearings insuch a way that energy could be taken from the lubricant film, which wouldhave given rise to instability.
2.3. Stability of a symmetric rotor-and-bearing system with a constant bearingload rotating synchronously with the shaft
2.3.1. Equations of motionIn sec. 2.2 we have examined the stability of a system in which the bearings
support a static load. Another case encountered in practice is that of a load ofconstant magnitude rotating with the rotor, for instance an unbalanced rotor.If we assume once more that the load is symmetrical with respect to aa' (cf,fig. 2.1) and that the bearing response is rotationally symmetrical, then thevectors E, denoting the equilibrium "position", will describe circles.It appears to be convenient to relate, as it were, the system X,Y,Z to the
vectors E so that this system rotates with angular velocity Q about the Z-axis.To find the equations of motion in X,Y coordinates we may proceed as fol-lows.
For the translational mode the equation of motion in the inertial space canagain be expressed by
(2.21)
With ej representing the base vectors of the X,Y,Z system, and withde,
X=Xjej, -=Q,xejdt '
we have
(2.22)
The vectors of (2.22) are still defined in the inertial space; in a space in whichthe system X,Y,Z is stationary, the equation of motion becomes
- 18-
MR (d2X dX )l\F = _.- -- + 2Qx-- Q2 X ,
2 dt2.. dt(2.23)
with Q = 1Q. I. Indicating the vectors of (2.23) by their components with respectto the X,Y,Z system we may write
For the conical mode the equation of motion in a space in which the U, V, Wsystem (defined with respect to X,Y,Z in the same way as in sec. 2.2.1) is sta-tionary, becomes once more
MR(d2X dY )iJFx= - -- - 2Q-- Q2 X ,2 dt2 dt
M (d2Y dX')/.JF. = ~ -- + 2Q__ Q2 Y .> 2 dt2 dt
dHT = - + (Qr + Q.)xH.
dtFor ()« 1 we have
i; I"::;i -iJFyL,
Tv I"::;i +iJFxL,
t; I"::;i 0;
2 (dY )a; +Qu I"::;iL -cit-.QX ,
Hu = J (.Qru + Qu),
u, = J (.Qrv + Qv),
u; = I (.Q,w + Qw).
From (2.25), (2.26), (2.27) and (2.28) it follows that
. 2J(d2X dY )iJFx = - -- (2- À)Q-- (1- À)Q2X ,£2 dt2 d!' ,
2J(d2Y dX )iJF =- -+(2-À)Q--(1-À)Q2y.
Y L2 dt2 dr
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
-19-
Using the given definitions of Wand À and introducing dimensionless quan-tities we may write, instead of (2.24) and (2.29),
1L1!x = W [i - (2- À) y- (1- À) x],
1 ..L1.t;, = --= [y + (2- À) x- (1- À) y].
W
(2.30)
Assuming again that the relations (2.12b) may be used, we obtain
~i + bxx x + ( axx - 1W À) x + ( bxy _ 2; À) y + aXY Y = 0,
~Y + byy Y + (ayy - 1;..1) y + (byx + 2 -;;~) x ~ ayX x = O.
Equations (2.31) correspond to eqs (2.13): the characteristic equation can befound in the same way as with (2.13).
(2.31)
2.3.2. Application to a rotor with smooth journal bearings
The response coefficients of (2.31) are defined in the rotating system X,Y,Z,and should normally be calculated in this system.With a smooth journal bear-ing, however, it is possible to deduce the coefficients in the rotating systemdirectly from the coefficients that apply in the stationary system of sec. 2.2.2.Let us regard fig. 2.6 and suppose that we know the coefficients in fig. 2.6a.
+)Q:
y
x xa) bl cl
Fig. 2.6. Transformation of a statically loaded bearing into a bearing with a rotating load.
It is permissible to superimpose an angular velocity Q on fig. 2.6a, withoutthe flow in the lubricant being affected, and hence the coefficients are unaffected.For smooth bearings it is of no consequence (with the usual assumptions onfilm flow) whether the bush or the shaft rotates, so that the situation in fig. 2.6cis identical with that in fig. 2.6b. In fig. 2.6c Q is negative with respect to theX,Y coordinates. Reversing of the direction of the Y-axis leads to a positive Q(assumed in sec. 2.3.1). This implies that the coefficients in the rotating system(indicated by the index rot) can be found from the coefficients in the stationarysystem (no index) in the following way:
/
- 20-
bxx rot = ,bx:obxy rot = -bxy,byx rot = -byx,byy rot = byy"
(2.32)
The stability-transition curves calculated for smooth journal bearings arerecorded in fig. 2.7. Here again an increase of the gyroscopic effect, and hencean increase in À, turns out to have a stabilising effect on the conical motion.Unlike the conclusions which can be drawn from fig. 2.5, the conical motionand the translation are not inherently stable in the range e > ca 0·7 and À = O.Only for À > ca 0·6 does the conical motion become inherently stable., For eccentricities of e < ca. 0·7 tbe value of Ww, defined by a stabilitycurve, increases but slowly, or decreases, with an increase in e. But becauseW increases rapidly with e, the net effect of an increase in e (or in W) on agiven rotor-and-bearing systemwill be an increase in the speed atwhich instabilitysets in. This is equivalent to saying that in this case an increase in the unbalanceof the rotor has a stabilising effect.r---------------~
Wwt 0·20
(..=0
~"."t.. unstable
1·0____ e
Fig. 2.7. Stability curves of a smooth journal bearing (BID = 1) with a load of constant
magnitude, W, rotating with the angular speed of the rotor. Ww = MR2L1~ Q2 for translation
(Ä = 0) and Ww = 2J~~~2 for conical motion (j. = ;).
0·10
;_ 21 - ,
REFERENCES2-1) B. Sternlich tand N. F. Rieger, Rotor stability, Conference Lub. and Wear, London
(September 1967) 182, part 3A, paper 7. " _2-2) R. Holmes, Instability phenomena due to circular bearing oil films, J. mech. Eng. 8,
419-425, 1966.2-3) J. W. Lund, The stability of an elastic rotor in journal bearings with flexible, damped
supports, Trans. ASME, J. appl. Mech. 32, 911-920, 1965.2-4) F. K. Orcutt and E. B. Arwas, The steady-state and dynamic characteristics ofa full
circular bearing and a partial are bearing in the laminar and turbulent flow regimes,Trans. ASME, J. Lub. Techn. 89, 143-153, 1967.
2-5) J. P. Reinhoudt, A grease-lubricated hydrodynamic bearing system for a satelliteflywheel, ASLE-ASME Lub. Conference, Houston, Texas (October 1969);J. Lub. Eng. 26, no. 3, paper 22, March 1970.
2-6) J. G. Malkin, Theorie der Stabilität einer Bewegung, R. Oldenburg, München, 1959.
,
- 22 ___;
3. LINEAR AND NONLINEAR ASPECTS OF THE STABILITYOF A SYMMETRIC ROTOR-AND-BEARING SYSTEM WITH RIGID
BEARING SUPPORTS AND ROTATIONALLY SYMMETRICBEARING RESPONSE
3.1. Introduetion
Let us now consider the motion, and in particular the stability, of the sym-metric "standard" rotor (fig. 2.1) for the special case that the bearing responseis rotationally symmetric with respect to the Z-axis of the X,Y,Z system shownin fig. 2.1. Such a situation occurs, for instance, with an unloaded bearing havinga rotationally symmetric geometry. The concentric position of the shaft, then,is the equilibrium position, and a rotation of the X,Y,Z system about the Z-axis does not affect the response coefficients.The rotational symmetry in a bearing simplifies the investigation of the mo-
tion of the rotor. This means, for one thing, that it is possible. to derive stabilitycriteria 'which can be easily applied in practice, and for another that an insightcan be gained into the behaviour at bearing displacements that are so large thatlinearisation of the bearing response is no longer permissible.Large bearing displacements occur, for example, when, on the basis of the
linear stability theory, a rotor becomes unstable. Some authors express the ex-pectation that nonlinear effects may ensure that the bearing displacements donot become steadily larger but remain limited. With nonlinear effects takeninto account, our analysis shows that for a rotationally symmetric response thebearing displacement will - under certain conditions - remain within bounds.
3.2. Types of bearings with rotationally symmetric response
A bearing with rotationally symmetric response will be considered to be abearing in which a rotation of the coordinate systemX, Y,Z (cf. fig. 2.1) throughan arbitrary angle about the Z-axis does not affect the response coefficients.In a number of cases a bearing appears to have a rotationally symmetric re-sponse.
Case 1A bearing which has a rotationally symmetric geometry with respect to the'
Z-axis has a rotationally symmetric response, too. Examples: unloaded smoothand helically grooved journal bearings and spherical spiral-groove bearingswithout radial load.
) On the assumption that linearisation is permissible the general relations(2.12b) are still true, but in the case of rotational symmetry it can be provedthat the following relations should exist between the response coefficients:
- 23-
(3.2)
Introducing the coefficients of rotationally symmetric response,
b = bxx = byy,
t5c b = aXY = -ayX'
b,= bxy = -byx,
we may write, instead of (2.12b):
iJ/x = -a x - b x - t5c by - br j,
iJ_{y = -a y - b j + t5c b x + b, X.
With respect to the parameter t5c it may be noted that, for a circular orbitabout the equilibrium position, t5c is the dimensionless angular velocity atwhich the bearing response is exactly radial. For this reason 'we shall call t5cthe characteristic angular velocity of the bearing. With smooth bearings it caneasily be proved that t5c = t; for the majority of other bearing types t5c R::i t.
(3.3)
Case 2A bearing with three or more identical parts that are equally spaced circum-
ferentially (cf. fig. 3.1) possesses (at the "concentric" position) a rotationally
-~--~-Cl
al ev
,
X
X
Fig. 3.1. Bearings with rotationally symmetric response in the concentric position. Ca) Three-lobed bearing; Cb) bearing with three identical parts; Cc) smooth bearing with equally spacedoil grooves.
- 24-
symmetric response to deflections that still allow of linearisation (cf. eq.(2.12b))., If, for one part, the relation (2.12b) is true, then, for the com-bination of all parts, the relation (3.3) applies and it can be proved that
(3.4)
with n = number of parts.
Case 3Various grooved bearings appear to have a rotationally symmetric response.
Examples are the helical-groove bearing (ORS bearing) and the spherical spiral-groove bearings (SOAT and SORS bearings), which will be considered inchapter 7. With these bearings the rotational symmetry is not limited to theconcentric position but is maintained up to relatively high eccentricities. Indeed,the symmetry is not exact and the relations (3.1) are only approximately true,but the approximation appears to be admissible in many practical applications.An averaging of the response coefficients will make a (rotationally symmetric)calculation of these bearings more accurate:
a = t (axx + ayy),
b = t (bxx + byy),
Oeb = t (aXY - ayx),
b, = t (bxy - byx),
(3.5)
3.3. A stability criterion of the case of rotationally symmetric response
Analogously to (2.13) the equations of motion become
1 ( . Ä)- x + b oX + a x + b, + - Y + oe by = 0,W W(3.6)
(3.9)
- 25-
The characteristic determinant and characteristic equation (cf. eq. (2.15)) are
1_S2 + bs + aW
-{ b, + ~) s- (je b1_S2 + bs + aW
in whichao = 1,
al = 2b W,a2 = 2 a W + (b2 + bt2) W2 + 2 bt It W + A2, (3.8)
a3 = 2 b W (a W + (je s, W + (je It),
a4 = (a2 + (je2 b2) W2.
For stability the conditions (2.19) or (2.20) should be satisfied. If we postu-late that the damping coefficient, b, the stiffness coefficient, a, and the char-acteristic velocity, (je, are positive, and further that (je b, < a (this is foundto be true for all the bearings that have been investigated), then the only con-dition that must still be met is
When this is worked out, it is found that the motion will be stable if thefollowing condition is satisfied:
(j2( A)(a+(j b) __ e_ 1--- >0c vr W (je (3.10)
ExampleThe response coefficients of an unloaded bearing with optimal radial stiff-
ness, known as an ORS bearing, are (cf. sec. 7.4.4)
a R:i 0·20, b R:i 0·59, (je b ~ 0·31, b, R:i O.
According to (3.10) the translational mode (A = 0) will be stable if
_ (je2 (1 - AJ(je)W> = 1·35.
a + (je b,
This result corresponds to the result found (in another way) in fig. 7.18. Fora given rotor, with 'YJ = 0·03 Ns/rn", R = 5.10-3 rn, L1R = 20 . 10-6 rn,MR = 100 kg, and (cf. eq. (2.14))
- 26-
2FoW=----
MRARQ2
the translational mode is stable if the angular speed satisfies the condition
Q < 415 rad/s.
The translation will be unstable if the speed exceeds ca 4000 rpm.
Remarks(1) In an unloaded smooth journal bearing we have a = bt = O. If in addi-
tion A. = 0, then (3.10) predicts that the bearing will be inherently unstable;in the presence of a gyroscopic effect the conical motion of the rotor isonly stable with A. > de. Because of de = 0·5 the motion will then be stablewith A. > 0·5. '
(2) For many bearings with rotationally symmetric response (chapter 7) it istrue that bt R:j 0, de R:j 0,5; if we then assume A. = 0, the motion will bestable if a > ca 1/4 W.
(3) Of major importance is the conclusion that for stability the radial bearingstiffness at the characteristic velocity, de, plays the leading part.
3.4. Physical interpretation
3.4.1. Physical meaning of the parameters
In this section we shall introduce a physical model to enhance our insightinto the behaviour of a rotor-and-bearing system at the transition of stability.That model should explain that at this transition (cf. eq. (3.10»
• d2 dA.(a+ de bt)- _e + _e_ = O. (3.11)
W WIn our model it is assumed that, at the transition of stability, the journals
circulate abouttheir equilibrium position, that the bearing response is °inequi-librium with the centrifugal and gyroscopic forces, and that the parameters in .(3.10) may be interpreted as follows:1
W,is the dimensionless rotor mass,
is the dimensionless angular velocity,
is the radial stiffness at the velocity dc,
is the stiffness induced by the centrifugal forces,
'is the stiffness induced by the gyroscopic forces.
1iJfx = W (x + AY),
1iJJ;,= W (ji - Ai).
\
(3.12)
- 27-
The left-hand side of (3.11) represents an effective stiffness: if this effectivestiffness is negative, the motion is unstable: and if it is positive, then the motionis stable. At the transition of stability the effective stiffness is exactly zero.
In the next section the motion will be described in terms ofpolar coordinatesand it will be shown that the proposed interpretation is correct.
3.4.2. Equations of motion in polar coordinates
According to (2.11) the dimensionless equationsCartesian coordinates read:
of motion expressed in
For the transformation into polar coordinates (see fig. 3.2) we make use of
x = e cos a, y = e sin a,da
w=-dr '
iJfx = iJfr cos a - iJfr sin a,
iJ!;. = áf; sin a + iJfr cos a.
(3.13)
We then find the equations in the following form:
d2e- + (A W - w2) e - W iJfr = 0,d.2
dw A- 2wde _ iJfr------W-=o ..dr e dr e
(3.14a)
(3.14b)
Let us now investigate the circumstances in which a circular motion can exist.If we assume that the journal does i~ fact describe a circle, it must be true that
xFig. 3.2. Transformation into polar coordinates.
- 28-
. d2e de dw- = - = -- = 0 and e > O.d.2 d.' dr
(3.15)
In conjunction with (3.14b) this gives
LJfr = O. (3.16)
From (3.3) it follows that
deLJfr = -b e w + oe be + b, - ,d. (3.17a)
deLIl!, = -a e- b-- b,ew
r d.' (3.17b)
and from (3.15), (3.16) and (3.17a) we obtain
(3.18)
With (3.17a) we can rewrite (3.14a) as
1 d2e Àw- w2( de )--+ e- -ae-bd.,.-brew =0,Wd.2 W •
and with (3.15) and (3.18)
[(a + oe br) - O~ (1- :e) ] e = 0;(3.15) and (3.19) imply that
(3.19)
which is the same as (3.11), and therefore we may draw the conclusion that ourmodel was correct and that it is indeed justifiable to attribute the proposedinterpretation to (3.11).
3.5. The effect of large bearing displacements
3.5.1. Thepossibility of a circularorbit
If the displacements of the bearings are "large", then it is not necessarily'permissible to Iinearise LIJ,. and LJfr (see (3.17». Nevertheless it is interestingto ascertain whether the bearing displacements can remain limited even if, invirtue of the criterion expressed by (3.10), the motion is unstable.
Ir = Ir (e, :: ' e w).
ft =ft ( e, ::' e w ).
(3.20)
-29 -
Guided by the results of the linear theory, let us examine whether for largedisplacements a circular orbit is possible and, if so, whether this motion isstable. We shall assume that the bearing response possess rotational symmetryalso in the case of large displacements and that linearisation of the equations'of motion (2.11) continues to be permissible (this is the case at all times withbearings having a rotationally symmetrical geometry; in the case of bearingswith three or more lobes it is a meaningful approximation). .From the Reynolds differential equation, briefly alluded to in chapter 2 and
derived in chapter 5, it follows that with boundary conditions that are, at themost, functions of e, de/d., ew, we may write:
Let us suppose that a circular motion is takingplace;inaccordancewith(3.14a)and (3.20) the displacement eo and the angular velocity Wo of this motion arethen defined by the formulae
(3.21a)
(3.21b)
(Ir may no longer be linearised; for this reason the prefix Ll is omitted). Be-cause of W/eo =1= 0, it follows from (3.21b) that
(3.22)
This means that Wo is equal to (je, the characteristic angular velocity at whichthe direction of the bearing response is purely radial. Calculations of smoothand grooved journal bearings (see also chapter 7) show that (je is virtuallyindependent of eo (almost invariably we find (je ~ 0·5).With (je known, eo can be found from (3.21a):
Ir (j/ ( A)- - (eo, 0, eo wo) - -=- 1- - = 0.eo W (je
(3.23)
Note. Normally (3.23) has a solution, eo > 0, only if the motion is unstable invirtue of (3.10) and if the "stiffness" increases with eo (fig. 3.3), i.e.
Ir- - > a + (je b..eO .• ' .(3.24)
- 30-
Radialresponse
1eo -e
Fig. 3.3. The "normal shape" of fr(e, 0, e öc)'
3.5.2. The stability of the circular orbit
If from' (3.21) a solution (eo, wo) is found, then it will be necessary to in-vestigate whether this solution implies stable motion. By studying slight mo-tions about this "equilibrium" solution we are able to investigate its "stability".
Let, in (3.14),
- e = eo + Lie, with ILlel« leo I,w.= Wo+ Llw, with ILlwl « IWol.
(3.25)
This yieldsLlë + (À WO - wo2) Lie + (,1,- 2wo) eo Llw +
("Dir "DJ,. • "DJ,. )-JP -Lle+-. Lle+-Llew =0,"De "De "Dew
(3.26a)
eo LI}v- (,1- 2wo) Llë +
("Dit "Dir "Dir )- JP -Lie +-. Llè+ -Llew = O."De oe "Dew
(3.26b)
From calculations it is found that in general
(3.27)
are negligible with respect to the other terms. With
(3.28)
where
one obtains
Llë + (À Wo- wo2)Lle + (,1- 2wo)eo Llw + JP(acLie + beLlë) = 0, (3.29a)
eoLlw- (,1- 2wo)Llë +.JPbfeo Llw = O. (3.29b)
(3.31)
- 31-
With p = Wae + (À WO.:' Wo2), the characteristic determinant and the char-acteristic 'equation become:
-(À- 2lVo)s
al = W(be + bf),
a2 = W2 b; bf + (À- 2WO)2 +p,
a3 = Wpbf·
For stability the following relations should hold:
al > 0, which is satisfied because W> 0, s,» 0, bf > 0;
a3 > 0, which is satisfied because p > 0;
al a2 - a3 > 0, in other words,b; [j.f2 b, b,+ (À- 2WO)2 +p] + bf [j.f2 b; bf + (À- 2WO)2] > 0,
(3.30)
which is satisfied, provided p > 0. From this and the definition given for p itfollows that the circular motion is stable if
p = Wae + (À WO- wo2) > 0,
or, in conjunction with (3.23) and Wo = c5e, if
(3.32)
(3.33)
If (3.33) is satisfied, then the circular motion is stable.Figure 3.4 shows examples of a stable and an unstable point of intersection.
Note. In the foregoing we have shown that a circular, stable motion is possibleif (3.33) is satisfied. This does not, of course, form a rigid proof that J10other motions can exist, although on the basis of experiments and of cal-culations with the aid of the analog computer it seems plausible to assumethat the circular orbit is usually the only practical possibility.
3.6. Conclusions and supplementary remarks
The assumption of the rotationally symmetric bearing response simplifies thetheory and provides a clear insight into the physical beha:viour of a rotor atthe boundary of stability. .A relatively large number of bearings exhibit a rotationally symmetric re-
sponse and can be treated by means of the simple theory given above.By including nonlinear effects in the theory, we were able to show that, under
-:- 32 -
Radialresponse
) -fr
I ~ ~(l-~e
.!~ stable intersec.tion a >.:!r.e ea)
)Radial
response
b)-e
Fig. 3.4. (a) "Stable" and (b) "unstable" points of intersection.
certain other conditions, circularly stable orbits of the bearing could exist. If,on the other hand, the bearing were then considered in terms. of the lineartheory, then the motion would be found to be unstable.On account of the "nonlinear" stability behaviour a bearing can yet function
properly if it is considered to be unstable according to the linear theory. How-ever, the journalof the bearing will vibrate to a greater or lesser extent, becausethe journal describes an orbit. But if such vibrations were acceptable, then itwould be' possible to design bearings with a far better "nonlinear" stability -than could be attained on the basis of only the linear theory.It is often very important to be able to treat bearings, which in themselves
are rotationally asymmetrie, as being rotationally symmetric by averaging theresponse coefficients; this permits of finding a practicable approximation ofthe stability parameter W by means of (3.10) and without the more èOmplicatedcalculJ,tions of chapter 2 (see also the examples in chapter 7).
--33 -
-!(
4. THE EFFECT OF FLEXIBLE BEARING SUPPORTS ox THESTABILITY OF A SYMMETRIC ROTOR~AND-BEARING SYSTEM
4.1. Introduetion
In the foregoing we have at all times assumed that the nonrotating part ofthe bearings is rigidly supported. This assumption is only permissible if thedisplacements of the support due to deflection are negligible with respect tothose of the bearing itself. In a number of cases, however, the support of thebearing is so flexible that this should be taken into account in a considerationof the stability.
As compared with a rigid support, a flexible one can have both a destabilisingand a stabilising effect. As a rule a decrease in stiffness of an undamped supportgives rise to instabilities at lower speeds, so that it has a destabilising effect. Butthe provision of damping in a flexible support improves the stability of therotor, sometimes even to such an extent that we may speak of a stabilising sup-port. Figure 4.1 shows an example of such a stabilising support, viz. the O-ringsuspension of the bearing bush in an aerostatic bearing. Kerr 4-1) has publishedexperimental results of air bearings supported by O-ring, from which the stab-ilising effect ofthe support becomes evident. Powell and Tempest 4-3) reportedthat experimental dental turbines equipped with a support of this type reachedspeeds of as high as 720000 r.p.m.
Some publications on the theory of gas bearings also deal with the effectof the support on the stability. Lund 4-5) has investigated the effect of a flexiblesupport, with dampers, on the stability of a symmetric rotor, and he has appliedthe results to gas bearings. He employed the zero-point method, took theflexibility of the shaft into account in a somewhat questionable manner (cf.chapter 1 of the present work), and confines himself to supports having rota-tional symmetry. Marsh 4-2) obtained the bearing coefficients by a linearisingof the Reynolds equation, but then he proceeded in essentially the same manneras Lund, i.e. by using the zero-point method. Tondl 4-6) investigated the effect,
Fig. 4.1. Principle of an aerostatic journal bearing with "O"-ring support.
- 34-
of the foundation tuning, but with very great simplification indeed of the bear-ings. Mori and Mori 4-:_.8) examined the effect of a support by means of anaerostatic sub-bearing in a greatly simplified system.Relatively little attention has been paid to flexible supports in the case of
liquid bearings. Yet the performance of a liquid bearing differs greatly fromthat of a gas bearing, and the results for gas bearings cannot simply be usedfor liquid bearings. This different behaviour is caused by, among other things,cavitation in liquid bearings and is due to the fact that, unlike gas bearings,liquid bearings have response coefficients that are frequency-independent.. In this chapter we shall deal with a symmetric, flexibly supported rotor with
liquid bearings. In secs 4.2, 4.3 and 4.4 the support is imagined to be built upfrom springs and dampers. In order to be able to draw a clear distinction be-tween the main effects we shall first consider rotationaIIy symmetric supports.In sec. 4.3.8 we shall discuss some effects due to rotational asymmetry in thesupport.
Section 4.5 will be concerned with a so-caIIed floating-bush bearing. Theouter bearing in such a system is, as it were, a flexible support of the innerbearing. However, the main difference is that the stiffness of the support de-pends upon the speed of the rotor, and that the support reaction contains whatare known as cross-terms. The stabilising effect of this type of bearing does notseem to have been worked out yet. In an article of Orcutt and Ng 4-4) thefollowing comment is made on "good" stability.
"There is no clear explanation for the latter attribute (i.e. good stability)since, unlike the tilting-pad or lobe bearings, it has no inherent antiwhirl char-acteristics. "
4.2. The expected behaviour of a flexible bearing support consistlng of springsand dampers
In chapter 3 it was shown that the proper stability criterion (3.11) could bededuced also by assuming the forces to be in equilibrium. at the transition ofstability. Analogous considerations will be used in this section to examine theeffects to be expected from a flexible bearing support, thus permitting of aconvenient interpretation of the results of the more exact calculations in thefollowing sections. In this we shall restrict ourselves to rotationally symmetricsystems with negligible gyroscopic effects (A ~ 0) and bearing supports con-sisting of springs and dampers.
First a remark should be made about the response coefficients. The (non-dimensionless) response.coefficients of the supports, Asxx, Asyy, Bsxx, etc., willbe constant in a given system.1f it is assumed that the (dimensionless) rç:sponsecoefficients of the bearings are functions only of the dimensionless static bear-ing deflection, e (e = 0 in this section), it follows from the definitions of thesecoefficients (cf. chapter 2), viz.
that the damping coefficients, Bbxx, etc., of a given bearing will be constant butthat the stiffness coefficients, Abxx, Abxy, etc., will be proportional to the speedof the rotor, Q. Hence the bearing stiffness will increase with the speed. \In chapter 3 we have seen that, with a rigidly supported bearing which is
just on the border of stability, the centre of the shaft can describe a circle aboutits equilibrium position with an angular velocity ~c. In this situation the (radial)bearing response counterbalances exactly the centrifugal forces acting on the"rotor mass", uw. According to (3.11) this balance of forces is expressed by
~c2
-eW '~centrifugal force
in which e (cf. fig. 4.2a) is the dimensionless bearing deflection.If the bearing is supported not rigidly but elastically, by means of springs
with a stiffness as (fig. 4.2b), then here again, on the border of stability, thecentre of the shaft can describe a circle at an angular velocity ~c. For equi-librium, then, the following should apply:
~c2
ar etot = W eto!>
- 35-~
QLlRbbxx = -- Bbxx, etc., _
Fowith
121] Q R4
Fo=----(LlR)2
(a + ~c br) e~radial bearing load
(4.1)
(4.2)
(4.3)
in which ar is the total stiffness and etot the total deflection (cf. fig. 4.2b). Sincear < a + ~c b., the "mass", 1jW, must be smaller on the border of stabilitythan in the case of fig. 4.2a. Hence we can expect the stability of the rotor todecrease with the stiffness of its undamped support.
By adding a (slight) damping to the springs of fig. 4.2b we obtain the situa-tion represented in fig. 4.2c. In the support a difference in phase arises betweenthe support deflection es and the force of reaction. For an "equilibrium situa-tion", then, it is required that the attitude angle CPa of the bearing be positive.This is only possible with Ww < ~c (ww is the dimensionless whirl velocity),so that the equilibrium must be established at a smaller angular velocity.In the equation
(4.4)
ar is bigger than in (4.3), due to phase shifts (/.and cP" and, in addition, owingto the fact that as a rule the bearing stiffnesswill increase with decreasing whirl
\
36 -
0 c)~bt ~~~bt)e'"( ~ ~-=-eW
~
bl
~
1.=1. + ..L~bt ' ês~~etat Or Os O+6cbt
1: .1 e &~ ~ot=~+es W iN etat,
~
cl
~I ~ ~et~t UJw<óc
~ ~Ww W eta!
d)
-(,JwFig, 4.2. The physical behaviour of a support;Ca) rigid support,Cb) flexible support without damping,Cc) flexible support with slight damping,Cd) flexible support with damping; predominating support deflection.
velocity, WW' In the situation shown in fig. 4.2c the mass, I/W, can be greaterthan that in the situation of fig, 4.2b. Hence we expect that the addition of slightdamping to an undamped flexible suspension will improve the stability.The support illustrated in fig. 4.2a is rigid, so that here the displacement of
the bearing predominates over that of the support. Let us now assume that thedisplacement of the support prevails, owing to the stiffness of the support being
. low and/or the stiffness of the bearing being high (for example at high Q). Wecan then conceive of a situation as given in fig. 4.2d. Here es ~ etot, CPa has abig value, almost n/2, and ewilldecrease with increasing rotor speed.ltis possibleto re-establish the equilibrium only if ct is smaller, hence at a smaller value ofWw and consequently at a bigger value of I/I¥. Accordingly we expect that, ifthe displacement of the bearing decreases in comparison with that of the sup-port, and hence if in the situation of fig. 4.2c the rotor speed is greatly increased,a situation will arise as is shown in fig. 4.2d; this is characterised by the factthat the stability is improved by increasing of the rotor speed.
- 37-
At very high values of the; damping, the support will act ~s if it were rigidagain. We can therefore expect that, at very high values of the damping of thesupport, we shall find the same values of the stability as with a rigid support.
4.3. Supports consisting of springs and dampers
4.3.1. Equations of motionA rotor with flexible bearing supports is shown in fig. 4.3 in which MR is
the mass of the rotor, and Ms that of the bearing bush; I is the moment ofinertia about bb', and J that about an axis in plane aa' passing through G;Q is the angular speed of the rotor, and L is the distance between the centresof the bearings.
We base our considerations on the following assumptions:- The system and the externalload are symmetric with respect to plane aa'.- The rotor is rigid and rotationally symmetric with respect to bb'.- The external load is constant.- The bearing support can be represented by linear springs and dampers.- The bearing reactions are linear functions, with constant coefficients, of the
relative journal displacement, Xb, and journal velocity, (djdz) Xb•
The coordinates of the centres of the journal and of the bush are given infig. 4.4. In this figure, OB and QB are the equilibrium positions of respectivelythe journal and the bush, while 0 and Q are the positions of respectively thejournal and the bush after a displacement from the equilibrium position. Thedynamic deflections of bearing and support are denoted by X, and X s:
The motion of this system is described by the following dimensionless equa-tions (cf. chapter 2).
Journal forcesLlft,x = -abxx Xb - bbxx Xb - abxjl Yb - bbXjl Yb'
Llft,jI = -abjIjl Yb - bbjljl Yb - abjlx Xb- bbjl.,<Xb'
(4.5)
x xFig. 4.3. Symmetric rotor-and-bearing system with flexible bearing support.
- 38-
IJIs" = -IJ};,,, - as"" Xs - bs"" i.,IJlsy = -IJfi,y - aSYy ys - bsyy Ys'
Equations of motion of the rotor
(4.6)
xFig. 4.4. Coordinates of the centres of bush and journal.QB = static equilibrium position of centre of bush;OB = static equilibrium position of centre of journal;Q = deflected position of centre of bush;o = deflected position of centre of journal.
Forces on bearing bush
1IJfi,,, = W (x + Ä y),
1 ..IJfi,y = W (y - Ä i).
(4.7)
Equations of motion of the bearing bush
(4.8)
In the above equations the dimensionless quantities are defined as follows:
dynamic parts of the forcesacting on the journal; (4.9a)
x,x =-s LlR'
Ysy =-s LlR
LlRabxx = - Abxx, etc.
Fo.Q LlR
bbXX= -- Bbxx, etc.Fo
LlRasxx = - Asxx, etc.
Fo.Q LlR
bsxx = -- Bsxx, etc.Fo
2Fo
--39 -
dynamic parts of the forcesacting on the bush; (4.9b)
deflections of the journal; (4.9c)
deflections of the support; (4.9d)
stiffness coefficients of thebearing; (4.ge)
damping coefficients of thebearing; (4.9f)
stiffness coefficients of thesupport; (4.9g)
damping coefficients of thesupport; (4.9h)
(translational mode)
(translational mode)
(translational mode)
(conical mode)2 J LlR.Q2
À ~ l 0
I
J
2Ms
MRm =,
MsL2
2J
(conical mode)
(conical mode)
, The reference force, Fo is chosen as follows (cf. chapter 2):
stability parameter (4.9i)
\ gyroscopic parame~,r; (4.9j)
mass of support; (4.9k)
total deflections of the joumal;
dimensionless time.
(4.9 I)
(4.9m)
12 'YJ.QR4
R -----0- (LlR)2
(4.9n)
- 40-
4.3.2. The choice of the dimensionless support parameters
, It was already mentioned in sec. 4.2 that the bearing response coefficients,abxx, abxy, bbxx, bbxy, etc., are to be considered solely as functions of the equi-librium position, e. The damping coefficients, bsxx, bsyy, are for a given systemconstants, too (cf. (4.9h) and (4.9n», but the stiffness coefficients of the sup-ports become functions of D when made dimensionless (cf. (4.9g) and (4.9n».
In representing the stability curves in a diagram it appears to be convenientto have (for a given system and a given e) only one parameter that depends onD, and for this reason we shall introduce a new parameter, k, that is independ-ent of D:
asxx L1R Asxx 1 MR (L1R)6Ic=-= =----AUi W Fo 288 'YJ2 RB sXX'
(4.10a)
Further let
(4.10b)
(4.10c)
(4.10d)
We can then rewrite (4.6) as
iJ!sx = -L1j"x- Ui k Xs- f2 xs,
iJ!sy = -L1j"y - , Ui k ys - g f2 Ys'
Now (4.11), (4.5), (4.7) and (4.8) describe the motion of the system, and fora given system and a prescribed equilibrium position, e (= QEOE), the stabil-ity is governed solely by the stability parameter, W. The latter can be inter-preted as the reciprocal value of the dimensionless angular speed of the rotor,since Ui is the only parameter still containing Q. For translation it followsfrom (4.9i) and (4.9n) that
(4.11)
(4.12a)
and for the conical movement:_ 6'YJR4U 1w= -
J(iJR)3 D(4.12b)
4.3.3. The characteristic equation if the mass of the support and the gyroscopiceffects are negligible
In many cases the mass of the support is small as compared with the mass
- 41
(or the reduced moment of inertia) of the rotor (ms « 1) and the influenceof gyroscopic effects is only slight (A ~ 0)._ Hence in what follows we shall assume the effect of the mass of the supportand the gyroscopic effects to be negligible and put
m, = A = 0. (4.13)
Itmay be noted here that from chapter 2 it appears that gyroscopic effectsusually have a stabilising influence, so that, as far as stability is concerned, oneerrs on the safe side in neglecting them. From (4.8) we derive
LI..fsx = LlJsv = 0.
With the usual substitutions (x, = ~ exp (s r), Yb = Yb exp (s r), etc.) thisleads to the characteristic equation of the system given by (4.11), (4.5), (4.7)and (4.8):
S2 + bus + all
b2lS + a21
S2
°
bl2S + a12 S2
S2 + b22S + a22 °° S2 + b33s + a33
S2 b43S + a43
°S2b34S + a34
S2 + b44S + a44
=0,
(4.14a)or
in which
all = WabXX' bu = W bbxx,
al2 = WabXV' b12 = W bbxv'
a21 = WabVX' b2! = W bbVX'
a22 = WabVV' b22 = W bbVV'
a33 = W2k, b33 = W (2,
a34 = 0, b34 = 0,
G43 = 0, b43 = 0,
a44 = C W2k, b44 = ~W (2.
(4.15a)
(4.15b)
The coefficients ao to a6 of the polynomial are given in appendix 11.Using the Routh-Hurwitz criterion we can determine from the coefficients
ao to a6 whether the motion is stable or unstable for agiven value oftheparam-eter W: on the border of stability and instability one of the roots of (4.14b)is purely imaginary. Utilising this property, this root can be evaluated from(4.14b), and Xb' Yt" etc., can be calculated from (4.l4a).
-42 -
4.3:4. Application to an ALG bearing and a support having rotationally symmetricresponse
The ALG bearing (axiallubricant groove bearing) is described in more detailin chapter 7; the lubricant is supplied to this bearing through an axial grooveat the location of the widest :filmgap; in this groove, at the two bearing endsand in the cavitation region the pressure is assumed to be zero. .In most cases small variations in shape and location of the lubricant grooves
have only slight effects on the load-carrying capacity and the bearing coeffi-cients, so that the ALG bearing may be considered as representative of manysmooth cylindrical bearings.Figures 4.5a to 4.5d show the stability curves of the system of fig. 4.3 in which
the support has rotational symmetry (C = ~= 1). Here the stability number,W, and the support damping, tb determine the positions of the stable and un-stable regions. The static bearing eccentricities, ex and ey, and the supportstiffness, k, are also used as parameters.From the foregoing it follows that, for translation,
2 r; 24 rJ R4 1W= = -,
MR Q2 LlR MR (LlR)3 Q
LlR Q (LlR)3'(! = -- B.xx = B.xx,
Fo 12 rJ R4
LlR Am MR (LlR)6le =---= AW Fo 288 rJ2 RB sxxs
LlR s.: J (LlR)3k =--= Ai72 rJ2 RB L 2 sxx»W Fo
(4.17c)
(4.I 6a)
(4.16b)
(4.16c)
ÎI.=o,
(4.16d)
(4.16e)
C = ~= 1,
and for conical motion:
FoL2W=---
2 J LlRQ2(4.17a)
(4.17b)
ÎI.=o. '
(4.17d)
(4.17e)
-43 -. .
/ In fig. 4.6a we may regard the stable and unstable regions, bounded by thecurves A and B, as the fundamental form, from which all the other figurescan be derived. This fundamental form can also be seen in fig. 4.5c at k = 0·1.On the left there is, with slight damping, an unstable region bounded by
curve A. At point 1 the stability is less than with a rigid support, due to thedestabilizing effect caused by the flexibility of the support at low damping. Aswe pass from 1 to 2, the stability is improved as the damping increases. Atpoint 3 the bearing is much stiffer than at 1, due to the higher speed (cf. sec.
, 4.2). Here the stability is enhanced by an increase in the speed (decreasing of W),and again the unstable region is reduced owing to the increase of the damping.If the damping exceeds that at point 4, then the rotor is found to become stablefor all values of Wand hence for all speeds.At high damping the support becomes rigid again, and consequently the
value of W at point 5 corresponds to that of W for a rigid support.A decrease in the stiffness of the support, k, causes curve A to shift upwards
and to the left (Al)' The inherently stable region is then widened. An increasein the stiffness of the support narrows the inherently stable region (A2). Thismay even give rise to a degeneration, as shown in fig. 4.6b (and fig. 4.5c withk: = 1). Curves A and Bwill then coalesce, a small stable region being left belowthe unstable region. With a construction having the line of action pq it is pos-sible, by a rise in the speed, to start at p, then pass through the unstable regionand arrive again at the stable point q.If the bearing itself has better stability properties because it operates at a
bigger static displacement, ex, then this also manifests itself in the stabilitycurves: at equal values of k and e the unstable regions appear to decrease withincreasing ex (cf. fig. 4.5).It is seen from fig. 7.7 that a rigidly supported ALG bearing is inherently
stable if ex > ca 0,8; if the support is flexible, the stability turns out to besatisfactory also: no instability was found in the region ex> ca 0·8. .At ex = 0 a rigidly supported ALG bearing is inherently unstable (cf. fig. 7.7),
but with a flexible support a stable region, bounded by the e-axis, appears stillto exist. Accordingly, a flexibly supported rotor with ALG bearings may, afterhaving been unstable at low speeds, become stable again by a sufficient increasein the speed (cf. fig. 4.5a).
4.3.5. Application to an ORS bearing and a support having rotationally sym-metric response
By an ORS bearing (cf. chapter 7) we understand a helical-groove journalbearing with a groove pattern such that the radial stiffness is optimum ate = [e]= O. This bearing, in contrast to an ALG bearing, possesses ade-quate stability properties even at [e]= 0, and with rigid supports (cf. fig. 7.18).'The results with a flexible support are given in fig. 4.7. At ex = 0 (fig. 4.7a), .
-44-
w
j;ble
I+unstable
.unstable
~ble
2-5 A /~ 0
2 ;=-004 Jl5 .i
I/
'5/al 10-3
e
w
·5
e -0·2~=O~·~-1
2·5
1·5
e
Fig. 4.5. Stability curves of a rotor with ALG bearings (BID = 1) and flexible bearing sup-ports. (a): ex = 0; (b): ex = 0'2; (c); ex = 0'4; (d): ex = 0·6.
- 45-
cl
w
J:ble
i'unstable
dl
I"'II
+unstable
.unstableI
tble
tunstable
I
kble
bl
stable
-46-
wt
worki ng line for a given rotor andsipport with a given beari ngxcentricity
al
wÎ
Fig. 4.6. Typical shape of stability curves for a rotor with flexible supports. (a) Lobe-shapedinstability regions; (b) degeneration of (a).
w
.rII
tunstable
t unstableI
Lble
al 10-3
Fig.4.7. Stability curves ofa rotor with ORS bearings (BID = 1) and flexible hearing supports.Ca): e", = 0; Cb): e" = 0'4; (c): e" = 0'8.
p, " ,,' ..
-47-
2
I·01
\·04
;;able
",unstable
I...unstableI
t'able
w
2.5 k='004
w
2~able
';unstable
...unstableI
1':ble
2·5
the stability is much better than that of an ALG bearing under the same cir-cumstances (fig. 4.5a). Because the response coefficients of an ORS bearing donot vary so greatly with ex, the increase in stability with increasing ex is muchsmaller !han with an ALG bearing.· .
-48 -
While fig. 4.7 gives the stability curves for an ORS bearing with BID = 1,in fig. 4.8 they are given for BID = 0·5. The stability with BID = 0·5 is dis-tinctly inferior to that of the bearing with BID =--= 1. This is a logical con-sequence of the smaller response coefficients.
w
II
e zOeX-QÇ·~·l
;able
';'unstable
f'unstabler;2·5 I
·01I
k=·OO4I•j
J/
'5/
2
1·5
p
Fig. 4.8. Stability curves of a rotor with ORS bearings (BID = 0'5) and flexible bearingsupports. (a): e.. = 0; (b): e.• = 0'4; (c): ex = 0·8.
-49 -
2
x~eI
+unstable
tunstableI
t:ble
2'5
~10/-:=::::. ...(l
cl
4.3.6. Example of t/ze application of the stability diagrams
The manner in which the stability curves can be used is explained by meansof the following example. .
Problem. It is required to ascertain whether a symmetric rotor-and-bearingsystem with a vertical rotor (unloaded journal bearings, cf. fig. 4.9) and withrigidly supported ORS bearings is stable, and if not, what flexible support willensure stability.The data of the system are:
mass of the rotor MR = 10 kg,moment of inertia of the rotor J = 0·033 kg m 2
,
bearing distance L = 0·20m,angular velocity Q = 2000 rad/s,radius of bearings R = 6 . 10-3 m,length of bearings B = 12. 10-3 m,radial clearance LlR = 20 . 10-6 m,viscosity of lubricant 'YJ= 5. 10-3 Ns/m2•Furthermore let it ~ 0; then J/L2 = 0,83, JIL2 «MR, and it follows fromthe definition of Hi that, with rigid supports, only the translational mode willassume importance. According to (4.9n),
12'YJ Q R4Fo = = 390 newtons,
(LlR)2
- 50-
L
Fig. 4.9. Symmetric vertical rotor.
and, with (4.9i),2FoW (translational mode) = = 0·98.
MRQ2 L1R
From fig. 7.18 (or fig. 4.7a, extrapolating to k = co) it follows that the rotoris unstable for W< ca I'3!Is it possible to stabilise the rotor by means ofrotationally symmetric, flexible
bearing supports if, on account of design considerations, the minimum supportstiffness is allowed to be 2. 105 NIm? Because a low stiffness favours thestability in this case we choose the given minimum value and obtain:
L1R Asxx 20. 10-6 2. 105k=---= x--~O·01.
W r; 0·98 390
Figure 4.7a then shows that, for a stable translational mode, the supportdamping must lie in the region of
ca 3 . 10- 3 < (! < ca 2.
For the conical mode, however, W(conical mode) = (MRL2IJ)xO·98 = 11·8.To prevent instability in this mode we can narrow the limits of e, so that thesystem operates in the inherently stable region, and put ca 10-2 < e < ca 1.
4.3.7. Possibilities of designing aflexible support
In principle a flexible support can easily be realised with the aid of an O-ring(fig. 4.IOa). This construction has the great drawback, however, that the sup-port damping depends on the stiffness, and vice versa: at a given stiffness thedamping is often insufficient to ensure stability (especially with liquid-lubricatedbearings). This disadvantage is absent in the construction of fig. 4.l0b, becausethe stiffness is brought about by the two O-rings and the damping is pre-dominantly effected by the squeeze bearing, which is capable of operating with
- 51-
/
b)Fig. 4.10. (a) Bearing with "O"-ring support. (b) Bearing with "O"-ring support and addi-tional squeeze bearing.
element
bearinggid support
Fig. 4.11. Bearing with flexible support.
the same lubricant as that of the main bearing. In the design of such a squeezebearing it is possible to select the damping within wide limits. .If the stiffness that can be reached with an O-ring support is insufficient, then
constructions of the type shown in fig. 4.11 can be used. Here the collar of thebearing bush represents the spring element. Damping is again accomplished bythe squeeze bearing.
4.3.8. Asymmetrie supportsIt will be clear that it is not always possible to rely on rotational symmetry
of a support. Supports without rotational symmetry also occur, and sometimesthey are certainly designed intentionally in that form. Restricting ourselves toa support as shown in fig. 4.3 we can express the asymmetry in the supportstiffness by the parameter C (eq. (4.10b)) and that in the support damping byç (eq. (4.10c)). Now the number of parameters determining the stability of arotor-and-bearing system becomes so great as to make it impossible to describeall the cases occurring in practice. Hence we shall confine ourselves to just oneexample, in which the effect of the introduction' of rotational asymmetry in anoriginally symmetric support will be considered.
- 52-
. .Figure 4.12a shows for an ORS bearing (BID = 1) with a symmetric sup-
port (l; = ~= 1), the effect of a reduction of the stiffness in the Y-directionand therefore a reduction in l;..Itwill be seen that reduction of t has a slight effect on the right-hand curve
and that initially the 1eft-hànd curve shifts to the left, whereby the inherentlystable region is widened. At l; ~ ·5 there arises a second lobe, which Impedesfurther extension of the inherently stable region.
al 10.3
W
.rII.unstable
.unstableIIt~
·11
p" """I " , ~oo
Fig. 4.12. Stability curves of a rotor with ORS bearings (BID = 1) with symmetric supportdamping and asymmetrie stiffness, ex = O. .
·- 53-
Figure 4.12b illustrates the effect of an increase in C. Here again the left-hand curve divides into two lobes, but an increase in C ultimately causes theinherently stable region to vanish.It is striking that both a slight increase and a slight decrease of C should
enlarge the stable region. This effect is likely to be due to the fact that themotions in the X- and Y-directions are uncoupled, impeding the transfer ofenergy and thus producing the same effect as that caused by the enhancementof the damping in a system with rotational symmetry.The fact that new lobes come into being shows that, for a certain asymmetry,
other forms of motion will start to play a role and govern the stability of thesystem. Inspection of fig. 4.13, which shows the effect of asymmetry in dampingon an ORS bearing, reveals that the fundamental form of the diagram remainsthe same and that the increasing or decreasing of g is of the nature of anincrease or decrease in the mean damping.
·1
·5
J;;able
...unstable
.unstable
'table
1-5
Fig. 4.13. Stability curves of a rotor with ORS bearings (BID = 1) with symmetric supportstiffness and asymmetrie support damping, ex = O.
4.4. Simplified method of calculation applicable if the bearings as well as thesupports have rotational symmetry
4.4.1. Analysis
If both the bearings and the supports have rotational symmetry, then" at theborder of stability, the centre of the journal can describe a circle about theequilibrium position. In that situation the responses of the bearing and of thesupport just balance the centrifugal forces acting on the rotor (cf. chapter 3
- 54-
and sec. 4.2). With the help of the equilibrium of forces the calculation of pointson the borderline of stability becomes a fairly simple matter.In what follows we shall again take m, - A = O.To the bearing response
of a bearing with rotational symmetry the following relations apply (cf. (3.3)):
iJhx = -a Xb - b Xb - (je b Yb - b, Yb'
iJfbl' = -a Yb - b Yb + (je b Xb + b, Xb'(4.18a)
or in complex farm
(4.18b)
with Zb the complex bearing deflection and LJ/" the complex dynamic force.Writing Zs for the complex support deflection and Z for the total journal
deflection, we may introduce complex amplitudes z", Z's, and Z, as follows (cf.fig.4.14):
support'reactims
__ G:lw
1centrifugalforce
Fig. 4.14, Forces and deflections at the boundary of stability.
(4.22)
- 55-
in which Ww is the dimcnsionless angular velocity. Using (4.19) and LlJi, ~LIJ;,exp (j ww.) we obtain from (4.l8b):
(4.20)
and hence we find
. (4.21)
Likewise for the support (assuming W> 0, k > 0, e > 0, Ww > 0) we find:
LIJ;, = (- Wk - j e ww) ;., (4.23)
1LJf;,1= Izsl [(W k)2 + (e ww)2]1/2, (4.24)
e Wwtana=--
Wk'(4.25)
and from fig. 4.14:. IZb I sin CPasm a = --,,-- .
Izsl(4.26)
It follows from fig. 4.14 that_ ILl);' I cos aw=---Izsl k '
(4.27)
and from (4.25) and (4.26)
Wk !Zb I sin CPae = -- A A. " •
Ww Jzsl [1- (IZbl sin CPaflzsJ)2]1!2
With Ä. = 0 we obtain from (4.7):
(4.28)
1 ..LlJi, = -z,W
(4.29)
(4.30)
and from (4.27) and (4.30):-
A I;bl COS CPa [(I~I COS CPa)2 ILlJ;,12Jl!2IZsl=- + +-- .
2 cos a 2 cos a k Ww2(4.31)
- 56-
Now the calculation of a point of the borderline of stability may proceed asfollows: .- Take IZbl = 1.- Select le and Ww (0 < Ww < c5e).- From (4.21) find ILO·bl.- From (4.22) determine cos CPa and sin CPa.- Estimate cos Cl. (usually cos Cl. R::I 1, provided that the damping is not too
strong).- From (4.31) find 1z.1.- From (4.27) calculate W.- From (4.28) calculate (J.
- By means of (4.26) check whether the estimate of cos Cl. was sufficientlyaccurate, and try again if this is not the case.
4.4.2.' Examples
In the following examples a number of borderline points of fig. 4.7a will becalculated with the above, simplified theory. In fig. 4.7a an ORS bearing withBID = 1 has been considered. From fig. 7.16 its follows that for that bearing,and with ex = 0: a R::I 0·2, b R::I 0·6, Oe R::I 0·5 and b, R:;! o.
Example 1Let us choose Ww = 0·05, le = 0·1.Then from (4.21): ILl};,1 = 0·338,
from (4.22): cos CPa = 0·59, sin CPa = 0·80.Take cos Cl. R::I 1.Then from (4.31): li.l = 21·4,
from (4.27): W = 0·16,from (4.28): (J = 1.2.10-2,from (4.26): sin Cl. = 0·037 => cos Cl. ~ 1.
The values of Wand (J thus found appear to correspond to those of fig. 4.7a.
Example 2Let us choose Ww = 0·2, le = 0·1.Analogously we find W = 0·69 and (J = 5·4. 10-2•
Example 3Let us choose Ww = 0·45, le = 0·1.Analogously we find W = 2·04 and (J = 6·7 . 10-2•
Note. The first example clearly shows the Iow angular velocity (ww = 0·1 c5e)at which the journal centre orbits at the lower end of the borderline ofstability.
- 57-
4.5. Bearing supported by an additional bearing (floating-bush bearing)
4.5.1. Equations of motion and characteristic equation
The floating-bush bearing is a more general form of a flexibly supportedbearing, in the sense that the force of reaction of the support is governed hereby eight response coefficients instead of the four found in the system of fig. 4.3.A rotor with floating-bush bearings is schematically represented in fig. 4.15.
Fig. 4.15. Rotor with floating-bush bearings.
Hele MR is the mass of the rotor and Ms that of the bush: I is the momentof inertia about bb' and J that about aa'; Q is the angular velocity of the rotorand Q2 that of the bush.We start from the following assumptions:
- The construction and the externalload are symmetrical with respect to aa'.- The rotor is rigid and has rotational symmetry with respect to bb'.- The external load is constant.- The bearing reactions are linear functions, with constant coefficients, of the
relative displacements X, and X, and of the velocities (d/dr) X, and(d/dr) X, (see fig. 4.4).
The coordinates of the centles of the journals can again be represented, infig. 4.4, by OE and QE in the equilibrium position and by 0 and Q after adisplacement from the equilibrium position.The motion of this system is described by the following dimensionless equa-
tions (cf. sec. 4.3.1).
Journal forces
1 1LI};,,, = -ab"" Xb - -- bb"" Xb _.: ab")' Yb - -- bb")' Yb'
. 1+ v 1+ 'jI
1 1LI};,)' = -ab)')' Yb - -- bb)l)l Yb - ab)'" Xb - -- bb)''' Xb·
1+ 'jI 1+ ')I
(4.32)
Forces on bearing bush
- 58-
(4.33)
Equations of motion of the rotor1
iJhx = W(x + Ity),
1 ..iJf"y = W (y - It x).
Equations of motion of the bearing bush
(4.34)
(4.35)
The dimensionless quantities are defined as follows:
, iJRlabxx -:- -- Abxx,FOl
Dl iJRlbbxx = Bbxx,
FOl
iJR2asxx = --;:;-Asxx,
L'02
D2 iJR2)bsxx":"'" . Bsxx
F02
iJFbxIJhx = --;:;-,
L'Ol
iJFsxIJJ.x = --;:;-,
L'Ol
XbXb=--'
iJRl
x,xs=--'
iJRl
YbYb = LlR
l
Ys
Ys = LlRl
etc.
etc.
etc.
dynamic parts of the forcesacting on the journal, (4.36a)
dynamic parts of the forcesacting on the bush, (4.36b)
deflections of the journal, (4.36c)
deflections of the support, (4.36d)
stiffness coefficients of theinner bearing, (4.36e)
damping coefficients of theinner bearing, (4.36f)
stiffness coefficients of theouter bearings, (4.36g)
damping coefficients of theouter bearing, (4.36h)
oI
J
Further
121]2 D2 R24
F02 = -(-Ll-R-2-)2-
F02 LlR!vc=---'
Fo! LlR2
(4.36i)
(4.36j)
(4.36k)
total deflections ofthe journal, (4.361)
dimensionless time. (4.36m)
reference force relating to the inner bearing, (4.36n)
reference force relating to the outer bearing, (4.360)
effective angular velocity of the inner beating, (4.36p)
effective angular velocity of the outer bearing, (4.36q)
(4.36r)
(4.36s)
If, analogously to sec. 4.3.3, we put m, = À = 0, the 'characteristic deter-minant and characteristic equation are given again by (4.14) and, instead of(4.15), we obtain . /'
all = WabXX'
al2 = WabXY'
a21 = WabYX'
a22 = WabYY,
- 60-
1bu = W--bbxX'
1+ v1
b12 = W--bbX)"l+v1 .
b21 = W--bbyx,1+ v .1
b22 = W--bbyy;l+v
(4.37a)
Vcb33 = «=«:
V
b34 =_ Vc
W-bsxY'V
b43 =_ VcW-bsyx,
V
b44 =_ VcW-bs)'y.
V
(4.37b)
4.5.2. Approximate torque balance for the determination of the angular velocityofthe bush
For a given construction the angular velocity ofthe bush, Q2, deterrnines thestiffness ratio of the inner to the outer bearing.Figure 4.16 shows a concentric floating-bush bearing. In this case, with
smooth bearings, the torques transmitted by the lubricating films are:
Fig. 4.16. Floating-bush bearing in the concentric position.
8,
(4.39)
- 61-
inner bearing:R13 BI
Tl = 2:n;1]1 -- (Q- Q2);L1Rl
R23 B2
T2 = 2:n;1]2--- Q2'L1R2
(4.38a)
outer bearing:(4.38b)
If the torques Tl and T2 alone act on the bush and if we assume that theabove formulae also yield a reasonable approximation for eccentric and groovedbearings, the speed ofthe bush can be obtained from (4.38) by putting Tl R:i T2:
(4.40)
Further
(4.41)
(translational mode),
61]1 R14 L2
J Q (L1Rl)3
(4.42)
(1 + 11) (conical mode).
Since W still contains the parameter 11, we cannot yet interpret Was the recip-rocal of the speed. We therefore introduce
241]1 R14
W* = MRQ(L1Rl)3
61]1 R14 L2
J Q (L1Rl)3
(translational mode),
(4.43)
(conical mode),
so that W* now represents the reciprocal of the speed of the rotor.
4.5.3. Application to ORS bearings
The stability diagram of fig. 4.17 gives the location of the stable and unstableregions for a rotor .in floating-bush bearings as a function of W* and L1Rl/L1R2.The floating-bush bearings are built up from two ORS bearings, with BID = 1and 1]1= 1]2' BI = B2' Rl R:i R2.
The figure shows that the stability of the rotor is optimum for L1RI/LlR2 =
-62-
ex=ey=O~/Bz-l ; R11R2"'1~1/72-1 ; 81101 -1
j;table205
III
2 + unstable
1·5 stable
·5unstable AR1
AR2
·5 1·5 2 2'5Fig. 4.17. Stability curve of a rotor with floating-bush bearing (torque balanced); innerbearing: ORS (BID = 1); outer bearing: ORS (BID = 1).
1·4. The poor stability at low values of LJR1/LJR2 is due to the fact that thediagram really indicates the stability as a function of LJR2 and of the recip-rocal of the speed, the other parameters being considered as constants: asmall value of LJR1/LJR2 implies a large LJR2, leading to slight stiffness and slightdamping of the outer bearing, so that on the basis of the previous results, thestability will be poor.
4.5.4. Floating-bush bearing with a freely chosen bush speed
In the preceding section the velocity of the bush was governed by the balanceof torques.In fig. 4.17 this means that the stiffness and the damping of the outer bearing
are :fixedfor a certain value of LJRdLJR2• We have seen, with the system of theflexible support with springs and dampers in sec. 4.2, that in particular theappropriate combination of stiffness and damping of the support yields op-timum stability. Therefore the coupling of damping and stiffness by the balanceof torques can have an unfavourable effect on attempts to reach optimumstability.Hence for the rotor in the preceding section the stability has been investigated
with Q2/Q as a freely adjustable parameter between the limits of 0 and 1. Theresult is given in fig. 4.18, which also shows the curve for (Q2/Q)k' representing
. the value prescribed on the basis of the balance of torques. This figure clearlyreveals that, especially at low values of Q2/Q (substantially lower than thoseobtained from the balance of torques) a satisfactory stability can be reachedin the range of ca 0·9 < (LJR1/LJR2) < ca 1·4.
:_ 63
ex=ey=O8,/~-1 ; R,/Ri"l7,/72"1 ; 8,/0, =1
,j;""I
~unstable
2·0
1'5x_x-X""7~Tx- ·6
_:J./ _---- ---.=.:::/'_-- _,_I....- ...-._.-1·0
'_'-._{.fu~.n..'k ._.0'5
k
·8
·4
·2
o 0·5 1·0 1·5 2·0 2·5 ~àR2
Fig. 4.18. Stability curves of a rotor with floating-bush bearing for various bush speeds;inner bearing: ORS (BID = 1); outer bearing: ORS (BID = 1).
Fig. 4.19. Floating-bush bearing with additional brake bearings for various bush speeds.
Low values of Q2/Q are well realizable in practice by artificial braking ofthe motion of the bush. A possible design is displayed in fig. 4.19. Mountedon both sides of the normal bearings there are bearings which are responsiblesolely for a reduction of the speed of the bush, without their contributing muchto the load-carrying capacity (they have therefore been given grooves).
4.5.5. Floating-bush bearing with smooth bearings
To smooth bearings, under no-load conditions, the relation
applies. This is to say that these bearings have no radial stiffness at all. Byanalogy with the preceding bearings, a few smooth floating-ring bearings havebeen investigated with respect to their stability. It has been found that thesebearings are inherently unstable under no-load conditions.
64 -
Obviously, the good stability reported by Orcutt and Ng 4-4) cannot beexplained by means of a linear theory.Analogously to the nonlinear theory, described in chapter 3, it is also quite
possible here that with a floating-bush bearing which was initially unstable, acertain deflection leads to such a great radial load-carrying capacity that thebearing and the bush are capable of describing stable circles. Some evidenceof this has been obtained also from exploratory calculations, but these will notbe dealt with here.
REFERENCES4-1) J. Kerr, The onset and cessation of half-speed whirl in air-lubricated self pressurised
journal bearings, Proc. Inst. mech. Engrs 180, part 3k, 145-153, 1965/66.4-2) H. Marsh, The stability of self acting gas journal bearings. with noncircular members
and additional elements of flexibility, Lub. Symposium, Las Vegas (1968) paper no.68-LubS-45. r
4-3) J. W. Powell and M, C. Tempest, A study of high speed machines with rubber stab-ilized air bearings, Trans. ASME, J. Lub. Techn. 90, 701-708, 1968.
4-4) F. K. Orcutt and C. W. Ng, The steady-state and dynamic characteristics of thefloating-ring journal bearing, Trans. ASME, J. Lub. Techn. 90, 243-253, 1968.
4-5) J. W. Lund, The stability of an elastic rotor in journal bearings with. flexible, dampedsupports, Trans. ASME, J. appl. Mech. 32, 911-920, 1965.
4-6) A. Tondl, The effect of an elastically suspended foundation mass and its damping onthe initiation of self-excited vibrations of a rotor mounted in air pressurised bearings,Gas Bearing Symposium (Univ. Southampton 1971), vol. 1, paper no. 1.
4-7) H. Mori and A. Mori, A stabilizing method of the externally pressurized gas journalbearings, Gas Bearing Symposium (Univ. Southampton, 1969), vol. 2, paper no. 29.
4-8) H. Mori and A. M 0 r i, A stabilizing method of the externally pressurized gas journalbearings, Gas Bearing Symposium (Univ. Southampton, 1971), vol. 1, paper no. 4.
- 65-
5. THE FINITE-ELEMENT METHOD FOR THE CALCULATIONOF SLIDING BEARINGS
5.1. Introduetion
If one wants to investigate the dynamic behaviour of a rotor-and-bearingsystem, then one must, in addition to the equations of motion of the rotor,.know the dynamic properties of the bearings used. For sliding bearings theseproperties can in principle be determined by solving the Reynolds differentialequation, which describes the pressure distribution of the lubricant in the filmgap. Since an exact solution has proved to be possible in only a few cases,mainly approximate methods have to be used. In such a method, use is oftenmade of calculation programs based on the well-known finite-difference meth-od 5-10).
At present the number of known types of sliding bearings (smooth bearings,grooved bearings; cylindrical, conical and spherical bearings; hydrodynamicand hydrostatic bearings; etc.), which frequently possess very different geom-etries, is so large that it is desirable to have the disposal of a calculationprogram which enables one to compute all these types of bearings. Hence thiscalculation program must be versatile. The versatility becomes even more im-portant if one wants to examine, for example in the design phase of a rotor-and-bearing system, the effect of a geometric modification, such as displace-ment of a lubricant groove to another place.A method which possesses this versatility to a high degree is that which is
known as the Finite Element Method (FEM). This method is currently em-ployed on a wide scale, for instance in calculations of the strength of construe-tions. It will become apparent from the present chapter that the finite-elementprinciple is applicable to the calculation of sliding bearings, too.The method takes its name from the fact that the region in which the dif-
ferential equation has to be solved is divided into a number of sub-regionsknown as elements. In these elements the function to be calculated is greatlysimplified, and one restricts oneself to solutions that are uniqu.ely defined bytbe functional values in the "nodal points" of the element. In order to findthe approximate solution for the whole region one does not employ theReynolds equation in its original form but transforms it into a specific integralknown as a variation integral. The function of the solution whicb minimisesthis integral is adopted as the "best approximation".In this chapter the variation integral associated with the Reynolds equation
will be derived. Minimisation leads to a set of linear equations which describethe solution, viz. the pressures in the nodal points. Next, for grooved bearings,a derivation-is given of a "generalised" Reynolds differenttal equation for the"smoothed pressure", which means that. the local pressure fluctuations above
- 66-
the grooves and ridges are smoothed. Again, by a procedure similar to that .for the "ordinary" Reynolds equation, the variation integral and the set oflinear equations for the nodal p~essures are found,
In sec. 5.5 a triangular element with a linear pressure distribution will bediscussed by way of an example.
Besseling 5-1) and Zienckiewicz 5-2) have pointed out that the FEM isgenerally applicable and-that, in addition to strength calculations of construe-tions, it can be applied also to problems of heat conduction. Reddi, for example,used the method for the incompressible 5-3) and the compressible 5-4) Reynoldsequations. Argyris and Scharpf 5-5), also, described a theory for the incom-pressible Reynolds equation. Vogelpohl 5-6) was probably the :first to use thevariation principle for the calculation of sliding bearings.
The derivation presented here, which differs from those of the authors justreferred to, is particularly useful for a clear insight into the proper approachand the way in which the boundary conditions must be handled.
Hirs 5-7), and Vohr and Pan 5-8) used the principle of the smoothed pres-sure in the calculation of grooved bearings. They assumed the number ofgrooves to be infinite, but this turns out to be an unnecessary limitation.
The combination of the FEM and the generalised Reynolds equation forgrooved bearings facilitates the application considerably; it does not, however,seem to have been published previously.
5.2. Principles of the method
One of the advantages of the FEM is that it makes a strong appeal to thephysical sense of the user. Because of this, errors can be avoided, and whenapplying the method one realises where the inaccuracies arise. Before a be-ginning is made with the derivations it is therefore relevant to outline theprinciples used for the case of incompressible flow under consideration.
In a calculation of a bearing the problem is to find that function, the pres-sure P(Xf, Yf), which in the region G (:fig. 5.la) obeys the Reynolds différen-tial equation
(è:J2p bP )f --,--, ....X}, Yf = 0è:JX/ sx,
and satis:fies the conditions at the boundary R. If the principles of variationcalculus are used, this can also be formulated in terms of :finding the functionP(Xf) Yf) which minimises the variation integral
al
bl
cl
- 67-
part of R whereRa boundary cand.
Ra applies
'.
Fig. 5.1. The steps taken in the FEM.
In looking for the minimum of (IJ we might in principle vary the pressure Pin every point of the region G, until the minimum is found. To this end wehave to vary the pressure in "002" points of G.The first step taken in the FEM is that, in determining the minimum, one
contents oneselfwith varying the pressure in afinite number ofpoints, the nodalpoints of G, and specifying the pressure between the nodal points (fig. 5.1b) bymeans of interpolation functions. Here we shall not go into the conditions tobe fulfilled by the interpolation functions but confine ourselves to the remarkthat in G the integrand must be sufficiently "smooth", so that by increasingthe number of nodal points we are able to approximate the exact solution toan arbitrarily close degree (cf. ref. 5-2).The second step to be taken consists of dividing G into sub-regions (the
elements Gk) and determining the pressure distribution within these elementsby means of the associated nodal points and interpolation functions (fig. 5.1c).A refinement of the distribution of elements increases the number of nodalpoint and thus improves the accuracy.
"----------------------------~------- -- - --
- 68-
The choice of the elements and associated interpolation functions is governedby the expected pressure distribution, which must admit of being representedsufficiently accurately by means of available degrees of freedom (the pressuresin the nodal points). If, after a preliminary calculation it turns out that this hasnot been the case, then it is possible to perform further calculations with adaptedelement distributions.
5.3. The finite-element method based on the "ordinary" Reynolds equation
5.3.1. The Reynolds differential equation
Figure 5.2 shows an element ABCDA'B'C'D' of a liquid film. Here ABCDis a part of the stationary surface and A'B'C'D' a part of the moving surface ..It is assumed that the flow is perfectly viscous and that the density and viscosityare constant.Using the Navier-Stokes equations, with the usual simplifications (Came-
ron 5-9)), it then follows for the sub-element abcda'b'c'd' thatoP 02U,,*-='YJ--oX oZ 2 'f f
oP 02Uy*-='YJ--oYf oZ/ '
(5.1)
oP--=0,ez,
Fig. 5.2. Element of the lubricant film. H is the thickness of the lubricant film at the locationof element ABCDA'B'C'D'.
- 69-
where P is the pressure and U;* and U,* are velocities in respectively theXr and Yrdirections.
WithUx* (Zf = H) = Ux,
Uy* (Zf = H) = Uy,
Uz* rz, = 0) = 0,
u;* (Z f = 0) = 0,
(5.2)
and
o(5.3)
it follows from (5.1) thatH3 bP H u,
Qx=---+-,12'YJbXf 2
H3 bP HUyQy=---+--.
12'YJbYf 2
(5.4)
To the element ABCDA'B'C'D' the equation for continuity is applicable in thefollowing form:
bQx bQy bH-+-+-=0.bXf bYf bt
Now dimensionless quantities are introduced as follows:
(5.5)
o, u,Ux=--; uy=--;
DoLo DoLo
x, Yfxf=-; Yf=-;
Lo i;H
(5.6)
h
Pp
-70 -
Here Qo (the reference flow), Ho (the reference film thickness), Lo (the referencelength), {Jo (the reference angular velocity), and Po (the reference pressure) canstill be chosen freely.
Equations (5.4) and (5.5) can now be rewritten as
Qo H02 Po bp---qx=- h3-+thux,{Jo Ho Lo 12 'IJ{Jo L02 bXf
(5.7)
Qo H02 Po bp---qy =- h3-+thuyo{Jo HoLo 12'IJ {JOL02 bYf
If we now choose,
(5.8)
Qo = {Jo Ho Lo,
12'IJDoL02Po=----
H02(5.9)
and pass into vector notation we may write:
q =-h3 VP + thu, (5.10)
'bh(\7 . q) + - = O.
b?:(5.11)
The relations (5.10) and (5.11) hold for any coordinate system. Eliminatingthe flow q from these equations one obtains the dimensionless Reynolds equa-tion:
M(\1 . (h3 \1p))- t Cv . (h u))- -= O.
b.(5.12)
5.3.2. Transformation of the Reynolds equation into a variation integral
The calculation ofthe pressure distribution in a hydrodynamic or hydrostaticbearing amounts to the solution of the Reynolds differential equation (5.12) ina region G with prescribed boundary conditions at its boundary R. It will beassumed that there will be a solution, p, with its derivatives, in the region G.Then it follows from the principles of variation calculus (cf. Courant 5-12)
that the relation
,M]) = 0 = J (cv. (h3 \7p)) - t (\1 . (h u)) - ~:) 6p dG (5.13). G
is equivalent to (5.12).
,,
-71-
For the following operations all that is required with respect to the variationof the pressure, öp, is that it can be sufficiently frequently differentiated in G;other restrictions, imposed on öp by the' boundary conditions, may be intro-.duced during the final stage of the solution of the problem:
Using Gauss' theorem we may convert (5.13) as follows:
J (\1 . (h3 \1p» öp dG= J (\l . (h3 öp \1p» dG- J (h3 \1p . \l öp) dGG G G (5.14)
= J(h3 sr», n) öp dR- ö J(!h3 \1p. v» dG,R G
joh JOh- -öpdG =-ö -pdG,o~ o~G G
(5.16)
- t J (\1 . (h u» öp dG = -t J (\1 . (h öp u» dG + t J (h u . \l öp) dGG G G
(5.15)
= -t J (h u. n) öp dR + t ö J (h u . \Jp) dG,R G
then (5.13) becomes:
j (h3 h Oh )Mp = 0 = -ö - \lp. \1p - - u . \1p + - p dG +2 2 O~G '
(5.17)
Here n is a vector with unit length normal to the boundary R and directedoutwards with respect to G.
In (5.17)
j (h3 h Oh )- \1p . \1p - - u . \lp + - p dG2 2 o~
G
is the variation integral. The region over which the integral is calculated maybe multicoherent. If this region is not coherent, the problem may be dividedin part pro blems.
5.3.3. Boundary conditions
An important point to investigate is how different boundary conditions can, be applied to (5.17). For this purpose let us consider the second term of the :right-hand side of that equation, meanwhile defining q)boundnry by means of
-72-
Ó(Pboundnr; = f ((-h3 v» + I~U).D) (Jp dR.R
(5.18) ,
Practical boundary conditions that may occur in different problems are thefollowing.
Boundary condition Ra
.The pressure at a section Ra of the boundary is required to have an prescribedvalue. Since (Jp is not permitted to affect the boundary conditions, the relation(Jp = 0 should hold at this boundary: .
(J(Pboundnry = (J(/jRa = o.Boundary condition Rb
The outflow at à section Rb of the boundary has a prescribed value. Thisoutflow per unit length is
(hU)qn = _h3 \lp +2 .D.
Hence, at this boundary, it is true that
(J(Pboundnry == J qn (Jp dR = (J J qnP dR.
An alternative is that the total flow, qtot n across the section of the boundaryRb is prescribed, it being required that the pressure is uniform everywhere atthis boundary (= Prb). We may then write:
(J(Pboundnry = J qn (Jp dR = qtot n (JPrb.Rb
Boundary condition R;
For part Rel of the boundary it is required that the pressure there is equalto that at part Re2' and that the normal outflow at Rel is equal to the inflowat Re2 (cf. fig. 5.3). This boundary condition is obtained if, for instance, acylindrical surface is cut open and flattened. Then we have:
(J(Pboundnry = J qn (Jp dR + J qn (Jp dR = O.Rel Re2
Boundary condition Rd
For a section Rd of the boundary R it is required that the normal flowacross this section of the boundary is proportional to the pressure on thatsection. Hence we may write:
-- 73-
Fig. 5.3. Boundary condition Rc.
Therefore
The restrictors in hydrostatic bearings may lead to such a boundary condition.
5.3.4. Introduetion of the finite elements
With (5.17) the integrals of the right-hand side ofthe equation are evaluatedboth over the whole region G and over the boundary R. We now split up (5.17)into integrals referring to sub-regions (the elements). Let m be the number ofelements Gk (with boundaries Rk); then we may write instead of (5.17):
In
~f(h3 h M )~rp = 0 = -~ L__; 2" v», \jP-"2u. v» + ()7/ dGk +k=l Gk
m
(5.19)
Within G the boundary integrals of two adjacent elements will cancel eachother, and the sum of boundary integrals will furnish a contribution only ifthe boundary Rk forms part of R (Rk ER). Because (5.19) is equivalent to(5.13) it still contains the exact solution.
1\t this point we start the approximations and apply the finite element prin-ciple. This means that we restrict the solution of (5.19) to those functions, p,which are completely defined for each element by the values they assume in thenodal points of this element, the region between the nodal points being describedby means of prescribed interpolation functions. The choice of the elements andinterpolation functions depends on the nature of the problem to be solved.The interpolation functions should be such that p and its derivation exist in the
- 74 -:-
(5.20)
sub-regions Gk and that the boundary integrals oftwo adjacent elements canceleach other.
For the pressure in the kth element we put,
n
Here the summation involves the total number of nodal points n in G. In con-formity with the requirement that the pressure in Gk is governed solely by thepressure in the nodal points of Gk, it must be true in respect of the interpola-tions functions that
Ikl = ° for nodal points: iE Gk,
and that
{IkV = [0,0, ... , O,lkr> 0, ... , O,lkSO 0, ... , O,lkt' 0, ... l
is the 1x n matrix of the interpolation functions for the pressure in elementGk with nodal points r, sand t.
Furthermore
{p} =
PIP2
Pn
is the n x 1 column matrix of the nodal pressures.For the pressure gradient \lp in Gk it follows from (5.20) that
'è'Jp
'è'JXf= [rkl {p},
'è'Jp{\lp} = (5.21) .
in which
"blkr Ülks «:0,0, ... ,-, ... ,-, ... ,-, ... ,0, ...
'è'Jxf 'è'JXf 'è'JXf
'è'Jlkr 'è'JlkS 'è'Jlkt0,0, ... , -, ... , -, ... , -, ... ,0, ...
'è'JYf 'è'JYf 'è'JYf
is the 2x nmatrix ofthe interpolation functions for the pressure gradient in Gk•
From (5.19), (5.20) and (5.21) it follows that
(5.22)
~75 -
III
111
III
where Rkb and Rkd denote the sections of the boundary R to which respectivelythe boundary conditions Rb and Rd are applicable.
WithIII
(5.23a)
III
(5.23b)
III
(5.23c)
III
(5.23d)
III
(5.23e)
it follows from (5.22) that
{Jr[J= 0 =-{J(1 {PY [a]{p}- {V}T {p} + {qsY {p} + {qnY {p} ++ t {p}T [al] {p }) =
=-(J Ct {pY [a]{p}- {pY {v} + {pY {qs} + {PY {qn} ++ t {pY [al] {p })
-76 -
or
c5{pV ([a] {p} - {v} + {qs}. + {qn} + [al] {p}) = D. (S.24a)
Notes
(1) [a] is symmetrical, since (h3 [rdT [rdY = h3 [r~y [rk];.[al] is symmetrical, since (Ä. {Ik} {/dT)T = Ä. {Ik} {IkV;
(2) [a] is positive definite, since }z3 (\1p. \1p) ~ 0;. [al] is positive definite, since Ä. (p . p) ~ O.
As is evident from (S.24a) the whole problem now resolves itself into avariation problem with a finite number of variables, namely the elements ofthe pressure matrix {p}. As a consequence of the boundary conditions appli-cable to the problem, not all the elements of the pressure vector are necessarilyunknown, and relationships may exist between the elements of {p} and c5{p},etc.
Since the pressure on the boundary is determined via the interpolation func-tions Ik!> advantages are offered by expressing a prescribed qn at R by meansof the Ikl in terms of nodal-point values of these quantities.A difficulty then arises which is due to the fact that the "influence" of a
nodal point extends in general to within the "influence" of other boundarynodal points. The influence of a boundary nodal point in fig. 5.4 extends towithin the two boundary elements. This means that the boundary conditionsapplied to two adjacent boundary nodal points merge into one another, andthis presents difficulties in the calculation of (S.23d) and (S.23e) at locationswhere two sections of the boundary, with different boundary conditions, adjoineach other.By arranging that the sections of a boundary with different boundary con-
ditions do not adjoin one another but are separated by an element, it is pos-sible to define (5.23d) and (S.23e) uniquely. In fig 5.4, RI and RIl as well asRv and RVI are therefore included when calculating boundary conditions, butRIV is disregarded.
boundary cond. A
boundary cond, B
Fig. 5.4. Diffusion of boundary conditions.
-77-
5.3.5. Application of the boundary conditions
Let us now examine how the boundary conditions of sec. 5.3.3 are to beapplied to the variation problem given by (5.24a). In doing so, we shall assumethe two different sections of the boundary to be separated by an element.
Boundary condition RaLet i be a nodal point on this boundary, with a given pressure PI' Of the
pressure variation in this nodal point it is then true to say that
(JPI = O.
Now the term of (5.24a) which corresponds to (JPI disappears. This means thatthe number of rows of, for instance, matrix [a] is then reduced by one (andalso that of [al], {v}, etc.). The number of columns of [a] can also be reducedby one if the.ïth column is multiplied by the known PI and the product addedto {v}. .
The same procedure is followed for the remainder of the nodal points of Ra·
Boundary condition Rb
(1) The local value of qn is prescribed.In that case account has to be taken of (5.23d) alone. As a modification of
(5.23d), instead of qn along the whole boundary, qnl in the nodal points i maybe prescribed, and the intermediate values can be determined with the helpof the interpolation functions fkl' If {qn} is the column matrix with elementsqni, where
qnt = 0 for i f/= Rb'qnl is prescribed for i E Rb'
then we have for element k: qn (x" Yt) = {fkV {qn}. Thus (5.23d) becomes:m
{q ny = ~ f {qnY {ik} {fkY dRk·k= 1Rkb
(5.23d')
(2) The total value of qn is prescribed for a section of the boundary at whichall the nodal-pressure variations are equal.
This implies that the terms of (5.24a) associated with these boundary nodalpoints may be added, the value qn, being inserted for the sum of the corre-sponding elements of {qn}. The number of columns of [a] and [al] can bereduced by addition of the columns associated with this boundary.
Boundary condition R;Let i and j be two corresponding nodal points on the two 'adjacent bound-
aries. We may then write
[a*] {p*}- {v*} + {qs*} + {qn*} + [al*] {p*} = o. (5.24b)
-78 -
{JPI = (JP).
This means that the coefficients of (5.24a) corresponding to ÖPI and (Jp) maybe added and that the corresponding rows in [a] and [al] may be added, too(and so may the "rows" of {v}, {qs}, and {qn}). The number of columnsof [a] and [al] is reduced by one, because of
PI =p),
and the ith andjth columns of [a] and [al] can be combined to a single col-umn by addition.
Boundary Rd
This boundary condition merely affects (5.23e). In this equation, A canfurther depend upon the coordinates xf and h. Just as we saw for qn in thecase of Rb' the column matrix {A} made up of the nodal-point values AI> canbe formed with
AI is prescribed, for iE Rd'
which results in
Thus (5.23e) becomes
'"[al] = ~ J {Ik} {IkV {Ä} {IkV dRk.
k= 1Rkd(5.23e')
If the boundary conditions are applied in the manner just referred to, theremaining pressure variations are independent and we obtain the solution ofthe pressure field from the set of equations *)
5.4. The finite-element method based on the "generalised" Reynolds equation
5.4.1. The generalised Reynolds equation
If to a viscous flow of liquid between two surfaces the Reynolds differentialequation is applicable and if one of the surfaces is provided with identicallyequal, parallel and equidistant grooves with a period Ig (fig. 5.5), then the pres-
*) The index * has been added to "distinguish the symbols used in (5.24b), from the corre-sponding symbols of (5.24a), where the boundary conditions still had to be applied.
sure distribution can be imagined as being built up of a smoothed pressure Psmand a superimposed fluctuating pressure with period 10 (fig. 5.6).
As it is just the smoothed pressure that proves to be of importance in thedetermination of the load capacity we shall, in what follows, derive a "gener-alised" Reynolds equation for that pressure.
From (5.10) and (5.11) and fig. 5.5 it follows that
bpqs =-Ch +hs)3 - +t (h +hs) us,
bs
-79 -
oq, bqs bh-+-+-=0,ör ös ö'r
_. _'_,' _._ ..J,,,,,
Fig. 5.5. Plan view and cross-section of a grooved bearing section.
Fig. 5.6. Smoothing of the local pressure p.
(5.25a)
(5.25b)
(5.26)
in which we have
q,= q, (r, s, r)
qs = qs (r, s, r)
U,= u, (r, s, r)
u,= u, (r,s, r)
p =p(r,s,.)
h = h(r,s,.)
hs = hls)
- 80-
flow in r-direction,
flow in s-direction,
velocity of grooveless surface in r-direction,
velocity of grooveless surface in s-direction,
pressure in film,
film thickness,
groove depth,
time,
period of grooves.
The above-mentioned parameters are dimensionless and are based on thesame reference parameters as used in (5.6).For the special case that
M M 'Oh- = - = - = 0 (5.27)'Or us u.
it can be readily shown that there is a solution of (5.25) and of (5.26), of thefollowing form:
qs = constant,
q,= qr(s),op- = function (s),usop- = constant.or
(5.28)
b b b
1 J qs ds 1 J up u, J dsI; (h + hs)3 = - I; us ds + 2 Ig (h + hsY'
a a a
(5.29a)
In the calculating of bearings it is often found that the boundary conditionsare su.c~that for a small area - for instance of width Ig - the relations of
. (5.28) are approximately satisfied. We now make use of this fact in the choiceof the integrands for the integration of (5.25) and (5.26) over an s-interval ofwidth Ig:
b b b
1 JIJ op u, J- q, ds = - - (h + lisP - ds + - (h + hs) ds,Ig Ig or 21g
a a a
(5.29b)
b1 J ('/)qr '/)qs M)- -+-+- ds e=D,19 '/)r '/)S '/)7:
a
(5.30)
- 81-
in which/g
a= s- -'2 '
We now define
the smoothed pressure:1 b
Psm = - J pds,19 a
(5.31)
the smoothed flow in the r-direction:
1 b
qsm r = - J qr ds19 a
(5.32)
and:
1 b
In = - J (h +hs)n ds./y
a
(5.33)
This yields:b
~J qsds =- '/)Psm + Us 1_2,/g (h + lls)3 (lS 2
a
(5.34a)
b
1 J '/)P u,qSIll r = - - (h + lls)3 -ds + -1+1'
/g '/)r 2a
(5.34b)
b b
'/)qsmrIJ oq, I J ()h--+- -ds+- -ds=O.ör 19 '/)s 19 '/)7:
a a
(5.35)
If we approximate the set of equations (5.34) and (5.35) by the followingset of equations:
(5.36a)
'DPsm u,qsmr =-1+3 -- +- 1+1'
'Dr 2
'Dqsmr ''Dq. oh--+-+-=0,'Dr ös 'Dl'
the solution for q., qsm.. and Psmobtainable from (5.36) and (5.37) is usable onlyif substitution in (5.34) and (5.35) gives merely a negligible error. This leads,for example (depending on the relative magnitude of the terms, other condi-tions can also be imposed), to the following requirements:
- 82-
,'-
I 'DPsm I Ib 3 'Dp I I 'DPsmI1+3---- (l1+h.) -ds = 82« 1+3--,'Dr Ig 'Dr 'Dr
a
bI 'Dq. - I 1 oq, I I 'Dq. I- -ds = 83« 1;''Ds Ig 'Dsa
I». 1 bM I I 'öh I-I -ds = 84« 'Dl' .'Dl' Ig 'Dl'
a
(5.36b)
(5.37)
(5.38a)
(5.38b)
(5.38c)
(5.38d)
Here 81 and 82 are the errors in respectively (5.36a) and (5.36b), the error in(5.37) being determined by 83 and 84'
The quantities q., qsmrand Psm in (5.38) are approximate values obtainedfrom (5.36) and (5.37). From these, and by the use of (5.25a) and (5.31), wecan evaluate the quotient 'Dp/'Dr in (5.38b). From (5.25a) follows:
'D2p 'D ( -qs U.)-=- +t---()s ör 'Dr (h + 11.)3 (h + hs)2
or, .as q. is known,()p- = C(r, 1') + funct (r, s, 1').'Dr
The integration constant C can be calculated from (5.31):
b b
1 1 'DPsm 1 f 'C(r, 1') = - C(r, 1') ds = _- - fer, s, 1') ds.Ig 'Dr 19
a a
(5.39)
(5.40)
(5.41)
- 83-
So far the boundary conditions have been left out of consideration. Theobvious course is to introduce the boundary conditions of the set of equations(5.36) and of (5.37) by way of the quantities q., qsm, and Psm. This, however,means that in general the solution obtained no longer satisfies the local bound-ary conditions. Let us assume, for instance, that at a boundary we have P = o.This implies that Psm= 0, too; hence:
P = 0 => Psm= 0,
but Psm= ° does not imply that P ...:..._0, for only if 10 ~ ° is it true thatP ~ Psm, and then P = ° <:> Psm = O.
To determine the pressure variation at the boundary and its effect on thepressure at a sufficiently great distance from the boundary, "the end effect",it is still necessary to solve the equations (5.25) and (5.26) in terms of localquantities (for the determination of the end effect, cf. also Muijderman 5-11».
At this stage we have to answer the question of whether it is permissible touse the smoothed pressure,psm, instead ofthe local pressurep. We are interested I
in the load capacity of a bearing, hence in integrals of the type '
'2 s2J Jp dr ds.
Consequently, in addition to the relations (5.38a) to (5.38d), we must imposethe following conditions:
1
1 bib I 11 b- J pds-- rPsmds = es« - J pdsl;~ ~ ~a a a
in other words,
IPsm - :0 lPsm ds I= es « IPsmi.a
(5.38e)
The relations (5.38) enable us to make an estimate of the accuracy, for verygeneral cases, of a solution of (5.36) and (5.37) and to ascertain whether it ispermissible to use the smoothed pressure thus found for calculations of theload capacity.
A particular case where (5.38) is satisfied is obtained if
10 ~ 0, ·hence if b ~ a and ek-+ ° (k = 1,2,3,4, 5).
This result has been used by, among others, Vohr and Pan 5-8) for the deri-vation of a generalised Reynolds equation, but with respect to (5.38) this isan unnecessarily stringent limitation.
-'84-
In vector notation, (5.36) and (5.37) can be written as
qsm = - C \7Ps", + t B u,
Oh\7 . qsm + - = O.
01.
(5.42)
(5.43)
In the coordinate system r,s (fig. 5.5) the vectors qsm and u can be represented(taking q. = qsm .) by
qsm: {qsm}r. = {:::Ju : {u}r. = {~J
(5.44a)
(5.44b)
and \7,' C and B by
\7:_\:,
{\7 }r. -/ ~ •
os
(5.44c)
B: [B]r. (5.44d)
C: [C]" (5.44e)
.The indices rs are used here to indicate that the (r, s)-system applies.
In the (xf> Yf)-system (see fig. 5.5) the components qf qsnu u, and \7 canbe obtained by changing (r, s) into (xf> Yf). The components of Band C can
. be deduced from
[B]XfYf = [T] [B]r. [T]-\
[C]XfYf = [T] [C]r. [T]-i,
with
[T] = [ C?s ct -sin ct J.S10 ct cos oe
On working this out we obtain:
- 85-
(1+1 - ~~:) sin a cos al1 '
1 . 2 -2 2 J+1 sm a +- cos aL3
(S.4Sa)
-(1+3- I~Jsin a cos al.
cos? a1+3sin- a+--L3
(S.4Sb)
By eliminating qsm one finds from (5.42) and (5.43) the generalised Reynoldsequation:
()Il\l . (C \l Psm) - t \l . (B u) - - = 0
()1:(5.46)
5.4.2. Transformation of the generalised Reynolds equation into a variationintegralThe derivation is similar to that of the ordinary Reynolds equation given in
the preceding section. Instead of (S.l3) we obtain:
MP = - f ( \l . (C \lPsm) - t \l . (B u) - :~) c5Psm dG = O. (5.47)G
Given that
I [\l . (C \lPsm)] c5PsmdG = I tv. (c5Psm C \lPsm)] dG- I (C \l Psm • \l c5Psm) dGG G G
R= I(C \lPsm' n) c5Psm dR- c51(1; C\lPsm' \lPsm) dG,
(S.48a)G
G G-t I [\l (B u)] L~7JsmdG=-t I [\l . (ÖPsmB u)] dG +t I (B u . \l c5Psm)dG
G
=-t I (B u . n) c5Psm dR +t c5I (B u . \lPsm) dG,R G (S.48b) ,
.: j()h- - ÖPsmdG = -c5 - PsmdG (S.48c)()1: ()1:
G G
·- 86-
and M .(j(/J = 0 = -(j f (C \l Psm . 'VPsm- tB u . 'VPsm + (:J1/sm) dG+
G
,- f [(-C \lPsm + tB u) • 0] (jPsmdR, (5.48d)R
we can now write, by analogy with (5.22):RI
(j(/J =-(j If(t {PY hY [Cl [rd {p}- t {uV [B] [rd {p} +k-l Gk 'öh )- + - {fdT {p} dGk +
'ö.111 •
(5.49)
and withRI
(5.50a)
k=l
RI
(5.50b)
k=l
RI
(5.50c)
RI
(5.50d)
k=l
RI
[al] =I£dÀ {Id {IkV dRk>k=l
(5.50e)
it follows that
(j {pV ([a] {p}- {v} + {qs} + {qn} + [al] {p}) = O. (5.24a)
(5.51)
./
- 87-
This equation is equal to eq. (5.24a) and the given boundary conditioris canbe introduced in the same way.
5.4.3. The integrals L3' L2' 1+ 1 and 1+3/or rectangular and triangular grooves *)
Rectangular-groove pattern (fig. 5.7)
We assume the groove pattern to be sufficiently fine, so that
h R:J constant }h
for 0 ~ s ~; t,h + s = hgr R:J constantthen
Fig. 5.7. Rectangular groove shape
Triangular groove pattern (fig. 5.8)
We assume that
h R:J constant for 0 ~ s ~ Ig.
It can easily be shown that the outcome of the integrals is independent of thegroove shape and is governed solely by the width and depth of the groove.
Hence, for the calculation, we start from a groove with the shape of a right':'angled triangle (indicated in fig. 5.8 by a dotted line).
*) Reference should be made also to Ir J. Bootsma, "Spiral groove bearings with arbitrarygroove shapes", Philips Res. Lab. Techn. Note No. 96/69.
We have
- 88-
19
Fig. 5.8. Triangular groove shape.
1_3 = _1_ (y h-3 __ h_gr_2_-_h2_),
1 + y 2 (l1gr- h)
1-2 = _1_ (y /z-2 __ hR_r-_l_-_h_-_l),1 + y I1gr- 11
1 ( 11 2_ h2 )1+1= -- y 11 + _gr _I+y 2(l1gr-l1) ,
+ I1g/- 114).
4 (l1gr- 11)
(5.52)
1+3 = _1_ (y 113I+y
5.5. Example of an element: the triangular element
A very simple element is a triangle with the angular points as nodal points,the interpolation functions being linear. Let Xfh xfj, Xj'k be the coordinatesof the nodal points (fig. 5.9). By means of interpolation functions the pressure(we shall not distinguish between pand Psm and just write p) in element e canthen be described by
Xii
Fig. 5.9. Triangular element.
P(Xf) =!elp(xfl) + lelP (Xfj) +lekp(xfk),
\lp =P(xfl) \llel + P (XfJ) \llel + P(xfk) \llek'
(5.53)
(5.54)
Introducing the new coordinates A and fJ-, we can rewrite Xf (cf. fig. 5.9) as
- 89-
(5.55)
It follows thatlel = 1- fl,- A,
lel = fl"
lek = A;
b/eh bfl, biel. bA--+---b,u bXf bA bXf
biel. bfl, bie" bA--+---bfl, by! bA by!
(5.56)
(5.57)
From (5.55):bA Yn - YII
bXf ap
bfl, Yfl- Yr«bXf ap
bA Xfl- XfJ
by! ap
bfl, Xfk- Xfl--bYf ap
(5.58)
with
ap = I Xfk- XII XfJ- XII I = 2x area ofthe triangle.Yr« - Yfl YfJ - y"
From (5.57) and (5.58):
(5.59)
ith column jth column kth column1 [0,0, ... ,0, (Yfk- YfJ),· .. (YII- Yfk)' ... (YfJ - YII)' ..• 0,... ]
[rkl = - .ap 0,0, ... ,0, (xfJ- Xfk)' ... (Xfk- Xfl)' ... (Xfl- Xfj), ... 0,...
(5.60)
bXf bXf
bA bp,det = = apo
bYf bYf-- -bA bp,
REFERENCES
(5.62)
- 90-
To calculate expressicris (5.23) and (5.50) it may be necessary to determineintegrals of the type
IG lJ(nm)= J (fel)n o.s: dGkGk
= 1,2,3,
j = 1,2,3,
n = 0,1,2, ,
m = 0,1,2, .
(5.61)IR lJ(nm)= J (fel)n (fel)'" dRk
Rk
The solution of these integrals is a simple matter provided that we use thecoordinates Aand u, For example:
J (fel)n (fel)m dGk = J J (fel)n (!el)'" dXf dYJ= J J (fel)n (fel)'" det dJ d«,Gk
in which
5-1) J. F. Besseling, The complete analogy between the matrix equations and the fieldequations of structural analysis, International Symposium on Analog and DigitalTechniques Applied to Aeronautics, Liège (1963).
5-2) O. C. Zienkiewicz and Y. K. Cheung, The finite element method in structural andcontinuum mechanics, McGraw-Hill Publ. Co. Ltd, London, 1970.
5-3) M. M. Reddi, Finite element solution of the incompressible lubrication problem,Trans. ASME, J. Lub. Techn. 91, 524-533, 1969.
5-4) M. M. Reddi and T. Y. Ch u, Finite element solution of the steady-state compressiblelubrication problem, ASLE-ASME Lub. Conference Houston, Texas (Oct. 1969).
5-5) J. H. Argyris and D. W. Scharpf, The incompressible lubrication problem, TheAeronautical Journalof the Royal Aeronautical Society 73, 1044, 1969.
5-6) G. Vogelpohl, Beiträge zur Kenntnis der Gleitlagerreibung, VDI-Forschungsheft386,edition B, vol. 8, Oct. 1937. .
5-7) G. G. Hirs, The load capacity and stability characteristics of hydrodynamic groovedjournal bearings, Trans. ASLE 8, 296-305, 1965.
5-8) J. H. Vohr and C. H. T. Pan, On the spiral-grooved, self-acting gas bearing, MTIreport no. 63TR52, Jan. 1964.
5-9) A. Cameron, Principles of lubrication, Longmans Green and Co. Ltd, London,1966.
5-10) O. Pinkus and B: Sternlicht, Theory of hydrodynamic lubrication, McGraw-. Hill Book Co., Inc., New York, 1961.5-11) E. A. Muijderman, Spiral-groove bearings, Philips Res. Repts Suppl. 1964, No. 2.5-12) R. Couran tand D. Hil bert, Methods ofmathematicalphysics, IntersciencePublishers,
1966.
- 91-
6. CAVITATION
6.1. Introduetion
If the pressure in the lubricant film of a liquid bearing falls to a certain value,which lies between the vapour pressure of the lubricant and the partial pressureof the dissolved gases, then a further drop in pressure is prevented by thebreaking of the coherence of the film, due to "boiling" of the lubricant orescaping of dissolved gases, or both; this phenomenon is called "cavitation".For the time being we shall confine ourselves to cavitation in a steadily loadedbearing.
The flow in the region of cavitation does not obey the Reynolds equation.Figure 6.1 shows this region of cavitation in the lubricant film of a journalbearing. Sometimes part of the boundary of the region of cavitation is fixeddue to the presence of a lubricant groove (e.g. Sassenfeld and Walther 6-4)),but the position of the boundary is often governed exclusively by both the flow
Fig. 6.1. Cavitation, pressure profile and film thickness in a plain journal bearing.
- 92-
in the full-film region and that in the cavitation region. The determination ofthis boundary between two regions with different flow equations complicatesthe calculation of a cavitating bearing very much. In the literature this problemis usually avoided by specifying such conditions for the cavitation region thatthe calculation can be made fairly straightforwardly and the situation in thecavitation region is then approximated 6-3/6). But these éavitation conditions,which can be handled easily enough in the calculation, frequently describe theflow in the bearing in a physically incorrect manner. By physically correct con-ditions we should mean a description of the flow in the cavitation region insuch a way as to satisfy at least the continuity law throughout the bearing. Itis precisely the disregarding of the continuity law that in some cases can giverise to substantial errors in the calculation. That is the reason why we put thislaw first and foremost as a cavitation condition.
Physically correct cavitation conditions have been established by Jakobsonand Floberg 6-1) (and referred to also by Horsnell 6-2). The author is fàmiliarwith only one paper 6-2) in which an attempt 'is made to use the Jakobsonand Floberg (J & F) condition in the calculation, but in a later article 6-5)the workers just' referred to have decided in favour of approximate cavitationconditions, among other reasons on account of the time-consuming computercalculations. In all these papers it is pointed out that the choice of the cavita-tion condition does slightly affect the load capacity, maximum pressure, andattitude angle. This is, indeed, often the case, but it does occasionally happenthat the deviations become impermissibly great. In what follows it will beshown that, when the J & F condition is applied to, for example, a smoothjournal bearing with circumferential oil grooves, the load capacity wil be afraction of that following from the - frequently used - data published bySassenfeld and Walther. Itmay be reassuring for the users of helically groovedjournal bearings, which are frequently fed fromjust the circumferential grooves,that these bearings, unlike smooth bearings, are far less sensitive to the supplypressure of the lubricant. In general it may be said that the application of anapproximate cavitation condition does not yield the true flow balance of abearing and will cause the largest deviations in those bearings that are sensitiveto this effect.
In the following we shall describe a method, called the dummy-flow method,which greatly facilitates the application of the J & F condition in the compu-tation of a bearing and which consumes little time. Thus the use of approximatecavitation conditions and the associated uncertainty as to the results can beavoided.
6.2. Cavitation conditions
6.2.1. Conditionsfor smooth journal bearingsFor the sake of clearness we shall first describe the J &F condition (condi-
Iql:S:;;Ithul. (6.3)
- 93 .:._
tion 4) and subsequently the conditions 1, 2 and 3. In the J &F condition aflow pattern is proposed, for the cavitation region, which guarantees continuityof the flow everywhere in the bearing and seems to approximate very closely thereal flow in a steadily loaded smooth journal bearing 6-7,8).
Condition 4, Jakobson and Floberg conditionsThe flow pattern which is in conformity with the J &F condition is as fol-
lows.From (5.10) it follows that the flow in the full film of a smooth bearing is
given by
q= -h3 \lp + th u. . (6.1)
At the leading edge of the cavitation region the pressure in the full film hasdropped to a level where cavitation sets in; this means that the component of\l p along u is negative or zero, and
Iql ;;:;:Ith u]. (6.2)
In the cavitation region it is true that
Assuming that h is a continuous function of the film coordinates, (6.2) and (6.3)imply that at the leading edge \lp = 0, and the conditions for the leadingedge become:
\l p = 0 and P = Pcny' (6.4)
At the trailing edge the pressure begins to rise again and the relation (6.3)holds for both the cavitation region and the full film. Now the full film maystart with \lp> 0, and this is determined by the continuity of flow: if weassume that for smooth surfaces the flow in the cavitation region is parallelto u, then the amount of lubricant entering the cavitation region via aa' (fig.6.1d) should be equal to the amount leaving across bb'. Experiments 6-7,8)
support this model of the flow.Let us next describe the approximate cavitation conditions.
Condition 1The pressure distribution in the bearing is calculated as though no cavitation
occurred, and hence as though the Reynolds equation did apply in the cavita-tion region. Next, by putting P = Pcny if it is found that p <Pcny, the cavitationregion is approximated.The handling of condition 1 is very simple. However, the pressure gradient
at the leading edge (saa'c in fig. 6.1d) is discontinuous, and hence the flow isdiscontinuous as well. Besides, the' flow entering the full film at bb' will ingeneral be unequal to the flow crossing at aa'.
-94 -
2Tl
Condition 2
For the leading edge saa'c of the cavitation region we haveI
\lP = 0 and P = Pony.
As this is equivalent to (6.4), the flow at the leading edge will be continuous.Furthermore the trailing edge is fixed at (]J = 2:n:,i.e. at the location of thewidest film gap. In general the flow at the trailing edge will then be discon-tinuous. If an axiallubricant groove is present at (]J = 2:n:,which compensatesdiscontinuities in the :flow, then condition 2 is physically correct. This situa-tion has been taken as the starting point in the calculations by Sassenfeld andWalther 6-4).
Condition 2 can readily be used if the field of pressure in the bearing is cal-culated by an iteration method (cf. ref. 6-5). Then the usual method is to putP = Pony if, in the iteration proces of nodal pressures, it is found that P < Poay'
Condition 3
For the leading edge as well as for the trailing edge it is assumed that
\lP = 0 and P = Poay'
The liquid flow at the leading edge is continuous. However, the flow leavingthe cavitation region via the trailing edge will generally be greater than theflow entering this region via the leading edge. This cavitation can easily beapplied in an iteration method of calculation, as also was condition 2.
p
-- condition 123
"
Fig. 6.2. Typical shapes of pressure profiles in the middle of the bearing, for the differentcavitation conditions.
The typical shapes of pressure profiles for the different cavitation conditionsare displayed in fig. 6.2. Here we have assumed that the reference pressure ateither end of the bearing (P = ± B12) is equal to zero.
6.2.2. Modification of the Jakobson and Floberg conditions for grooved bearings
For smooth surfaces the J& F conditions are supported by experiments 6-7.8)
which show that the :flowfrequently crosses the cavitation region of a steadily
(BU.U)(J.c = arccos 18 ui X lul . (6.8)
- 95-
loaded bearing in ribbon-shaped flow paths (streamers) parallel to u. If weassume that, in the streamers, the liquid makes contact with both surfaces and\lp = 0, then (6.1) also forecasts a parallel flow:
q=-thu (in a streamer). (6.5)
Our experiments with grooved surfaces show that in this case the flow is notentirely parallel. This makes a modification of the J &F conditions for groovedsurfaces necessary and we shall base the modification on the equation, corre-sponding to (6.1), for grooved surfaces:
qsm = - C \l Ps," + 1- B u. (6.6)
Assuming that \lP = 0, (6.6) reduces to
(6.7)
and the angle «; between the flow vector, qSlm and u (fig. 6.3) becomes
In the modified J & F condition it is therefore assumed that the flow in thecavitation region makes an angle a; with the direction of u!
Note. Using (6.8) for smooth surfaces we find (J.c = 0, as was to be expected.
Fig. 6.3. The characteristic flow path in the cavitated film of a plain cylindrical bearing Ca)and a helically grooved cylindrical bearing (b).
- 96 ___
6.3. The dummy-flow method
We shall now discuss the manner of including the modified J & F conditionin the calculation program.Assuming that there is no cavitation in the lubricant film the set of equations
(5.24b) can be generated. Boundary condition R; will be ignored for this case,so that we obtain:
[a*] {p*}- {v*} + {qs*} + {qn*} = O. (6.9)
By means of (6.9) the unknown nodal pressures may be determined for the casethat cavitation is suppressed. The ith equation, corresponding to the ith un-known pressure, may be written as follows (p/ = Pj): .
n·
~ aij*Pi>: v,* + qs,* + qn,* = O.j=1
(6.10)
Equation (6.10) states that the net outflow at nodal point i should be zero.Introducing the symbol q, for the net outflow:
n·
(6.11)
and equation (6.10) can be written as
q, =0. (6.12)
Let us now consider the same lubricant film, but with cavitation. In the cavi-tation region the equation of Reynolds does not normally apply; by means ofan artifice, however, we can make the Reynolds equation hold also in thisregion.For this purpose we make the nodal points leaky, as it were, in the cavita-
tion region, and we keep the pressure in this region equal to Pcnv by supplyingor drawing off a sufficient quantity of liquid at the location of the nodal points.In this case q, =1= 0, and the flow that has to be supplied to the nodal pointwill be called the dummy flow (fig. 6.4).
pressure
tL\~"'i=1 ==l=l=Pc=av~==(-_"Ijl'~I
regionFig. 6.4. Dummy flow in characteristic flow path of a cavitated film.
- 97-
The film diverges in the vicinity of the leading edge and the duminy flowwill be positive (fig. 6.4). Near the trailing edge the dummy flows are negative.The dummy flows, together with the main flow, will move along a characteristicflow path (fig. 6.3) to the trailing edge, which is where the sum of the dummyflows along the flow path equals zero.
In case of cavitation an iterative solution of (6.10) is required. In the full-film region a new approximation of the pressure in the nodal point i is foundfrom the surrounding nodal-point pressures:
(6.13)
If it is found that P <Pcav, then we are obviously concerned with a nodalpoint in the cavitation region and we put PI = Pcav. With the help of (6.11)the dummy flow for this nodal point is subsequently calculated.
Calculating "downstream" in the cavitation region we again find, near thetrailing edge:
PI> Pcav·However, if
I:dummy flows > 0 (6.14)
is applicable to a characteristic flow path, then the generation of positive pres-sures, and hence that of the trailing edge, should be suppressed and we canput once more PI = Pcav and calculate the dummy flow of i. Thus in a specificflow path, the trailing edge cannot begin until the dummy flows just cancel out.
Note. If it is possible for the modified J& F method to be applied in the cal-culation program, then 'use can again be made of the conditions 1,2 and 3.For condition 1 this is trivial.Condition 3 is obtained if the generation of the trailing edge is not sup-pressed as long as (6.14) is still true.With condition 2 the trailing edge is :fixedand can thus be processed asthe "natural" edge.
From the foregoing it follows that in our calculation we have to follow thedummy flow in the cavitation region on its road along a flow path and that wemay again accept positive pressures, hence allow the full film to start, onlywhen the dummy flow is fully absorbed again. .If, in a smooth bearing, 'we adopt a rectangular distribution of nodal points,
in which, for instance, the columns of nodal points are parallel to the vector u,then these nodal-point columns will be capable of functioning as characteristicflow paths (since etc = 0) and it will be possible during the calculation to keep
- 98-
a record of the total dummy flow per column of nodal points. With a groovedsurface, on the other hand, the flow path will then make an angle ('4with thecolumns. As an approximation the total dummy flow, qd' can then be suppliedto the next row of nodal points, according to the scheme shown in fig. 6.5.
total dummyflow qd
~~,~ aI Jèl+b qd
" b +--row
tcolumn
Fig. 6.5. Distribution of total dummy flow qd, over the next row of nodal points.
6.4. Examples
6.4.1. Effect of the parameter Pc av on the load capacity of a smooth journal bear-ing with circumferential feeding
In this section we shall consider an example of a bearing which is highlysensitive to the cavitation condition used. The load capacity of this type of
. bearing may be largely overestimated if the J & F condition is not used. Theconclusion that can be drawn from this example is that the use of the J & Fcondition is at all times preferable, particularly because, with the dummy-flowmethod, this takes but little extra computing time.The bearing in question is sketched in fig. 6.6. This smooth journal bearing
rotates, as it were, in a medium of lubricant with a (reference) pressure equalto zero. This pressure governs the boundary conditions at either end of thebearing. If the pressure anywhere in the bearing falls below the parameter Pcnv,
then cavitation will occur. The value of this parameter, which is deterrninedby the vapour pressure, the partial pressure of the dissolved gases, the tem-perature, etc., must be given in order to make the calculation of the bearingpossible. This parameter can be calculated from the bearing data, for example.
cross-section cc'
Fig. 6.6. Cavitation in a smooth cylindrical bearing with circumferential feeding.
- 99--
pressure,Let us assume that a bearing cavitates at a pressure of 1 bar under the ambient
With'YJ= 0·08 Ns/m", Q = 100 rad/s, R/iJR = 103,
Pcav becomes:r.: r.; (LI R)2 _Pc.v=--=--- -- =-10 3.Po 12'YJQ R
Figure 6.7 is a plot of the load capacity, calculated with the finite-elementmethod, together with the J & F cavitation condition (see also chapters 5 and 7),of the bearing given in fig. 6.6 as a function of the bearing eccentricity, e;:ç, fordifferent values of the parameter Pc.v' Furthermore the load capacity accordingto Sassenfeld and Walther 6-4), which is often taken as the basis for the cal-culation of the load capacity of such a bearing, is drawn in this figure (ALG).
Figure 6.7 shows that for small values of Pc.v the load capacity can be afraction of that calculated by Sassenfeld and Walther (S &W).
The results do agree with the S 8iW data only if IPc.vl ~ 0·01. At highvalues of IPcavl, i.e. IPcavl= 0,1, the load capacity exceeds that ofS &W be-cause cavitation is suppressed.
f
i
p"",=-{)()()l
//'_'
·5 -e)Fig. 6.7. The load capacity of a smooth cylindrical bearing with circumferential feeding(BID = 1,e~=0).
- 100 - \
Figure 6.8 gives the pressure in the various nodal points of a bearing halffor ex = 0·8 and different values of PC'V' The nodal points where the filmcavitates are marked by.the symbols vvvvvvvv.·
The nodal-point coordinates are defined in fig. 6.1. It is clearly seen that thecavitation region increases with decreasing IPc.vl : the bearing looses, as it were,its lubricant. This can also be understood physically, if it is realised that it isprecisely the pressure difference (0 - Pcnv) which provides for the filling ofthe bearing at its edge. Itwill become more difficult to compensate the leakageon the high-pressure side as the pressure difference (0:- Pcnv) decreases.If Pcnv = 0, the, filling mechanism ceases to function, so that the load capac-
ity is found to fall to zero.
Note. An ALG bearing (see chapter 7) with a lubricant groove at the locationof the widest film gap is filled in quite another way. Here, independentlyof Pc.v, the full film starts at all times at the widest film gap, and it ispractically impossible for the bearing to "loose its lubricant". In thiscase Pcnv has far less influence. Sassenfeld and Walther used for their'calculations Pcnv = 0, a value at which, in the bearing just described,a complete loss of load capacity does in fact result.
o ____. ¥ 8/20- -.,070.2 ••2910. 2 -.24'0- 2 -.1760. 2 -.91'0. , ••0000- 0
+.60'0. 2 +.,8'0. 2 +."'0. 2 +.'970- 2 +.2260_ 2 _.0000_ 0+.1,60- 1 +.1490. 1 +.",.- 1 +.998•• 2 +.,61'0-2 -.000.- 0+.262.- U +.2" •• 1 +.220.- 1 +.167•• 1 +.940.- 2 ••000.- 0+.'90•• 1 +.'7'.- 1 +.'26.- 1 +.249•• 1 +.140.- 1 -.0000- 0+.5500. l' +.'280- 1 +.4610- 1 +.35'"- 1 +.1980. 1 ••000.- 0+.7,40. 1 +.7260- 1 +.6'90- 1 +.490.- 1 +.2760. 1 -.0000- 0+.10'0.0 +.9910- 1 +.8740. 1 +.677.- 1 +.,890. 1 _.000._ 0+.1410.0 +.1,6.- 0 +.1200- 0 +.9"0. 1 +."'0. 1 ••000.- 0+.19'0.0 +.1890- 0 +,1680. 0 +."'0. 0 +.76,0- 1 ••000.- 0+.2740.0 +.26,.- 0 +.2'70- 0 +.1870. 0 +.1110- 0 -.0000- 0+.,860.0 +.'7'0- 0 +."70- 0 +.2700. 0 +.1620_ 0 ••0000- 0+.,400. 0 +.'270- 0 +.4790. 0 + '870. 0 +.2'70. 0 -.0000. 0+.7280. 0 +.712.- 0 +.6"0- 0 +.", •• 0 +.", •• 0 -.000 •• 0+.877•• 0 +.8".- 0 +.78'0. 0 +.647.- 0 +.4090. 0 -.0000- 0+.80'0- 0 +.77'0. 0 +.6990. 0 +.'7'.- 0 +.,600. 0 -.000._ 0
1\-+.'9'0.0 +.", •• 0 +.,020. 0 +.226._ 0 +.1220.0 ••0000- 0-.,4'0- 1 ••", •• 1 -.7,40- 1 -.947. -1 -.9770· 1 -.0000. 0~ ~ ~'Y "'I'VV'VV¥VYY y •••••••• -.000 •• 0yyyVvyyyy y •• vv •••• yyyYYvvvv ~y •••••••••••••••• -.000•• 0'Y"IfV'VVVVVY Y'Y ••••••• ;,n it ........ " •••••• " ••• '.'.'V'TY -.000 •• 0VVVYVYYYV ~ ••••••••••••• , ••••••••••••• _.000_. 0~v ••• v.v ••••••••••••••••• , ~ •••• v. -.732 •• t -.000 •• 0YVYYYYVYV " •••••••••••••••• '" ..,....;.t ••••• -.607.· 1 -.000 •• 0
~
•••••••••• ~v ••••••••••••••• ;898.- 1 -.'0'0. 1 ••0000- 0., •••••• , ••••••• "•••••••••• -.72" •• 1, -.399.- 1 -.000.· 0""" ....., ••981.-. 1 -.7870. 1·_.,64•• 1 ••'08.- 1 '••000.- 0-.760•• 1 -.69a.- 1 ••'80•• 1 -.42'0. 1 ••2".- 1 -.000.- 0••'22•• j •• 492.- 1 -.418•• 1 -.'09.- 1 -.169.· 1 ••000 •• 0
~
-.'57•• 1 ••,40.- 1 ••2910. 1 -.216._ 1 ••1190. 1 ••000._ ~-.2300. 1 ~.220 •• 1 ••189•• 1 -.141.- 1 ••77'•• 2 -.000.- 0••125.- 1 -,119 •• 1 -.10'.· 1 -.76,•• 2 -.417.· 2 ••000.- 0
al~x ~y ~ ATT.AIIa!.!:
+.,8486•• 0 +.4685' •• 0 +.606•• 0 +.88,
Fig. 6.8a. The pressure in the various .noda! points of a bearing half; ex = Q'8, ey = 0,Penv= -0,10.
o ___. w0- VVVVVVVVV VVVYVVVVV VVVVVYVVV VVYVYVYYV -.491'- 2
VVVVVYVVV VVVVVVVTV V"fV'V"Vvvyy YVYVV'VVYV ... 428.- 2VVVVVVVVV VVVVVVVYY VVVVVVVVV -.7270; 2 -.28).- 2VVTVVVVVV VVYVTYYYY VvvvYYY'rY _.4200_ 2 -.824.- )TYVVVTVVV YYVYYTYVV -.992.- '2 '-.1920- ) +.220.- 2YTVVYvvvv -.7880- 2 +.4)50- 2 +.9290- 2 +.7490- 2+.1990- 1 +.2290- 1 +.26,.- 1 +.2,420- 1 +.156.- 1+.5750- 1 +.5740- 1 +.544.- 1 +.4480- 1 +.272.- 1+.10)0- 0 +.1000- 0 +.91'.- 1 +.7)).-' 1 +.4)6.- 1+.1620- 0 +.1580- 0 +.1420- 0 +.11'0- 0 +.67'0- 1+.24'0- 0 +.2)80- 0 +.2150- 00.1710- 0 +.102.- 0+.)600- 0 +.)510- 0 +.)18.- 0 +.2560- 0 +.1550~ 0+.5190- 0 +.'070- 0 +.462._ 0 +.)76.- 0 +.2~10- 0+.71)0- 0 +.6980- 0 +.6400- 0 +.5260- 0 +.)290- 0+.87)0-' 0 +.8520- 0 +.782.- 0 +.6470- 0 +.410.- 0+.820.- 0 +.7860- 0 +.716.- 0 +.5890- 0 +.'720- 0
TI-+.4,60- 0 +.)970- 0 +.)470- 0 +.2720- 0 +.1'9'- 0+.)710- 1 +.1950- 1 +.197.- 2 vvvvvyvvv ~VVVY"VVVY"I' 'YVVVY'If"If'Y ~ VTYTVVYVV 'YVVVVVYYV'V'VV'VVV"'II' ., •••.. ~ "I'T"f"VYTTY VVYYVVVVV Y'VYY'V'YYYT•• ~.¥¥~ VYTYVVYTV .y¥ •••••• VYYVTYYYV •••••••••"',. •••••• 'Y"Y'YYYY'rYV YVYV'VV'V'YV ~ ~V"If"YV"IIVY 'V'VVYTYV'\"T VVTYVVYV'f" "f'Y¥""" v VVVVV"r'Y"r'I
8/2-.0000- 0-.0000- 0_.000._ 0_.000 ... 0_.000._ 0-.0000- 0-.0000- 0_.000._ 0-.0000- 0-.0000- 0-.0000- 0-.000.- 0~.OOO.-°-.0000- 0-.0000- 0-.000.- 0-.000.- 0-.'0000- 0-.0000- 0_.000._ 0-.000.- 0_.000._ 0-.000.- 0_.000._ 0-.0000- 0-.000.- 0-.0000- 0-.000.- 0-.0000- 0-.000.- 0-.0000- 0-.0000- °
101
lVTVVVVVVV VVVTYVYTT """"""'" ••••••••• """"""'"~ ~ YTVV'Y'VVVT YVVYVY"tV'Y ~
VYV'V"I'VYVY V'Y'V'Y'YVVVV TVVYTVTYY YV'YY'If'V'YV'f "f"Y'VY'YV'V'~ V'Y"'fTVV"Y" ~., ••• " VV"IYV"fTVY ~~ =:~,=::fiF:1
~ YYVVYYYVY V"rVYYY"nY ""..... .. .. .. .. • - .. 5790- 2YVVYV'Y"I'YV V'VVTTTV'n" YYYVVYVVY VYVVVVVY"If .... !SJ J.~' 2
blfX fy f 4Tr.4I1GLE
+.46228.- 0 +.)10510- 0 +."70- ° +.591
Fig,6.8h. The pressure in the various nodal points 'of a bearing half; e" = 0'8, ey = 0,Penv = -0,01.
o ___. 'i' 8/2O-~ TVYV"f""t"'Y TTTTYV'Y'YV .•••••••••••••••••• +.000.- 0•••• f' •••••••••••• ~•••••••• " •••••••••••••••• -.000.- 0'VVV"t"VV"YY 'VYV"f'Y"VV'I "......... " ••• ij , • •• "........ 000... 0YVTVVVYVV •••• , ••• ,. VV"tf"II"'YV ••• ,.".. 'YVVV'Y'VVV'Y 000... 0VV'YY'YV"t'VY •• " ••••• ~ YTYTYTVYV ••••••• ", -.000.- 0•••••••••• " •••••• YYYVTTYV'f' '."'f'f' ••••••••• -.000.- 0~ ••••• , ••••••••• , ••••••••••••••• "'"••••••••• 000 •• 0"',•••••••••••••• ~ ~ •••••••••••••• "'•• ~ •• "'''~-.000.- 0v ••• , , •n Yf'VTYVVVV " ••• ,.,.. 'YYYVYVYY"t" VYY't"'V'VYVV -. 000.-. 0YVVYYYYVV ", ••,••v ~"., •••• " •••••• v +.10t •• 1 _.000._ 0::m=1 ~ ::5;4;~ ::72';:'; ::;~~:: ~ ::=::~+.218.- 0 +.220.- 0 +.220.- 0 +.20a.- 0 +.1'1.- 0 -.000.- 0+.4490- 0 +.4480-'0 +.429'- ° +.'770- 0 +.Z~,o- 0 _.0000_ 0+.6540- ° +.6450- ° +.608.- 0 +.52'0_ 0 +.,410- 0 -.oooo~ °+.660.- 0 +.6'70- 0 +.592e- 0 +.5010- 0 +.)27.- 0 -.0000- 0
T1-+.)48.- 0 +.'170- 0 +.28'.- 0 +.2'20- 0 +.140.- 0 -.0000- 0+.11' •• 1 ••••••••• ~ ••••• , ••••••••••• ", ••• ,•• _.000 •• 0'¥"'f"'YVTVTT , •••••••• ".".,., •••••••• , , •• ".,., -.000.- 0YVVVYYTTT """~ •• ", ••••••••••• ,••• ,."'rrY -.000 •• 0TV"VV'Y'YV"If •••••• vvv •••••• ,,.. ,., ••• " ••••••••• , -.000.-.0'Y"Y'YVYY'V'YV .. ,., ••••••••• , •••••• , •••••• ~ -.000.- 0rt ••• ,v •••••••••• , v., •••• " ••••••~ ~ -.000 •• 0
~
.~ VVTTT'f'VVV , v ,., ~ •••••••• _.000._ 0v.v •• v ••• TTTTTYYVY .,.,.,., ••• ,., , -.000.- 0VV"'I"Y'Y'Y'V ." •• i/"'It'YY ....,.." •••••••• ,."v ••• , •••••• -.000.- 0VVVVYVVVT TYYTTYTVY •••••••••• " •• "" ~ -.000 •• 0T'V'Y'TVVVYV 'Yrf"ITVVVY ~,-yyy 'f'YYYVYVVV 'V V , • • • • •• -.000.. 0"V'VTVVV'YVV V'Y'VV'YV"t'TY 'Y'YYVVVVYV •• , •• ,... • ••••• " ,. _.000._ 0
ih vvvvvvvvv ....,..••• " •• ,.", •• , ..,...,..••••••• , •• ~ -.000.- 0r VVVVTVTTV TYYYYYVTV ~ •••••••••.• ,,,••• ,,•• -.000.- 0YVYVYVVYV ~ .,••••••• ~ •••••• v -.000.- 0
~x fy ~ 4TT.ANGLEcl +.)14100- 0 +.127090- 0 +."90- 0 +.)84
Fig. 6.8c. The pressure in the various nodal points of a bearing half; e" ;. 0'8, ey = 0,Pe.v = -0.001.
- 102-
6.4.2. Effect of the parameter Pcav on a helical-groove bearing
Figure 6.9 shows a bearing similar to that of fig. 6.6, but now with helicalgrooves. Grooves of this kind can be provided, for instance, to improve thestability under certain load conditions. It is to be expected that with such abearing - as compared to the grooveless bearing shown in fig. 6.6 - thepumping effect of the grooves will assist the filling action from the edges.
That this is in fact the case may be seen from fig. 6.10. Here the total radialload capacity,!, of a grooved bearing is plotted as a function of the eccentricityex, with the grooved part Lg/B of the bearing surface as parameter. Taking Pc. v= 0, we find the emptying effect to be entirely suppressed if only 20% of the bear-ing surface is grooved. Similar results are found for different groove parameters.
Fig. 6.9. Cavitation in a helically grooved cylindrical bearing.
f
i
~-8 -
·2.,1.
Fig. 6.10. The load capacity of a helically grooved cylindrical bearing (BID = 1, ey = 0).
103 -
6.5. Dynamically loaded bearings
.The previous discussion on cavitation has been restricted to statically loadedbearings. For a given bearing, then, the region of cavitation is uniquely definedby the position of the shaft and the parameter Peay'
With a dynamically loaded bearing, at a certain point in time the region ofcavitation is not uniquely defined, but the form and position of the region ofcavitation basically depend upon the whole previous history of the bearingmotion.In general this creates unsurmountable problems for the calculati~n. In cer-
tain circumstances, however, we have a quasi-static situation, i.e. the dynamiceffects are dominated by the static effects. Then the cavitation region may becalculated for the static load, and the influence ofthe small dynamic excursionsaround the static equilibrium position on the shape of the cavitation regionmay be neglected.When the bearing-response coefficients are calculated (chapters 2,3,4 and 7)
the assumption of such a quasi-static situation has to be made, too. This is a con-sequence of the fact that for the linear stability theory it is necessary that theexcursions, from a stationary equilibrium position, be so small that linearisa-tion of the bearing response is permissible.
REFERENCES6-1) B. Jakobson and L. Floberg, The finite journal bearing, considering vaporization,
Trans. of Chalmers Univ. of Technology, Gothenburg Sweden, no. 190, 1957, reportno. 3 from Inst. of Machine Elements.
6-2) R. Horsnell and H. McCallion, Predictions of some journal bearing characteristicsunder static and dynamic loading, Proceedings of the Lub. and Wear Conv. 1963, paperno. 12, pp. 126-138.
6-3) C. Y. Chow and J. H. Vohr, Helical-grooved journal bearing operated in turbulentregime, ASLE-ASME Lubr. Conference Houston, Texas(Oct. 1969),paper no. 69-Lub-28.
6-4) H. Sassenfeld and A. Walther, Gleitlagerberechnungen, VDI-Forschungsheft 441,edition B, vol. 20, 1954.
6-5) A. J. Sm aIl ey, T. Lloyd, R. Horsnell and H. McCallion, A comparison of per-formance predictions for steadily loaded journal bearings, Proc. Inst. mech. Engrs180, 133-144, 1965/66.
6-6) F. K. Or.cu tt and E. B. Arwas, The steady-state and dynamic characteristics of a fullcircular bearing and a partial are bearing in the laminar and turbulent flow regimes,Trans. ASME, J. Lub. Techn. 89, 143-153, 1967.
6-7) J. A. Cole and C. J. Hughes, Oil flow and film extent in complete journal bearings,Proc. mech. Engrs 170, 499-510, 1956.
6-8) G. Vo gelpohl, Beiträge zur Kermtnis der Gleitlagerreibung, VDI-Forschungsheft 386,edition B, vol. 8, Oct. 1937.
- 104-
7. ACCURACY OF THE FINITE-ELEl\'IENT METHOD AND RESULTSOF CALCULATIONS OF LOAD CAPACITY, RESPONSE
COEFFICIENTS AND STABILITY OF VARIOUS TYPES OF BEARINGS
7.1. Introduction
The theory developed in chapters 5 and 6 will be used here for the actualcalculation of bearings. It is impossible, within the scope of this chapter, togive a complete survey of all the various kinds of bearings, and therefore thepossibilities of the element method will be demonstrated by means of the cal-culated load capacity and bearing coefficients of a small number of bearingtypes that' are 'of importance in engineering. .
T~~ versatility of the method will become clear as soon as it is realized thatbearings with greatly differing geometries and boundary conditions can becalculated with the help of essentially the same basic program.
The outcome of a numerical computation is of practical use only if one knowsthe accuracy of the method. Now the finite-element method has the additionaladvantage of making a strong appeal to the sense of physics, so that greatinaccuracies can readily be discerned.
If linear elements (see chapter 5) are used, then the inaccuracy increases if,in the case of a uniform distribution, the pressure gradient differs greatly fromplace to place. The approximation to the real pressure variation, and hence theaccuracy, can be improved by making the elementary distributions finer; butthen the requisite calculating time becomes longer. The choice ofthe elementarydistribution will therefore have to be a compromise between calculating timeand accuracy.
In the first part of this chapter the accuracy of calculation is analysed bycomparison of the numerical results with cases which have been computedexactly as well as with generally accepted data from the literature which havebeen verified by experiments. In doing so, we shall adopt the criterion that, inview of the effect of the manufacturing tolerances, temperature, etc., on theperformance of the bearing, an accuracy of calculation to within about 10%is adequate.
In the second part the load capacity and response coefficients of various kindsof bearings are calculated. To permit of an insight into the "stability" of acertain type of bearing, the stability of a symmetric rotor-and-bearing system,in which both the rotor and the bearing supports-are rigid (theory of sec. 2.2),has been calculated. For some bearings it is shown that the assumption ofrotationally symmetric response in the stability analysis (theory of chapter 3)leads to results that are sufficiently accurate.
- 105 - .
7.2. Short description of the computer program
In appendix III a: computer program is given for the calculation of cylindricalbearings, with and without (rectangular) grooves. The main procedures of thisprogram arè used also in similar programs for flat and spherical spiral-groove bearings.With this program it is possible to calculate bearings which have different
groove patterns in three regions (fig. 7.1). The groove angle ct, the groove depthh., and the ridge-to-groove ratio y are represented in the program by therespective symbols fit, ho and gam.The position of the three regions is determined by the values of Pm1n, Pi'
"P2 and Pmnx (in program: psimin, psil, etc.).The deflection ofthe journal, relative to the bush, is determined by the dimen-
sionless static deflection e (= EjLJR) and dynamical deflection x (= XjLJR).In the program the components of e are represented by ex and ey and thecomponents of the velocity, X, by eptx and epty.The nodal points are chosen as a rectangular grid; the number of rows (along
the circumference) equals 2x nfi and the number of columns in the regions 1,2 and 3 is specified by nps1, nps2, and nps3.In the first part of the program the coordinates of the nodal points are gener-
ated. Next, by means of the el procedure of appendix In, triangular elements(see sec. 5.5) are formed and the contribution of each element to the matricesof (5.50) is determined with the genmatcilinder procedure.The set of equations (5.24b), obtainable after application of the boundary
conditions, may be resolved either by a direct method, the cholbd procedure,or by an iteration method, the iterate procedure. The solution then obtained,
qF-:""X
Fig. 7.1. Cylindrical bearing with different groove patterns in the regions, 1, 2 and 3.
106 -
the field of nodal pressures, is used in the genkracht procedure to calculate thedimensionless bearing reaction forces fx and J;,. Normally the input dataeptx = epty = 0 are used and then the forces fx and J;, represent the staticreaction forces. By adding a small dynamical deflection to ex, etc., and repeat-ing the calculation, we can find the data needed for the calculation of theresponse coefficients.
7.3. Inaccuracies of the finite-element method
7.3.1. Sources of errors and principles for determining the inaccuracy
. The results obtained by a numerical method, such as the FEM, are of prac-tical value only if the possible inaccuracy can be established. In this methodseveral sources of inaccuracies c~n be distinguished.
Error type 1. The generalised Reynolds equation is, as a rule, applied tobearings with grooved surfaces. In sec. 5.4 mention is made of the errorsresulting from the use of the generalised Reynolds equations instead of thenormalone. Roughly speaking we may say here that, for example with arectangular groove pattern, the error increases with increasing deviation fromlinear pressure profiles across the grooves and across the ridges.
Error type 2. When using the FEM the real pressure is approximated inthe elements, for example, by a pressure which is a linear function of the coor-dinates (sec. 5.5). The accuracy that can be attained depends on the shapeof the real pressure profile and on the possibility of approximating this pressureprofile with the chosen functions: with linear functions the inaccuracy increasesin general as the discontinuities, at the elementary boundaries, become greater.(but at a discontinuity in the film thickness the actual pressure gradient will bediscontinuous and then an error is not necessarily introduced if the elementaryboundary coincides with the discontinuity).
Error type 3. The FEM yields a set of linear inhomogeneous equations,from which the unknown nodal pressures have to be obtained. Depending onthe method of solution used, which can be direct or iterative, an error mayagain be introduced here.
In predicting the inaccuracy of a numerical method it is common practiceto compare the results that have been obtained from an exact calculation, orresults that have been verified by measurements, with the results ofthat numeri-cal method. The experience gained in this way makes it possible to estimate theerror in the remaining cases. For this reason we shall now deal with some casesin which the results of the FEM can easily be checked. Here it should be notedthat, on account of the great influence of the inevitable deviations in the geom-etry of a bearing in practice (manufacturing tolerances) on the pressure dis-
- 107-
tribution, the demanded accuracy should not be put too high, and an error ofabout 10% should be regarded as acceptable.
7.3.2. Comparison with the Sommerfeld and the Reynolds-solutions
For an infinitely long bearing (B = co) exact solutions are known for twotypes of boundary conditions: with the Sornmerfeld conditions the assumptionis made of P = 0 at if> = 0 and at if> = 11: (fig. 6.1), while with the Reynoldsconditions we assume P = 0 at if> = 0 at the beginning of the pressure hill,and at its end it is required that P = 0 and dP/dif>= 0 (Pinkus et al. 7-1)).In figs 7.2 and 7.3 the exact pressure is represented by a fully drawn curve;
the values numerically calculated with the FEM are indicated, for 16, 32 and64 nodal points in the circumferential direction, by respectively black circles,crosses and open circles.
p=.E 11.2Po
1·1
1-3• = 15 divisionsincircumferential direction(If>)( = 32 divisions in ciraJmferential direction(')D = 64 divisiof'!Sin circumferential direction(')
f\·9 / \B/D= """·8 e = 08
B - 12?JfiR2 -/·7 0- (.o.R12
·6 /·5
-t;
'3 /·2
·1 -:R/2 R
-'f=~/RFig. 7.2. Influence of the number' of grid points on the calculated pressure; Sommerfeld'sboundary condition.
1·2
.e 1P-.Po 1.1
- 108-
1·3
• = 16 divisions in circum ferential direction 1'1')
o = El. divisions in circum ferentiel directiont\'
1-0B/O=co
·9 e",0·8Po= 121~R2
'8 c.R
·7
·6
·5
·3
nJ2 TC
Fig. 7.3. Influence of the number of grid points on the calculated pressure; Reynolds' bound-ary condition.
The greatest departures occur at the maximum pressure, where the pressuregradient shows the greatest rate of variation. In any case, even with the coarsedistribution of 16 grid points, the deviation is less than 10%. Owing to theaveraging effect, due to the integration of the pressure, the error in the loadcapacity will be even smaller. This is seen from table 7-1, in which the loadcapacity and attitude angle are recorded.
7.3.3. Comparison with the results of Sassenfeld and Walther
The data on the load capacity of smooth cylindrical journal bearings pre-sented by Sassenfeld and Walther 7-2) are frequently used for engineeringpurposes and have been verified experimentally. They can therefore be used forchecking the accuracy of the FEM.Table 7-11embodies the results of Sassenfeld and Walther and those obtained
with the FEM. In the FEM programme the distribution with 32 nodal points
- 109-
TABLE 7-1
Load capacity and attitude angle of a section with length R of an infinite journalbearing. Influence of the number of nodal points on the accuracy;
12 '1]Q R4F=fFo, Fo=----
(LlR)2
Sommerfeld Reynoldsnumber of conditions conditionsnodal pointsalong circum- load att. load att.
ference capacity angle capacity angle
f CPa (rad) f CPa (rad)
16 1·52 1·57 1·18 ·7432 1·57 1·57 1·23 ·7464 1·58 1·57 1·24 ·74
exact solution 1·59 n/2 1·26 ·74
eccen-tricity
·8·8·8
·8
TABLE 7-11
Comparison of the results of Sassenfeld and Walther with those of the FEM.B/D = 1; 32 nodal points in circumferential direction; 6 nodal points in axialdirection for a half bearing; computer program: appendix Ill;
12'1] Q R4Fo-----
- (LlR)2
Sassenfeld- Walther finite-el. methodeccentricity
e load cap. att. angle load cap. att. anglef CPa (rad) f CPa (rad)
0·1 0·039 1·38 0·039 1-380·2 0·084 1·29 0·083 1·290·3 0·139 1-19 0·134 1-190·4 0·203 1·08 0·200 1·090·5 0·297 0·98 0·290 0·990·6 0·441 0·87 0·428 0·880·7 0·693 0·75 0·660 0·770·8 1-18 0·63 1·13 0·640·9 2·78 0·44 2·62 0·46
- 110-
along the circumference and 6 nodal points along the axis is used. On accountof the symmetry, only a half bearing had to be calculated. The departures arefound to increase with the eccentricity. This can be explained by the fact thathere FEM uses linear elements, which are susceptible to the strong changes inpressure gradient occurring at large eccentricities (the difference method utilisedby Sassenfeld and Walther is a "quadratic" approximation). Since the deviation,even at an eccentricity of 0·9, is less than 6%, a distribution of 32x 6 is con-'sidered to be sufficiently "fine".
7.3.4. The accuracy ofthe pressure build-up in a grooved, centricjournal bearing
If a grooved bearing (fig. 7.1) is in the concentric position, the pressure profileon the groove pattern is linear. This means that the linear elements used canrepresent the real pressure variation exactly, and the effect of type-l and type-2errors vanishes.We have made a comparison with the analytical results of Muijderman 7-3):
if the direct resolution method (cholbd procedure) is used, the error is found tobe negligibly small « 0·1%), irrespective of the number and shape of theelements.
7.3.5. Comparison with analytical results of a spherical spiral-groove thrust bear-ing
The geometry of this type of bearing is much more complicated than that ofa journal bearing. Nevertheless the same basic procedures can be used in thecalculation by means of the FEM. In sec. 7.4.5 this type of bearing will betreated in greater detail and for the definitions of the geometric parametersand illustrations the reader is referred to that section.In the present section an impression of the accuracy of the calculation has
been gained by comparison of the FEM results with the exact value obtainedby Muijderman 7-3) in the particular case of the concentric position of thesphere in the cup (ex = e" = ez = 0) and an infinite number of grooves(kg = 00). The results are listed in table 7-111.For the groove parameters, usewas made of the values which were optimum for the axialload capacity 7-3).It was assumed in the FEM calculation that the sphere was smooth from
lJf = 0 to lJf = lJfmln and grooved from lJfmln to lJfmax. This departs from ref.7-3, where lJfmln was taken to be zero, but this of course cannot be realised inpractice. From table 7-III it can be deduced that lJfmln has a relatively slighteffect on the load capacity. Even for Pmln = 0·6 rad (case No. 1) the devia-tion is as little as 7%. This is in agreement with expectations, bearing in mindthat the groove pattern at the "south pole" rotates at a comparatively slowspeed and contributes but little to the pressure build-up and the load capacity.The fact that for lJfmln = 0·1 a slightly lower value for fz is found is attributableto the fact that the shape of the elements at low values of lJfmln becomes more
-0( = 0·28 rad,lIs = 2,6,y = 1,Ü-Pmln: grooveless,
Fz =L Fo;
Pmln-Pmax: grooved,ex = e, = ez = 0,ko = 00,
- 111 _:_
TABLE 7-III
Discrepancy between calculated axial thrust and exact value for a concentric, spherical thrust bearing.
12'YJ Q R4F. -----0- (LlR)2
nfi = number of nodal points on a parallel of latitude,npsi = number of nodal points from Pmln to Pmaxs
number nfiI
npsiPmln Pmax fz(rad) (rad)
1 32 4 0·6 1·57 0·0893
2 32 6 0·4 1·57 0·0918
3 32 6 0·2 1'57 0·0919
4 32 6 0·1 1·57 0·0915
5 32 8 0·1 1·57 0·0928
6 32 10 0·1 1·57 0·0934
7 exact value (Muijderman) 0·0954
unfavourable, so that the error will increase. From No. 4, No. 5, and No. 6 inthe table it is seen that a distribution with 6 nodal points from Pmln to Pmax
provides sufficient accuracy. The requisite number of distributions in the cir-cumferential direction is difficult to evaluate with this concentric position, asit is precisely the changes in the pressure gradients (which increase in the caseof an eccentric position) that introduce inaccuracies into the calculation. Thechoice of 32 nodal points in the circumferential direction for each calculationis therefore based on the insights gained from experience withjournal bearings.
7.4. Load capacity, response coefficients and stability diagrams of some types ofbearings
7.4.1. Determination of the response coefficientsIn what follows, the load capacity, the response coefficients calculated with
the FEM, and the stability curves of several types of bearings calculated withthe theory of chapter 2 are given.
-112 -
The calculation of the dynamic coefficients requires some furhter explana-tion. It will be inferred from (2.12) that, for example, axx and avx allow ofbeing determined by a displacement of the bearing through a small distancex from the equilibrium position x = 0: this results in increments Lllx and LIJ;,of respectively Ix and J;" so that we have:
Lllx LIJ;,axx = - - ; avx = - - .
x x
Analogously avv and axv are determined with the aid of a displacement y, bxxand bvx are determined with the aid of a velocity X, bvv and bxv are determinedwith the aid of a velocity y. .The displacements x and y should be so small that the effect on Ix and J;,
remains linear, in other words dfx/dx, dJ;,/dx, etc., will be adequately approxi-mated by Lllx/x, LlJ;,/x, etc.The Reynolds equation implies that/x andJ;, are linear functions of x and y,
so that x and y do not have to be small to yield an accurate result. It is in factbetter to take them fairly large in order to arrive at the desired numerical accu-racy.The occurrence of cavitation complicates matters: the shape and size of the
cavitation region will then also depend on x, x, etc. In a dynamically loadedbearing the cavitation region depends on the instantaneous situation (deter-mined by x and :ie) as well as on its whole history.In the publications with which the author is familiar this problem is solved
by the assumption of a quasi-stationary situation when determining the regionof cavitation. Then x and :ie must be small. On the assumption, however, thatthis quasi-stationary situation does in fact govern the cavitation region, it isalso justified to adopt the situation x = :ie = 0 for the iteration of the cavi-tation region and, when axx and so on are determined, subsequently to fix theboundary of the cavitation region (with p = Pcnv as the boundary condition).Thus the calculation is speeded up, since the recurring iteration of the cavi-tation region, for each increment, can be omitted. Besides, :ie need no longerbe small.
7.4.2. Smooth journal bearing with axial lubricant-supply groove (ALG bearing)
This configuration has been used by Sassenfeld and Walther in their calcula-tions 7-2). It is assumed that this type of journal bearing possesses an AxialLubricant-supply Groove (ALG) at the location of the widest film gap (C/J = 0).
j In this groove, in the cavitation region, and at the edges of the bearing P is putequal to zero (fig. 7.4). At the same time this groove, which supplies the lubricantrequired for continuity of flow, forms the trailing edge of the cavitation region.
- 113-
oil supP,ly
!I
oilleaKage.
Fig. 7.4. An ALG bearing.x
c.. tb•.•
bxx cxx l.
Çna"i.-ay b..0
3eta
5
ey=O III/ji//
./"
Fig.7.5. Response coefficients ofanALGbearing (BID = 1,Pcnv= 0).
Figures 7.5 and 7.6 give, for respectively BID = 1 and BID = Q'5, the re-sponse coefficients needed for the calculation of the stability. With Fo =121] Q R4/(iJR)2 these coefficients conform to
iJR LlRaxx= -Axx; aXY = -Axy;
Fo Fo
2
axxi bxxi. II :-ay
i f· II
· II·II
o ·5 - .ex 1
Fig.7.6. Response coefficients of an ALGbearing (BID = 0'5, Pcnv= 0).
etc.
Figures 7.7 and 7.8 show the load-capacity and the stability curves, relatingto the rotor described in sec. 2.2, for the bearings of figs. 7.5 and 7.6.
This bearing is not rotationally symmetrical on account of the presence ofthe groove, so that it cannot be proved that the bearing is inherently unstable
w1
-114 -
2 W' 10
f Î1
-exFig. 7.8. Stability curves and load-carryingcapacity of an ALG bearing (BID = 0·5,static load).
Fig. 7.7. Stability curves and load-carryingcapacity of an ALG bearing (BID = 1,static load).
at an eccentricity ex = 0 (and A = 0), but in practice this is certainly the case.For A = 0 and ex > O·Sthe bearing is inherently stable (see also chapter 2),as it is also for A > ca 0·5. The load capacity in the two figures is in completeagreement with ref. 7-2.It appears that both the load capacity and the dynamic coefficients remain
relatively small at low values of the eccentricity (ex < O·S)and increase rapidlyat higher values of ex, which explains the good stability at the last-mentionedvalues.
7.4.3. Smooth journal bearing with circumferential oil supply (CLG bearing)
This type of bearing is in fact an ALG bearing, but without an axial groove(see fig. 7.9). At its edges (P = ± B/2) the lubricant has a pressure P = o.If the lubricant cavitates at a pressure-Pcav, then the pressure difference repre-sented by O-Pcav is the determining factor affecting the oil supply to the bear-ing.A description of an investigation of the effect of the cavitation pressure on
the load capacity has already been given in sec. 6.4.1. In fig. 7.10 theresponce coefficients are plotted against ex for Pcav = Pcav/Po = -0·01. Thedifferences from the ALG bearing (cf. fig. 7.5) are not very great. The stability
115-
curves also give a similar picture (cf. fig. 7.11). But atpcav = -0·1 (figs7.12 and7.13) the picture is quite different: the occurrence of cavitation is now stronglysuppressed, causing the coefficients axx and ayy for small values of ex (ex < 0·4)to tend to zero (fig. 7.12), which affects the stability curves in such a way thatfor ex < 0·4 the bearing virtually becomes inherently unstable (fig. 7.13). Thiswas to be expected, since according to the theory given in chapters 2 and 3 anoncavitating smooth bearing is always unstable (for example in a bearing withrotationally symmetric response: a = 0).
a"11O "xx 0xx W2
b.. S} 1 rB
ey=OPcav=-'Ol
5bx ;b yx
Fig. 7.9. A CLG bearing.
Fig. 7.10. Response coefficients of a CLGbearing (BID = I,Pcav=-O.OI).
-exFig.7.11. Stability curves and load-carryingcapacity of a CLG bearing (BID = I, staticload).
Note. An exact determination of the cavitation region is all-important, sincethe stability of a rotor in grooveless bearings (chapters 2 and 3) can beachieved solely by cavitation. If, in the case Pcnv = -0'1, the data ofthe ALG bearing were used, then a wrong result would be obtained.Hence the cavitation region has a considerable influence not only onthe load capacity (see chapter 6) but also on the response coefficients.
7.4.4. Groovedjournal bearing with optimum radial stiffness (ORS bearing)
The previously discussed grooveless bearings possess poor stability at smalleccentricities. This is bound up with the fact that the "attitude angle", CPa(fig. 7.14), is large. The x-component of the bearing reaction consequentlybecomes small and, on that account, a"" and ayy become small. Because a"xand ayy are the predominant factors determining the stability (compare withthe "a" of chapter 3) this leads to poor stability.The large value of CPa is due to the fact that the smooth bearings derive their
load capacity from the pressure build-up in the wedge-shaped gap (in the ab-sence of cavitation such a pressure field implies that CPa = nI2). This phenom-enon we shall call the pressure field due to the "wedge effect" (fig. 7.l5a).As the eccentricity increases, CPa then becomes smaller, owing to cavitation, and
a..b..
_ex
Fig. 7.12. Response coefficients of a CLGbearing (BID = I).
116 -
ey=OPcav=-"
Fig. 7.13. Stability curves and load-carryingcapacity of a CLG bearing (BID = 1,staticload).
· - 117-
F
Fig. 7.14. Helically grooved journal bearing.
x
-..--"-'" ·filmreactionforce
al blFig. 7.15. The wedge and groove effects. (a): Pressure field due to wedge effect; (b): pressurefield due to groove effect.
the stability improves. It is also a well-known fact that the pressure field dueto the wedge effect vanishes if the vector E rotates at a velocity D 12.
With a grooved bearing (fig. 7.14), besides the wedge effect a "groove effect"can be distinguished also. The pressure field due to the groove effect (fig. 7.15b)is produced (in the case of a correct design) because of the fact that the groovesin the narrow gap induce, at <I> = st, a larger pressure gradient in the axialdirection than they do in the wide gap at if> = O. Thus a pressure field profilewith a shape as shown in fig. 7 .15b is generated inside the bearing. As this pres-sure distribution is more or less symmetrical with respect to if> = n, theforce of reaction is practically opposite in direction to the displacement Ex;this is precisely the situation which favours stability (a grooved journal bearingis, as it were, "hydrodynamically prestressed").If a grooved bearing whirls at the characteristic frequency, (Je, this means
that E rotates with an angular velocity of ca DI2 ((Je R:i 0.5) and also then thewedge effect will be small: the load capacity will be generated by the grooveeffect. For the system of chapter 3, in which the stabitity is governed by theload capacity at the characteristic frequency, we may say that this stabilityis almost exclusively determined by the groove effect.
Chow and Vohr 7-4) have determined the geometric parameters of a bearingwith an Optimum Radial Stiffness (ORS bearing) in the case of a concentricposition of the shaft. This type of bearing (fig. 7.14) has two grooved ends anda smooth part in the middle.
- 118 - •
The optimum parameters are, for the case of ko = 00:groove depth ho = 1·1,groove angle oe= 0·5 rad,ridge-to-groove ratio y = 1,fraction grooved LolB = 0·75.The boundary conditions are:
P = 0 for 1JI = ± B12,Ponv = Ponv = O.
"
As found above, the choice of the cavitation condition has little influenceon the load capacity. That this is also applicable to the type of bearing underdiscussion may be seen from table 7-IV, which gives the load capacity and theattitude angle for cavitation condition 1 (also used by Chow and Vohr 7-4))and cavitation condition 4. Table 7-IV in addition shows that the responsecoefficients are slightly different. In figs 7.16 and 7.17 the response coefficientsare plotted against ex for condition 4. Figures 7.18 and 7.19 show the load-capacity and the stability curves in respect of a rotor for which the theory ofchapter 2 is applicable. The difference between the two cavitation conditionsbecomes manifest only at ex = 0·6 to 0·8.There is a remarkable difference between the stability curves for a smooth
bearing (for example fig. 7.7) and those for a grooved bearing (fig. 7.18). The
TABLE 7-IV
The response coefficients, attitude angle and load capacity of an ORS bearing(BID = 1)
cav. Ib"" byy b"y I by" fcond. e" a"" ayy axy i ayx 'Pa
1 0 +·200 +-200 +-301 1-.301 +·591 +·591 -·004 +·004 ·980 04
1 0·2 +·217 +·206 +-306 -.3181 +·638 +·603 -·001 +-007 ·980 ·0744
1 +·313 +·213 +-265 -·225 +·668 +-358 -·055 -·048 ·939 ·1540·4
4 +·289 +-234 +·265 -·242 +·674 +·388 -·018 -·010 ·903 ·156
1 +·526 +-255 +·212 -·295 +-787 +·361 -·153 -·144 ·828 ·2440·6
4 +·504 +·259 +-288 -·299 +-831 +·364 -·143 -·134 ·745 ·250
1 +1·05 +-347 +·193 -·460 +1-30 +·410 -·314 -·297 ·709 ·3920·8
4 +1-04 +·338 +·236 -·470 +1·41 +·383 -·314 -·297 ·620 ·398
32
ey=o
,t'~" 2 ·1
II
, unstable
a"1b..
_exFig. 7.16. Response coefficients of anORS bearing (BID = 1).
~~~ __ ~~~~~o1-ex
Fig. 7.18. Stability curves and load-carryingcapacity of an ORS bearing (BID = 1, staticload) -- x -- cavitation condition 1;- - - 0 - - - cavitation condition 4 (J & F).
- 119-
-:b..
·5
- bxy.-byx
o 1_exFig. 7.17. Response coefficients of an ORSbearing (BID = 0'5).
w " '2 f
Î t
Fig. 7.19. Stabilitycurves and load-carryingcapacity of an ORS bearing (BID = 0'5,static load).
120 -
smooth bearing is inherently unstable at small values of the eccentricity,whereas the grooved bearing exhibits satisfactory stability properties even ifex = O. But at ex> 0·3 .the smooth bèaring (e.g. A = 0) is distinctlysuperior. The same applies to the load capacity.The designer can apply the following criterion to good advantage:
- if a bearing has to work under the influence of a static load at small eccen-tricities (say ex < 0'3) then it is better, in view of the load capacity andthe stability, to select the grooved bearing (fig. 7.14);
- if, on the other hand, the bearing has to work at bigger eccentricities, thena smooth bearing (fig. 7.4) is preferable.
Note. In the case of a limit cycle (chapter 3) being employed, the groove pat-tern referred to above will not be optimum and a new optimisation ofthe parameters should be performed. .
7.4.5. The spherical spiral-groove bearing
One of the most interesting features of a spherical spiral-groove bearing isthat it is capable of taking up a radial load as well as an axial load. Althoughthe radial load capacity is less than the axial load capacity, the former cannevertheless be up to 25% of the latter, provided that the bearing is properlydesigned. Another favourable property of this bearing is that it does not haveto be aligned because it can rotate, as it were, as a ball joint about the centreof the sphere.In fig. 7.20 the geometric parameters of the bearing are defined. The angle ct
between the grooves and the parallel of latitude is measured in a plane tangentto the sphere.
z
Ij!min
Fig. 7.20. Geometric parameters of a spherical spiral-groove bearing (drawn in the coneen-tric position).h. = groove deptb/radial clearance (= H./LJR);ex = angle between grooves and parallel of latitude;l' = ridge-to-groove ratio;kg = number of grooves;ex = dimensionless displacement in X-direction (= Ex/LJR);ey = dimensionless displacement in Y-direction (= Ey/LJR;e: == dimensionless displacement in Z-direction (= Ez/LJR);boundary conditions: P = 0 at '1' = l[f ma. = n/2; Pcay = O.
- 121
In the computer program, which resembles very much the program of.appendix Ill, the groove parameters can again be given in three regions, viz.-from Pm1n to Pl, from Pl to P2 and from P2 to Pmax. .
The location of the centre of the sphere is measured from the concentricposition and denoted by Ex, E, and Ez, viz. the displacements in respectivelythe X-, Y- and Z-directions. The sphere rotates about the Z-axis in the positivesense with an angular velocity Q.The calculation is based on the generalised Reynolds equation. Hence the
influence of the so-called "end effects" (see chapter 5) is left out of accounthere.
The forces of reaction of the bearing in the X-, Y- and Z-directions are de-noted by Fx, F, and Fz, and the dimensionless equivalents by lx, J;, and!z.Again it is true that
etc.; and12'YJ Q R4
F, -----0- (LlR)2
7.4.6. Spherical spiral-groove bearing with optimum axial thrust (SOAT bearing)
Muijderman 7-3) determined that geometry of a spherical bearing' which pro-vided an optimum thrust in the concentric position (such a Spherical bearingwith Optimum Axial Thrust we shall call a SOAT bearing). He considered agrooved sphere that worked, from its south pole to its equator, in conjunctionwith a smooth spherical cup. The grooves were also supposed to extend fromsouth pole to equator. For an infinite number (kg = (0) of rectangular groovesthe optimum values of the parameters are:
IX = 16°, hs = 2·6, y = 1.
With the FEM it has become possible to calculate the reaction forces of sucha bearing also in eccentric positions of the sphere. Results are given in fig. 7.21.Because, in practice, the grooves cannot begin at the south pole, the beginningof the grooves, Pl, was fixed at 0·2 rad. The choice of Pl is not very criticaland we found, for the concentric position, nearly the same result as Muijderman(sec. 7.3.5). The maximum axialload capacity occurs when the south poletouches the cup and is 2·3 times higher than the concentric load capacity.Figure 7.21b shows that r, slightly depends on ex for the case that e, = ez = o.The attitude angle (= arctan (J;,/fx)) of this bearing is small, about 45°, andthis indicates good stability properties. Assuming a minimum allowable filmthickness of 0·2X R, we may state that the allowable radialload may be about8% of the allowable axial load.
0)
-,2 -·4 -,6 -,8 -1-ez
·15
- 122-
x __f_ p
fZi·25r----------~r---------_, ·25~~ o/~
·20 / -fx r+fy
'0.15
'10
'05
,(' Iy
·010 ----K- -/o-IX. ---.c___ ...,..._'"~ ---M tz
~'005 //
~./.l
·05
o ·6 ·8
Fig. 7.21. Load capacity of a spherical bearing with optimum axial thrust (SOAT bearing);boundary conditions: P = 0 for 'I'= n/2, Pen. = O.
In fig. 7.22 the response coefficients are given as a function of the staticdeflection e; for ez-values of -0'5,0 and +0'5. In fig. 7.23 the stability curvesfor a symmetric system, with rigid rotor and rigidly supported SOAT bearings,are given. Regarding the response coefficients we may say that up to high valuesof the eccentricity e", the bearing behaves rotationally symmetrically, because
b",,,, R::i bJlJl = b,
a"'JI R::i -aJl'" = t5c b,
b"'JI R::i -bJl'" = b, (see chapter 3).
Moreover the response coefficients do not vary very much with e",; this meansthat linearisation around the position e", = 0 is also permitted up to relativelylarge displacements.
If we approximate as follows:
a = t (a",,,,+ aJlJl),
b = t (b",,,,+ bJlJl)'t5cb = t (a"'JI - aJl",),
b, = t (b"'JI - bJl"')'
. then the theory-for a rotationally symmetric bearing (chapter 3) gives almostI
the same results as those shown in fig. 7.23; for example,
0·1__ .bxx
0·1--._._--=:.::::::.:._._._. byy
._._._bxx
•__ • __ .byy
-,ayxOyyuxy
Oxx
-bxy,-by
0 ·5 -ex 1 0al bl
a..b..
a"1b ..
0·1
_.....bxx_ .._ ...-, -,-=..:::::":. _. _b yy
- 123-
a ..b ..
Fig. 7.22. Response coefficients ora SOATbearing. (a): ez = -Q·5 ; (b): ez =,0;(c): e: = 0·5.cl o
- 124-
W 110
• "~ ,.~}I
unstable ..
al ·5 e 1x-
J'b"I
unstable t
cl
w
i
bl
of a SOAT3 Stability curv~\ load.Fig.. 7.2 . ith a static rad~. (c): e: = 0.5.bearmg_w_o.s; (b): e: = ,(a): ez _
~ 125-
(i) with A = ex = el'=e, = 0 it follows from fig. 7.22 that a= 0,03; b =0'064; fie b = 0'024; b, = O.Application of (3.10) predicts a stability transition at W = 4·7 (fig. 7.23indicates W = 4,6); .
(ii) with A = 0·2 and ex = el'= e, = 0 we obtain W = 2·2 (fig. 7.23 indi-cates W = 2'1);
(iii) with A:_ el'= ez = 0, ex = 0,8: a = 0,018; b = 0,072; be b = 0,029;b, = 0; W = 8·9 (fig. 7.23 indicates W = 8·7).
. . Wé may conclude that, for practical applications, the assumption of a rota-tionally symmetric response gives sufficiently accurate results. If the sameassumption is used in more-complex systems, for instance, with flexible rotorsor flexible supports, the analysis is also considerably simplified.
7.4.7. Spherical spiral-groove bearing with optimum radial stiffness (SORSbearing)
In some applications of spherical bearings the axialload is not crucial, butit is precisely the radial load capacity which is inadequate. The FEM offers thepossibility to optimise a bearing for certain characteristics. This quality of theFÈM was used to determine the parameters for which the radial stiffness (axx)in the concentric position was optimum. We shall call such a Spherical bearingwith Optimum Radial Stiffness a SORS bearing.
For a bearing, in which the lubricant film extends from the south pole tothe equator and the grooves from PI = 0·2 rad to the equator, the optimumgroove parameters for an infinite number of rectangular grooves (kg = 00) are:
hs=l'l, ')1=1.
Figure 7.24 gives the load capacity as a function of ez (ex = el'= 0) and asa function of ex (el'= ez = 0). It transpires that this bearing is in nearly everyrespect superior to the SOAT bearing: the radialload capacity is about threetimes higher and the axialload capacity for ez= -1, is 20% higher. With anallowable film thickness of 0·2 X R the allowable radialload is about 25 % ofthe allowable axialload (8% for the SOAT bearing).In fig. 7.25 the response coefficients are plotted versus ex. At higher values
of ex the coefficients are more "asymmetrie" than those of the SOAT bearing.This is attributable to the fact that for this bearing and under the conditionsused (Pcav = 0) cavitation sets in at ex > ca 0·4.
Figure 7.26 shows the stability curves of a symmetrical rotor-and-bearingsystem with SORS bearings for the case that both the rotor and the bearingsupports are rigid (cf. chapter 2). It may be surprising that, in spite of thehigher radial stiffness of the SORS bearing, the stability at ex = 0 is onlyslightly better than that of the SOAT bearing. This can be accounted for by
126 ._
·35 '035 ·35
fz t efefO ef ez:O ;"/ '30
·3e z ·030 //
=: /fy lfz'25
-fxf25 I ·25/+fy /
'20 ·020 / '20/
fZ//
'015r
·15 / ·15
// //
. ·le ·010 / ·10~x r fz
-7 - ....... -_--------'05 ·005 I. ·05;,
- ·2 -'4 -,6 -·8 -1 0 ·2 '4 ·6 '8-ez _ex
Fig. 7.24. Load capacity of a spherical bearing with optimum radial stiffness (SORS bear-ing); boundary conditions: P = 0 for IJl = 1J12 = n/2, Pcav = O.
a.. ib..
-bxy-byx
o -- ex 1Fig. 7.25. Response coefficients of a SORSbearing; boundary conditions: P = 0 for IJl =1J12 = n/2, Pcav = O.
w
Î10.------------,
•5
o
J:abl~
II
t unstable
Fig. 7.26. Stability curves of a SORSbearing with a static radial load.
- 127-
the fact that the characteristic whirl frequency, (Je, is much higher for thisbearing, which appears from the following example.
Example. With A = e", = el' = ez = 0 it follows from fig. 7.25 that (Je b = 0'081,b = 0,163; this gives (Je = 0·50 (SORS bearing), while from fig. 7.22, (Je b =0'024, b = 0·064 and (Je = 0·37 (SOAT bearing).
The fact that in (3.10) (Je occurs in a quadratic form has such a strong effectthat, if the stability is considered, the high stiffness of the SORS bearing isalmost compensated by the high characteristic whirl frequency, (Je' This makesit clear that, if instead of the radial stiffness, the stability is to be optimum,then parameters such as (Je should be considered also.If in the same way as in the preceding section the theory of rotationally
symmetric bearing response is used for the SORS bearing, stability data can becalculated that are sufficiently accurate for practical purposes.
ExampleWith A = el'= ez = 0, ex = 0·8 we find:
a = t (axx + apy) = 0,084,
b = t (bxx+ b)l)l) = 0,174,
(Je b = t (ax)l - a)lx) = 0,072,
b, = t (bx)I - b)lx) = 0
and (3.10) predicts a stability transition at(J/ (1- Af(Je)w= = 2·1.a + (Je b,
With the theory of chapter 2, where the asymmetry of the bearing responsewas taken into account, we found (fig. 7.26) W = 2·5. In practical applicationssuch a difference will as a rule be negligible.
REFERENCES7-1) O. Pink us and B. Sternlicht, Theory of hydrodynamic lubrication, McGraw-Hill
Book Co., Inc., New York, 1961.7-2) H. Sassenfeld and A. Walther, Gleitlagerberechnungen, VDI-Forschungsheft 441,
edition B, vol. 20, 1954.7-3) E. A. Muijderman, Spiral-groove bearings, Philips Res. Repts Suppl. 1964, No. 2.7-4) C. Y. Chow and J. H. Vohr, Helical-groove journal bearing operated in turbulent
regime, ASME-ASLE Lub. Conference, Houston, Texas (Oct. 1969),Paper No. 69-Lub.28.
,.' - 128 -
\ 8. DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS OFFURTHER RESEARCH
8.1. Discussion
In the previous chapters a number of problems concerned with the dynamicsof rotor-and-bearing systems, particularly in respect of the stability, have beenconsidered. On the one hand the results obtained can be applied directly to thedesign of rotor-and-bearing systems, and on the other hand the knowledgeacquired opens new avenues of further research and development of suchsystems.The work described in this report has been concerned mainly with rotors
having sliding bearings, principally because of the good prospects which thesebearings offer in the realising of high-speed rotors. Since the mastery of thedynamics in the application of such bearings resolves itself mainly into masteryof the stability, it is this aspect which has been treated in greater detail.Excepting the general introduetion and the present chapter, this work can be
divided into three parts. In the first part, chapters 2, 3 and 4, the dynamics andparticularly the stability of a number of rotor-and-bearing systems have beenexamined, the assumption being made, for the moment, that the sliding bear-ings used are elements with familiar properties.In the second part, chapters 5 and 6, a technique, the finite-element method
(FEM), has been developed for the calculation of the data of the "sliding-bearing" element which are needed for the rotor dynamics of the first part.In the third part, chapter 7, the results have been presented of calculations
by the finite-element method of a number of important types of bearings.Furthermore, inter alia by means of the theory in chapter 2, an indication hasbeen given of the "stability" of those bearings.The rotor-and-bearing systems that have been treated are assumed to be
"symmetrical". A symmetrical system is here taken to mean(a) that the rotor is rigid as well as rotationally symmetric;(b) that the entire system, viz. rotor, bearing, bearing support, and external
load, is symmetrical with respect to a plane perpendicular to the axis ofrotation (symmetrical with respect to aa' in fig. 2.1).
These specifications of the symmetry do represent a limitation, but they clearlyshow also the influence of an effect such as the gyroscopic effect. .
8.(1. Outline of chapters 2-7
Let us now turn to a brief discussion of chapters 2 to 7.
Chapter 2In practice we often find that both the rotor and the bearing supports may
-129 -
be considered to be rigid. Here these conditions have been examined in respectto a "symmetrical system" with sliding bearings.In the first part of chapter 2 we have assumed for our starting point that the
externalload is static, so that the two bearings will have equal static deflectionsin the equilibrium position. Unlike the approach familiar from the literature,gyroscopic effects have been taken into account in our considerations of thestability. It turns out that there are two modes of motion, and that the stabil-ities associated therewith can be represented in a single diagram. An importantpoint is that gyroscopic effects occur only in the "conical" mode and then theyhave mostly a stabilising influence; in the case of rotors with a large polarmoment of inertia with respect to the axis of rotation, instabilities in this modecan even be suppressed completely. A stability diagram has been constructedfor rotors with plainjournal bearings; this makes it possible to calculate whethersuch a rotor will be stable or unstable.In the second part of the chapter the corresponding case, in which the_load
of the rotor is not stationary but rotates in synchronism with the rotor, hasbeen considered. In the equilibrium position the centres of both journals thendescribe circles around the centres of the bearing bushings. Such a loadingcondition occurs if, for example, a vertical rotor in plain journal bearings isloaded only by a force of unbalance. The conclusion was reached that an un-balance had a stabilising effect. '
Chapter 3In many cases it is justifiable to assume that the bearings used have a rota-
tionally symmetrical response. Then it appears that the theory developed inchapter 2 can be simplified considerably and the stability of a symmetrical .system with rigidly supported bearings can be described by a single equation(3.10).Examples of bearings with a rotationally symmetrical response are, of course,
rotationally symmetrical journal bearings, but unloaded bearings with three ormore lobes come into this category, too. A number of grooved bearings, suchas the helical-groove journal bearing with optimum radial stiffness and thespherical bearing of chapter 7 have, to a first approximation, a rotationallysymmetrical response, even with a "static" radial load in the resulting eccentricequilibrium position. In all these cases the stability can be determined quicklyand easily by means of (3.10).The simplification achieved on the assumption of rotationally symmetrical
response makes it possible also to obtain an idea ofthe effect of "nonlinearities"in the bearing response, which begin to play a role in the case of big excursionsfrom the equilibrium position of the bearing. Then it is found for an unloadedrotor, that in a case in which instability and unconstrained build-up of theexcursions are predicted by the linear theory, the nonlinearities do in fact cause
130 -
the excursions to remain limited, so that the bearing describes a "limit cycle"around the equilibrium position. For a rigid rotor with rigidly supportedbearings, as that considered here, this means that the speed can be raised abovethe "linear stability limit" if the vibrations of the bearing due to the motionalong the liÎnit cycle are acceptable.
Chapter 4Here the starting point is that the bearings 'Of the rotor treated in the first
part of chapter 2 are supported not rigidly but flexibly. Such a support canconsist of "springs" and "dampers", but also of a second bearing (floating-bush bearing);
In the first part of this chapter a support with springs and dampers has beenexamined. Stability diagrams have been calculated for a few important typesof journal bearings. These stability diagrams (figs 4.5,4.7 and 4.8) have acharacteristic form:
for a given eccentricity of the bearing and a given rigidity of the support'the stability with low damping of the support (which means' that for agiven rotor the instability region starts at lower speeds) is poorer than inthe case of a rigid support; an increase in the damping leads in the firstinstance to an improvement in the stability, and in conjunction with suf-ficiently low rigidity the support can make the rotor inherently stable, i.e.stable at all speeds; at very high values of the damping the support behavesas if it were completely rigid .
.When designing high-speed rotors in particular, the inherently stable regionin the stability diagrams appears to be of the utmost importance. Since anappropriate choice of the support parameters can be critical, an analysis likethat presented here is indispensable.
In the second part ofthe chapter the "floating-bush bearing" has been treated.It was known already that these bearings in their existing forms possessed goodstability. From the stability theory presented there, a new insight is obtainedinto the design of a floating-bush bearing with optimum stability; this may leadto new designs which will depart considerably from current conceptions.
Chapter 5This chapter has laid the basis of the Finite Element Method for calculating
the pressure variations in liquid-lubricated sliding bearings. FEM is familiarfrom mechanics and finds ever wider application, among other things for thecalculation of stresses. Here the method is used for the calculation of the flowof a "Newtonian liquid" in a lubricant film. The starting point in the first partof the chapter is the "ordinary" differential equation of Reynolds, applicableto flow between smooth surfaces; in the second part a "generalised" Reynolds
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differential equation, applicable to flow between a grooved and a groovelesssurface, is derived and theri used in.the FEM.The method thus developed is .of such general applicability that, provided
that the lubricant film is completely filledwith lubricant (cavitation being treatedseparately in chapter 6), almost all types of sliding bearings can be calculatedwith one kind of computer program. . 'In the present work the method is used only for those data of grooved and
grooveless cylindrical bearings and of grooved spherical thrust bearings whichare needed for an analysis of the stability, but it is equally possible to calculatethe load capacity, attitude angle, response coefficients, etc., of other kinds ofbearings such as conical bearings, spiral-groove thrust bearings, and lobedbearings. Thè flexibility of the method in respect to !he calculation of the ~ffectsof geometrical changes in a bearing makes the method an important aid in thedesign of new types and shapes of bearings.
Chapter 6This chapter forms an extension ofchapter 5 in the sense that the method of
calculation given in chapter 5 has been supplemented to cover bearings which,because of the occurrence of cavitation, are not filled completely by lubricant.After all, the phenomenon of cavitation does occur frequently in liquid bearings,and the examples given here show that it may have an important influence onthe load capacity and the response coefficients.
Chapter 7In this chapter the accuracy of the method of calculation 'developed in chap-
ters 5 and 6 has.been checked by comparison with, among other things, exactresults, and the data required for the stability calculation of several types ofbearings have been determined. In addition, after the response of a bearing hasbeen determined, the stability of a symmetric rotor-and-bearing system in 'whichsuch bearings find application has been calculated by means to the theory ofchapter 2. The following types of bearings have been treated in turn:
- plain bearing with axial lubricant groove (ALG bearing);_ plain bearing with circumferentiallubricant supply (CLG bearing);_ helical-groove bearing with optimum radial stiffness (ORS bearing);_ grooved spherical bearing with optimum axial thrust (SOAT bearing);_ grooved spherical bearing with optimum radial stiffness (SORS bearing).
The fact that spherical bearings can be calculated at all forms an illustrationof the potentialof the method with respect to the calculation of bearings witha complicated geometry. An example of a case in which FEM has been usedfor the design of a bearing with certain desired characteristics (for exampleoptimum radial stiffness) is the SORS bearing.
/
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8.1.2. Conclusions
On the basis of the above, the following conclusions appear justifiable:(1) The finite-element method (chapter 5) and its extension to bearings subject
to cavitation (chapter 6) yield a unified approach for calculating all typesof sliding bearings lubricated by a Newtonian liquid, with almost the samenumerical procedure. Moreover, the ease with which the effect of geo-metrical modifications of a bearing can be calculated makes it possible todevelop new types of bearings that have certain desirable characteristics.
(2) The stability of "symmetrical" rotor-and-bearing systems with rigid rotorand with rigidly supported bearings can be determined by means of thetheory of chapter 2; whenever the reaction of the bearing may in additionbe assumed to be rotationally symmetrical, it becomes possible to use thesimple theory of chapter 3 and thus to take into account nonlinear effects.
(3) The stability of "symmetrical" rotor-and-bearing systems havingrigid rotorsand flexibly supported bearings can be determined by means of the theorygiven in chapter 4.
(4) Chapter 7 contains basic data of several familiar types of self-acting bear-ings which are useful for stability calculations.
8.2. Final remarks
8.2.1. Flexible rotors
In the above it has invariably been assumed that the rotor is rigid, in otherwords that the distortions of the rotor are negligible.
For the theoretically more complicated "symmetrical" rotor-and-beatingsystems in which the rotor is flexible it is possible to utilize the fact that at theboundary between stability and instability a flexible rotor, with mass MR'behaves in the translational mode as a rigid rotor of "apparent" mass MR*.For MR*, we then have
(8.1)
in which Qw = angular frequency of whirl at the stability-instability bound-ary,
Qllt = angular frequency of the bending vibrations in the transla-tional mode of a rotor rigidly supported at the position ofthe bearings. .
If in the conical mode the gyroscopic effects may be neglected we obtainanalogously:
JJ*=-----
1- (Qw/Q"cf '(8.2)
- 133-
in which J = moment of inertia of the rotor (taken perpendicularly to theaxis of rotation and through the centre of gravity),
J* = apparent moment of inertia,Qnc = angular frequency of bending vibrations in the conical mode.
The derivation of the above equations and the manner in which they can beused to obtain the data of flexible rotors from stability diagrams applicable to ,rigid rotors will not be considered further in this report.
8.2.2. Asymmetrie rotor
The theories of chapters 2, 3 and 4 have been developed for "symmetrical"rotors, so that it has become possible to resolve the motion of the system intotwo modes and to determine the stability of each mode separately. When sucha resolution into modes is not permissible, a stability analysis becomes muchmore complicated, because a characteristic determinant twice as extensive hasthen to be examined.In a number of cases it is possible to apply the theory of symmetric systems
to asymmetrie systems. An example, of fairly frequent occurrence, is the onein which the asymmetry of the rotor is compensated by the asymmetry of thebearing. Here we might consider the rotor of fig. 8.1 with centre of gravity G,mass MR' moments of inertia J and I, and rigidly supported bearings. If Fxlis the force acting on bearing 1 in the x-direction, Xl the displacement of thejournal, etc., then the equations of motion are
(8.3)
(8.4)
L
L2
1_]'4.J& ;;;r-~._. G._ -._._. rr'
'7? /' . T,7,/'MR !
I
Fig. 8.1. Asymmetrie rotor-and-bearing system.
- 134-
where à = L~/L2 (for L, and L2' see :fig.8.1).Provided that the following choice can be realised:
1- Cl.--=-----:- ... =--= ....=-_.,.AXX2 AXY2,. BXX2 Cl.
where AXXl' ••• , BxYl are the response coefficients of bearing 1, and AXX2' ••• ,
BXY2 those of bearing 2, such "asymmetrie" bearings compensate the asym-metry of the rotor and then the motion can still be resolved into a translationalmode and a conical mode. The former mode is described by (8.3), in whichXl = X2, and the latter by (8.4), with Cl.X2 = -(1 - a)Xl: The analysis ofeither mode can then be performed separately by means of the theory ofchapter 2.
Even in cases in which (8.5) holds only approximately, a resolving into twomodes often leads to a result which is still sufficiently accurate for design pur-poses. Relations of the form of (8.5) are encountered more or less as a matterof course whenever, on account of a gravitationalload, bearing 1 of the rotor'shown in fig. 8.1 must in any case be given a heavier construction than bear-ing 2.
(8.5)
8.2.3. Expertmental verlfication
.Many of the results of chapters 2 to 7 can be checked by means of data knownfrom the literature. Thus the results of load-capacity ealculations by means ofFEM, among others those in chapter 7, have been compared with data obtainedpreviously by others by different methods, the correctness ofwhich had alreadybeen established. Thus the response coefficients of smooth bearings calculatedby means of FE,M (cf. figs 2.3 and 2.4) are in good agreement with data fromthe literature if allowance is made for the various ways in which cavitation istaken into account. Furthermore the stability curve of the translational modein fig. 2.5 differs bl:lt little from the results of Holmes B-2). /
Even though experience with rotor-and-bearing systems (flywheel for space'vehicles B-3), .experimental gyroscopes, etc.) does confirm the theory, experi-mental verificatiou of a number of results is still desirable. In particular thestability of rotors with flexible supports (chapter 4) should be investigated ex-perimentally, above all because of'.the greatimportance of such designs innumerous applications. Such e~pe~i~~!1~s,though they are being prepared to:fillin these gaps, will not be treated in .further detail within the scope of thepresent report.
- ~~~-. . _. _ .. - -~ ..8.2.4. Computer programme for transforming the characteristic determinant into
the characteristic equation '
A serious objection to the Routh-Hurwitz method, frequently used in thiswork for stability analyses, has always been that the "characteristic determi-
135 -
nant" had to be converted into the "characteristic equation" (cf. chapter 1).As thenumber of parameters describing the motion of a system increases, theorder of the determinant becomes greater, and if the characteristic equation has .to be found by manual multiplication, insuperable difficulties soon arise ..This has now changed because lately computer programs have become
available by means of which algebraic operations can be performed on a com-puter (e.g. FORMAC-IBM). The forming of the characteristic equation canthen be programmed and, even with big systems, no longer presents a problem.
8.3. Possibilities of further investigations and development
8.3.1. Extension of the above theories. Obviously it is good sense to extend the above theories which have provedto be a reliable basis, in the direction of systems of greater complexity. In par-ticular those systems of which the asymmetry precludes a resolution into modesof motion ought to be investigated in more detail. Further analysis is desirablealso in respect of the behaviour of flexible rotors, particularly for high-speedapplications.Partlyon account of the fact that the characteristic determinant can now be
transformed easily into the characteristic equation (cf. previous remarks) the, analysis of more-complicated systems can be performed with the same tech-
niques as used above, and this will not give rise to any novel difficulties. Thedistilling and generation of regular patterns, such as the existence of an optimumdamping of the bearing support in the cases dealt with in chapter 4, is here ofthe greatest importance.
8.3.2. New forms of sliding bearings and of rotor-and-bearing systems
The better insight into the dynamics, and particularly into the stability, ofconventional rotor-and-bearing systems, in conjunction with the better possi-bilities of calculation, may well prove helpful in creating novel designs of high-speed rotor-and-bearing systems. Thus chapter 4 shows the potentialities of aflexible bearing support, and a few possible designs have already been men,-tioned in that chapter. Further investigation into the potentialities of bearingsupports appears to be desirable, however, and may yield worthwhile rewards.In rotor-and-bearing systems for high speeds it is necessary to keep the dissi-
pation of energy in the (sliding) bearings as low as possible. Since the principaldimensions of a bearing are usually determined by factors external to the bear-ing (e.g. resonance frequencies of the rotor), one is often forced, in the case ofthe conventional forms of sliding bearings, to choose a lubricant with very lowviscosity, for example a gas, so as to obtain a low dissipation for a certain loadcapacity. This does not, however, lead to bearings of very high efficiency, i.e.those in which the specified load capacity or stiffness are obtained with the
- 136-
least possible dissipation. Since it is the application of a highly viscous lubri-cant, in particular, that can lead to efficient bearings it becomes desirable, bymeans 'of the theories of chapters 5 and 6, to develop new bearing geometries .which, for specified principal dimensions and lubricants, such as oil, do provesuitable for high speeds.
FEM can therefore be used here, as has already been done in the case of thespherical bearing with optimum radial stiffness(SORS bearing), to find bearinggeo'metries that have the desirable characteristics.
REFERENCES8-1) J. W. L~nd, The stability of an elastic rotor in journal bearings with flexible, damped
supports, J. appl. Mech., Trans. ASME 32, 911-920, 1965.8-2) R. Holrnes, Instability phenomena due to circular bearing oil films, J. mech. Eng. 8,
419-425, 1966. .8-3) J. P. Reinho ud t, A flywheel for stabilizing space vehicles, Philips tech. Rev. 30, 2-6,
1969.
- 137 --
APPENDIX I
From (2.15)"it follows that the characteristic determinant and the charac-teristic equation for the translation of a rigid symmetric rotor with rigid bear-ing supports are given by
IS2 + !(bxx S + axx)
W (byx S + ayx)(ALl)
(AL2)
In the so-called zero-point method it is assumed that, at the stability-instabil-ity transition, there is a pair of purely imaginary roots for s. These roots canbe written as
s = jco (co real).
The equation (ALl) can be divided into
(AI.3)
(AlA)
If a4 =1= 0, then co =1= 0; in that case (AL3) and (AI.4) lead to the equation forthe value of W at the stability - instability transition, WtTanSltlon:
(ALS)
When WIT.nSltlon is found from (ALS), the two roots are defined by (AlA) andthe remaining pair of complex roots can be solved from (AL2). Here, itshould be emphasised that, for a transition from the stable state to the unsta-ble state, these roots should have no positive real parts, otherwise there is notransition at all; the motion is unstable.The results obtained with a rigid rotor having rigid bearing supports can be
used to determine the stability of a symmetric rotor with a flexible shaft and/ora flexible bearing support if the shaft and support are rotationally symmetricand undamped.According to (4.14) and (4.15) the characteristic determinant for the trans-
lation is:
S2+ W(bxxs+axx) W (bxy s + aXY)S2 0
W (byx s + ayx) S2 + W(byys + ayy) 0 S2=0.S2 0 S2 + W2k 0
0 S2 0 S2 + W2k
(AL6)
,.Here k represents the total stiffness of shaft and support.For S2 =1= 0,
. S2 +W2 k =1= 0,W2 k =1= 0,
equation (AI.6) can be transformed into
IS2 + Wf (bxx 8 + axx) Wf (bxy S+ aXY) I = °
Wf (byx 8 + ayx) S2+ Wf (byy S+ ayy) , (AI.7)
- 138-
with
(AI.8)
Comparison of (AI.1) with (AL7) reveals that both equations have the samepairs of purely imaginary roots at the transition points, and
WtranSitlon (from (ALl)) = Wf transition (from (AI.7)). (AL9)
If SI' 82' 83 and S4 are the roots of (ALl) and 81*, S2*, 83*, 84*, 85* and 86*
those of (AI.7), we may put, at a transition point:
SI = 81*,82= 82*'
the imaginary roots (AI.10)
Also S3*, 84* lie in the same half-plane as 83' 84 becauseIim 83* = 83'k-HO
(ALl I)
and S3*, 84* remain, for finite values of k, in the same half of the complexplane; otherwise (AL7) would have more transition points than (ALl).What about the pair S5*' 86*? We shall now see that, for "normal" sliding
bearings, and large values of k this pair has negative real parts (as should bethe case for a transition). If this can be demonstrated they remain, just as83*, S4*, in the left-hand half of the complex plane.From (AI.6) we obtain:
S6--- (bxx byy - bxy byx) +(k W)2
+~ (bxx + s; + _1_ (bxx ayy ••• )) +kW kW
-139 -
(AI.12)
+ S2 ( Wayy + W ': + W2 «: byy - W2 s; i.;+ k IW (... ) ) +
+ s (Wbxxayy •.. ) ++
As k tends to infinity the characteristic equation (AI.12) reduces to the char-acteristic equation (AI.2) and the roots ss* and S6* tend to infinity; these rootscan be found from
With
01:1 = bxx byy - bxy byx,01:2 = bxx + byy,
and
for "normal" sliding bearings, it is true that
Further
(AI.13)
(AI.14)
(APS)
(AI.14) and (AI.1S) imply that ss*, S6* have negative real parts, which was tobe demonstrated.
-140 -
APPENDIX Ir Coefficients of the Characteristic Equation
aQ = -b43Xb34-b21Xb34+b33Xb44+bllxb44+b22xb33-bl~b43-bl2Xb2l+bllXb22
al z -b22xb43xb3~11xb43Xb34-c43xb34-c21Xb34-b43xc34-b21Xc34+b2a~b33xb44+bllXb33xb44+c33xb44-b2lxbl2Xb44+bllXb22Xb44+cllXb44+b33xc44+bllxc44-b21Xbl2Xb33+bllXb22Xb33+c22Xb33+b22xc33-c12Xb43-bl2xc43-c21Xb12-b21Xc12+cllXb22+bllxc22
B2 = +b21Xb12xb43Xb34-bllXb22xb43xb34-c22xb43xb34-cl1Xb43xb34-b22xc43xb34-bllxc43xb34-b22Xb43xc34-bllxb43xc34-c43xc34-c2lxc34-b2lxbl2xb33Xb44+bllXb22Xb33xb44+c22xb33xb44+cllXb33xb44+b22xc33xb44+bllxc33xb44-c2lxbl2Xb44-b2lxcl2xb44+cl1Xb22xb44+bllxc22xb44+b22xb33xc44+bllxb33xc44+c33xc44-b2lxbl2Xc44+bl1Xb22xc44+cllxc44-c21Xbl2Xb33-b2lxcl2xb33+cllxb22xb33+bllxc22Xb33-b2lxbl2xc33+bl1xb22xc33+c22xc33-c12Xc43-c2lxc12+cllxc22
a3 = +c21xbl2Xb43xb34+b2lxcl2Xb43xb34-cllxb22xb43xb34-b1lxc22xb43Xb34+b21Xb12xc43xb34-bllxb22xc43xb34-c22xc43xb34-cllxc43xb34+b21Xbl2Xb43xc34-bllXb22Xb43xc34-c22xb43xc34-cl1xb43xc34-b22xc43xc34-b1lxc43xc34-c21Xbl2xb33xb4~2lxcl2Xb33xb44+cllXb22xb33xb44+bl1xc22xb33xb44-b21Xbl2xc33xb44+bllXb22xc33xb44+c22Xc33xb44+cl1xc33xb44-c21Xcl2Xb44+c11xc22xb44-b21xbl2xb33xc44+b11Xb22xb33xc44+c22xb33xc44+c11xb33xc44+b22xc33xc44+b11xc33xc44-c21xbl2Xc44-b21xcl2xc44+c11xb22xc44+b11xc22xc44-c21xcl2Xb33+c11xc22xb33-c21Xb12Xc33-b21Xcl2xc33+c11Xb22xc33+b11xc22xc33
a4 = +c21xc12xb34xb43-c11xc22xb43xb34+c21Xbl2xc43xb34+b21xcl2xc43xb34-cllxb22xc43xb3~11xc22xc43xb34+c21xbl2Xb43xc34+b21xc12xb43xc34-cl1xb22xb43xc34-b11xc22xb43xc34+b21Xbl2xc43xc34-b11xb22xc43xc34-c22xc43xc34-c11xc43xc34-c21xcl2xb33xb44+c11xc22xb33xb44-c21xb12xc33xb44-b21xcl2xc33xb44+c11xb22xc33xb44+b11xc22xc33xb44-c21xbl2xb33xc44-b21xcl2xb33xc44+cllxb22xb33Xc44+bllxc22xb33xc44-b21xbl2xc33xc44+bl1Xb22xc33xc44+c22xc33xc44+c 11xc33xc44-c21 XC 12xc44+c11xc22xc411-c21xcl2xc33+c11xc22xc33
a5 = +c21xcl2xc43xb34-cl1xc22xc43xb34+c21xcl2Xb43xc34-c11xc22xb43xc34+c21Xbl2Xc43xc34+b21xcl2Xc43xc34-cllxb22xc43xc34-b11xc22xc43xc34-c21xcl2xc33xb44+cllxc22xc33xb44-c21xc12xb33xc44+c11xc22xb33xc44-c21Xbl2xc33xc~21xcl2Xc33xc44+c11Xb22xc33xc44+b11xc22xc33xc44
a6 = +c21xcl2Xc43xc34-c11xc22xc43xc34-c2lxcl2xc33xc44+c11xc22xc33xc44
...:._141 -
APPENDIX III Computer Program for a Cylindrical Journal Beari~
~ sgmment j p reinhoudt,box 57berekening cilindrisch glijlager met elementen methode,en algemene vrgl van reynolds.de berekening van de matrix en de krachten is gelineairiseerd 22-1-71;~6S~ n,nloop,nkmax,prkn;nkmax :=read;nloop:=read;!2~ n:=l .§j;~p 1 Y!:!l!J.nloop gg~!l~21 ex,ey,eptx,epty,eptz,om,gam,hOl,h02,h03,hO,dfi,dps,psimin,psil,psi2,
psimax,fit,fitl,fit2,fit3,sfit,cfit,pcav;!!!~ 1,J,k,n,nk,m,swO,nfi,nps,npsl.,nps2,nps3,t;!!-~~:z: pSi,fi[ 1:nkmax];
~!9£~Y~~ cholbd(n, m, dec, a, b, fail);~ly~ n, m, dec; !D~~g~~ n, m, dec; ~r!!-:Z: a, b; 1!!-~~J.fail;~!lB!!l!D~~6S~ k, ml , i, j;
ml := m;!! dec = 0 V dec = 2 V dec = 4 V dec = 6 ~b!l!:!
fg! k := 1 .!l~~ 1 YD~ll n gg~!ls!!!!! alk, 0] S 0 ~b!l!!gg~g fail;
ark, 0] := sqrt(a[k, 0]);ti m > n - k ~b!l!:!ml := n - k;fg~ i':= 1 .!l~!lP1 Y!!~!J.ml gg ark, i] := ark, i]/a[k, 0];fg~ i := ml + 1 .!l~~P1 Y!l~!J.m gg alk, i] := 0;fg! j := 1 .!l~~P1 Y!lHl ml ggfg. i := 0 .!l~P 1 y~g ml - J !LIlark + j, i] := alk + j, i] - a[k,j] X alk, j + i]
~g k;
ml r= m;!! dec = 0 V dec = V dec = 4 V dec = 5 ~b!l!l
f~ k := 1 §:!i!ll?1 Y!l~!J.n gg~gi!l b[k] := b[k]/a[k, 0];
if m > n - k :!ib!l!lml := n - k;fgr j :=.1 §~!lP 1 Y!l~!1 ml ggb[k + j] := b[k + j] - alk, j] X b[k]
.!l!!!J. k;ml := m;!! dec < 4 ~b!l!l
!fir k := n !i!~!l~- 1 Y!:!~!11 gg .~!lB!!!b[k] := b[k]/a[k, 0];
!f k<m+l~b!l!! ml :=k-l;!gr j := 1 .!l~~P1 Y!!~!1 ml ssb[k - j] := b[k - j] - alk - j, j] X b[k]
!l!!!l-k!l!!gcholbd;
o
-142-
procedure Bchuif(i,J,k)j~r i,J,kj~ ~ Bjs:=iji:-JjJ:=kjk:-s ~ j
procedure el(n,!Ml~pt)j~ n,BwOjprocedUre ptj~~mjm m:=2 §j;,ÇR I Ymll npe.\'!g~D i:=npex(n-l )+mj
J:=npex(n+swO)+mjk:=npex(n+swO)+m+ljt:=ijptji:=npsx(n-l )+mjJ :=npex(n+swO)+m+lj ..k:=npex(n-l)+m+ljt:=ijprj
ex:=readj ey:=readj eptx:=readj epty:=read) eptz:=read) om:=read)hOl:=read) h02:=readj h03:=readjnpe I:Bread) npB2: =readj npe3: =readjfitl:=readj fit2:=readj fit3:=read)nfi:=readjpeim1n:=read)peil:=readjpsi2:=read)peimax:=readjgarn:=readjpcav:=readjnlcrjprinttext(~ex ey eptx epty eptz om hol ho2 ho3 nps I nps2 nps3 fitl fit2 fit3nfi ps fmän pei 1 pei2 peimax gam pca'/.l»jnlcr)fixt(0,1,ex)jfixt(0,1,ey)jfixt(0,2,eptx»)fixt(0,2,epty)jfixt(O,2,eptz»)fixt(2,1,om»)fixt(2,1,hOl»)fixt(2,1,h02)jfixt(2,1,h03»)absfixt(3,0,npsl»)absfixt(3,0,nps2)jabsfixt(3,O,nps3»)fixt(1,3,fitl»)fixt(1,3,fit2»)fixt(1,3,fit3»)absfixt(3,0,nfi)jfixt(1,2,peimin)jfixt(1,2,peil»)fixt(1,2,pei2)jfixt(1,2,pe1max)jf1xt(1,2,garn»)flot(2,2,pcav»)
Mi :=3. 1416/nf1)fi[2] :=0) pei[2] :=psiminj swO :=1)n:=1;
nps:=npsl+npe2+nps3+1)191::!:!l!" m:=2 §:!;~ I !!!l:li1!. npa ggbegin I~~ltest)test:=pei[(n-l)xnps+m]+~j
.fi[ (n-I )xnps+m+l] :=fi[ (n-l )xnps+m]jif test<psil :!;~ dps:=(psil-psimin)/npsl
~1~ ;!.:!: test<psi2 ~D dps :=(psi2-psi I )/nps2~1.!!~ 1:1: test<psimax:!;~ dps:=(psimax-pei2)/nps3else exitj'sir (n-l )xnfl"+m+l]:=pei[ (n-l )xnps+m]+dpe ~ j
n:=n+l;
l! n$!><nfi ~ ~fi[(n-l)xnps+2]:=fi[(n-2)xnps+2]+swOx(n-l)XMijpei[ (n-l )xnps+2] :":rsi[(n-2)xnps+2]j
SwO:-SwOi ~ 181 ~ i
-143 -
m:=2xnps+l; nk:=2xnfiXnps+l;nlcr;pr1nttext(~=»;f1xt(3,O,nk);
begin m a[ 1:nk,O:m],v,p,:frac[ 1:nk);~ :fx,:fy,:fz;
~~ array randpt[l :nk);
procedure genkracht;
begin ~ ap,aa,ab;aa:=:f1[k)-:fi[1);ab:=fi[j)-:fi[i);
if abs(aa»l ~ aa:=aa-6.2832xsign(aa);
if abs(ab»l ~ ab:=a.b-6.2832xsign(ab);
ap:=abs(aax(psi[j)-psi[i)-abX(psi[k)-ps1[1))/6;
:fx:=:fx-apx(p[i)xcos(:fi[i)+p[j)xcos(fi[j)+p[k)xcos(:f1[k));
:fy:=ty-apx(p[i)xsin(:fi[i)+p[j)xsin(fi[j)+p[k)xsin(:fi[k)»;
~ genkrachtcylinder;
procedure printdruk;
~ inte~ t; real x;
nlcr;
w j :=2xnf1 step~..lli 1,1step 2Jmlli 2xnfi-l~
begin nlcr;~ i:=lstep luntil nps~
begin x:=p[i+l+npsx(j-l »);if x=pcav~ printtext(~ »else :flot(3,2,x)~ ;
SJ!!;nlcr;nlcr;
~ pr1ntdruk;
procedure iterate;
~m real :fout,rnxp;array c[2:nps+l);
integer tel;tel:=O;
~ 1:=1+npsstep ~luntil 2!!2 c[i) :=0;
12::fout:=o;
tel:=tel+l;
f2;r k:=2~~ 2XnpSl!!l:tllnk,nk-nps+l§j;gp -2xnpsJ!!!j;ll nps!!o
~ i:=k+nps-lstep -luntil k~
begin ~ x,y;
n:=1-(i-2)lnpsxnps;
if randpt[i):t~ begin c[n) :=O;goto r~nd ;
x:=v[i)+c[n);
~ j :=i-m§j;gp luntil i-l!!2 1f j>O~
x:=x-a[j,i-j)Xp[j);
~ j:=i+l§EP 1l!!l:tll i+m~ 1f j$nk:t~x:=x-a[i,j-i)xp[j); y:=p[i);
p[i):=1.7xx/a[1,O)-.7Xp[i);
if p[i)<pcav ~ begin IS!!:! x;x:=v[i)+c[n );p[1) :=pcav;
~ j:=1-mstep 19ntil i~ if j>O~ x:=x-a[j,i-j1xp[j);
f2r j:=i+lstep 1l!!lll! i+l'!!!21f j:Snk!:~ x:=x-a[1,j-i)Xp[j);
mxp:=frac~i);
- 144-
it n>21ll!m ~ c[n] :=(1-mxp)XX,ÇJ,.!!!: c[n]:=x)
~~ c[n]:=o;f'out:=f'out+abs(y-p[i]»)rp:~ ;rnxp:=O)~ i:=2m;p 1ll!!llink!!g rnxp:=mxp+abs(p[i]»)f'out:=f'out/rnxp) f'lot(3,2,f'out»)it f'out>.OO1.t~ ~.J..?)~ iterate)
procedure gcnmatcilinder)~ ~ ap,hg,hptg,hgr,hgr3,bt11 ,bt12,ct11 ,ct12,ct22,us,test)
~ m,n,i1,J1,k1,i2,J2,k2,Bw)&~ r1,r2[1 :3])
test:=psi[t]+n-8)it test<psi1 ]b~D ~~giD hO:=h01)f'it:=fit1~~
~~ it test<psi2 .tb~D ~~siDhO:=h02)f'it:=fit2~Dg~ it test<psimax]b~D ~siD hO:=h03)fit:=f'it3~ng )
cfit:=cos(fit»)sf'it:=sin(f'it»)!2I n:=1,2,3 gg ~~r1[n] :=psi[k]-psi[j])r2[n] :=f1[j]-f'i[k];it abs(r2[n]»1~D r2[n]:=r2[n]-6.2832xsign(r2[n]);it abs(r2[n]»1.tb~D printtext(~elfout~»)sChuif(i,j,k) ~g ;ap:=r2[2]xr1[3]-r2[3]xr1[2];hg:=1-eXX(cos(f'i[i]) + cos(fi[j]) + cos(fi[k]»/3-eyx(sin(fi[i])+sin(fi[j])+
sin(f'i[k]))/3;hptg:=-eptxx(cos(fi[i])+cos(fi[j])+cos(fi[k]»/3-eptyx(sin(fi[1])+sin(fi[J])+
sin(f'i[k) )/3;hgr:=1+hO/hg;hgr3:=hgrxhgrxhgr;us:=om;ct11:=(hgr3+gam)/(1+gam)XCf'itxCf'it+(hgr3+gamxhgr3)/(1+gamxhgr3)xsf1txsfit;ct12:=«hgr3+gam)/(1+gam)-(hgr3+gamKhgr3)/(1+gamxhgr3»xsfitxcf1t;ct2~:=(hgr3+gam)/(1+gam)Xsfitxsf'it+(hgr3+gamxhgr3)/(1+gamxhgr3)xcfitxcfit)bt1.1:=(hgr+gam)1 (1+gam)XCfitxcfit+(hgr+gamxhgr3)1 (1+gamxhgr3 )xsfitxsfit;bt12 :=((hgr+gam)1 (1+gam)-(hgr+gamKhgr3)1 (1+gamxhgr3) )xsf'itxcf1t;i1 :=i2:=1; .11:=.12:=2; k1 :=k2:=3;!2I n:=i,J,k gg ~v[n]:=v[n]+.25xaign(ap)xusXhgx(bt11xr1[i1]+bt12xr2[i1)-abs(ap)xhpts/6;~ m:=i,j,k gg begin it ~n ~a[n,m-n]:=a[n,m-n]+hgxhgxhgx.5/abs(ap)x(ct11xr1[i1]xr1[i2)+ct12xr1[i1]xr2[i2]+
, ct12xr2[i1 ]xr1[i2]+ct22xr2[i 1)xr2[ i2]);sChuif'(i2,J2,k2) ~ ;schuif(i1,J1,k1) ~;~ genmatcilinder alg vrgl van reynolds;
- 145-
procedure arfrac;~ IW hg,hgr,hgr3,btl1, bt12,cf1t,a1'1t, teat;
m 1:",1~ lJ.lDlll nkà2~ teat:xps1[1]-.-B;
.1! test<ps11 ~ ~ hO:=hOl;f1t:=f1tl~
~~ 1! test<ps12 then ~s1D hO:=h02;f1t;=1'1t2~
~~ 1£ teat<ps1maxthèD ~ hO:=h03;f1t:=1'1t3~g ;Cf1t:=Cos(f1t);sfit:=sin(fit);hg:=1-exxcos(fi[i]}-eyxsin(fi[1]);
bgr:=1+hO/hg;hgr3 :zhgrxhgl'Xhgr;btll:=(hgr+gam)/(1+gam)xcf1txcfit+(hgr+gamxhgr3)/(1+gamxhgr3)xsf1txsfit;
bt12 :=( (hgr+gam)/ (1+gam)-(hgr+ga$(hgr3)/ (1+ga$(hgr3) )xsfitxcfit;
.1! i-(i-,2) ..npsxnps>2~frac[i]:=bt12/btllXdfi/(ps1[i]-psi[1-1]) ~ frac[i]:=O;
~;!2r j :=2xnfi§~p -,2~ 1, l.§j;llll2!!!1ID 2xnfi-lggbegin nlcr;fg,t i :=l§~llP l!!!1:ll;Lnpsgg fixt(0,3,frac[1+1+npsX(j-l)] hmg
~ arfrac;
swO:=o;
lit:
!gE i :=1 §~llP l!!!1ID nk gg !lll6!!! t!2I j :=0 ~ll l!!!1ill mgg
a[i,j]:=O;v[i]:=O;randpt[i):=~ä!l ~;
a[l,O):=l;!2I n:=2!U;.!lP 2!!!1lli 2xnfi-2 gg el(n,1,gennatc1l1nder);
!gE n:=3!U;.!lP 2 lID:!<U2xnf1 gg el(n,-3,gennatc1l1nder);
el(1,0,genmatc11inder);
n:=2xnf1;el (n,-,2,genmatc11inder);
t!2I 1:=nps+l§l!lp nps~ll nkgg !llls1D p[i):=O;randpt[,i):=~.!Wl;
tg]; j :=i-rn§~.!lPlY!l:lll i!!g :!! j>O~llll!! ~s1!!v[j):=v[j)-p[i)xa[j,i-j);a[j,1-j):=OllDg;
!gE j: =i+1!U;lllllY!llill i +m!!g :!! j$nk~llll!! !llls1!!v[ j) :=v[ j )-p[ i)xa[ i,j-i); a] i, j-i) :=o~ ;
a[i,O) :=1;v[1) :=p[i];~ ;
it swO=o~ll!lD!lll6!D cholbd(nk,m,0,a,v,end);awO:=3;
t!2I i:=l~.!lP lY!lli!l nk!!g :!! v[i):::pca~!! p[1):=v[ikl,§!l !l~!!
p[i):=pcav;swO:=lllD!! ll!!g ll!§llif swO=l~llll!!~6!D arfrac; iterate; s\I0.:=2!l!l!! llm
J.f S\lO>l~!l!! ~s!!!f!2I i: =1~llP 1!!!!:ll1nkà2 :!! p[i) =pcav~ll!l!!!lllS!D
f!2I j :=i-lll§~ l~ i gg it j>O~!l!!!llls!!! v[j):=v[j)-p[i]xa[j,i-j);a[j,i-j):=O~;
, !gE j:=i+l~ )~!l i+m!!g1! j~llll!!!llls1!!v[j]:=v[j)-p[i)xa[i,j-i);a[1,j-i]:=0~g ;
a[i,O] :=l;v[i) :=p[i];S!!S ;
146
cholbd(nk,m,O,a,v,end);!2r 1:~1~~R 1YD~il nkgg p[1]:=v[1];nlcr;pr1nttext«ex ey eptx epty»;awO:=swO+1;nlcr;fixt(1,2,ex);f1xt(1,2,eY);fixt(1,2,eptx);fixt(1,2,epty);~ns;pr1ntdruk;
fx:=fy:~O;
!~.n:~ §!~~ 2 YD!il 2xnfi-2 gQ el(n,1,genkracht);i~~ n:=3 ~ 2 YD~il 2xnf1 gQ el(n,-3,genkraeht);el(1,O,genkracht);n:=2xnf1;el(n,-2,genkraeht);
~~pr1nttext« fx fy f att.angl~);nlcr;flot(5,2,fx);flot(5,2,fy);flot(3,2,sqrt(fxxfx+fyxfy));f1xt(1,3,aretan(fy/fx»; nler; new page;if awO=1 ~b~D gg~Q lit;if swO~2 ~b~D gQ~Q lit;if awO=3 ~b~D h~g!D ex:~ex+.01;!! swO~4 ~b~D h~giD ex:~ex-.Ol;ey:~ey+.01; gg~Q lit ~Dg;if awO=5 ~b~D h~g!D ey:~ey-.Ol;eptx:~eptx+l; gg~Q lit ~Dg;if swO=6 lQ~D h~S!D eptx:=eptx-l;epty:=epty+1;gg~lit ~Dg;~n9 genkraehtblok;eM:~g nloop~Dg·programma
List of symbols
General remark
-147 -
. .Quantities denoted by' an upper case symbol are as a rule nondimensionless;quantities denoted by a lower case symbol are in general dimensionless.
N.B. The following definitions do not apply to the floating-bush bearing. The'quantities used in that case have to be defined in a slightly different way (cf.text).
Mathematical symbols
E
=>
~symbol or symbol
[symbol]
[symbol]"
{symbol}
mxn matrix
belongs to
does not belong to
implies that
is equivalent to
approaches
vector
matrix
transposed matrix, [symbol]
column matrix
matrix with m rows and n columns
gradient operator
divergence operator
variational operator
. d(symbol) = - (symbol) dimensionless time derivative
dl"
Reference parameters (and how they are normally chosen in this work)
t.; (R)
Ho (LlR)
Do (D)
p (12'YJ D R2)o (LlR)2
reference bearing dimension
reference film thickness
reference angular velocity
reference pressure
F. (121] D R4)a (LlR)2
-148 -
reference force
Qa (D LlR R)
Note: In chapter 2, Fa = W, the radial bearingload, was chosen.
reference flow
Coordinates
EE e=-, LlR
EsE., es=-
LlR
etot = e + es
XsX., Xs=-
LlR
XX, x =- = Xb + x,
LlR
x, Y,x,y
cpCP, cp=-
Larp
lJf,'ljJ=_La
equilibrium position of journal with respect to the
bearing bush
equilibrium position of support with respect to the
frameequilibrium position of journal with respect to theframe
length of e
x-components and y-components of e and es
dynamic deflection of journal with respect to the bush
dynamic deflection of support with respect to the frame
dynamic deflection of journal with respect to the frame
components of X, and Xb
components of X, and x,
components of X and x
local coordinates of the lubricant film
(global) coordinates of the film region
- 149-
Quantities related to the dynamics of the system
I
.J
J*
Ms
Q
QwQ..,ww=--
Qo
t,7: = Qo t
LF
F,f=-Fo
L\FL\F, L\f=-
FoL\Fb
L\Fb, L\fb = --Fo
L\FsL\Fs, L\fs =-
FoLlf"x, Llj,,y> LI!sx, etc.
rxs
mass of the rotor
apparent mass of the rotor
moment of inertia of the rotor in relation to the axisofrotationmoment of inertia of the rotor in relation to an axisthrough the centre of gravity and perpendicular to theaxis of rotation
apparent moment of inertia
mass of the support
angular speed of the rotor
angular whirl velocity
time, dimensionless time
distance of bearingsforce acting on journal when rotor is in equilibrium
position
dynamic force acting on journal, rigid support
dynamic force acting on journal, flexible support
dynamic force acting on the bush
components of the vectors L\fb, L\fs, etc.
phase angle of the support response
LlRaXY = - Axy, etc.
Fo
stiffness coefficients of bearing,
rigid support
etc.
stiffness coefficients of bearing,
flexible support
- 150-
bLlF;" bLlFxBxx = - b(dXfdt)' Bxv =- b(dYjdt) ,
etc..Q LlR .Q LlR
bxx =-- Bxx, bxJ'=-- BxJ" etc.Fo Fo'
bLlFbx bLlFbJ'Bbxx =- b(dXbjdt)' BbxJ'=- b(dYbjdt)
etc..Q LlR .Q LlR
bbxx =-- Bbxx, bbXJ'=-- Bbxv' etc.Fo Fo
2Fo
FoL22J LlRQ2
A= I~,2Ms
MRms=
MsL2
2J
damping coefficients of bearing;
rigid support
damping coefficients of bearing,
flexible support
Istiffness coefficients of support
!damping coefficients of support
stiffness coefficient of support
damping coefficient of support
asymmetry in support stiffness
asymmetry in support damping
(translational mode)
(conical mode)
(translational mode)
(conical mode)
(translational mode)
(conical mode)
\ stability parameter
!gyroscopic parameter
lmass of support
- 151
Quantities related to the bearing
D
R
B
LlR
'YJ
diameter
radius
length
radial clearance
viscosity of lubricant (nondimensionless)
period of grooves (in chapter 5)
length of the grooved part of a cylindrical bearing(in chapters 6 and 7)
depth of grooves
groove angle
ridge-to-groove ratio
number of grooves
attitude angle
dimensionless bearing reactions in
in X-, Y- and Z-directions
dimensionless (radial) load capacity
Quantities related to the lubricant film and the FEM
n number of nodal points
number of elements
film region
boundary of G
sub-region, element
boundary of Gk
section of R with prescribed pressure
section of R with prescribed outflow
section of R with "periodicity"
section of R with outflow proportional to pressure
dimensionless outflow per unit length (Rb)
k
G
R
PP,p=-
Por.;
Psm,Psm=-Por.;
Pcnv Pcav = --. Po
1{P}, {p} = - {P}
Po
QQ,q=-
QoQx, qx, etc.
Q,mQsm, qsm= Qo
Qsm x, qsmx, etc.
[a]
{v}{qs}{qn}[al]
- 152-
dimensionless total outflow (Rb)
reciprocal value of flow resistance at Rd
local pressure
smoothed pressure
cavitation pressure
column matrix of nodal pressures
elements of {P} and {p}
local flow
x-components of Q and q, etc.
smoothed flow
x-components of Qsm and qsm, etc.
x- and y-components of local velocity in the film
velocity of moving surface
x-components of U and u
column matrix of surface velocities in system Xf> Yf> Zf
matrix of interpolation functions of element Gk
element of {[k} related to nodal point i
2 X n matrix of the interpolation functions 'of the pres-sure gradients in Gk
nX n fluidity matrix
nX 1 surface-velocity matrix
n X 1 squeeze matrix
n X 1 boundary matrix
nX n matrix, denoting normal flow proportional toboundary pressure