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2. THEORY
The shape of the surfaces of the teeth of the gears is obtained from a ‘cut and mesh’ worm gear contact model developed by Fish and Munro [3]. This analysis considers a series of positions of the worm gear and for each position the wheel is brought into contact with the worm in the geometric model. This establishes the point of contact under zero load and the direction of the common normal at the contact point. The position of a series of points on the surface of the worm and the wheel tooth is obtained and these are fitted with a high order polynomial of the tangent plane coordinates. These polynomials are taken to describe the shapes of the contacting surfaces for the subsequent EHL analysis. The EHL mechanism requires that the Reynolds equation for the lubricant's hydrodynamic action, and an elasticity equation relating the lubricant pressure to the elastic deflection are solved simultaneously. The high sliding in the contact generates heat by lubricant shear and a thermodynamic energy equation is also specified for the lubricant film and for each of the contacting solid bodies. The means by which these equations are solved is described in detail in [1,2] and is therefore not repeated in the current paper. A thermal EHL solution for each of a series of mesh positions can be obtained and these result in detailed distributions of pressure, lubricant film thickness, lubricant temperature and contacting body surface temperatures.
When the contact occurs near the root of the wheel tooth its behaviour is essentially Hertzian and the area of dry contact can be established from a semi-infinite body analysis, as is inherent in the EHL solution scheme referred to above. As the contact moves towards the tip of the wheel tooth, however, the area of contact becomes longer and narrower than that predicted on the basis of a semi-infinite body elastic analysis. This is due to the fact that tooth flexure under load influences the stiffness of the tooth in a non-uniform way with the tooth being stiffest at its centre and least stiff at the limits of its face width. A correction to the geometry is introduced to ensure that the semi-infinite body treatment of elastic deflection that is embedded in the EHL analysis calculates the contact area correctly. This correction is a curvature change in the axial direction of the tooth which is adjusted for each mesh position so as to obtain dry contact areas that coincide with those calculated based on tooth stiffness ideas described in [3, 4, 5].The model adopted to calculate the rate of material removal by the wear process (expressed in m3/s) is based on the Archard wear formula [6, 7]
In this formula W is the load, Us, is the sliding velocity, H is the hardness of the material, and the wear coefficient, k, is a dimensionless parameter that can be expected to vary according to the nature of the sliding contact, and is to be determined empirically for any given system. For the calculation of wear on the surface of the wheel tooth it is clearly necessary to distinguish between locations that are subject to different loading and there is also a need to introduce a lubricant film thickness parameter to distinguish between regions experiencing thick and thin lubricant films in comparison with the surface roughness.
In Eq. (2) p is the local pressure, us is the local sliding speed, Ra is the Centre Line Average of the composite surface roughness, h is the local film thickness, and the wear rate is the change of wear depth per unit time. Power n is a disposable parameter that controls the extent to which film thickness in the form of 'lamda' ratio h/Ra influences the wear rate. In the overall wear calculation included in relevant standards [8] this power is specified as 2.24 which causes a wear rate when h/Ra = 0.2 to be some 40 times that when h/Ra is unity. When h >> Ra the surfaces are separated by a full lubricant film which prevents wear. As h is reduced to the order of Ra and below, localised direct interaction of the surfaces becomes increasingly likely to occur with corresponding increased wear of the surfaces. In the current paper the emphasis is on wear patterns experienced by the softer bronze wheel, and the Ra value is taken to be that of the hard steel worm component.