Influence of gear loss factor on the power loss prediction

In order to accurately predict the power loss generated by a meshing gear pair the gear loss factor must be properly evaluated. Several gear loss factor formulations were compared, including the author’s approach. A gear loss factor calculated considering the load distribution along the path of contact was implemented. The importance of the gear loss factor in the power loss predictions was put in evidence comparing the predictions with experimental results. It was concluded that the gear loss factor is a decisive factor to accurately predict the power loss. Different formulations proposed in the literature were compared and it was shown that only few were able to yield satisfactory correlations with experimental results. The method suggested by the authors was the one that promoted the most accurate predictions.


Introduction
According to Kragelsky et al. (1982) tribology is an important field in engineering which can contribute to develop more reliable and efficient mechanisms like gearboxes.
According to Höhn et al. (2009) the power loss in a gearbox consists of gear, bearing, seals and auxiliary losses.Gear and bearing losses can be separated in no-load and load losses.No-load losses occur with the rotation of mechanical components, even without torque transmission.No-load losses are mainly related to lubricant viscosity and density as well as immersion depth of the components on a sump lubricated gearbox, but it also depends on operating conditions and internal design of the gearbox casing.Rolling bearing no-load losses depend on type and size, arrangement, lubricant viscosity and immersion depth.
Load dependent losses occur in the contact of the power transmitting components.Load losses depended on the transmitted torque, coefficient of friction and sliding velocity in the contact areas of the components.Load dependent rolling bearing losses also depend on type and size, rolling and sliding conditions and lubricant type (SKF, November 2005).
At nominal loads the power loss generated in a gearbox is mainly dependent of the gears load power losses, which puts in evidence the importance of the evaluation of the gear loss factor.
This work shows the influence of the gear loss factor formulation (considering different gear geometries) in the prediction of the power loss.The gear loss factor formulations will be compared with experimental results previously published by Fernandes et al. (2015).
2 Load dependent power loss in meshing gears Ohlendorf (1958) introduced an approach for prediction of the load dependent losses on spur gears.The power loss generated between gear tooth contact can be calculated according to Eq. ( 1), (1) H V is the gear loss factor.Originally Eq. ( 1) was obtained assuming a constant coefficient of friction (µ mZ ).This was a simplification of the problem.
Equation (1) can be used to calculate the average power loss between gear teeth, given the correct gear loss factor H V .Despite considering β b the Eq.(2) initially proposed by Ohlendorf (1958) is mostly valid for spur gears (Wimmer, 2006).
Published by Copernicus Publications.C. M. C. G. Fernandes et al.: Influence of gear loss factor on the power loss prediction The classical formulas for gear loss factor (Eqs. 3 and 4) consider a rigid load distribution, and a constant coefficient of friction, but tooth profile modifications are disregarded.In depth details about these formulas can be found in the classical works of Niemann and Winter (1989) and Buckingham (1949).Niemann and Winter (1989) proposed the gear loss factor that is shown in Eq. ( 3).
Buckingham (1949) also introduced a Eq. ( 4) for the gear loss factor of a meshing gear pair. where The more recent approach of Velex and Ville (2009) includes the effects of profile modifications, keeps the constant coefficient of friction assumption, but no a priori assumptions about the load distribution are made.Velex and Ville (2009) which did no a priori assumption on tooth load distribution by using generalized displacements, in order to calculate the efficiency of a meshing gear pair, obtained a closed form solution for the efficiency of a meshing gear pair (constant coefficient of friction was assumed) as presented in Eq. ( 6).It turns out that Eq. ( 4) suggested by Buckingham is an approximation of the one suggested by Velex and Ville (2009) when µ 1.
The load distribution (force per unit of length along the path of contact) disregarding elastic effects can be calculated dividing the total normal force F n = M i r bi by the total length of the lines of contact along the path of contact.
The total length of the lines of contact along the path of contact can be calculated with the algorithm presented in Appendix A. The load distribution per unit of length along the path of contact can then be calculated according to Eq. ( 8).An example of the load distribution in a helical gear is presented (Fig. 1).
The gear loss factor can now be calculated according to Eq. ( 9) proposed by Wimmer ( 2006) To solve Eqs. ( 8) and ( 9) the total length of contacting lines should be known at each point along the path of contact.To perform this task, an algorithm was developed and implemented (Appendix A).

Average coefficient of friction
Several authors (Ohlendorf, 1958;Eiselt, 1966;Naruse et al., 1986;Michaelis, 1987;Schlenk, 1994;Doleschel, 2002) have introduced different formulas to calculate the average coefficient of friction between gear teeth for different gear geometries.Due to the complexity of the problem, these equations are usually based in experimental results, and naturally, the results yielded by these models vary for the same operating conditions.In this work, instead of calculating the coefficient of friction yielded by these formulations, a value is calculated from the experimental procedure used in a previous work (Fernandes et al., 2013) and then compared to the models.
Assuming that P V Z0 , P V L and P V D are correctly calculated the power loss generated by the meshing gears can be obtained according to Eq. ( 10).The rolling bearing, seals and load independent gear losses were discussed in previous works of Fernandes et al. (2013Fernandes et al. ( , 2015)).
Considering the power loss generated by the gears in the gearbox (Eq.10) an average coefficient of friction (µ exp mZ ) can be calculated.It can be calculated according to different approaches: 1. From Ohlendof's approach (Eq.11).
V is the gear loss factor which can assume various forms, depending on the formulation that is used.Four H V were defined according to Eq. (2) H Ohl V , Eq. ( 9) H num V , Eq. 3 H Nie V , Eq. 4 H Buc V .2. Considering the average power loss generated between gear teeth along the path of contact according to Velex and Ville (2009), µ exp mZ can be obtained solving Eq. ( 12) to find µ exp mZ .
The coefficient of friction extracted from the gear mesh power loss obtained with Eq. ( 10) will be dependent of the formulation that is used to calculate the gear loss factor.In order to decide which gear loss factor formulation is better suited for the authors study, this factor was calculated for seven different gear geometries, in which, spur, helical and low loss gears are included (Table 1) (Fernandes et al., 2015).
The gear loss factor was also calculated based on the results obtained with the commercial software KissSoft which accounts for elastic effects.
Figure 2 shows the comparison between the different gear geometries as a function of the k 0 (Eq.5) parameter.There are clearly two groups of results that diverge at a certain point.A deviation is found in the solutions proposed by Buckingham (1949), Niemann and Winter (1989) and Velex and Ville (2009) because Eq. ( 5) is expected to yield values between 0 and 0.5.which means that it is not suitable for gears with profile shift.
The H501 and H951 geometries were previously tested for power loss in an FZG test rig (Fernandes et al., 2015).The results presented were collected for FZG load stages with a lever arm of 0.35 m, i.e.K5 = 105, K7 = 199 and K9 = 323 Nm applied on wheel.Changing from H501 to H951 resulted in a dramatic power loss reduction (Fig. 3), which was attributed to the H951 gear geometry (everything but the gear geometry was kept the same).These experimental results suggest that the gear loss factor of the H951 must be lower than that of the H501.The trends shown by the gear loss factors obtained with KissSoft, the author's method and Ohlendorf are in agreement with the experimental observations of Fernandes et al. (2015).The gear loss factors obtained with Eq. ( 9) are close to those obtained with the ones derived from the KissSoft computations.Aiming for simplicity and fast computing the gear loss factor was calculated using Eq. ( 9).
Following Fig. 2 it becomes clear that Buckingham, Velex and Winter's approaches are not suitable for all gear geometries and can only be applied over a limited range of the k 0 parameter.

Validation with experimental results
In order to validate the gear loss factor that was proposed, Schlenk's (Schlenk, 1994) coefficient of friction was used (Eq.13).The lubricant parameter (X L ) was previously determined with a spur gear geometry (C40) for different wind turbine gear oil formulations (Fernandes et al., 2013).Alternatively, experimental results obtained with H501 and H951 gear geometries were presented in Fig. 3 (Fernandes et al., 2015).The gear loss factors calculated according to different approaches for the C40, H501 and H951 gear geometries are presented in Table 2.In Fig. 4 the absolute error of the power loss model prediction using the KissSoft, Ohlendorf and Author gear loss factors is presented.The results suggest that the gear loss factor presented by the authors in Eq. ( 9), considering the rigid load distribution, present a much lower absolute error for the prediction of a mineral wind turbine gear oil power loss for with helical gears, previously published by Fernandes et al. (2015).
Schlenk's Equation should be valid for both helical and spur gear geometries, also H Ohl V is mostly valid for spur gears.This means that using the lubricant parameter X L extracted from experimental results with spur gears and applying it to helical gears resulted in excellent correlations between numerical and experimental data when using H num V .Rotational speed [rpm]

Figure 1 .
Figure 1.Load distribution of a helical gear with an applied torque of 320 Nm.

Table 1 .
Geometrical parameters of the gears.

Table 2 .
Gear loss factor calculated according to different approaches.