This paper presents a physically grounded calculation method to determine the efficiency of worm gear drives. This computation is based on the Institute of Machine Elements, Gears, and Transmissions (MEGT) tribological simulation, which can determine the local tooth friction coefficients (Ref. 1). With this knowledge other power losses such as the bearings, oil churnings and seals power losses can also be calculated.

Introduction

Worm gears belong to the cross axis drives; with them a very high gear ratio can be realized in one stage. But this advantage is coupled with a high sliding velocity between the meshing gear teeth. Therefore the tooth friction and also the tooth friction power losses are higher than with other gear variants. By the construction of gear solutions it is important to know their future efficiency. Previous calculations can help engineers to compare the different solution principles with each other and to choose the best variant. In this case, a singleworm gearbox competes with a complex multi-stage, helical gear/bevel gear transmission. By the last variant, more responsible empirical equations exist to predict the power losses in all stages (Refs. 2–4). Unfortunately the standardized empirical calculation method (Ref. 5) for determining the efficiency of worm gear drives is not useful if the gear ratio differs from 20.5. Another disadvantage of this calculation is that it ignores the variation of lubricant oil and surface roughness. These omissions complicate the prediction of the efficiency of worm gear drives. At the University of Kaiserslautern, a calculation method to determine the tribological behavior of worm gear drives has been developed. This method is able to calculate the local tooth friction and thereby the efficiency of the gear meshing. Taking into consideration the other loss components in the gearbox, the efficiency of the complete gear unit can also be calculated. This calculation method is presented in detail in this paper.

Power Flow in Worm Gear Drives

Worm gear units are very compact; a single gearbox includes only two shafts with the coupled worm and worm wheel, bearings, seals, and oil sump. These machine elements cause power losses in the transmission. Therefore the losses in worm gear drives can basically be traced back to four reasons; tooth friction Pltf; oil churning Plchur; bearings Plbear; and shaft seals Plseal losses. The potential loss sources in a worm gearbox and the power flow assigned to the Shafts 1 and 2 in a worm gear drive are illustrated in Figure 1. To determine these power losses it is necessary to know the local loads of every machine element. The loss sources can be divided into load-dependent and no-load-dependent components. Load-dependent power losses are the tooth friction power losses and the bearing power losses. But the latter has also a no-load-dependent part. Other no-loaddependent power losses are the churning and the shaft seals power losses. In the next section the determination of power losses is described.

Calculation of Power Losses in Worm Gear Drives

Tooth friction power losses. To determine the tooth friction in worm gear drives, a tribological calculation method has been developed at the Institute of Machine Elements, Gears, and Transmissions (MEGT), University of Kaiserslautern (Ref. 1).This simplified tribological model of the tooth meshing by worm gears is shown in Figure 2.

The first step of this calculation is determining the contact. Worm gear drives have a line contact. Several points of the current contact line are calculated by a solution of the “equation of meshing.” According to the equation of meshing, the normal vector of the common surface is perpendicular to the relative velocity vector of the bodies at the contact point (Ref. 6). Between two calculated contact points of a single contact line — according to Niemann — the tooth flanks are substituted by rolls whose radius coincides with their reduced radius of the curvature, and the rolls perform rotational motion of the same or opposite direction as the velocity valid for the given contact point. If the pressure mound above the flattening was approached according to Hertz, then the oil film thickness between the pairs of rolls can be calculated analytically (compare Figure 2). Because worm gear drives are usually working under mixed lubrication, it is necessary to determine the local proportions of the dry and fluid friction mechanisms. It is estimated with pre-computed division curve, which represents the rate of boundary lubrication depending on the dimensionless film thickness. The generation of this curve is based upon the statistical description of the representative roughness profiles of the tooth flanks and the contact mechanics (Ref. 1). With knowledge of the film thickness and the rate of boundary lubrication, the load distribution along the contact lines can be calculated iteratively. This technique was developed by Bouché (Ref. 7) and is based on the special wear properties of worm gears with full contact pattern. In the next step of the simulation a mixed friction coefficient between a pair of rolls is recorded. The friction between the teeth generates heat rise. Therefore both the contacting surfaces and the oil film between them are heated by friction. The calculation of the surface temperature is based on the numerical solution of Fourier’s law for heat conduction. Our solution-technique is founded on Plote’s method with Fourier integrals (Ref. 8). The oil temperature in the contact is calculated according to the simplified energy equation of the oil film. To calculate the surface temperatures and the oil middle temperature, the Fourier and the energy equations are solved simultaneously. Knowing the distribution of the oil temperature, an oil viscosity and an oil film shearing can be calculated as well. This shearing is two-dimensional in the gap by worm gear drives. By integrating the shear stress along the penetration surface, the frictional force arising from hydrodynamic lubrication can be elaborated. In mixed lubrication, the friction coefficient consists of a component arising from boundary lubrication and one arising from hydrodynamic lubrication. These components are weight by the abovementioned division curve and so the coefficient of the mixed friction is already known. As this value was freely chosen at the beginning of the thermal calculation, the last computation steps must be repeated in an iteration loop until the error is acceptable. After this calculation the local parameter — for example, oil film thickness, contact pressure, surface temperatures and the coefficient of the tooth friction — are known.

To analyze the efficiency of the worm gear drives the average coefficient of the tooth friction was determined in every meshing position. In this paper a ZK profile worm gear drive with a = 100 mm center distance and i = 20.5 gear ratio was analyzed. Figure 3 shows the calculated local tooth friction coefficients above the meshing field and their average value above the meshing position. In this case the input driving speed was n1=1,500 1/min and the output torque T2 = 500 Nm. Mineral oil lubrication was used with viscosity class ISO VG 150 by sump temperature ?s = 60°C.