Viewpoints regarding the influence of temperature on determination of the oil limit viscosity in hydrodynamic bearings.

Motomancea, Adrian ; Popa, Constantin

1. INTRODUCTION

The researches in the field of dynamics of sliding bearings generally start from the well known equations of lubricant film (Constantinescu et al., 1980) the integrations of which, numerical integrations, as a rule, offer a quite complete image on the bearing dynamic behavior. The number of parameters taken into account as influencing this behavior type, complicates or simplifies, as the case may be, the approach. Depending on the calculation methods used (long finite bearing method, short bearing method, one-dimensional method etc.), results are obtaining proceeding from the running of more or less evolved numerical integration software (Bonneau et al., 1989)).

The non linear overall model introduced by the first author (Motomancea et al., 2004) offers a more simple method and an analytical solution for determining the critical amortization, starting from the theory and study of stability of the equilibrium solutions of the differential equations system that describe the motion of the spindle in the bearing.

In principle, the non linear overall model method starts from the premise of non coincidence of the centre of the spindle with the centre of the sleeve during the operation of the bearing, i.e. upon a displacement r of this point and consequent upon this the occurrence of two interstices, one of minimum size and one of a maximum size, between the spindle and the sleeve. Assuming that in the moment of coincidence of the two centers, the uniform running play in the bearing is S as it is shown in fig.1, the plays S-r and S+r corresponding to the minimum interstice and the maximum interstice respectively shall occur diametrically opposed.

The model relies on the assessment of the forces occurring in the points of tangency with the shaft in these extreme interstices evidenced in figure 2 and then on application of the general theorems of dynamics in scalar projection on the axes: the mass center motion theorem and the theorem of kinetic moment with respect to the rotation axis.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [V.sub.A] respectively [V.sub.B] are the speeds in the specified points, [epsilon] a dimensional coefficient, [OMEGA] angular rotation speed of the spindle, and [omega] the derived function with respect to time of the angle [theta] between OC and the fixed direction Ox, and R the spindle radius.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The differential equations that describe the shaft motion under the influence of the forces described above, of a elastic type equation with the form [F.sub.e] = kr and of an amortization force of the form [F.sub.a] = c[??] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where:

k/m = [p.sup.2]; c/m = 2[alpha]; 2[epsilon]/m = [[epsilon].sub.1]; 2[mu]R/J = [[epsilon].sub.2] (3)

J--shaft's mechanical inertia moment, m--mass within the bearing of the spindle, k--elastic constant of the bearing, c--viscous amortization coefficient.

2. MATHEMATICAL CONDITIONS RESULTED IN CONSEQUENCE OF THE STUDY OF STABILITY OF THE EQUILIBRIUM SOLUTIONS

The study of the equilibrium solutions of the differential equations system (2) leads to the determination of two solutions triplets. One of them is uninteresting, it being unconditionally met (Motomancea et al., 2004). The other triplet is:

r = [square root of R[delta]([OMEGA] - p)/p; v = 0; [omega] = -p (4)

The study of stability of this solution by means of small perturbation method after the right member of the third equation in the system (2) is previously transformed by Taylor's series development and the equation proceeding from the determinant of the own values of the Jacobi matrix and the application of Routh-Hourwitz stability criterion is written, leads to the following inequation:

4[[alpha].sup.2]B + 2[alpha][B.sup.2] - 2[r.sub.0]pA > 0 (5)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

From the inequation (5) the expression of critical amortization coefficient results:

[[alpha].sub.[alpha]] = - [B.sup.2] + [square root of [B.sup.4] + 8AB[pr.sub.0]]/4B (7)

This critical amortization coefficient calculated by the formula (7) offers results within an error range satisfactory with respect to the classical methods that use the integration of the lubricant film equations in conditions specific for the studied bearing type, for the running and operating conditions (Motomancea et al., 2007). Starting from this expression of critical amortization coefficient, the critical viscosity, respectively the operating critical temperature of the bearing will be determined.

3. DETERMINING THE CRITICAL VISCOSITY AND THE CRITICAL TEMPERATURE IN THE BEARING

There is introduced below the link relation between the coefficient [alpha], which defines the amortization, and [mu]--dynamic viscosity. From the dimensional analysis of the two quantities, it may be written:

[mu] = [alpha] m/[delta] (8)

With respect to the dependency of viscosity on temperature, the relation usually accepted in the specialty literature will be used (Constantinescu et al., 1980):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[mu].sub.0] is viscosity at the reference temperature [T.sub.0] and [kappa] - a coefficient. From the expressions (8) si (9) the formula of dependency of the amortization temperature is easily deducted:

T = [T.sub.0] - 1/[kappa] ln [alpha]m.[delta][[mu].sub.0] (10)

The variation of viscosity with temperature within a broad temperature interval may not be represented with a sufficient accuracy by a simple mathematical relation. None of the numerous proposed mathematical relations permits the examination of variation of oils viscosity with temperature by an only value. Therefore, varied conventional systems have been employed, the most used being the one based on Dean Devis viscosity coefficient. This coefficient uses as reference the paraffin oils of Pennsylvania with a small viscosity- temperature coefficient (ratio between viscosity in saybolt seconds at 100 and 210 F) considered to have a V.I. = 100, and the naphtene aromatic oils of Gulf Coast with a high viscosity- temperature coefficient considered to have a V.I. = 0 For a series of oils of the same origin the inclination of the viscosity- temperature curve deepens as the viscosity increases. Therefore, empiric equations have been employed to express the viscosity at 100 F (37.8 [degrees]C) depending on the viscosity at 210 F (98.9 [degrees]C) for each of the two series of oils chosen as reference.

The viscosity index is calculated based on the cinematic viscosity at 37.8 [degrees]C and 98.9 [degrees]C or at 40 or 100 [degrees]C. The formulas (9) or (10) satisfactorily approximate the calculus of critical dynamic viscosity, respectively of the critical temperature of operation of lubricant film accomplished by varied empiric methods accepted by the scientific community.

Likewise, upon the simple view of the formula (9) showing the dependence of viscosity on temperature, it is noticed that the said viscosity dramatically decreases with the temperature increase, which lead to the idea that the bearing operation viscosity requested by the designer should be maintained by a strict control of the temperature in the lubricant film.

4. CONCLUSION

The results of the above described study will be hereinafter applied to a practical example: a hydrodynamic bearing that equips a RFC 320 grinding machine without centers, which made the research and re-designing subject matter for the authors.

The technical data of the bearing are as follows: Spindle diameter D=150 mm (radius R=75 mm). Shaft mass (the grinding stone included) = 450 Kg. Shaft speed n=1200 rpm.

Calculating the coefficients A, B (implicitly [[epsilon].sub.2] which depends on [epsilon]), [r.sub.0], using specific nomograms for calculation of the quantities k, c etc. (Constantinescu et al., 1980), the problem is to assess the coefficient s. Analyzing the relations (1), it can be noticed that this coefficient has the dimension of an impulse. Let's consider the spindle impulse

H = mv m[OMEGA]r (11)

where v is the spindle speed.

As a rule, this type of grinding machine uses in its bearings oils of which cinematic viscosity varies between 7-10 cSt. Whatever oil type is used, the viscosity value is kept constant by control of temperature. Likewise, an increase of the temperature with 20 [degrees]C over the ambient temperature is not permitted by the usual regulations. Therefore, the temperature variation in the bearing is usually placed in the range of 20-50 [degrees]C.

Using the calculations introduced before and the calculated value of the critical amortization coefficient [[alpha].sub.cr] for this type of bearing and a value of the oil viscosity of 8 [[alpha].sub.St], a value of the critical temperature in the bearing (the temperature that maintains the viscosity constant) of about 38[degrees]C is obtained. The value obtained by probe measurement at the bearing level indicates the temperature of 35[degrees]C, which means that the error introduced by the use of this method is of about 7.8%, which is remarkable taking into account the absence of some accurate formulas in the specialty literature in this field.

5. REFERENCES

Constantinescu, V.N., Nica, A., Pascovici, M.D., Ceptureanu, G., Nedelcu, S., (1980), Sliding Bearings, Technical Publishing, Bucharest, C.Z. 621.82

Bonneau, O., Frene, J., (1989), Comportement dynamique de rotor monte sur paliers fluide- Etude non lineaire, 9-me Congres Francais de Mecanique, Metz 5-8 sept. 1989, Actes-tome 2-Mini Colloques, pp.358-359

Motomancea, A., Dragomirescu, C., Predoi, M.V., Bugaru, M. (1999), New- non-linear model on hydrostatic bearings. The study of the instability zones, European Conference on Computational Mechanics ECCM '99, Aug. 31-Sept. 3. Munchen, Germany, published on CD-ROM, with resume in ABSTRACTS, pg. 578

Motomancea, A., Predoi, M.V., Hadar, A., Constantinescu, I., (2004), Considerations about the using of Non-Linear Global Model for Hydrostatic Bearings, 1st IC-SCCE, Athens, Greece, 8-10 Sept. 2004

Motomancea, A. (2007), Considerations about the different behaviors of the hydrostatic and hydrodynamic bearings using non-linear complex models, 18-th International DAAM Symposiun, 24-27 oct. 2007,Croatia, Annals of DAAAM 2007 & Proceedings pp.483-484, ISSN 1726-9679 ISBN 3-901-509-58-5. (ISI Thomson Scientific Proceedinngs-Thomson Gale