Development of a multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives

This article describes a new multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives. The typical steps of the multiblock schemes have been investigated in depth to obtain a fast and simple way to mesh planar sections of gear teeth, allowing local mesh refinement and minimizing the appearance of distorted elements in the mesh. The proposed procedure is completed with two different mesh quality enhancement techniques. One of them is applied before the mesh is generated, and reduces the distortion of the mesh without increasing the computational time of the meshing process. The other one is applied once the mesh is generated, and reduces the distortion of the elements by means of a mesh smoothing method. The performance of the proposed procedure has been illustrated with several numerical examples, which demonstrate its ability to mesh different gear geometries under several meshing boundary conditions

Contribution of the deflection of tapered roller bearings to the misalignment of the pinion in a pinion-rack transmission

The misalignment of the gears is one of the main causes of premature failure in gear transmissions. This misalignment can be caused by different phenomena where the deformation of the elastic elements of the transmission usually has a major importance. In this paper, a pinion and rack transmission is used to investigate the misalignment of the pinion produced by the deflection of both shaft and bearings, focusing in the contribution of the bearings to this misalignment. For this task, a design space with 261 different cases of pinion and rack transmission has been explored. For each case, a realistic 3D FEM model (including gears, shaft and roller bearings) has been created and the coupled structural and contact problem has been solved. From the results of the FEM models, the different parameters of misalignment of the pinion have been computed and analyzed through five studies. These studies revealed the high importance of the elasticity of the bearings in the misalignment of the pinion and, in general, the important error that can be assumed when the bearings are not included in the structural models of gear transmissions.The authors express their deep gratitude to the Spanish Ministry of Science and Innovation (MICINN) for the financial support of research project ref. DPI2013-47702-C2-2-P

A 2D finite element based approach to predict the temperature field in polymer spur gear transmissions

This article describes a new numerical approach to determine the temperature field of polymer spur gears during their operation. The approach is based on an uncoupled procedure in which a mechanical problem is solved to determine the amount of heat that is generated by friction during the meshing of the gears, and then this heat is considered as a thermal load to perform a thermal analysis of a finite element model of the transmission. The amount of heat generated by friction is determined from the results of a numerical loaded tooth contact analysis of the transmission, which is based on the finite element method. The generality of the finite element method enables this approach to be applied to any kind of spur gear transmission, regardless of the geometry and the material of the gears and lubrication conditions. The resulting approach is applied to determine the temperature field of a spur gear transmission where polymer and metallic gears are combined, under several different operating conditions. The results obtained from this approach are compared to those obtained from experimental analyses, showing a good degree of similarity between them

An adaptive mesh refinement approach for solving nonHertzian elastic contact problems

Semi-analytical methods are a common way of solving non-hertzian contact problems when designing mechanical components. These methods require of the discretization of the domain into a set of pressure elements and their accuracy and computational cost are related to the number of elements in which the domain is discretized. But, while the accuracy increases as the pressure element mesh is refined, the computational cost increases quadratically with the number of pressure elements. So in the great majority of the cases, a commitment between accuracy and computational cost must be achieved. In this work, a new approach has been developed to improve the performance of semi-analytical methods for solving contact problems. This approach uses an adaptive mesh refinement strategy, based on the quadtree decomposition of the domain. As a result, the computational cost decreases, while the accuracy of the method remains constant

Adaptive Mesh Refinement Strategy for the Semi-Analytical Solution of Frictionless Elastic Contact Problems

Semi-analytical methods are commonly used to solve contact problems. These methods require the discretization of the domain into a mesh of pressure elements. In general, it can be said that their accuracy increases as the pressure element mesh is refined. However, the refinement of the pressure element mesh also implies an increase in their computational cost. So, in the great majority of the cases, a commitment between accuracy and computational cost must be achieved. In this work, a new approach is presented, whose main purpose is to improve the efficiency of the semi-analytical methods that are used to solve contact problems. To do so, an adaptive refinement of the pressure element mesh is implemented. This strategy allows for a reduction of the computational cost of the method, while its accuracy remains unaffected

A simple procedure for generating locally refined 2D quadrilateral finite element meshes of gears

This article describes a new procedure for automated generation of two-dimensional locally refined quadrilateral meshes of gear drives. In this new procedure, a base mesh is generated using a multiblock meshing procedure. Then, selected elements of the base mesh are subdivided to obtain a refined mesh in certain parts of the gear teeth. The proposed procedure is completed with a mesh quality enhancement technique, which is based on an optimization-based smoothing. It also includes strategies that allow to automatically identify and refine those areas of the gear that are typically subjected to elevated stress gradients. The performance of the proposed procedure is illustrated with numerical examples, and it is compared to other existing meshing procedures, both in terms of mesh distortion and accuracy of the results

Convective heat transfer modelling in dry-running polymer spur gears

Heat convection is an important phenomenon in the process of cooling polymer spur gears running in dry conditions, which ultimately affects the strength of the gears. In order to gain some insight into this phenomenon, a numerical heat convection model for polymer spur gears is proposed in this work, which is based on a detailed CFD simulation of the gears in operating conditions and it allows us to investigate the heat convection through their external surfaces. The performance of this numerical model is illustrated with several examples, in which a parametric study has been conducted to observe the variation of the heat transfer coefficients with the face width and the angular speed of the gears. The results obtained from this parametric study are compared to those obtained from a representative classical heat convection model, observing that the relative differences between them in terms of heat transfer coefficients can be as high as 125%. Finally, a new optimized heat convection model for polymer spur gears running in dry conditions is proposed, in which the convective heat transfer coefficients for the external surfaces of the gears are calculated from empirical equations based on the Newton’s law of cooling. This optimized model has lower computational cost than the numerical one, while it provides an important increase of the accuracy of the classical heat convection models, reducing the maximum relative differences to 10%.Funding for open access charge: CRUE-Universitat Jaume

A new analytical model to predict the transversal deflection under load of stepped shafts

Shaft deflection is a common phenomenon in machine design that has an important influence on the behavior of many transmission elements supported by the shafts, like gears, pulleys, sprockets, etc. This deflection can be estimated efficiently by using 1D models associated to beam theories, but machine shafts are usually stepped shafts and it has been demonstrated that classical beam theories do not predict accurately the deflection of this type of shafts. Thus, in this work, an equivalent model of the shaft to be used in conjunction with the Timoshenko beam theory has been proposed to improve the accuracy of the computed deflections. The new model substitutes the steps of the shaft by a linear variation of the diameter of the cross-section, removing the discontinuities caused by these steps. The slope of the linear variation provides new variables that are optimized locally and globally for the best coincidence with the deflection obtained with a realistic finite element model of the shaft. Then, two different approaches (with locally and globally optimized slopes) are proposed and their accuracy is investigated with 56 cases of study and with the analysis of a realistic gear drive. The results demonstrate the higher accuracy of the proposed model and the improvement with respect to the direct application of the Timoshenko beam theory

Numerical tooth contact analysis of gear transmissions through the discretization and adaptive refinement of the contact surfaces

The tooth contact analysis (TCA) is an important resource for the design of gear drives. This widely used analysis provides the contact pattern, contact path and the function of transmission errors that are directly related to the performance of the gear set. In this work, a new geometric approach for the TCA is proposed. This approach is general, deterministic and independent from the type or alignment status of the gears. It is based on the discretization of the contact surfaces of the reference teeth pair and on a geometrically adaptive refinement to solve the contact problem and to compute the instantaneous contact area for each position of the gear set along the gearing cycle. The new algorithm demonstrated to be versatile, robust and efficient through different test cases, obtaining accurate results with a relatively low computational cost.The authors express their deep gratitude to the Spanish Ministry of Science and Innovation (MICINN) for the financial support of research project ref. DPI2013-47702-C2-2-P

Determination of the ISO face load factor in spur gear drives by the finite element modeling of gears and shafts

The face load factor is a common coefficient used in gear design standards that takes into account the uneven distribution of load across the face width of the gears caused by the mesh misalignment. In this paper, a finite element model that includes the gears and the corresponding shafts is proposed. The results obtained from the application of finite element analysis to this model are compared with those obtained from application of the ISO Standard 6336 coefficient-based method (Method C). The influence of the length of gear shafts, the face width of the gears, the relative position of the gears over their shafts, the ratio between the pitch radii of the gears and the radii of their shafts, and the relation between the mesh misalignment and the face load factor, have been investigated.The authors express their deep gratitude to the Spanish Ministry of Science and Innovation for the financial support of research project refs. DPI2010-20388-C02-01 and DPI2010-20388-C02-02