Ellipsoid contact analysis and application in the surface conjugate theory of face gears

The machining principle and realization method for the continuous generative grinding face gear by a worm wheel are introduced. Based on a five-axis linked CNC grinding machine, a new method is presented to deprive the equation of face gear error tooth surface by assuming the tool surface as the error surface, where actual tool installation position error is introduced into the equation of virtual shaper cutter. Surface equations and 3-D models for the face gear and the worm wheel involving four kinds of tool installation errors are established. When compared, the face gear tooth surface machined in VERICUT software for simulation based on this new method and the one obtained based on real process (grinding face gear by using a theoretical worm wheel with actual position errors) are found to be coincident, which proves the validity and feasibility of this new method. By using mesh planning for the rotating projection plane of the face gear work tooth surface, the deviation values of the tooth surface and the difference surface are acquired, and the influence of four kinds of errors on the face gear tooth surface is analyzed. Accordingly, this work provides a theoretical reference for assembly craft of worm wheel, improvement of face gear machining accuracy and modification of error tooth surface.

The tooth surface of a face gear is recognized as un-developable ruled surface in tradition, which result in the problem that face gear generation using simple cutter with high efficiency and accuracy puzzled people for a long time. Applying the traditional shaping method for face gear generation in derivation of the tooth surface position vector, a space straight line equation was established by using this vector, to illustrate that the tooth surface of the traditional face gear could be considered as developable ruled surface constituted by space lines family, the position vectors of the developable ruled surface were established respectively based on involute, rack and considering the spatial position relationship in meshing. Elaboration and proposition of the principle and method that were used to generate the face gear with the developable ruled surface by straight edged cutter in line contact, and representation of the generated surface model. Furthermore the developable ruled surface was compared to the traditional surface, the normal difference between the two kinds of surfaces is on micro-scale. Numerical examples shows that the normal distance is no more than 1/70 of modulus, the developable ruled surface to which was performed simple longitudinal crowning is able to mesh with the involute surface of the pinion correctly, and demonstrates acceptable geometric meshing performance.

A new geometric spiroid worm drive has been designed and later produced, i.e., spiroid worm drive having arched profile in axial section (Bodzas (2014), Dudas et al. 2012). With the knowledge of the two-parametric vector-scalar function of the conical helicoidal surface having arched profile in axial section, the aim was to determine the tooth surface of the spiroid face gear and to make a common modeling process with which any tooth surface of the face gear which is mashing with spiroid worms can be produced. After the definition of the tooth surface, CAD model of the worm drive pair the production tool has been produced and the real physical model of the drive pair and the tool has also been made by rapid prototyping (RP) technology.

A new closed-form calculation of envelope surface for modeling face gears

In order to further improve the grinding efficiency of face gear, this paper presents a novel method for grinding face gear along contact trace using disk CBN wheel. Firstly, the grinding principle of face gear is studied, and the tooth surface equations of rack and gear shaper cutter are derived, thus formulating the tooth surface equation of disk CBN wheel. Secondly, considering the structure of the face gear grinding machine tool and the impact of the grinding wheel parameters on the face gear tooth surface, the disk CBN wheel configuration is proposed, and the calculation method of the tool position of the face gear grinding is established. Thirdly, according to the structural characteristics of the grinding wheel, the tooth surface contact trace of the face gear is designed, and the face gear development grinding processing method along the tooth surface contact traces is presented. Finally, a test of grinding spur face gear with disk CBN wheel is carried out on a laboratory-made five-axis face gear processing machine. The test results verified the correctness of the face gear grinding. Compared with the dish wheel for grinding face gear, the grinding method proposed in this paper can further improve the grinding efficiency of face gear. This method will break new ground for the engineering application of the face gear.

This paper proposes a modification method for tooth surface of face gear drive with a grinding worm on a numerical grinding machine. The surface equation of grinding worm is derived, and the coordinate System of generating the worm is established. Tooth contact analysis (TCA) is performed to investigate the performance of face gear drive before and after modification, and the alignment error is considered. This method can obtain a more stable bearing contact in contrast to the method by increasing tooth number of shaper. The longitudinal bearing contact on the face-gear tooth surface has been obtained which will increase the contact ratio. By modification the edge contact at surface edges of the gears can be avoided and the modification magnitude can be controlled freely.

Face gear drives have been well applied in high-speed and heavy load applications. Due to the strict requirements of machining accuracy and quality for those applications, face gears are mainly manufactured by worm grinding method, of which a dressing wheel is applied to generate the worm surface as its enveloped surface. Based on this manufacturing process, the profile of the dressing wheel should be well defined to make sure the final meshing performance of the face gear drives. In this work, we firstly investigate the mathematical model of dressing wheel with a general profile modification. The worm surface is the envelope to the family of dressing wheel surfaces. Specially, the result is obtained as a closed-form. Subsequently, the face gear tooth surface is calculated as the envelope surface of the worm surface. With the tooth surface models of both face gear and pinion, the tooth contact analysis (TCA) can be computed. According to this method, different parameters for the profile modification of the dressing wheel are compared to improve the working performances by finding the minimal transmission error without edge contacts. The proposed method is validated with simulations.

Calculation of line and point contact ratio for orthogonal spur-face gear drive

In order to process the hardened face-gear, the worm for face-gear is studied. The research includes designing and dressing method of worm. According to the meshing relationship of face-gear, worm and shaper, coordinate systems of face-gear, worm and shaper are established. By envelope theory, the face-gear tooth surface are obtained, worm tooth surface and shaper tooth surface. The solid models of face-gear, worm and shaper were built, and the approach of dressing was simulated for hardened face-gear processed.

Identification of crucial geometric errors of face gear grinding machine based on geometric error-tooth surface normal error model considering worm dressing

To further develop the advantages of face gear drives and improve their anti-sticking performance and wear resistance, a novel method using laser cladding as the primary process is proposed; here, grinding processing is used as the secondary process, i.e., achieving tooth surface strengthening of the orthogonal offset face gear using two types of materials to form a material gradient based on a laser cladding technology. Then, grinding processing is performed using a five-axis grinding machining. First, according to the forming principle of the face gear tooth surface, we establish the coordinate system of the orthogonal offset face gear and the gear cutter coordinate system; we also derive the tooth surface equation of the virtual gear cutter and the tooth surface equation of the orthogonal offset face gear. Second, we research the principles of the laser cladding and establish a laser cladding test bed for the orthogonal offset face gear. Based on the analysis of the design of the cladding layer and the process parameters of the face gear, the selection of the scanning path, the heat accumulation during the laser cladding process, and the accuracy control of the scanning position, we perform a laser cladding experiment on the offset orthogonal face gear. Third, we establish the machining coordinate system of the face gear based on the machine tool structure and conduct a grinding experiment for the face gear after the cladding process. Finally, we analyze the morphology and hardness of the face gear after grinding. A bench test verification of the axle face gear transmission system is completed. The results show that the precision and surface hardness of the tooth surface are significantly improved after grinding. Meanwhile, the accuracy and feasibility of the laser cladding surface of the face gear are verified.

PurposeThis study aims to research the time-varying mesh stiffness (TVMS) model for orthogonal face gear drives considering elastohydrodynamic lubrication (EHL) and provide a theoretical basis for understanding the dynamic characteristics of face gear drives.Design/methodology/approachConsidering EHL, a novel model is proposed to calculate the TVMS of orthogonal face gears using the deformation compatibility condition. First, the tooth surface equations of orthogonal face gears are derived according to the tooth surface generation principle. Then, the oil film thickness on the tooth surface of face gears is obtained by solving the governing equations of EHL. Furthermore, the proposed model is used to calculate the TVMS of face gears along the mesh cycle and is verified. Finally, the effects of module, tooth number of shaper cutter and pressure angle on mesh stiffness are analyzed.FindingsThe results indicate that when the contact ratio is greater than 1 and less than or equal to 2, the TVMS of face gears exhibits a phenomenon of double-single tooth alternating meshing where sudden changes occur. As the module increases, the overall mesh stiffness of face gears increases, and the magnitude of the sudden change at the moment of single-double tooth alternating meshing gradually increases. As the tooth number of shaper cutter and pressure angle increase, so does the TVMS of face gears. When the effect of oil film is considered, the calculated TVMS of face gears slightly increases overall and the increase in average oil film thickness leads to a rise in the TVMS. This study provides a theoretical basis for understanding the dynamic characteristics of face gear drives.Originality/valueThis study’s originality and value lie in its comprehensive approach, which includes conducting analysis based on loaded tooth contact, considering the influence of elastohydrodynamic lubrication, proposing a novel analytical–finite–element model, calculating TVMS of face gears, verifying the proposed model and analyzing the effects of typical structural parameters and oil film thickness.

To solve the problem of generalized design and batch processing of face gear, a method of tooth surface design and side milling of ruled line face gears is proposed in this paper. First of all, the discrete points of the pitch cone surface of the face gear are calculated, and the straight-line cluster and section curve of the tooth surface is constructed to analyze the deviation of the ruled line face gear. Secondly, the tooth surface model of the ruled line face gear is constructed, and the curvature and contact trace are calculated. Thirdly, the curvature and tangent of the ruled line face gear tooth surface are analyzed, the feed plane of the equidistant surface and the guideline are obtained, the torsion angle of the tooth surface and the distribution of the tangent vector along the generatrix are calculated, and the tool axis position based on the two-point offset method is studied. Finally, the side milling processing test of the face gear is carried out, and the tooth surface of the face gear after side milling is measured. The measurement results verify the correctness of the tooth surface design and the side milling processing method of the ruled line face gear.

Face gear transmission is a kind of space-meshing mechanism that is mainly used in the field of aviation. Compared with traditional transmission, it has the advantages of stability, reliability, low noise, and strong carrying capacity. However, owing to its complex tooth surface, there are no means to accurately model the face gear. Likewise, research based on the geometry is difficult. Therefore, the tooth surface equation of the face gear is derived in this article based on the meshing theory. Based on the equations, the point cloud of the face gear tooth surface is calculated, the complex tooth surface is generated, and the face gear is accurately modeled. Moreover, taking tooth surface friction excitation into consideration, a multi-degree-of-freedom nonlinear dynamic model of face gear transmission system is established, using the adaptive variable step length Runge-Kutta method. As shown in the results, the bifurcation diagram, phase diagram, time history diagram, and Poincaré section diagram are combined to analyze the influence of tooth surface friction and meshing frequency on the dynamic characteristics of the system.

Based on the general mathematical model of Illés Dudás which is appropriate for mathematical modelling of production technology methods we have worked out a model for resharpening analysis of conical hob. After the hob resharpening using numerical calculations the determination of the tooth surface of face gear by cutting edges is necessary for the analysis. Based on this methods we could calculate the permissible critical angle of the hob and the profiles of the hob and the face gear in axial section. The permissible critical angle of the hob is the critical angle the hob cutting edge of which manufactured face gear profile is situated in the permissible profile error tolerance. We have worked out a new geometric conical worm gear drive that is the conical worm gear drive having arched profile. Using this mathematical model we have done resharpening analysis for the hob having arched profile and determined the permissible critical angle.