1 Problem

An electric powertrain—also known as eDrive—consists of the electric machine, the gearbox and the power electronics (Fig. 1a). It can be applied in fully electric vehicles or in axle-split hybrid electric vehicles [1]. The gearbox represents a linking element between electric machine and drive shafts to the wheels.

Fig. 1
figure 1

Example of eDrive solution and gearbox interior a eDrive solution in desired installation space (transparent blue and turquoise) and gearbox-specific installation space (highlighted in transparent turquoise) b Generic gearbox model with corresponding desired installation space (left) and schematics of interior (right)

Full size image

In the gearbox design process of a new eDrive (Fig. 2), several input parameters have to be considered [2], e. g.:

  1. a.

    transmission ratio,

  2. b.

    (range of) offset between input- and output shaft,

  3. c.

    load spectrum and service life to guarantee reliability,

  4. d.

    desired installation space.

Fig. 2
figure 2

Gearbox design problem

Full size image

Various gearbox variants are eligible to fulfill these requirements; each being characterized by gearbox-internal design parameters. They affect the main components (Fig. 1b) found in a gearbox, which are:

  1. a.

    gear wheels,

  2. b.

    shafts,

  3. c.

    bearings,

  4. d.

    a housing supporting this interior.

Each of these components influences the outer shape of the gearbox and therefore the package as well as the efficiency and production costs; also favorable noise, vibration and harshness (NVH) behavior is a key design goal (although not yet covered by the presented approach). The high variability of gearbox design parameters, conflicting design objectives and packaging-related aspects lead to a complex problem in the design process.

2 Importance of system perspective

The conventional design synthesis process implements heuristic synthesizing strategies at component level. Machine elements commonly found in gearboxes are well examined as isolated components. Acknowledged reference works such as [3] give detailed design recommendations on how to select the design parameters of a single component in order to achieve a certain design goal. This requires the application of design strategies at component level, which might lead to suboptimal results because of the limited capabilities of the applied synthesizing strategies. Many components coexist in the respective supersystem and their interactions also dictate the system properties. The synthesizing process must therefore not only consider isolated components but also their interactions in the system context. An illustrative example includes an exemplary efficiency optimization of a gearbox. An efficient gear design will generally favor high helix angles [3]. As a consequence, such a design induces high bearing loads, which in turn leads to high frictional losses in the bearings. This exemplary conflict outlines the importance of well-considered synthesizing strategies at component level, which can be eliminated by smart optimization algorithms supporting the design synthesis at the respective supersystem level.

3 Approach

The present contribution introduces a computer-aided gearbox design optimization process supporting the early development phase of gearboxes in eDrives. For given load, lifetime and desired installation space, the differential evolution-based [4] process (Fig. 3) delivers design parameters for shafts, gears, bearings and their arrangement to handle efficiency, package and costs in a multi-objective manner. In general, any objective derivable from the gearbox properties can be used as an optimization objective.

Fig. 3
figure 3

Gearbox optimization as closed-loop of design parameter determination and gearbox analysis

Full size image

Since this methodology is focusing on the early development stage, specific detailed design characteristics, e. g. lubrication, deformation of the housing, acoustic behavior (NVH), rotor stability and park lock integration are not covered by this design optimization. Nevertheless, these influences are essential for the design of successful gearbox solutions. They must be covered in a subsequent process where the expertise of an experienced gearbox manufacturer is vital to deal with the challenges. The introduced approach for holistic gearbox design is demonstrated on a single-speed, two-stage helical transmission with integrated differential drive, which represents a common configuration (Fig. 1b). Nevertheless, the introduced method can be applied for other gearbox architectures as well.

4 Literature discussion

The application of optimization algorithms to automate gearbox design has been addressed in previous research work. In 2000, Chong and Lee [5] used a genetic algorithm to optimize the gearing parameters of a two-stage gear train. Their choice of a stochastic algorithm is still state of the art for high-dimensional problems, since deterministic algorithms perform poorly on such problems and tend to converge to local optima. Furthermore, the design parameters of a gearbox require a mixed integer optimization (e. g. discrete number of teeth and real-valued face widths), for which the application of derivative-based methods is not sufficient [6].

The layout process of a gearbox for electric vehicles must consider multiple design objectives, such as minimizing cost as well as maximizing efficiency. Chong and Lee [5] and also the more recent work of Chandrasekaran et al. [7] treated the optimization problem as multi-objective. They used a weighted-sum-method to transform the multiple objectives into one objective function—thus transforming a multi-objective problem into a single-objective one. As pointed out by Deb and Jain [8], a weighted-sum-approach can be highly restrictive to the optimization process as the weights have to be defined a priori. There is no way to determine if even a slight change in weights might have resulted in a more favorable solution. A legit multi-objective optimization therefore always aims to find the so-called Pareto front [8] of the optimization problem. The Pareto front, named after Italian engineer and economist Vilfredo Pareto, consists of all possible sets of design parameters that result in a non-dominated solution (Pareto dominance). Non-dominated in this context refers to solutions for which no single objective value can be improved without worsening at least one of the others.

Regarding the system analysis process, most published works focusing on gearbox optimization use simplified calculation schemes for machine elements. This generally lowers the computational effort for the optimization, but at the same time it seriously lowers the quality of the result. This behavior is in strong conflict with the aim to provide a solid basis for low-risk design decisions. In this context, the system analysis process, as a state-of-the-art design method as described in [9], must implement up-to-date calculation schemes based on current standards and guidelines like ISO 6336 for the load capacity calculation of gears.

Another aspect that is rarely considered in published works is a holistic approach to the optimization problem. Gearing parameters only are considered in an exemplary manner for the optimization in [5, 7] and [8], omitting influences from the bearing selection. However, a holistic design method must consider all relevant design goals as objectives and all design parameters that significantly influence those objectives as optimization parameters. In recent work, a holistic optimization approach for electrified powertrains is presented by Albers et al. [10] that focuses on battery, power electronics, electric motor and drivetrain. It outlines a general methodology on how the system interactions can be modeled. However, a specific design method and optimization strategy is not discussed there.

Package restrictions are critical in the design of automotive components. However, published works on gearbox optimization neglect this aspect or use only simple geometrical models to estimate the package influence.

In contrast to the published work, the presented approach enhances the state of the art by

  • applying a proper multi-objective optimization (instead of a transformation to a single-objective optimization by weighted-sum),

  • synthesis on gearbox system level to find global optima,

  • using industry-standard component calculation schemes with high level of detail to ensure reliable results and

  • applying 3D geometrical models for components as well as the complete gearbox system to deal with package restrictions.

5 Methodology

The introduced optimization process consists of a closed loop of gearbox design parameters selection and subsequent gearbox analysis (Fig. 3). The gearbox design parameters (Fig. 4 and Table 1) fully define a specific gearbox variant. These parameters are set by a stochastic differential-evolution algorithm. Therefore, there is no explicit strategy on how to deal with the interactions of the individual gearbox components. This is contrary to conventional, manual design guidelines, which recommend a certain (usually recursive) sequence of macroscopic layout and subsequent detailed component design. Such explicit, sequential design strategies may lead to suboptimal results, since the variety of design options is strongly reduced. Instead, when using the optimization approach, the feasibility of given design parameters is checked by the gearbox analysis model and, if positive, their impact on the objectives is evaluated. The set of feasible solutions is compared regarding the multiple objectives in form of a Pareto-front, to show the tradeoffs from which the decision-makers can choose the best suitable solution. So in multi-objective optimization, there is no need to explicitly balance optimization targets in form of weighting factors. The objectives for optimization are minimization of

  • the losses (corresponding to maximization of efficiency),

  • the package metric (described in the following section) and

  • the overall costs.

Fig. 4
figure 4

Visualization of gearbox design parameters; variables description in Table 1

Full size image
Table 1 Gearbox design parameters
Full size table

Depending on the problem-specific requirements, additional objectives can be added, as any property derivable from the gearbox can be used as optimization objective. In the demonstrated case study, this will be axial width, ground clearance and lateral width of the gearbox.

Constraints to the optimization problem are the

  • requirements regarding the total center distance and the total gear ratio as well as

  • sufficient safety factors and lifetime of machine elements (shafts, gears, bearings) with respect to the given load spectrum,

  • geometrical feasibility (see following section ‘Internal collision checking’) and

  • adherence of basic design guidelines for gears in automotive applications to accomplish smooth running (see following section ‘Gear geometry’).

6 Gearbox analysis

The gearbox design parameters serve as inputs for the gearbox system analysis, which implements a successive calculation of system properties and checking of constraints. The idea behind this is that invalid gearbox variants (invalid due to constraint violation) are identified as early and with the least computational effort as possible. The aim here is to reduce the number of extensive calculation steps, e. g. of gear load capacity or package-related metrics to increase optimization performance. A flow chart representing the major steps of the system analysis process is shown in Fig. 5. The applied calculation schemes of system properties determination include:

  1. a)

    Gear geometry

    The gear geometry is calculated according to Köhler/Rögnitz [11] and Linke/Börner/Heß [3], which are both based on the industry standard DIN 3960. The width of the gear wheels \(b\) for each stage is determined by a predefined overlap ratio \(\varepsilon_{\mathrm{\beta}}=2\), which is recommended for smooth operation [11]. The gear wheel width is thus not implemented as an independent gearbox design parameter but calculated by the normal module \(m_{\mathrm{n}}\) and the helix angle \(\beta\) (Eq. 1).

    $$\varepsilon_{\mathrm{\beta}}=2\rightarrow b={\varepsilon _{\mathrm{\beta}}}\cdot \pi \cdot m_{\mathrm{n}}/\sin (\beta )$$
    (1)
  2. b)

    Shaft geometry

    The shafts are designed as hollow shafts with a fixed ratio of inner and outer diameters. The outer diameters depend on geometrical restrictions resulting from the gear design and the bearing selection, which are both determined by the design parameters. In particular, the gear’s tooth root diameter and the bearings’ inner diameter are considered to ensure geometrical compatibility of the shaft with the gear and the bearing, respectively. The shafts’ axial lengths are determined by the design parameters \(w_{1}\) and \(w_{2}\) and the widths of the gears and the bearings (Fig. 4). The mechanical torsional strength of the shafts is checked, although this is not critical in most cases.

    The radial contour of the output shaft also contains the shape of the differential drive, which is selected off-the-shelf depending on the required transmittable torque.

  3. c)

    Shaft arrangement & statics (bearing reaction forces)

    The center distances of the first and second stage result from gear-related design parameters, while the total center distance between input- and output shaft is explicitly defined as a design parameter (Table 1). Depending on the design problem definition, the total center distance is defined to be constant or within a specified range. A Boolean design parameter decides in which direction the intermediate shaft is aligned (see Fig. 2 for an illustration of both variants), which affects the resulting forces and the gearbox package. Shafts are modeled as rigid bodies and pressure angles of conical roller bearings are considered for the calculation of the resulting forces.

  4. d)

    Bearings’ static safety factor and lifespan

    The bearings’ static safety factor and lifespan is determined according to ISO 281 and the bearing manufacturer scheme [12] considering the load spectrum.

  5. e)

    Internal collision checking

    During the search for beneficial gearbox solutions, the optimization algorithm varies the gearbox design parameters. Certain value combinations can lead to a gearbox system that is physically not meaningful. An example is that the bull gear on the intermediate shaft might be too large and collide with the contour of the output shaft. In this way, invalid suggestions are identified by evaluation of polygon intersections of the radial contour of the shaft assemblies.

  6. f)

    Gear load capacity

    The gear load capacity is calculated according to ISO 6336 considering tooth root fatigue and pitting.

  7. g)

    Efficiency

    The modelled gearbox efficiency results from load- and speed-dependent gear- and bearing losses for the given load spectrum. The gear efficiency is calculated according to Linke/Börner/Heß [3] and bearing losses are determined according to the bearing manufacturer scheme [12].

  8. h)

    Costs

    For shafts, gears and housing, a mass-specific cost factor is applied for each component type, considering the applied material. The bearing costs are taken from a supplier catalogue [13, 14].

  9. i)

    Package

    To evaluate the package integration, a parametric 3D-model of the gearbox including the housing is applied [15]. It automatically generates the geometrical shape of the housing as a function of shaft- and gear wheel dimensions as also of the bearing arrangement. The package integration of the gearbox is then quantified by a specific package metric, depending on two cases:

    1. a.

      If the gearbox solution completely fits inside the desired installation space, the negative value of the minimum clearance between the gearbox and the installation space is measured (Fig. 6a).

    2. b.

      If the gearbox solution violates the desired installation space, the violating volume is measured (Fig. 6b).

      Fig. 5
      figure 5

      Major steps in the gearbox system analysis process

      Full size image
      Fig. 6
      figure 6

      Visualizaton of the package metric for a no package violation, b violation of desired installation space

      Full size image

    This definition enables the coverage of both cases with a single scalar value, where smaller values are beneficial.

7 Results

7.1 Case study

The proposed gearbox design method is exemplarily demonstrated in the context of a complete eDrive development challenge. Focussing on the gearbox layout process, the electric machine and inverter setup are kept constant. The requirements on the gearbox are derived from electric machine properties and required driving performance (Table 2). The layout process considers a given load spectrum for bearings and gears as also for the evaluation of efficiency characteristics.

Table 2 Main requirements of the demonstration gearbox design problem
Full size table

The desired gearbox installation space is highly challenging (Fig. 1b), as even the benchmark design does not fully comply with it. Later it will be shown in Fig. 8 that there is no feasible solution for the predefined material selection of gears and shafts (Table 2). Therefore, it is intended to find the best solution in terms of violating the predefined volume. The solutions found by the optimization algorithm are compared with a benchmark gearbox design (Fig. 1a). For a detailed evaluation of the packaging concept, the parameters of the benchmark solution are used to parametrize the generic housing model (Fig. 1b). This model is also used to evaluate the solutions found by the optimization algorithm and therefore comparability between the optimization results and the benchmark solution is provided.

It is important to keep in mind that the benchmark gearbox represents a completely engineered solution, which also considers specific detailed design characteristics, e. g. lubrication, deformation of the housing, NVH, rotor stability and park lock integration. These detailed aspects are not considered in the introduced conceptual gearbox layout method. Therefore, the comparison can only give an indication of the layout and objective values must be compared with care.

The optimization targets for this study are formulated to minimize

  1. a)

    costs,

  2. b)

    losses,

  3. c)

    axial width of the gearbox,

  4. d)

    gear tip radius on the output shaft (related to ground clearance, later called ‘ground clearance metric’) and

  5. e)

    lateral width of the gearbox (related to the eccentricity of the intermediate shaft).

After the main optimization process, a post-processing step evaluates the introduced

  1. f)

    package metric. The available degrees of freedom regarding gearbox installation positions and angles are examined by an independent optimization procedure to find the optimal arrangement for each gearbox variant.

This sequence of process steps boosts optimization performance, since the packaging-related optimization targets c) to e) are calculated with analytic expressions independently from the desired installation space. With this method, the process of packaging investigations, which requires high computational effort in a 3D-CAD software, is only executed for the Pareto-optimal results regarding the objectives a) to e). Furthermore, the analytic package-related expressions allow the decision-maker to prioritize certain locations of package violations.

8 Selected results

The benchmark gearbox considered would appear to be designed as a good tradeoff between ground clearance, violating installation space volume and costs (‘REF’ in Fig. 7).

Fig. 7
figure 7

2D-subsets of the Pareto-front representing ground clearance metric (tip radius of gear on output shaft) as a function of a package metric and b costs

Full size image

In the top right corner of Fig. 8, the 3D-Pareto front regarding package metric, costs and losses is depicted. When analyzing the Pareto subset of the package metric and the costs (Fig. 8), several improvement options arise (Table 3). Compared to the benchmark solution, both costs and package metric could be improved.

Fig. 8
figure 8

Optimization result: 2D-Pareto subset (circles) of 3D-Pareto front (top-right) regarding package metric and costs compared to benchmark solution (‘REF’)

Full size image
Table 3 Selected design options compared to benchmark design (all 20MnCr5)
Full size table

The Pareto-front of package and costs is strongly influenced by the selection of bearings. Fig. 8 shows that grooved ball bearings lead to cost-effective but larger solutions, cylindrical roller bearings and conical roller bearings are more expensive but allow a more compact design. In Fig. 9 the extrema of cost-optimal and package-optimal solutions are depicted.

Fig. 9
figure 9

Comparison of package situation for a benchmark solution (20MnCr5); b cost-optimal solution (20MnCr5); c package-optimal solution (20MnCr5); d package-optimal solution (high-strength material 18CrNiMo7‑6)

Full size image

However, the results indicate that with given requirements it is not possible to fully comply with the desired installation space. To address this issue, the best suitable solutions may be used to discuss an adaptation of the desired installation space with the customer or other involved parties to initiate another loop of optimization with changed requirements. Another option is to extend the search space by including high-strength materials for shafts and gears, which results in more compact solutions for this case study (Fig. 9d).

9 Outlook

The approach is presented for the topology of a single-speed, two-stage helical gearbox with an integrated differential drive. Future work will consider expanding the methodology to more general topologies, especially allowing more than two gear stages and other shaft arrangements.

Currently, to demonstrate the methodology, a mass-specific cost model is used for shafts, gears and housing. A more detailed cost model could further improve the validity of cost tradeoffs.

The gearbox is part of the electric powertrain system, also containing the electric machine and the power electronics. To address the multi-objective optimization design problem on powertrain level, a synthesis process based on component-specific Pareto solutions can be applied [2].

10 Summary and conclusion

A computer-aided gearbox design optimization process is introduced to support decision-making in the early development phase. To effectively deal with the conflicting objectives, a gearbox system perspective is applied to ensure well-balanced decisions in the design of each sub-component. The gearbox design parameters and system analysis procedure are described by an example of a single-speed, two-stage helical gear with an integrated differential drive. Well-established industry standards are applied for detailed component modelling and a generic CAD-model is used to estimate and quantify the space requirements of each gearbox variant.

The introduced methodology is demonstrated on a real eDrive application scenario and compared with a benchmark solution. The results indicate that a further improvement of the benchmark solution is possible, especially in view of the tradeoff between packaging and costs. By this means the introduced methodology is able to support layout processes of gearboxes taking complex and conflicting requirements into consideration. Nevertheless, the expertise of an experienced gearbox manufacturer is vital in the development of competitive gearboxes in terms of reliability, efficiency, NVH and costs. The core competences in these areas can be complemented by the proposed optimization methodology to significantly reduce development time and improve result quality in early development stages.