Comparison of Models for Rolling Bearing Dynamic Capacity and Life

2013 STLE Annual Meeting & Exhibition May 5-9, 2013 Detroit Marriott at the Renaissance Center Detroit, Michigan, USA Comparison of Models for Rolling Bearing Dynamic Capacity and Life Rolling-Element Bearings Authors: Pradeep K. Gupta 1, Fred Oswald 2, Erwin V. Zaretsky 2. Institutions: 1 PKG Inc. 2 NASA Glenn Research Center, Cleveland, OH, USA. The widely used Lundberg-Palmgren rolling-element bearing fatigue life model was developed in the 1940 s based on Weibull analysis and on life data for bearings made from the materials existing at the time. In the 1980 s, Zaretsky proposed a life model also based on Weibull analysis, but eliminating the Weibull slope in the critical shear stress exponent and the term involving the depth to the critical shearing stress. Over the years, significant advancements have been made in materials, lubricants and manufacturing techniques. As a result, life-modeling algorithms have advanced in two areas: modification of the basic life prediction model itself, and algorithms to calculate life modification factors for Lundberg-Palmgren life predictions. Existing life models have the elastic properties embedded in the model constant; therefore, there is no provision to vary these properties in the analysis. The objectives of this paper were (1) generalize the model formulation to allow arbitrary elastic properties of the materials to be inputs to the model, (2) correlate model predictions with experimental data to derive the model constants and (3) compare life predictions of the original and modified Lundberg-Palmgren and the Zaretsky models. The modified analysis considers the variation in elastic modulus with temperature. Introduction The Lundberg-Palmgren model (1), (2) relates rolling-element bearing fatigue life to subsurface shear stress, the volume of material stressed, and the depth below the surface where the cyclic orthogonal shear stress is a maximum. A simplified analytical formulation of this model incorporates all material properties in the model constant and makes certain simplifying assumptions to relate the subsurface stresses to contact loads. In the more recent Zaretsky model (3), based on high-cycle fatigue pertinent to more modern materials, the variation of life with the depth to the maximum subsurface shear stress is eliminated; in addition, the shear stress exponent in the life model is changed to be independent of any scatter in the life data. Also, the Zaretsky model bases the life prediction on maximum shear stress in comparison to the orthogonal shear stress in the Lundberg-Palmgren model. The model presented by Ioannides and Harris (4) is almost identical to the Lundberg-Palmgren model except that it assumes a critical shear stress, below which the bearing life is infinite. However, the actual existence of such a stress limit has been a controversial subject. Material properties, manufacturing techniques and cleanliness of modern bearing materials are different from the materials of the 1940 s. Furthermore, as the operating temperature of bearings

change, material properties, including the modulus of elasticity, will vary. Thus, a realistic implementation of any model for current applications requires enhancement of the analytical formulation so that appropriate material properties for the operating conditions may be implemented in the life model. This paper outlines a generalized analytical formulation of both the Lundberg-Palmgren (1), (2) and Zaretsky (3) models and shows the result of implementing these models in the bearing dynamics computer code, ADORE (5). The formulations developed in this paper provide load and stress relationships where the model constants are independent of material properties in both the modified Lundberg-Palmgren and the Zaretsky models. This allows for arbitrary variation of these properties in the analysis. The basic (unmodified) life, as computed from the subsurface shear stresses was corrected for improved materials, manufacturing processes and lubrication effects by means of life correction algorithms developed by Tallian (6). The model constants were computed by a least-squared deviation analysis of model predictions from experimental life data (7), (8) for a thrust-loaded angular-contact ball bearing. Bearing life predictions were then parametrically evaluated as a function of operating speed and load. Life predictions from both the modified Lundberg-Palmgren and Zaretsky models were compared to the original Lundberg-Palmgren model and to experimental data for a high-speed bearing. Nomenclature c shear stress-life exponent h exponent for depth to critical shear stress L life, millions of inner-race revolutions L 10 10-percent life: life at which 90 percent of a population survives, millions of inner-race revolutions e Weibull slope or modulus Q normal force (load) between rolling element and raceway, N (lb) Q c dynamic radial load capacity which results in an L 10 life of 1 million revolutions, N (lb) V stressed volume, m 3 (in 3 ) z o distance below surface to maximum orthogonal shear stress due to Hertzian load, mm (in.) τ m maximum shearing stress, Pa (psi) maximum orthogonal shearing stress, Pa (psi) τ o Subscripts LP refers to Lundberg-Palmgren life model Z refers to Zaretsky life model Enabling Equations In 1947 Lundberg and Palmgren (1) related rolling-element bearing life to the magnitude of the maximum subsurface orthogonal shearing stress τ o, stressed volume V and depth to the maximum orthogonal shear stress z o. c / e 1 1 h / e L [1] LP ~ ( zo) τo V The exponents c, e and h were chosen to fit experimental data available at that time. The rationale for the term involving z o, the depth to the maximum orthogonal shear stress, was that a significant 1/ e

portion of the fatigue life represents the time required for a crack to propagate to the surface and produce a fatigue spall. Zaretsky et al. (3) modified the Lundberg-Palmgren life equation [1], to better fit post-1960 life data for bearings made from vacuum processed steel, which have much longer lives, particularly at light loads where τ m is the maximum shear stress. L Z 1 1 ~ τm V In order to compute actual life from equations [1] or [2], appropriate proportionality constants are required. These constants are derived by correlating the model predictions to experimentally measured life. In order to carry out such a correlation, the shear stresses, stressed volume and depth to the critical shear stress are all computed relative to the applied contact loads and the elastic properties of the interacting materials using the fundamental elastic contact solutions available from the theory of elasticity. Thus equations [1] and [2] are transformed into load-life relationships. The commonly used term, dynamic load capacity, Q c, is defined from these relations such that at an applied load of Q c, a life of one million revolutions of the rotating race will be achieved Lundberg and Palmgren (1) related the fatigue life, L 10 of a radially-loaded bearing to the ratio of the load capacity of the bearing, Q c and the applied load, Q to the power p. Based on test data, they established that the exponent p = 3 for point contact on ball bearings. L Q = Q p [2] c 10 [3] The generalized expressions developed in this paper also result in p=3 when the values of the exponents c, h and e are set to 31/3, 7/3 and 10/9 respectively, as suggested by Lundberg and Palmgren. The Zaretsky life equation (or model), Eq. [2], which does not include the term involving the depth to the critical shearing stress, c and e are respectively set as 9.3 and 10/9 respectively in this paper, although the generalized formulation allows for any arbitrary values of these exponents. This leads to a value of p=3.7. The Zaretsky model therefore provides a larger load-life exponent. The original Lundberg-Palmgren life model implicitly included fixed material properties based on the lives of test bearings made from pre-1940, air melt AISI 52100 steel and based on the manufacturing techniques and lubricants available at the time. Since these properties are embedded into the model constant there is no provision to vary these properties. In the generalized relations developed in this paper the model constants are independent of material properties. The properties may be arbitrarily varied for both the modified Lundberg-Palmgren and the Zaretsky models. Life modification procedures to allow for improved manufacturing processes, lubrication of ball/race contacts and other operating variables are identical in both the modified and original life models. The load-life relationships for the original and modified Lundberg-Palmgren life models and the Zaretsky life model are compared schematically in Fig. 1. The modified Lundberg-Palmgren model predicts longer life than the original model but with the same load-life exponent, p = 3. The greater c 1/ e

life for the modified model is attributed to the difference in elastic properties of the interacting materials. The Zaretsky model with p = 3.7 predicts shorter life at high load but much longer life at lighter loads. Experimental Bearing Life Data The Lundberg-Palmgren life model was benchmarked to unpublished experimental bearing life data acquired prior to 1950. For this paper, we have chosen published test data from (7) and (8) that reports life tests performed at four temperatures on ABEC-5 grade, split inner-race 120- millimeter-bore angular-contact ball bearings with a thrust load of 25,800 N (5,800 lb), having a nominal contact angle of 20 degrees and operating at 12,000 rpm (1.44 million DN). The ratio of radius of curvature to ball diameter for the inner- and outer-race (conformity) was 0.54 and 0.52, respectively. The balls and races were made from CEVM (Consumable Electrode Vacuum Melted) AISI M-50 bearing steel, with Rockwell C hardness of 63 at room temperature. The surface finishes were approximately 0.05 to 0.75 μm (2 to 3 μin) rms on the races and 0.025 to 0.05 μm (1to 2 μin) rms on the balls. The bearings were lubricated with synthetic polyalphaolefin oil. Model constants for both the modified Lundberg-Palmgren and Zaretsky models were derived from a least-squared deviation analysis of model predictions and the above experimental data. Once the model constants are derived and the life models are fully defined, the analytical model predictions were compared with another set of experimental data obtained from high-speed, angular-contact ball bearings made from double vacuum melted, VIM-VAR AISI M-50 steel (9). Results and Discussion The bearing analysis code ADORE (5) was modified to compute bearing life according to the modified Lundberg-Palmgren and the Zaretsky models, in addition to the original Lundberg- Palmgren model. A least-squares deviation analysis was used to estimate the model constants for the modified Lundberg-Palmgren and the Zaretsky models. Once the model constants were determined for both the modified Lundberg-Palmgren and the Zaretsky models, the load capacities were calculated separately for the inner and outer races of the test bearing, where the load capacity, Q c is defined as the load that results in an L 10 life of one million inner-race revolutions. Predicted values of the load capacity are shown for the original and modified Lundberg-Palmgren models and for the Zaretsky model in Fig. 2. Note that Q c for the Zaretsky model is less than for the other two models. This should be expected for high load, short life conditions where the greater load-life exponent of the Zaretsky model results in short life. Life predictions from the three bearing life models are compared in Fig. 3 with another set of experimental data (9) from high-speed, 120 mm ball bearings similar to the bearings used in the database above except they had a nominal contact angle of 24 degrees, were made from double vacuum-melted VIM-VAR AISI M-50 steel and lubricated with a MIL-L-23699 tetraester lubricant. The bearings were operated at 25,000 rpm (3 million DN) with a thrust load of 22,240 N (5,000 lb) and at temperature of 218 o C (425 o F). The elastic modulus and material hardness values used for the analysis described in this paper are from the online database MatWeb (10), which provides properties for VIM-VAR AISI M-50 steel at three different temperatures. We obtained the values shown in Table I by interpolation. For example, at the operating temperature of 218 o C (425 o F) the elastic modulus of VIM-VAR AISI M-50 steel drops to 166 GPa, an 18 percent reduction from the room temperature value. The original

Lundberg-Palmgren model used room-temperature modulus values. For the purpose of this paper, the material property values in reference (10) are assumed to be correct and to apply regardless of material processing. However, these properties have not been verified by us. In addition to the change in elastic modulus the analysis implements a change in two life modifying factors, as discussed by Tallian (6); a processing multiplier, which is 0.077 for Carbon Vacuum Degassing (CVD) or Consumable Electrode Vacuum Melted (CEVM) for through hardened bearing steel, and a contamination factor, which is set to 1.0 for single vacuum melt bearing steel. For double vacuum melted (VIM-VAR) steels, Tallian (6) pointed out that the processing and contamination factors could be as low as 0.003 and 0.10 respectively. Consistent with such a recommendation, the processing and contamination factors for the VIM-VAR AISI M-50 bearings are arbitrarily set to 0.005 and 0.25 respectively. Identical life factors were applied to both the original and modified Lundberg-Palmgren, and the Zaretsky models. With the above life modification factors and the corrected elastic modulus in the modified Lundberg-Palmgren the predicted lives with the three models are shown in Fig. 3 along with the experimentally observed life. The Zaretsky life is nearly identical to the experimental life of 1600 hours; while the modified Lundberg-Palmgren life is much higher, about 2040 hours. This difference is primarily attributed to the higher load-life exponent in the Zaretsky model because the outer race contact loads and the resulting stresses are quite high at the high operating speed of 25,000 rpm (3 million DN). The original Lundberg-Palmgren model without life factors predicts a significantly lower life of about 290 hours. The change in elastic modulus at the operating temperature of 218 o C (425 o F) increases the ball to race contact area, resulting in lower contact stress, and therefore a reduced subsurface shear stress. Since the life varies inversely with the 9.3 power of the shear stress, a small reduction in stress can cause a large difference in life as seen in Fig. 3. There is also a small reduction in hardness over the temperatures analyzed, which reduces life. However, this causes a much smaller effect than the change in modulus. A parametric variation of life as function of operating speed for the three life models is shown in Fig. 4. The modified Lundberg-Palmgren and the Zaretsky models give similar results, with the Zaretsky model predicting a slightly higher life at lower speed and lower life with increasing speed. This is primarily due to a higher load-life exponent in the Zaretsky model (p=3.7) in comparison to the Lundberg-Palmgren model (p=3). The difference in life between the original and the modified Lundberg-Palmgren is a factor of almost 7. The effect of applied thrust load on bearing life is shown in Fig. 5, where the L 10 life, as obtained with the various models, is plotted as a function of load on logarithmic scales. This figure is similar to the conceptual plot in Fig. 1, which shows that life is inversely proportional to load to the power p. The Zaretsky model gives a significantly higher life at light loads and a much faster drop off as the load increases. At high load, the Zaretsky model life is slightly less than that predicted by the modified Lundberg-Palmgren model. Summary of Results The Lundberg-Palmgren and Zaretsky rolling-element bearing fatigue life models were revised to separate material and geometry constants that were embedded in the original models so that these properties could be independently varied to account for modern bearing materials or for variation in material properties, such as the modulus of elasticity. This analysis, for the first time, considers

the effect of temperature on the elastic modulus of the material. A bearing life database from four sets of published angular contact ball bearing life tests was used to develop new model constants that do not rely on external life factors. The analysis was applied to another set of published life data for high-speed aircraft engine bearings. The following results were obtained. 1. The modulus of elasticity can vary with temperature in contrast with the assumption of a constant modulus in the both the original Lundberg-Palmgren and original Zaretsky rollingelement bearing life models. At higher temperatures, the modulus can decrease, producing a lower Hertz (contact) stress and a longer bearing life than previously calculated. 2. The modified Lundberg-Palmgren model predicts a seven-fold increase in life as compared to the original model at higher operating temperatures. This is primarily due to the reduction in elastic modulus. 3. The modified Lundberg-Palmgren and the Zaretsky models give similar results, with the Zaretsky model predicting a slightly higher life at lower speed and lower life with increasing speed. This is primarily due to a higher load-life exponent in the Zaretsky model in comparison to that in the Lundberg-Palmgren model. 4. The Zaretsky model predicts a significantly higher life at light loads and a much faster life reduction as the load increases. At high load, the Zaretsky model life is slightly less than that predicted by the modified Lundberg-Palmgren model. References 1. Lundberg, G.; and Palmgren, A. (1947), Dynamic Capacity of Rolling Bearings, Acta Polytechica, Mechanical Engineering Series, 1, 3, Stockholm, Sweden. 2. Lundberg, G.; and Palmgren, A. (1952), Dynamic Capacity of Roller Bearings, Handlingar Proceedings, No. 210, The Royal Swedish Academy of Engineering Sciences, Stockholm Sweden. 3. Zaretsky, E. V., Poplawski, J. V. and Peters, S. M., (1996), Comparison of Life Theories for Rolling-Element Bearings, Tribology Transactions, 39, 2, pp. 237-247. 4. Ioannides, E., and Harris, T.A., (1985), New Fatigue Life Model for Rolling Bearings, J. Tribol. Trans. ASME, 107, 3, pp. 367 378. 5. Gupta, P.K., (1984), Advanced Dynamics Of Rolling Elements, Springer-Verlag,, ISBN 3-540- 96031. 6. Tallian, T.E., (1999), "A data-fitted rolling bearing life prediction model for variable operating conditions," STLE Tribology Transactions, 42 (1), pp 241-249. 7. Bamberger, E.N., Zaretsky, E.V. and Signer, H. (1970), Effect of Three Advanced Lubricants on High-Temperature Bearing Life, ASME Journal of Lubrication Technology, 92 (1), Pp. 23-31. 8. Zaretsky, E.V. and Bamberger, E.N., (1972), "Advanced Airbreathing Engine Lubricants Study With a Tetraester Fluid and a Synthetic Paraffinic Oil at 492 K (425 o F)", NASA-TN D-6771. 9. Bamberger, E.N., Zaretsky, E.V. and Signer, H. (1976), Endurance and Failure Characteristic of Main-Shaft Jet Engine Bearing at 3 106 DN, ASME Journal of Lubrication Technology,, 98, 4, pp. 580 585. 10. MatWeb: Online Materials Information Resource, Available at: http://www.matweb.com/ (accessed March, 5 2013).

Table I Variation of Elastic Modulus and Hardness of AISI M-50 Steel (data from ref (10)) Temperature 20 o C (70 o F) 204 o C (400 o F) 218 o C (425 o F) 260 o C (500 o F) 316 o C (600 o F) Modulus, GPa (ksi) 203 (29,000) 169 (24500) 166 (24,000) 158 (23,000) 148 (21,500) Hardness, RC 62 60 60 59 57 Fig. 1 Comparison of theoretical load-life relationships for original Lundberg-Palmgren, modified Lundberg-Palmgren and Zaretsky life models

Fig. 2 Comparison of bearing load capacities, Q c for inner and outer races calculated according to the original Lundberg-Palmgren, modified Lundberg-Palmgren and Zaretsky life models. Fig. 3 Comparison of L 10 bearing lives for a 120-mm, thrust-loaded angular-contact ball bearing with 24 degree contact angle at 25,000 rpm (3 million DN) and lubricated with a MIL-23699 oil. Experimental life values were taken from ref (12). Note, the Zaretsky model life matches the experimental life of 1600 hrs.

Fig. 4 Effect of inner-race speed on life of a high-speed, 120-mm, thrust-loaded angular-contact ball bearing with 24 degree contact angle and lubricated with a MIL-23699 oil as predicted by original Lundberg-Palmgren, modified Lundberg-Palmgren and Zaretsky life models. Fig. 5 Effect of thrust load on life of 120-mm angular-contact ball bearing at 25,000 rpm (3 million DN) as predicted by original Lundberg-Palmgren, modified Lundberg-Palmgren and Zaretsky life models. Keywords: Rolling bearings; Ball bearings;, Life prediction; Stress analysis