The simulation program WTplus was extended to automatically analyze the efficiency and heat balance of various designs of worm gears. For this, new approaches for the calculation of load-dependent and no-load losses, as well as new algorithms for nodalization and node-linking, were developed and implemented. Moreover, essential formulas describing the thermal resistances were customized. Simulation results were validated with measurements from research and industry showing very close alignment for various operating points and gear designs.
If torque conversion with high gear ratio, compact installation space and 90-degree axis-crossing angle is needed, often worm gears are used. Due to their high power density and sliding speeds within the tooth contact, frictional heat and thermal stresses are higher compared to helical, bevel and hypoid gears, and thus the thermal load capacity of worm gears is lower [24]. Therefore, the prediction of the heat balance and component temperatures of gearboxes containing one or more worm gear stages is very important, especially during the design phase.
In this study, an automatic simulation method for analyzing the efficiency and heat balance of various design of worm gears is developed and integrated in WTplus. First, suitable methods and calculations regarding the efficiency and heat balance calculation of worm gears are shown. Its integration into the simulation program WTplus is described afterwards. Finally, simulated efficiency and heat balance results of various worm gearboxes are compared to measurements from research and industry.
Niemann [23] and Weber [40] mathematically modelled the tooth contact of worm gears. Wilkesmann [41] performed elastohydrodynamic lubrication (EHL) calculations for different worm tooth geometries. Predki [29] carried out parameter studies and developed relative key figures, which form the basis of DIN 3996:2019-09 [9]. Bouchà [3] formulated a physics-based model for the calculation of the coefficient of friction under mixed friction for worm gears. Magyar [20] investigated the dynamics of worm gears and derived a tribological calculation model for the calculation of the coefficient of friction, which is the basis for a new standardizable approach for the calculation of worm gear efficiency [25].
Monz [22] and Mautner et al. [21] investigated the load capacity and efficiency of worm gears lubricated by consistent grease. They used a specific TNM for heat balance calculations, which correspond closely to the measurements. Further approaches to using TNMs for heat balance and temperature calculations with regard to gearboxes can be found in [26] for worm gears, [14] for hypoid gears, [4, 11, 19] for spur gears, [6, 42] for planetary gears and [38] for helical gears.
Although there are several approaches for the efficiency and heat balance calculation of worm gears, none of them uses an automatic approach to building the TNM. They either abstract their investigated gearbox as an isothermal system for which no temperature distribution can be calculated, or they build the TNM statically and specifically for an experimentally considered worm gearbox.
In terms of an efficiency calculation, values for every single one of the named forms of power loss are needed in as much detail as possible. Thus, a lot of research focuses on the formulation of calculation models to quantify load-dependent and no-load losses. The following two subsections present common and recent calculation models for predicting load-dependent and no-load gear losses of worm gears.
Since worm gears show different gear losses, depending on the direction of the power flow, the calculation of the meshing efficiency \(\eta_\mathrmz\) must be considered separately. When the worm shaft is driving, according to DIN 3996:2019-09 [9], Eq. (4) is used:
The mean coefficient of friction \(\mu _\mathrmmz\) represents the complex friction characteristic of meshing tooth flanks by one single mean value. In terms of worm gears, there are currently two different approaches and calculation models available. DIN 3996:2012-09 [8] describes a simpler, empirical model, while Oehler et al. [27] present a more detailed, semi-analytical one. The latter was standardized in DIN 3996:2019-09 [9], replacing the simpler approach in DIN 3996:2012-09 [8] very recently.
Currently, no specific, validated calculation model is available for the no-load gear losses \(P_\mathrmVZ0\) of worm gears. Even though DIN 3996:2012-09 [8] offers an equation for calculating the overall no-load loss of gearboxes with worm gears, it does not differentiate between the different power loss portions, as there are the gears, bearings and seals. Therefore, from a more gear component-specific perspective, this does not meet the requirements of a detailed analysis of the efficiency and heat balance of gearboxes with worm gears. This is in accordance with DIN 3996:2019-09 [9], where this approach was removed.
Calculating the no-load bearing losses, as well as the seal losses, and subtracting them from calculated overall no-load loss according to DIN 3996:2012-09 [8] does, theoretically, lead to the no-load gear loss of worm gears, but in practice, this is not useful. Also, calculations show that depending on the operating point, this may result in a negative no-load gear loss due to high calculated no-load bearing losses, which does not make sense.
Oehler et al. [27] points out that using this model may lead to uncertainties and minor miscalculations. For lack of a better solution, this may currently be the most precise calculation model for no-load gear losses of worm gears.
For example, the bearing manufacturers SKF [36] (cf. Eq. (14)). and Schaeffler/INA/FAG [32] (cf. Eq. (15)). provide simple empirical calculation models specifically for their bearing designs. Both models are based on the addition of no-load and load-dependent bearing losses.
Since temperature influences oil viscosity, which greatly affects the power loss of a gearbox, a temperature calculation model is required for an automatic and precise efficiency calculation. Since a gearbox shows local differences in temperature, it is reasonable to not only calculate a mean temperature for the whole gearbox but also specific local temperatures of the single components. This local heat balance analysis not only provides an opportunity to predict the thermal load capacity, but also to detect hot spots inside a gearbox. Using a TNM makes it possible to determine component temperatures in gearbox systems.
When expressed as a matrix, Eq. (20) contains \(n-1\) linearly independent equations and a single boundary condition, making it suitable for numerical solution. This formulation of an efficient, suitable thermal network is needed when it comes to an automatic and precise calculation of the efficiency and heat balance of gearboxes.
The efficiency and temperature calculation described in Sects. 3 and 4 appropriate to gearboxes with worm gears is customized and implemented in the simulation program WTplus [16], which is currently applicable to gearbox systems containing cylindrical and bevel gears. WTplus uses routines for the calculation of the efficiency and heat balance as shown in Fig. 2. Initially, a routine reads the input data followed by the macro geometry and parameter calculation according to [7, 8]. Where necessary, data is automatically complemented. WTplus then calculates the efficiency (blue) and heat balance (red) iteratively. If the calculation results in enough exactitude, an output file containing all relevant data is generated. The efficiency and temperature calculation, as well as the required extensions for gearboxes with worm gears, are described in the following Sects. 5.1 and 5.2.
Next, with the torques and speeds known, forces caused by the tooth system of worm gears can be calculated according to DIN 3996:2019-09 [9]. Subsequently, the tooth system forces are put into shaft-bearing context and thus the simulation program determines the reactive bearing forces. Then, oil data as viscosity and density is calculated to determine the tribological factors (cf. Sect. 3.1.1). Considering these values, the specific power loss portions of gears, bearings and seals are computed (cf. Sect. 3.1, 3.2 and 3.3). Lastly, the simulation program calculates all torques and speeds again, but this time it takes into consideration power losses that reduce the torques (lossy). Since these reduced torques change the tooth system forces, this leads to different bearing forces and thus changed power losses. Therefore, an iterative solution must be considered, comparing the output torques of two subsequent iterations. If the deviation between those results is below a given limit, efficiency is considered solved and the temperature calculation begins.
It is notable that the thermal network is built up fully automatically, abstracting the gearbox by suitable nodalization, linking those nodes and calculating necessary thermal conductances. The following explains the process of the abstraction for worm gears and shows solutions for the calculation of thermal conductances.
Regarding shafts, Geiger [11] shows the need to divide long narrow bodies suitably into several isothermal sections in order to minimize calculation errors and preserve compact network size. In terms of axial distance, the width of an isothermal section is set accordingly as less than or equal to its shaft diameter. Furthermore, the simulation program generates a new isothermal section wherever a component (bearing or gear) or a diameter change of the shaft is located (cf. Fig. 3).
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