Aspects regarding the lubricant film optimization for the helicoidally shaped thrust pad bearing.

Tarca, Ioan Constantin ; Tarca, Radu Catalin ; Hule, Voichita Ionela 等

Abstract: Significant differences were noticed between the theoretical optimum pad slope (Chisiu et al., 1981; Constantinescu et al, 1980) and experimental values prescribed in catalogues (http://www.waukbearing.com; http://www.kingsbury.com). This paper proposes a numerical method that allows the calculus of the optimum slope of the pad, in respect with the maximization of the load and the minimization of power loss through friction. Helicoidally shaped thrust pad were taken into account due to the simplicity of the lubricant shape.

Key words: thrust pad, optimization, helicoidally shaped pad, slope

1. INTRODUCTION

During some author's researches, noticed were made upon the way the axial pad used zones appears. As seen in figure 1 on almost identical shaped thrust pads, different wear zone shapes developed. Studying the distribution of pressure inside the lubricant film a noticed was made: Reynolds equation's simplifications conduct toward the idea that the errors of the geometry of the film influence the functioning of the bearing in a smaller way than in the real case.

Taking into account the inertia forces and the variation of tangential speed with radius in the Reynolds equation, a new ratio was introduced, p/r, and its influence on lubricant film behavior was studied. Also, optimum pad slope was searched for given functioning conditions, and comparisons with "classical" Reynolds equation were made.

[FIGURE 1 OMITTED]

2. GOVERNING EQUATIONS

Mainly, simplifications considered in the paper are the same used relative to the lubricant flow in bearings, except for those regarding the curvature of the film, i.e. inertia forces and the variation of tangential speed with radius.

In these assumptions, Reynolds equation becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 2 OMITTED]

Considering that the surface of the pad is helicoidally shaped, it is obvious that the height of the film h is constant in radial direction, given by:

h = (1 + m)[h.sub.z]-a.[theta] (2)

where a = [h.sub.1]-[h.sub.2] / [DELTA][theta] = [mh.sub.2] / [DELTA][theta] = r x tg[alpha] = const. and m = [h.sub.1] / [h.sub.2] -1,

according to figure 2.

Pressure distribution inside the gap was studied independently along [theta] and r axes. In [theta] direction ([partial derivative] / [partial derivative]r = 0), the following equation was achieved:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where and b = [rho]x[[omega].sup.2]x[r.sup.2] and c = 6x[eta]x[omega]x[r.sup.2]

A unique real solution is possible for the equation [partial derivative]p([epsilon])/[partial derivative][epsilon] = 0 which offers the maximum pressure along [theta] axes (a fourth degree equation in a), as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [epsilon] = h/[h.sub.2]. The solution of the eq.(4) should be searched inside the interval (1, m + 1), and can be computed through numerical methods.

Along r axes (for a given value of [theta], i.e [partial derivative]p/[partial derivative][theta] = 0) the modified Reynolds equation (eq.1) becomes:

[[partial derivative].sup.2] / [[partial derivative][r.sup.2]p(r) 1 [partial derivative / r [partial derivative]r p(r)+ p(r)/[r.sup.2] = b - c (5)

where b = [rho][[omega].sup.2], and c = 6[eta][omega]a/[h.sup.3].

Maximum pressure in radial direction develops at the point

where [partial derivative]p/[partial derivative]r = 0 which has the solution:

3. OPTIMUM CONDITIONS

Two conditions were taken into account in order to achieve the optimum functioning point for the thrust bearing:

* Load maximization

* Friction minimization

The first assessment condition means that for given load conditions (W), velocity (V), and dimensions [R.sub.1], [R.sub.2], [DELTA][theta], maximum pressure should be obtained modifying the slope of the pad (m). A program developed in MATLAB was used to plot pressure variation in circumferential and radial direction, (fig. 3) noticing that both pressure equations (3) and (5) can be written as function of m, [epsilon] and r, as follows: p(m,[epsilon]) in circumferential direction, and p(m,r) in radial direction. Plots were drawn for both cases, using "classic" Reynolds equation presented in (Carafoli, E., Constantinescu, V. N., 1981; Chiiu, Al., et. al., 1981; Constantinescu, V. N., et. al., 1980), and equation (1).

The second assessment is equivalent to the minimization of the friction coefficient, [mu] (Halling, J., 1975):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [bar.W] = 1/m [ln(1 + m) / m - 2 / 2 + m] represents the dimensionless load, [F.sub.0,h] friction forces at 0, respective h height inside lubricant film and W the load. One can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[FIGURE 3 OMITTED]

Minimum value for this coefficient develops when [partial derivative][mu](m)/m = 0, the solution of this equation being m = 2.80448.

4. CONCLUSION

Analyzing the influence of pad slope m on pressure, it can be noticed that significant differences occurs between the classical model of Reynolds equation and the equation (1). Sensible higher values were noticed considering eq.(1), together with the fact that the slope for the diagrams corresponding to equation (1) is more acute, so it can be concluded that in this case the point of maximum load is very sensible relative to pad slope. Maximum pressure is achieved for m=1.4 in classical assumptions, while considering the real shape of the pad, the value for m is between 1 and 1.1, in good concordance with producer's catalogues (http://www.waukbearing.com; http://www.kingsbury.com).

Combining results for m found using eq.(8) and in numerical integrations of eq.(1), it can be noticed that an optimum value for m is included inside the range [1, 2.804], the value 1 corresponding to maximum load, and 2.804 for minimum power loss through friction.

5. REFERENCES

Carafoli, E., Constantinescu, V. N. (1981) Dinamica fluidelor incompresibile, Bucuresti, Editura Academiei

Chisiu, Al., Matiesan, D., Madarasan, T., Pop D. (1981) Organe de masini, Bucuresti, Editura Didactica si Pedagogica

Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., Nedelcu, St. (1980) Lagare cu alunecare, Bucuresti, Editura Tehnica, 1980

Halling, J., (1975) Principals of Tribology, The Mac Millan Press

A General Guide to the Principles, Operation and Troubleshooting of Hydrodynamic Bearings (2004), Available from: http://www.kingsbury.com, accessed on 2004.03.15

Tilting Pad Radial Bearings. Metric Range. Catalog (2002), Available from: http://www.waukbearing.com, Accessed: 2004-03-05