For high-speed applications, gears of large dimensions and high-power density are used. Temperature distribution in those rotors is much different in operation as compared to manufacturing. Therefore, the tooth contact as it can be validated by blue ink during assembly is not only affected by distortion and bending under load but also by non-uniform thermal growth. As power density and specific load are continuously increasing over time, for highly sophisticated applications, this influence should be accounted for with suitable lead modification, as it is demanded by the latest version of API 613.

For many years, RENK-MAAG has been using empirical methods for thermal lead corrections based on measurements and experience. Lately, the authors carried out complex finite element calculations to numerically investigate the influence of temperature distribution on tooth contact. This kind of detailed finite element modeling for tooth contact analyses requires a strong effort with respect to the corresponding finite element meshing as well as extended computation time. Therefore, the numerical method was further enhanced. As a result, a simplified approach for quick and reliable heat analyses for thermal lead correction of single-stage double helical high-speed gears was developed. The paper describes the theoretical background and gives a comparison of the results with the different calculation approaches.

### Introduction

Under operation, a part of the transmitted mechanical power is lost as thermal energy. Therefore, heat generation in a gearbox is related to the overall power loss consisting of gear mesh, bearings, seals, and auxiliary losses [1]. The total gear-power loss can be further distinguished in load-dependent losses and load-independent losses. The first category is related to frictional rolling and sliding energy dissipated during power transmission, whereas the latter one considers dissipative effects due to the rotational movements of the wheel bodies; in detail, windage power loss and oil power losses due to oil injection impacts, oil squeezing during gear meshing, and oil acceleration. It is obvious that, especially for high- speed applications with larger dimensions, these power losses and their resulting heat generation can become significantly high. As a result, thermal distortion of the gear blanks needs to be considered in addition to bending and torsional deflections when designing profile modifications for high-speed gears. A schematic draft of the mechanical deflections and the thermal distortion of a pinion shaft is illustrated in Figure 1.

Already in the 1970s, MAAG published its philosophy on how to compensate for thermal deflection by specific lead modification [2]. The determination of the steady-state temperature distribution in a gear blank was investigated experimentally in [3,4] and simulated numerically in [5] and [6]. RENK-MAAG, a subsidiary to RENK AG, adopted these methods and has, for many years, been using thermal modifications, continuously improving the application through experience. The computational approach in [7] also focuses on the radial deformation of the gear teeth. In this article, a simplified calculation strategy is presented to determine the thermal condition of the gear wheel bodies as well as the resulting thermal profile correction along the path of contact in the normal direction relative to their corresponding tooth flanks. The underlying computational model of this technique considers only the pinion and wheel body as a basic cylinder volume without an explicit description of the gear-teeth geometry. The quality of this approach is tested against a detailed finite element analysis of the same gear type and is finally compared to design values of experience.

### 1 Theoretical Background of Thermal Load Analysis

The computational approach to determine an optimum thermal lead correction basically consists of three sequential steps:

- Calculation of temperature distribution in pinion and wheel.
- Calculation of thermal growth due to the temperature change in pinion and wheel.
- Derivation of thermal lead corrections by superimposing the results of pinion and wheel.

In the first step, the thermal situation in the pinion and wheel body is determined by solving the heat equilibrium equation [8] using the thermal boundary conditions defined in the next section

where

T is the current temperature [K].

t is the time [s].

λ is the material conductivity [W/m/K].

c is the specific heat capacity [J/kg/K].

ρ is the material density [kg/m^{3}].

q· is an external heat flux [W/m^{2}].

Then, the gained result — in form of a temperature distribution — serves as the boundary condition for the structural analysis in the second step. Here, the thermal expansion u_{a} and u_{r} in axial and radial direction, respectively, caused in a cylindrical body by a temperature change is given as follows [9]

where

r is the radial coordinate [mm].

z is the axial coordinate [mm].

υ is the Poisson’s ratio [-].

b is the gear width [mm].

r_{a} is the tip radius [mm].

α_{th} is the thermal expansion factor [1/K].

T_{0} is the initial temperature [K].

Finally, the deformations occurring on the active flanks of the pinion and wheel can be determined. As schematically illustrated in Figure 2, a subsequent superposition of these resulting thermal expansions delivers the effect of actual thermal distortion on the total gear meshing. As a result, a two-dimensional thermal lead correction field along the gear flank and the path of contact — from point A to point E — can be derived.

### 2 Determination of Boundary Conditions of Thermo-Mechanical Model

As the focus lies on the temperature balance of the pinion and wheel body, only the two corresponding components of the complete gearbox are considered in all computations. Moreover, these subdomains can be further reduced with respect to their axial- and longitudinal-symmetric design, see Figure 3. Any further effects of thermal growth such as impact on gear backlash, center distance, bearing load distribution, and alignment are not considered in this article.

In this specific area, the main heat generation consists of the power loss of the gear mesh and the power loss of the adjacent bearings. The thermal heat input related to the bearings is here assumed to primarily dissipate through the housing and hence no additional temperature exchange at the shaft surfaces **F**_{s} takes place in the following investigations. As a result, the only heat input into the wheel bodies is the corresponding heat flux acting on the respective surface **F** due to the gear power loss Pv, which consists of two parts [1]: a load-dependent power loss P_{z} and a load-independent power loss P_{z0}

The load-dependent power loss P_{z} is the result of the frictional losses along the path of contact during power transmission, where rolling as well as sliding friction occurs during the gear-meshing process. As sliding friction dominates, the rolling friction is neglected and the power loss P_{z} is finally given [10,11]

where

p_{et} is the transverse pitch [m].

F_{N} is the normal force [N].

μ is the frictional coefficient [-].

v_{g} is the sliding velocity [m/s].

The frictional coefficient μ is determined according to [12,13]. The calculation of the normal load distribution F_{N} along the path of contact is based on [14].

The load-independent power loss P_{z0}, related to the rotational movements of the wheel bodies consists of windage-power loss and oil-power losses caused by oil injection impacts, oil squeezing, and oil acceleration. Therefore, the amount of the power loss P_{z0} increases with the gear dimensions and the rotational speed and can thus move significantly into the foreground for high-speed applications. It can be determined by a computational fluid dynamics (CFD) analysis [15] or by an approximation according to [10] and/or [16]. The latter method is the approach of choice in this article.

Finally, the total power loss P_{v} serves as the basis for the thermal heat input into the surfaces **F** for the pinion and for the wheel, respectively. Under consideration of the gear ratio u = z_{2}/z_{1}, the respective portion of the pinion power loss P_{v,1} and of the wheel power loss P_{v,2} can, as an approximation suggested by the authors, be calculated by the following equations:]

Besides the heat absorption of the gear cylinder surfaces **F**, both the pinion body and the wheel body are in constant thermal interaction with the surrounding air/oil mixture. In detail, the front side surfaces **F**_{f}, the gear cylinder surfaces **F**, and the gap surfaces **F**_{g} between the helices. All interactions at these particular areas are specifically described by means of forced convection heat-transfer coefficients. While the cylinder surfaces **F** and **F**_{g} are approximated by flat plates with longitudinal flow [17,18], the front side surfaces **F**_{f} are interpreted as a rotating disk in a stationary fluid according to [6, 19,20,21]. Therefore, an individual calculation is required for the corresponding Nusselt numbers N_{u}.

where

R_{e} is the Reynolds number [-].

P_{r} is the Prandtl number [-].

k is an exponent-constant according to [19] [-].

With the definition of the physical properties of the surrounding air/oil mixture, the forced convection process at the two-wheel bodies is initially described. Hereby, a complete atomization of the injected oil is assumed due to the high circumferential velocities in high-speed applications. As a result, the specific fluid properties φ_{mix} of the homogenous two-phase flow inside the gear housing are calculated with the following formula

where

φ_{mix} is the fluid property to be determined.

φ_{air} is the corresponding property of air.

φ_{oil} is the corresponding property of oil.

ξ is the oil/air-ratio [-].

Since the oil-to-steel contact in the gear meshing means a further significant heat transfer for the two-wheel bodies, this cooling process is also considered by an additional forced convection heat-transfer coefficient interacting with 100 percent oil at the top surfaces **F** and **F**_{g} in the mesh sector of the circumference. In addition, a representation of the fling-off effect according to [22] is considered.

As the intermediate surface **F**_{sym} represents the symmetric split of the double helical gear, symmetric boundary conditions are applied for all thermal and structural analyses.

Finally, the thermal boundary conditions can be summarized in a basic manner as follows:

where

n is the unit vector normal to its corresponding surface [-].

h_{disk} is the convection coefficient according to rotating disk theory [W/(m^{2}K)].

h_{plate} is the convection coefficient according to flat plate theory [W/(m^{2}K)].

h_{flo} is the convection coefficient according to fling-off theory [W/(m^{2}K)].

h_{oil} is the convection coefficient considering thermal output during gear meshing [W/(m^{2}K)].

T_{amb} is the ambient temperature of the air/oil mixture [K].

T_{oil} is the oil temperature [K].

q· is the external heat flux due to the gear power losses [W/m^{2}].

### 3 Application and Results

The procedure from Section 2 to determine the thermal lead correction, under consideration of the boundary conditions defined in Section 3, is now carried out by means of an example. The computation is done once by applying the finite element method and then again by using a highly simplified analytical approach. The obtained profile modifications of these two methods are finally compared with each other to validate the analytical model on the basis of the finite element calculations. In addition, a reference is made to the results of an empiric formula developed and proven over the years at RENK-MAAG.

In all the investigations covered by this article, only the steady-state temperature distribution of the two wheel bodies is taken into account, and, hence, the time dependency in Equation 1 does not apply. All calculations and results refer to the gear data and the material properties listed in Table 1. While the ambient temperature inside the housing is here assumed by experience, the integral temperature is calculated as suggested in [13].

As indicated in Figure 4, the finite element approach maps the complete wheel body in three-dimensional space, including their exact tooth geometry. Therefore, the boundary conditions of the previous sections can be applied directly to the corresponding surfaces — for example: the heat input due to the gear power losses is directly applied to the active flank surfaces. Accordingly, the thermal expansions of the active tooth flanks, in normal direction, also are determined automatically by the finite element analysis.

In contrast, the analytical approach of the present work computes the temperature distribution and the structural deformations of a wheel body by describing its complete geometry as a simple cylinder without gear teeth. For this reason, the thermal boundary conditions are adjusted to take account of the difference in the surface area. In a subsequent step, the resulting thermal expansions of the cylinder — Equation 2 and Equation 3 — are transferred to the thermal expansions of the active gear flanks in normal direction by basic geometry operations.

Once the thermal expansions of the active flanks are available for pinion and wheel, the thermal lead correction from the superimposed thermal distortion between pinion and wheel is derived identically for both calculation approaches. The results for one half of the double helical gear are shown in Figure 5 for the finite element approach and in Figure 6 for the analytical approach, respectively. The colors in the figures correspond to equivalent scales. For this reason, Figure 5 and Figure 6 can be put directly in relation to each other.

It can clearly be seen that basically the finite element analysis aligns with the analytical approach. Both thermal lead corrections propose their highest peak at about 80 mm in longitudinal direction.

Furthermore, they have their minimum value located in the area of the tooth gap of the double helical gear. However, minor deviations show up along the path of contact at the beginning and at the end of the gear meshing. This can probably be referred to the fact that the exact tooth geometry is not considered with the analytic approach. Figure 7 shows an extract of these results — Figure 5 and Figure 6 — along the longitudinal axis at pitch radius. In addition, a thermal lead correction design based on the empirical formula as described in Section 1 is depicted in Figure 7.

As it can be expected, the curves are not identical. However, the general shape and amount of modifications align sufficiently well. The first validation of finite element approach and analytic method with the empiric values covered by experience is successful. Applying and continuously improving the calculation methods will even increase the reliability and provide confidence to even extrapolate the results beyond the actual area of experience.

### 4 Conclusion

In addition to mechanical distortion, the temperature distribution in the gears demands for a specific thermal lead correction for high performance high-speed gears. Sophisticated calculations with the finite element method are compared to a much easier-to-apply analytical method developed by the authors. The results not only align satisfactorily with each other but also with empirical formulae that have been used for decades and match well with experience.

Therefore, the newly developed calculation method is considered suitable for actual gear designs even if they extend the scope of coverage of the existing empirical methods. Further research and validation by practical experience will give the basis for further improvement of the calculation approach.

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^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2020 at the AGMA Fall Technical Meeting. 20FTM02}