Advances in Tribology

Volume 2016, Article ID 2957151, 8 pages

http://dx.doi.org/10.1155/2016/2957151

## A New Integrated Approach for the Prediction of the Load Independent Power Losses of Gears: Development of a Mesh-Handling Algorithm to Reduce the CFD Simulation Time

^{1}R&D Department, Bonfiglioli Mechatronic Research, Via Fortunato Zeni 8, 36068 Rovereto, Italy^{2}Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 1, 20156 Milano, Italy^{3}Department of Energy, Politecnico di Milano, Via La Masa 1, 20156 Milano, Italy

Received 30 October 2015; Accepted 6 March 2016

Academic Editor: Michael M. Khonsari

Copyright © 2016 F. Concli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To improve the efficiency of geared transmissions, prediction models are required. Literature provides only simplified models that often do not take into account the influence of many parameters on the power losses. Recently some works based on CFD simulations have been presented. The drawback of this technique is the time demand needed for the computation. In this work a less time-consuming numerical calculation method based on some specific mesh-handling techniques was extensively applied. With this approach the windage phenomena were simulated and compared with experimental data in terms of power loss. The comparison shows the capability of the numerical approach to capture the phenomena that can be observed experimentally. The powerful capabilities of this approach in terms of both prediction accuracy and computational effort efficiency make it a potential tool for an advanced design of gearboxes as well as a powerful tool for further comprehension of the physics behind the gearbox lubrication.

#### 1. Introduction

More and more severe requirements in terms of efficiency have encouraged the gearbox manufacturers to increase the investments for power dissipation reduction. An energy efficient solution, besides a pure power consumption reduction, shows lower operating temperatures and therefore has a higher reliability. The losses are dissipated in form of heat through the housing and shafts. While the exchange area/volume is often limited by external constrains, the internal design of a gearbox is completely in charge of the gearbox manufacturer. Considering that in industrial gearboxes the power losses are more or less equally shared between the bearings, the gear meshing, and the interaction with the oil, it is clear that at least this main contribution (there are other contributions related, e.g., to the seals) should be considered when a new gearbox is developed. Being able to correctly model the power losses enables the capability to predict also the operating temperatures under each operating condition and consequently the so-called thermal limit. While for bearings and for the gear meshing losses accurate models already exist [1–5], for the oil churning/sloshing power losses only basic research was carried out. All the experimentally derived models results are accurate only as long as the operating conditions and geometrical configurations are similar to those used in the original experiments. One of the first works on this topic was carried out by Mauz [6]. Mauz tested several geometries and operating conditions. From the authors experience [7] this model is not capable of taking into account very important parameters like the helix angle. Other studies in this field were carried out by other researchers such as Dawson [8] who concentrated on windage, Seetharaman and Kahraman [9] who concentrated on churning, and several other authors [10–14]. The authors maintain that a deeper understanding of the physical phenomena can only be achieved by means of numerical techniques. Different contributions have already shown the goodness of the CFD approach for such purposes [15, 16]. In the past the authors approached the problem using commercial software [17–22]. The main limitation to the wide diffusion of such approaches is related to the computational effort, the complexity of setting up a model with a general-purpose software, and the costs for the licenses. In this scenario the authors started studying a simplified configuration [23, 24] with the open-source code OpenFOAM [25]. This approach allows both to overcome the license costs (GNU license) but also, thanks to the specificity of the developed algorithm, to reduce the setup and calculation times. In this paper a mesh generation and handling algorithm that enables the simulation of the gear engagement is presented. The developed code was tested and the results show good agreement compared with experimental data [17].

#### 2. Power Loss Classification

According to [5] the power losses of gears can be subdivided into load dependent and load independent losses and again according to the mechanical component that is responsible for their generationThe subindexes , , , and stand for gears, bearings, seals, and other generic components (like clutches or synchronizers). The subindex indicates the load independent losses. The focus of this research is on the load independent gearing power losses . It results from the interaction of the rotating (or in general moving) mechanical components and the lubricant. This load independent contribution of the gears can further be subdivided into windage, churning, and squeezing/pocketing effects. Windage and churning as in external aerodynamics concern the main interaction of the components with the oil/air or with the oil-air mixture. The pocketing effects instead are related to the volume variation of the gap between the teeth in the mating region and the additional fluxes that take place.

#### 3. Objective of the Study

The main objective of this study is to find a reliable method in order to calculate the load independent losses of gears. Beside the accuracy of the results the models should also be manageable in terms of computational times so as to enable systematic studies in reasonable times, a requirement that is particularly stringent for industrial applications.

#### 4. Numerical Modeling

##### 4.1. Conservation Equations

For a generic volume it is possible to write five conservation equations: the averaged mass conservation equation for no-stationary incompressible flows and the averaged momentum equation (and eventually the energy conservation equation):where represent the Cartesian coordinate and is the velocity component, the pressure, the density, and the Reynolds term.

Solving these equations numerically it is possible to describe the fluid behavior in terms of macroscopic properties such as pressure, velocity, and their space and time derivatives. Simulations were run adopting a transient incompressible pressure-velocity coupled solver. In particular, the implementation of the pimpleDyMFoam solver of OpenFOAM (PIMPLE solver, a flexible implementation of a transient solver that allows operation in both PISO [26] and transient SIMPLE mode) was adopted. In this work, the temperature of the lubricant was assumed as uniform in the domain and a priori known; therefore the energy conservation equation was not considered. In a similar way the density and the viscosity values were set on the basis of the temperature.

##### 4.2. Boundary Condition

In order to correctly move the gear boundaries according to the kinematic laws, a new boundary condition was developed. The relation on which the boundary condition is based isin which represents the angular position of the gear at the actual time step, the angular position at the previous time step, the rotational speed of the gear, and the time step. In this manner it was possible to move the gear boundaries according to the kinematic law.

##### 4.3. Mesh

OpenFOAM is capable of handling the domain deformations by moving the mesh. The objective of the mesh motion is to accommodate externally prescribed boundary deformations by changing the positions of mesh points (Figure 2). During the motion the mesh must remain geometrically valid. The motion is calculated considering the Laplace smoothing equation (4) and the pseudo-solid equation (5) (linearization of the motion equations for small deformations) where represents the diffusivity, the position of the nodes of the mesh, and the actual time.

This approach is effective only as far as the deformation does not affect too much the quality of the mesh which should be updated in this case. Different from many commercial codes, OpenFOAM does not allow local mesh regeneration to circumvent the mesh quality degeneration.

In order to simulate the complete rotation and meshing of the gears, an approach which employs a multiple number of meshes was applied [27, 28]. In this framework, each mesh is generated for one specific rotational angle and has a certain angular validity. Once a mesh is created, motion is imposed at the boundaries, whereas the inner points of the grid are moved according to the solution of the Laplace equation in order to adapt to the boundary motion. Usually, when the mesh is deformed, its quality in terms of skewness, nonorthogonality and geometrical/topological validity decreases. As a consequence, when the deformation becomes excessive a new mesh must be created and the solution mapped from the old mesh onto the new one resorting to field interpolation techniques. In this case, the interpolation of the computed flow field from one mesh to the next one is performed by means of a second-order, inverse distance weighting method. The conservativeness of the approach was tested in order to verify that the errors induced by interpolation were negligible both in terms of integral qualities (fluid mass) and velocity profiles.

In OpenFOAM two main meshing tools are available. The first one, blockMesh, is capable of creating parametric meshes with grading and curved edges; blockMesh is however not suitable to handle complex geometries. SnappyHexMesh, another utility of OpenFOAM, generates 3-dimensional meshes containing hexahedra and split-hexahedra automatically from triangulated surface geometries in stereolithography format. The mesh approximately conforms to the surface by iteratively refining a starting mesh (generated with blockMesh) and morphing the resulting split-hex mesh to the surface. The surface handling is robust with a prespecified final mesh quality but the procedure is also very time consuming and not suitable for the purposes of this study. For this reason, a new and more efficient approach was tested. The geometry is generated with SALOME [29]. A python script allows the parametrization of the geometrical entities that are defined through an analytical formulation. In a second step, the domain is meshed. In order to reduce the computational time a special strategy was implemented. The domain (a box with two mating gears) is cut by a plane that lies on the gear sides. The generated internal faces (Figure 1) are meshed with Netgen [30]. Netgen follows a top down strategy. It starts by computing the corner points. Then the edges are defined and meshed into segments. Next the faces are discretized by an advancing front surface mesh generator. A fast Delaunay algorithm generates most of the elements, but sometimes it fails for the last elements, so a slower back-tracking rule-base algorithm takes over. Basically the Delaunay algorithm subdivides the edges into segments according to the local seed prescribed element size (Figure 3(a)). Then, the surfaces are seeded with internal points (Figure 3(b)). These were connected (triangulation) together. The results of such procedure is often a surface (or volume) mesh that does not match the boundary mesh (Figure 3(c)) so the outer and the intersecting elements are removed (Figure 3(d)).