Innovative tooth contact analysis with non-uniform rational b-spline surfaces

A major focus in the development of gear boxes is the optimization of the gears. The aim is to increase the transferable power as well as the efficiency and load capacity of those components. At the same time, the demands on the design process are growing in terms of development time and financial effort. The use of simulation tools can shorten the design process and significantly reduce the need for costly gear tests.

BECAL is a comprehensive simulation tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and cylindrical gears, developed at the IMM (TU Dresden) in cooperation with the Institute of Geometry at the Faculty of Mathematics (TU Dresden). It allows the calculation of local stress on the tooth flank as well as in the tooth root. Derived from this, the local load capacity regarding flank and root can be determined. In addition, this simulation tool enables the consideration of the efficiency as well as an analysis of the acoustic excitation in the tooth mesh. The BECAL calculation process is shown in Fig. 1 [1].

The flank and root geometry of the gear is provided by a manufacturing simulation, a geometry import or measurement data. The geometry is thus represented by a discrete 3D point cloud. However, in order to perform the stress calculation, it is necessary to determine the coordinates, normals and curvatures for each of these points. In addition, for tooth stiffness determination and tooth root stress calculation (BEM method [1, 2] or FEA method [3]), it is necessary to identify arbitrary points in the tooth root.

To do this, the discrete data points are transformed into a mathematically closed surface representation. At present, this is an approximation with Bézier tensor product surfaces. Depending on the tooth geometry, this surface representation can result in significant deviations between the approximated surface and the given points. Especially local modifications, such as tip and root relief or local flank damage (e.g. micro pitting, wear) can be insufficiently represented with this approach.

Thus, surface deviations can have a direct impact on the calculation of flank and root stresses. For example, the curvature on the flank has a significant influence on the Hertzian pressure, which in turn is the basis for the load capacity against pitting and micro pitting. The same occurs with the representation of the root geometry, which can have an effect on the root stress and, accordingly, on the tooth root capacity. Finally, such deviations are compensated for by an increased safety factor in the gear design, which means that the potential of the gears cannot be completely utilized.

In this context, this paper focuses on the surface representation with non-uniform rational B‑spline surfaces (NURBS) with the aim of significantly improving the approximation of the flank and root geometry. For this purpose, both the Bézier and NURBS approach are compared based on their mathematical properties. Furthermore, the results of the representation accuracy, the local stress for flank and root, and the local load capacity are used to demonstrate the potential of the alternative NURBS approach for an improved gear design process.

There are several simulation tools for the tooth contact analysis (TCA), which differ in the range of available gear types and in the degree of complexity and thus in the computational effort and time required. In the following section, two additional tools are presented to give a classification of the BECAL TCA.

RIKOR, developed at the FZG (TU Munich), is a TCA program for the calculation of spur, helical and double helical gears. It is able to consider tooth flank modifications but also to determine shaft deformations and bearing deflections of the surrounding system and to take these into account in the tooth contact simulation. The geometry of the cylindrical gears is described purely analytically from the involute profile according to DIN 3960. Hence, the determination of the contact distances in the tooth mesh, which is necessary for the further stiffness and stress calculation, is also conducted analytically and therefore requires less computing time. However, by using the theoretical geometry from DIN 3960, the actual geometry resulting from the manufacturing process cannot be taken into account. Furthermore, the geometry of bevel and hypoid gears cannot be calculated with this analytical approach due to the multiply curved flank and root geometry [4].

ZaKo3D, developed at the WZL (RWTH Aachen University), is another TCA tool that is comparable to BECAL in terms of manufacturing simulation and calculation of stiffness and stress, but has a significant difference in the representation of flank and root geometry. Since in BECAL the flank surface is approximated in a manner that the contact distance between the flanks in the mesh can be determined analytically, the contact analysis is accurate and very fast [1, 5]. In contrast, ZaKo3D uses an interpolation algorithm based on Bézier surfaces or a triangulated mesh to achieve the highest accuracy with respect to the output data from the manufacturing simulation. Both approaches make it necessary to determine the contact distances with numerical algorithms, such as the Newton scheme. The advantage of the interpolation lies in the consideration of detailed manufacturing-related micro geometric flank deviations, such as feed marks and profile section deviations. However, this detailed flank description requires a high computational effort [6, 7].

In summary, BECAL offers a compromise between a sufficiently accurate description of the flank geometry (including the manufacturing process) and a high computational speed. For this reason, the surface approximation concept will be pursued. In order to increase the surface accuracy, the use of an alternative surface approach with NURBS is further investigated.

For the approximation of the tooth geometry in BECAL, both, the flank and the root geometry are represented with separate surface descriptions by Bézier tensor product surfaces. For the purpose of surface approximation, the target points, e.g. as provided by the manufacturing simulation, must first be parameterized. In BECAL, the parameterization is based on spherical coordinates, as shown in Fig. 2. Furthermore, the parameters are limited to a range \(\mathbb{R}\in [0,1]\) to fit the polynomial surface representation. The approximation, which determines the Bézier surface coefficients, is done by solving a linear minimization problem (least square fit) [1, 5].

To describe the flank surface X, the spherical coordinates R and θ are used as parameters for the representation of φ according toshown in Fig. 2. This form of surface mapping has a major advantage: In order to determine the contact distances in the load-free tooth contact simulation, the tooth is divided into spherical sections along the face width. Based on this, the contact line and the contact distances along this line are calculated. Thus, the representation as defined in Eq. 1 has the benefit that the spherical sections as well as the contact distances can be determined directly and precisely. This avoids the use of complex contact determination algorithms and therefore enables high processing speeds [1, 5, 8].

The description of the root surface is also based on spherical coordinates. In contrast to the flank, an additional parameter u is introduced. This parameter is needed to describe the potentially strongly curved root geometry. The representation of the root surface is defined as

Following the flank description, spherical sections can also be calculated with a constant R along the u parameter, which is direct and fast [1].

To understand the differences between Bézier and NURBS surfaces and the resulting advantages of NURBS, their mathematical description is explained and contrasted, and possibilities for approximation with these approaches are shown.

Finally, both surface approaches are used in the BECAL TCA and compared with respect to their influence on the results of stress, load capacity and acoustic excitation. Therefore, two different gear types (acc. Fig. 7) are analyzed. The first example shows a hypoid gear set with a linear tip relief and a spherical root relief on the drive side of the pinion. The second example describes a forged differential gear characterized by a strongly curved tooth root geometry.

The flank deviation and stress results of the first example are summarized in Fig. 8. As shown in the upper left figure, the flank deviations of the Bézier surface range from −5.45 µm to 3.3 µm with a root mean square value of 1.42 µm. The difference in curvature, especially at the tooth tip at the transition from the involute profile to the linear tip relief, causes the Bézier surface to oscillate. This results in two local maxima of Hertzian stress along the tooth profile.

In contrast, the NURBS surface has an overall deviation of 2.35 µm, which is strongly concentrated on the transition line at the tooth tip. In addition, the surface is less oscillating, resulting in flank deviations of less than 0.1 µm for the most of the flank. The root mean square value is 0.21 µm. Thus, a more realistic pressure distribution can be achieved with the NURBS surface, while the maximum of the Hertzian stress is located at the tip. Table 1 lists the minima and maxima of stress and load capacity.

It is apparent that the maximum Hertzian stress is underestimated with the Bézier surface. Accordingly, the safeties calculated for the flank, especially for pitting and micro pitting, are too high compared to the results obtained with the NURBS surface. Contrary to expectations, the tip relief does not result in lower stress at the tooth tip in this example. This can be explained by the high total contact ratio of \(\varepsilon _{\gamma }> 2.5\), which leads to a smooth pressure distribution along the path of contact and thus to a low transmission error. Hence, the common effect of tip relief, to reduce the transmission error, is not given. Instead, it leads to a limitation of the contact pattern and therefore to an increase of the contact stress. However, the higher accuracy of the approximation surface allows for more detailed design of the micro geometry, such as applying higher profile crowning or using a spherical relief to reduce the stress at the tip.

Another important criterion for the gear design, is the acoustic excitation. In Fig. 9 the excitation spectrum for the transmission error under load is shown. By comparing Bézier and NURBS, two different spectra can be obtained. In particular, the first three harmonics differ significantly in the level of excitation, resulting in calculated volume levels of 94.3 dB (Bézier) and 89.5 dB (NURBS).

The results for the second example are shown in Fig. 10 and Table 2. The calculation of the root tensile stress is performed using FEM influence vectors according to [13]. Similar to the results from the first example, the deviations at the tooth root can be reduced by a factor of 3.7 by means of root mean square. Accordingly, Bézier gives deviations in the range of −78.9 µm to 114.24 µm, while the NURBS surface only shows deviations in the range of −43.08 µm to 24.52 µm. It can be seen that both surfaces are primarily oscillating at the transition from the toe to the straight midsection and from there to the heel. However, unlike for the flank surface, these deviations have no significant influence on the tensile stress in the tooth root and thus on the safety against tooth root fracture.

For the tooth contact simulation in BECAL, it is necessary to transfer the point cloud-based tooth geometry into a mathematical description. This is performed by an approximation of the target points, which, in contrast to an interpolation, enables an efficient and fast TCA. In order to improve the mapping accuracy of the flank and root geometry, only the use of a higher-order approach compared to the presently used Bézier tensor product surfaces must be considered.

For this purpose, NURBS and Bézier surfaces are analyzed and compared with each other. The comparison concludes that NURBS provide a significantly higher degree of freedom to describe complex surfaces. In particular, this results from the independence of polynomial degree and number of control points as well as the additional weights.

Finally, two different gear geometries are used to demonstrate that NURBS surfaces can improve the mapping accuracy by a factor greater than 3. In particular, the flank representation shows a significant influence on the pressure distribution as well as the results of the flank load capacity. Since the correct curvature is primarily relevant here, deviations in the range of a few microns already lead to a false statement regarding the flank load capacity and acoustic excitation. In comparison, deviations of the geometry in the tooth root are found to be less relevant for the calculation of the tooth root load capacity.