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Dieser Artikel geht auf die analytische Effizienzberechnung fettgeschmierter Schneckenräder ein, ein Thema, das aufgrund der spezifischen Vorteile, die Fett in bestimmten Anwendungen bietet, von wachsendem Interesse ist. Die Studie konzentriert sich auf die Hauptunterschiede zwischen Fett- und Ölschmierung, insbesondere auf das nicht-newtonsche Verhalten von Fett und seine Auswirkungen auf Reibung und Effizienz. Es stellt eine detaillierte Methode zur Bestimmung der lokalen Zahnreibung, einschließlich Grenz- und Flüssigkeitsreibungskomponenten, vor und führt einen neuartigen Ansatz zur Berechnung der Schichtdicke unter Verwendung des effektiven Viskositätsmodells von Morales-Espejel et al. Der Artikel beleuchtet auch die experimentelle Validierung des analytischen Modells durch Tests auf einem Schneckengetriebeprüfstand, bei denen theoretische und experimentelle Effizienzergebnisse verglichen werden. Die Ergebnisse zeigen eine gute Übereinstimmung zwischen dem physikalischen Modell und den experimentellen Daten, wobei geringfügige Abweichungen auf vereinfachende Annahmen und Wärmeerzeugungseffekte zurückzuführen sind. Diese Arbeit bildet die Grundlage für eine genauere Auslegung und Optimierung des Getriebes bei der Verwendung von Fettschmierung und adressiert die aktuelle Lücke in physikalisch basierten Berechnungsmethoden für fettgeschmierte Schneckenräder.
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Abstract
Lubricating oils are the predominant choice for lubrication in many commercially utilized worm gearboxes. However, specific requirements and environmental conditions have driven a growing demand for oil-free drive systems. In certain applications this can be achieved using lubricating greases. Substituting lubricating oil with grease, and the consequent alteration in lubricant properties, introduces new challenges in the design, analysis, and tribological characterization of these gearboxes. In this context an analytical model to determine the efficiency of grease lubricated worm gears was developed including the rheological properties of greases and the tribological description of a grease-lubricated tooth contact which helps to take advantage of greases in worm gears by predicting the friction in these applications. Experimental results from different tribometers are used to determine lubricant parameters as well as tests on worm gears to validate the model. A simulative and experimental study for various operating conditions was conducted to investigate the influence of different operating modes on the efficiency of grease lubricated worm gears.
For the evaluation of the measured surface data the software Digital Surf—MountainsMap was used.
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1 Introduction
For worm gearboxes, oil is primarily used alongside splash lubrication. In contrast, consistent lubricants like greases are less common and are mainly applied in cases where sealing the housing is difficult or not feasible [1]. Examples include positioning gears and power gears with strict sealing requirements, such as those used in the food industry. In addition, greases provide advantages over conventional oil lubrication under certain operating conditions. For instance, in positioning worm gears with low rotational and sliding speeds, grease can help reduce friction and wear more effectively than oil due to the effect of the thickeners [2].
On the other hand grease lubrication has a negative thermal impact on the worm, leading to higher operating temperatures compared to oil operation [2]. However, in positioning gear applications, this effect is less significant, making the benefits of grease lubrication outweigh its drawbacks.
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The use of grease lubrication also imposes different requirements on the design and calculation of gearboxes compared to oil lubrication. This affects both the housing and the gearing design. Rheologically, grease behaves as a non-Newtonian fluid, meaning its viscosity changes depending on the shear rate. Several models, such as the power law and the Bingham model, describe this behavior by assuming that grease acts as a solid at low shear rates and as a viscous fluid at higher shear rates [3].
Unlike oil lubrication, there are currently no established calculation methods for predicting friction in grease-lubricated worm gears. Developing a physics-based friction calculation method for grease-lubricated gear contacts would allow for more precise gearbox design and optimization, particularly in reducing friction. This work forms the basis for calculating total friction in worm gear tooth contacts, which is the key distinction between efficiency calculations for gearboxes using oil versus grease lubrication. The study considers all relevant friction components, including boundary friction, fluid friction, and their combined effect under mixed lubrication conditions. A key aspect of the analysis is the calculation of film thickness, which influences the transition between these friction regimes. The proposed calculation methods will be demonstrated using a specific grease as an example in combination with one gear ratio. To validate the analytical approach, experiments are conducted on a worm gear test bench, providing empirical data to support the calculation method.
In summary, this paper presents an analytical approach to efficiency calculation for grease-lubricated worm gears by developing a friction calculation method and validating it through experimental testing. The shown approach provides a basis for calculating gearbox performance and efficiency when using grease as a lubricant instead of oil.
2 State of the art
Since oil lubrication is still the predominant choice in today’s worm gear boxes, this field is broadly researched. Based on magyars dynamic model described in [4], oehler developed a physical model to simulate the tribological contact of worm gears [5]. This calculation method allows the determination of friction and the resulting efficiency for oil lubricated worm gear boxes.
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Currently, no such physically based calculation method exists for determining friction in the grease-lubricated tooth contact of worm gears. Such a method is essential for application-oriented gearbox design and optimizing friction reduction. Due to this lack of design tools, most gear manufacturers rely on gearbox designs intended for oil lubrication and simply replace the oil with grease. While this approach meets the requirement for high sealing performance, it overlooks potential optimization opportunities.
In contrast to worm gears, grease lubrication in the context of roller bearings is widespread and broadly researched [3]. Even though the standardized film thickness calculation to determine the rating life of grease lubricated bearings according to [6] is using a simplified approach only regarding the base oil properties, there are more detailed methods available to describe grease film thickness. morales-espejel et al. present an experimental correction factor to describe an effective viscosity of the grease [7]. Another alternative calculation method is presented by cousseau et al. where the properties of the bleed oil of lubricating greases are used to calculate the film thickness [8]. The approaches show promising results for the calculation of grease lubricated contacts in general and will be examined for their suitability for worm gears.
In addition to the film thickness the rheological behavior of the lubricant regarding fluid friction is an important difference between oil and grease. Common models to describe the non-Newtonian behavior of grease are for example the power law [9] and the Bingham model [10].
3 Determination of the local tooth friction
The local friction in the tooth contact consists of the boundary friction and the fluid friction µF combined with the solid load-carrying portion ψ, which describes the share between both types of friction. To determine the solid load-carrying portion the film thickness in the contact is a key value since it describes the amount of lubricant which separates the contact surfaces.
The local tooth friction coefficient µZ can therefore be described using the boundary friction coefficient µG, the fluid friction coefficient µF and the solid load-carrying portion ψ [5]:
The boundary friction coefficient µG describes the friction in the areas of the contact where no lubricating film is present. Since the simulative quantification of the effects which are responsible for boundary friction is very difficult, it is a common method to use tribometer tests to determine the boundary friction [5].
The tribometer tests presented here were performed on a two-disc test bench, which represents the line contact in the worm gears, based on the principle for oil lubrication described by oehler. A detailed description of test bench and the execution of the experiments is shown in [11] and [12]. The test was performed with a small lubricant present and the parameters were chosen so that the film thickness ratio is λ < 1 to ensure conditions in the boundary friction regime. The result of the tribometer test is a traction for every examined operating condition as shown exemplary in Fig. 1. By investigating different operating points by varying the pressure p, temperature ϑ (measured inside of the disc near the contact surface) and slide-to-roll ratio SRR, characteristic maps for boundary friction can be determined. Depending on the given conditions in the investigated tooth contact in the simulation the boundary friction can then be calculated through inter- or extrapolation using the determined maps.
Fig. 1
Traction curves for p = 386 MPa and 582 MPa at ϑ = 20 °C [12]
The fluid friction µF results from the shear stress of the lubricant in the tooth contact. This shear stress can be calculated using a rheological model for the given lubricant and so in contrast to the boundary friction the fluid friction can be analytically determined. A common model to describe lubricating oils is the one proposed by bair and winer in [13]. However, this method is not applicable to grease lubrication since it describes a Newtonian fluid. From the various models available to describe the behavior of lubricating greases the one introduced by sisko in [14] was chosen. The main upside from this model is the wide range of applicability especially in the area of high shear rates, which typically occur in the worm gear contact. The model describes the shear stress and shear viscosity of a fluid depending on the shear rate. The non-Newtonian behavior is represented by the yield point τy. In Eqs. 2 and 3τ describes the shear stress, γ the shear rate, ηb the viscosity of the base oil and η the shear viscosity of the grease. The parameters τy, K and n are dependent on the lubricant and need to be determined by characterizing the specific grease. This characterization was done for the investigated grease at different temperatures (25, 40, 60 and 80 °C) using a rotational viscometer with a plate-plate geometry. The measurements were performed at ambient pressure and shear rates between 0 and 1500 1/s. These conditions present a compromise between the capability of the viscometer and the typical conditions in a worm gear. The yield point τy can then be determined using the evaluation according to DIN 51810 [15]. The base oil viscosity ηb can be obtained from the data sheet of the examined lubricant.
Afterwards the grease consistency index K and shear thinning parameter n can be fitted to the measured shear viscosity of the grease in dependence of the shear rate using Eq. 3. This procedure is, analogous to the fit of the boundary friction, performed for various temperatures, so that a range of operating conditions can be evaluated. With these values the shear stress for this specific grease can now be determined for any shear rate in the tooth contact inside of the boundaries of the sisko model and the examined temperature range.
3.3 Determination of the film thickness and the solid load-carrying portion
The solid load-carrying portion ψ can be determined using the approximation according to zhou and hoeprich [16]:
The parameters BZH and CZH can be determined by using a contact simulation of the measured surfaces of both contact partners using the half space theory. The dimensionless film thickness parameter λ is obtained by calculating the quotient of the film thickness hfilm and the square center roughness of both contact partners Sq12 [5].
The basic difference between oil and grease lubrication in regards of the calculation of the solid load-carrying portion is the determination of the film thickness hfilm, since the surface roughness related parameters are not affected by the lubricant. These parameters are solely dependent on the surface of the two contact partners since they describe the contact between roughness peaks, where no lubricant is present.
To find the best fitting approach for calculating the film thickness tests on a ball-on-disc tribometer were performed with examined grease and afterwards compared to the mentioned calculation methods proposed in [7] and [8]. The procedure of the tests and the comparison is described in [12] and an example is shown in Fig. 2. This exemplary result shows the measured film thickness over the sum velocity in the contact at a temperature of ϑ = 60 °C. In addition to the measurement the theoretical film thickness for this operating point is shown as well. This value is calculated by using venners equation for the central film thickness of point contacts proposed in [17]. This calculation is combined with two different approaches for calculating the viscosity of the grease. One being the approach from morales-espejel et al. with an effective grease viscosity which consists of the base oil viscosity and a velocity dependent parameter to account for the effect of the thickener. The other approach is using the standardized oil calculation but with bleed oil properties. Analogous to the other performed characterization tests, the film thickness measurement was also done at multiple temperatures resulting in similar results for every operating point. The proposed model of an effective viscosity for the grease in [7] shows the best fit to the measured behavior. One important factor is the mapping of the increased film thickness at low speeds which is a known advantage of grease compared to oil lubrication. This effect occurs thanks to the thickener component of the lubricant which is necessary to produce a grease from base oil. Based on these results the approach from morales-espejel et al. will further be used in combination with venners equation to calculate the film thickness in the grease lubricated worm gear contact.
Fig. 2
Comparison between measured and calculated film thickness
4 Analytical and experimental efficiency determination
With the implementation of the friction model for the grease lubricated tooth contact into oehlers calculation method the resulting losses inside the tooth contact can be calculated and the overall efficiency can be determined. The determination of the remaining losses is briefly shown in the next section before describing the experimental efficiency determination to validate the calculation method.
4.1 Analytical efficiency calculation
In general, the efficiency of a gear box is defined as the quotient from outgoing power Pout to incoming power Pin. This can also be described using the power loss PL which is the difference between the two mentioned powers [18].
The total power loss is described by the load dependent and load free toothing losses PLZ and PLZ,0, the loaded and load free bearing losses PLB and PLB,0, the sealing losses PLS and other losses PLX which occur for example in clutches of shifting gears and can be normally be neglected in worm gears [5].
An upside regarding the design of the housing of the gearbox is the possibility of omitting the seals and there the sealing losses can be neglected next to PLX. Another difference to oil lubrication is the calculation of PLZ,0, which is mainly driven by splash losses when the gear is rotating through the lubricant. This effect is very difficult to describe for grease lubrication due to the non-Newtonian behavior of the lubricant and therefore the splashing losses will not be considered for simplification of this calculation method. It is assumed that the gears dig themselves free in the grease after a while. This reduces Eq. 6 to the determination of load dependent toothing losses and the bearing losses, which can be calculated using provided methods from gearing manufacturers. The load dependent toothing losses are determined using the local tooth friction coefficient µZ and the local present forces.
4.2 Experimental efficiency determination
To validate the physical model experimental tests are carried out on a back-to-back worm gear test rig which is shown in Fig. 3. The setup in general and the testing procedure is inspired by oehlers validation performed for his physical calculation model [5]. During the test the gear box is driven by a certain input speed and on the output a defined breaking torque is applied. The defined operating condition is then run continuously for 5 min without dedicated temperature control. Nonetheless the mass temperature of the worm wheel is tracked during the test to gain information about the heat generation due to friction. This is achieved with the wireless temperature telemetry at the output shaft of the gearbox as shown in the sketch. Additionally, the speed and torque are measured on both the input and output side of the gear box and send to the monitoring computer. With these values it is afterwards possible to determine the efficiency of the system based on torque and the constant gear ratio [18].
$$\eta _{\mathrm{G}}=\frac{T_{2}}{T_{1}\cdot i}$$
(7)
Fig. 3
Sketch of the test bench setup for the experimental efficiency determination
The torque measurement systems are connected with metal bellows couplings to the shafts of the gear box. This flexible design allows a wide range of different worm gearboxes to be tested with the setup. The efficiency of one test is afterwards calculated by creating a mean value over the duration of 5 min and every operating point is repeated three times. Based on the three resulting efficiency values the average efficiency for one operating point is determined.
4.3 Comparison between theoretical and experimental efficiency
The resulting gear box efficiency was theoretically and experimentally determined for a worm gear with center distance a = 50 mm and a gear ratio of i = 6.75. The test was performed at an input speed of n1 = 500 rpm and three values for the output torque T2. The values for the output torque were selected on the basis of the maximum torques specified by the manufacturer. Since there is no temperature control inside for the gear box the temperature starts at ambient level and increases during the tests due to heat development. The amount of temperature increase is dependent on the power transmitted at the respective operating point. To take this effect into account, the theoretically determined efficiency was calculated for the average temperature of the respective experiment. The comparison for both results is shown in Fig. 4 including the error bars for the experimental measurements based on the three performed repetitions which were mentioned before.
Fig. 4
Comparison between analytical and experimentally determined efficiency for i = 6.75 and n1 = 500 rpm at ambient temperature
In the plot the experimental results are represented in blue marked “Experiment” and the theoretical values are labeled “Tribosim” in green. The comparison shows a good alignment between theoretical and experimental values especially for the middle torque value investigated. The calculation for the lowest output torque overestimates the efficiency while the value for the output torque of T2 = 54 Nm is slightly underestimated by the Tribosimulation but still close to the experimental values. Overall, the physical model shows a good approximation for the efficiency of grease lubricated worm gears especially when the simplifying assumptions are considered. The visible difference between the calculation and the experimental results could possibly be due to the neglection of the load independent losses in case of T2 = 13.5 Nm. Another factor is the use of the average temperature for efficiency calculation, which could explain the greater difference at the operating point with the highest torque. Since this operating point also provides the greatest overall power, it has the biggest heat generation and the influence of the inaccuracy in temperature should be higher. This is something that needs to be further investigated in the future.
5 Conclusion
The presented physical model provides a method to calculate the efficiency of grease lubricated worm gears. This is achieved by using a calculation method for oil lubricated worm gears and adapting the description of the friction in the tooth contact. The single components of the friction calculation are adapted using available models for lubricating greases. In the process of choosing the best fitting model various approaches were analyzed and validated with different tests resulting in the decision for the best fitting model to describe the tooth contact in worm gears. Finally, the complete efficiency calculation method was validated with experimental results on a worm gear test bench. The comparison showed a good fit between theoretically and experimentally determined efficiency values with slight differences. The differences show room for improvement regarding the load independent losses and the effect of heat generation. One source of deviation could also be the difference in the analyzed contact to validate the calculation of the film thickness. This topic will be further investigated in the future to validate the model with a line contact setup on the tribometer. Another point which was not considered in this work is the grease distribution inside of the gear box. The assumption was that there was a sufficient amount of lubricant in contact at all times, which is somewhat contradictory to the theory of neglected splash losses. This is an important factor since it is possible that there is not enough grease available to fully lubricate the tooth contact which leads to starvation which is a known problem in grease lubrication.
In addition to that there will be further tests with different load cases in the future to get a wider comparison between experiment and calculation for the validation of the model.
Funding
This work was supported by the German Federal Ministry of Economic Affairs and Climate Action (IGF 22365 N) within the framework of the Forschungsvereinigung Antriebstechnik e. V. (FVA project 962 I).
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