1 Introduction

Modern products are becoming more and more integral and are often realised as cyber-physical systems (CPS). The development of CPS strongly depends on different disciplines such as mechanical, electrics/electronics and software [3, 4]. Model-based systems engineering (MBSE) is an approach to cope such a cross-domain development [5].

For developing technical systems as the drive train of an electrical vehicle, generic guidelines provide a methodological framework for a function-oriented product development process [6, 7]. However, in mechanical engineering, a function-oriented development is difficult, due to the methodical breaks in the transfer of functions into components [1]. To master these difficulties, fast functional and virtual testing provides a solution.

Virtual product testing is used to reduce development costs and time [8, 9], for instance utilising of engineering models. Those models are used to investigate the system under development concerning its physical behaviour and performance such as efficiency, loads, fatigue etc [10, 11]. Reusing already existing models helps to reduce model development time and thus, reduce development time [9].

In an existing model landscape, difficulties occur to identify a suitable engineering model out of the variety of models that exist over different solutions and different phases of the development process. Engineering models are often not structured and stored well enough to be identified for reuse.

MBSE-architectures as described in [1, 12] with the central element “solution element” contain a principle solution, domain models and workflows. By a classification scheme, domain models are classified and integrated into the solution element. This solution element provides a meaningful framework for structuring engineering models in detail. Furthermore, the solution element is the smallest and a function-oriented model unit that makes reuse during a development process possible. Function-oriented in this context means orientation towards the function of a technical system that fulfils a required customer requirement.

However, in the case of developing a drive train of an electrical vehicle, those solution elements are not identified yet. By defining solution elements, engineering models can be structured and allocated for reuse purposes. The goal of this paper is to identify mechanical solution elements as a basis to structure models systematically.

The article is structured as follows: Sect. 2 presents the state of the research of modelling a systems architecture and integrating domain models by classification schemes. In Sect. 3, the research question is defined and the solving hypothesis is stated. Section 4 describes the system of interest, which is the electro-mechanical drive train of an electrical vehicle. The approach of identifying solution elements is described in Sect. 5, whereas Sect. 6 concludes the work.

2 State of research

Systems Engineering is an interdisciplinary approach for developing and realising technical systems [13]. Model-based systems engineering (MBSE) is the formalised application of modelling to support system requirements, design, analysis, verification and validation within the development domain [14, 15]. A central system model is utilised and whereas the description of the system of interest is formalised. In context of MBSE, the System Modeling Language (SysML) has become established [16]. SysML is an extension of the Unified Modelling Language (UML) [17].

Several modelling methods exist to build SysML system model during product development, such as SYSMOD [18], FAS4M [19], MagicGrid [20], mecPro2 [21], OOSEM [22], SPES XT [23] and motego (short cut for Model-Test-Go, formerly mse architecture) [1, 12, 24,25,26]. All methods focus on the product description via requirements, functions and logic up to the product (RFLP). The approaches differ mainly in the representation of the logic layer. Only the motego method provides an adequate modelling depth for the development of contact models.

The MBSE method motego is an approach for developing cyber-physical systems with a strong focus on physical system behaviour. The approach connects the requirements level, functional level and solution level. Especially at the solution level, low fidelity up to high fidelity behaviour physical models are integrated and utilised. In Fig. 1, an example of the electrical motor within those layers is illustrated.

Fig. 1
figure 1

Typical system architecture based on motego [1, 31] with a detailed view of the solution element containing the principle solution, domain models and execution workflows

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In the requirements layer, requirements are modelled or detailed (refine), e.g. as a text. Subsequently, the requirements are transferred into a functional architecture and are linked via satisfy connection. In the function layer, the functional architecture is defined and decomposed down to elementary functions. The elementary function is a basic function, which cannot be decomposed meaningfully and was introduced by Koller [27].

In the solution layer, specific solutions are defined concerning the previously defined functions and thus represent special cases of functions. The system solution is a specification of the functional architecture and the solution element is a specification of the elementary function. The solution element consists of the principle solution that realises an elementary function. This realisation consists of the physical effect, the active surfaces and the associated material. Furthermore, the solution element consists of domain models and workflows for model execution. The system solution is a higher-level element and contains domain models (e.g. production model or engineering models) and workflows (e.g. validation or design workflow) as well as solution elements. Engineering models are used in the development domain to execute virtual product tests such as stress tests, efficiency tests, load tests, etc. of the system under development [10, 11, 28,29,30].

The different engineering models are integrated into the solution element or system solution. For integration, the engineering models are characterized by a classification scheme based on Stachowiak [32]. The classification scheme was applied to engineering models by Jacobs [1] and is characterised by the three following essential properties:

  • System scope—describes which part of the product is mapped to the model;

  • model purpose—describes the objective of the model and

  • modelling fidelity—describes the abstraction of the real context.

Examples of possible attributes are illustrated in Fig. 2.

Fig. 2
figure 2

Classification scheme of engineering models [1]

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In sum, the solution element is defined as the smallest element that realises on one hand a specific function (elementary function) and on the other hand integrates engineering models to test the behaviour of a specific solution of a system under development. Furthermore, the solution element offers a structuring approach to organise engineering models.

The solution element provides a function-oriented structure of engineering models for reuse and linking of engineering models with development artefacts. As a result, the reuse of solution elements shall help to increase the efficiency in product development.

However, there is no method existing to identify and define the scope of solution elements.

3 Research question and hypothesis

As seen in Sect. 2, promising approach for the mechanical domain is to structure and organise engineering models in function-oriented solution elements [1]. By that, engineering models can be identified and structured inside the element to enable efficient reuse for other solutions. However, as identified in the state of research the identification and definition of meaningful scopes for solution elements are missing.

Therefore, the research question of this paper is defined as: How can solution element be identified and defined systematically? The underlying hypothesis to answer this research question is: The interaction between two active surfaces in a contact serves as basis to identify solution elements.

Firstly, the scope of this research lies in the mechanical domain and only focuses on mechanical elements such as machine elements. Interactions within the electric motor or within the cooling system are based on magnetic fields and heat transfer and differ from mechanical interactions. Therefore, these interactions shall be investigated in a separate research.

4 System of interest: Electro-mechanical drive train

In the current political and public discussion about the future of transport, the focus is on technical transport solutions. In passenger mobility, the focus is primarily on the e‑car and various hybrid drive systems [33]. Due to the latest relevance, the use case of an electro-mechanical drive train of an electrical urban car was chosen.

The drive train consists of different sub-systems which realise specific functions. In this paper, the focused parts of the electro-mechanical drive train are the mechanical elements. However, for the purpose of completeness, the drive unit is also introduced briefly.

The drive train is a system solution for the function provide controlled mechanical rotational energy and consist of the following subordinate system solutions:

  • Drive (realisation of the function provide mechanical rotational energy);

  • gear system (realisation of the function change torque and rotational speed);

  • bearing system (realises the function conduct or isolate mechanical energy—depending on the desired degree of freedom);

  • clutch system (realisation of the function conduct or isolate mechanical rotational energy);

  • sealing system (realisation of the function isolate oil flow);

  • component connection (realisation of the function conduct mechanical energy) and

  • spring system (realisation of the function save mechanical energy).

Figure 3 shows the structure of the system hierarchically with the subordinate system solutions. The top system solution level is oriented towards the underlying technical functions of the systems, which is also a common categorisation in the literature on machine elements and drive technology (e.g. [34]). This figure applies to the specific drive train presented in this publication. However, it also applies in general to possible subordinate (mechanical) system solutions of the drive train. That means, these subordinate system solutions occur in most (mechanical) drive trains, but only differ in the characteristics or specialisations. For example, plain bearings can be used instead of roller bearings in specific applications, but this only represents a specialisation of the bearing system.

Fig. 3
figure 3

Identification of all necessary system solutions to build the system model of the electro-mechanical drive train

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The illustrated system solutions in Fig. 3 can also be contained in other system solutions. For example, the bearing system can be integrated in the gear system as well as in the electrical machine for the shaft bearing. Here, we will not list multiple system solutions, but rather list the subordinate system solutions only once.

5 Approach to identify solution elements

The aim is to identify and define solution elements systematically. The solution element is useful for reuse due to the combination of the description model (standardised description of requirements, function, etc. in a SysML-model) and the behaviour model (physical behaviour) that can be stored in SysML-model libraries. To derive solution elements, it is proposed to use the interaction of the active surfaces.

Active surfaces are surfaces of a component that participate in e.g. forces conduction. They can have different shapes and are essential for realising the system’s function. Also crucial for realisation of the function are the interactions between active surfaces. These interactions are described in physical effects that are used to achieve a certain effect.

5.1 Analysis of electro-mechanical drive train

As mentioned, the contact analysis in this paper is limited to the mechanical parts of the electro-mechanical drive train as shown in Fig. 4. Using the concept of the energy and force flow from the electrical machine to the wheels, multiple contacts (interactions between active surfaces) can be identified.

Fig. 4
figure 4

Mechanical contact of the electro-mechanical drive train, circles indicate contact

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Beginning from the electrical energy supply, the electrical contact of the electrical plug with a mechanical contact for voltage transmission can be identified. The electrical machine consists of a rotor and stator with an electromagnetic field in between.

The rotor is rotatably mounted by two ball bearings. These ball bearings consist of several mechanical contact points (between the balls and the ring as well as the press fit between the ring and the shaft). A spring washer within the rotor system has the function to induce pre-load force into the rotor bearing system. Also, here a mechanical contact exists between the surrounding parts and the spring.

The connection between the rotor and the gear system is realised by a spline. This connection contains several contact points to conduct forces between two systems.

Within the gear system, also bearings are located and several gear stages. For the gear system including two spur gear stages, the contact between the pitches is a mechanical contact. Also, a contact point is the press fit between the middle shaft and the gear wheel. The differential is also a specific gear system, but with other main functionally as the spur gear stage. However, the same tooth contact can be also identified.

The differential contains several bearing systems: a needle bearing and a taper roller bearing. The differential pinions are connected to the differential casing by a press fit. Here, a planar contact is given.

The connection between the output gear wheel and the shaft is realised by a spline. Following the output shaft of the differential, two sealing systems (radial shaft seal) with a mechanical contact can be identified. A radial shaft seal can be also located at the shaft between the electrical machine and the gearbox.

Following the force flow to the wheels, the homokinetic hinges contain multiple mechanical contacts.

The drive train contains several screw connections (between the gear wheel and differential case, electrical machine cover, etc.). The screw connection consists of two planar contacts (between screw head and component as well as between the two components) and several contacts within the screw thread. Figure 4 illustrates the result of the contact analysis.

In the next section, the identified contacts will be classified, in order to derive solution elements.

5.2 Derive solution elements

By the contact classification procedure for solution element derivation, the essential characteristics (e.g. the shape of active surfaces or the main physical effect) of the contact are investigated and revealed. The classification procedure provides a logical analysis to derive solution elements from the identified mechanical contacts.

Therefore, the structure of the solution element is analysed in order to define a classification procedure for the solution element derivation. The first attribute of the solution element is the realisation of an elementary function with specific input and output flows. According to the motego architecture, specific function flows are modelled by the material flow, energy flow, or signal. The transformation between an input flow and an output flow is modelled by the elementary operation (e.g. conduct or isolate energy).

A specific function is realised by the main physical effect, including the active surfaces and the material. Based on the attributes of the existing solution elements structure, the classification procedure is created. However, two attributes are missing, to have a full description of a solution element: information about an existing medium between the active surfaces and the relative movement between the active surfaces. By that, the main functional characteristics of a contact can be described and classified [35].

Figure 5 illustrates the classification procedure. The input for the procedure is the identified mechanical contact and the output is a classified and derived solution element.

Fig. 5
figure 5

Contact classification procedure in a decision tree

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If the identified contact is a single contact, the analysis of the elementary function and elementary flow is conducted. Subsequently, a detailed analysis of the active surfaces and the main physical effect (based on design catalogues such as from Koller [27] or Roth [36]) for functional realisation is performed.

In the case of multiple contacts as an input into the classification procedure that cannot be separated into single contacts, a system solution of Fig. 3 should be selected and assigned.

Following the activities in the classification procedure, we propose to define the solution element’s name as: [medium] [contact type] [act. surface]-[movement] ([physical effect]).

It should be mentioned, if the contact type is classified as mechanical contact, it implies a solid-solid contact, but will not be listed in the solution element. However, this differentiation may be necessary, when further interactions disciplines such as solid-fluid are considered (not considered in this paper).

The classification procedure as it is applied to the example of all tooth contacts in Fig. 5 in the input-process-output representation is shown in Fig. 6. The numbers in the process represent the intermediate steps from Fig. 5 and illustrate the intermediate results. In the output (Fig. 6 on the right) all interim results are summarised in the name of the solution element.

Fig. 6
figure 6

Example of applying the contact classification procedure in Fig. 5

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Applying the classification procedure of Fig. 5 to all identified mechanical contacts in Sect. 5.1 makes it obvious, that the same solution elements are valid for many contacts. That means finally, only a few different solution elements can be derived (Fig. 7).

Fig. 7
figure 7

Solution elements derived from the contact classification

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The defined solution elements are the functional solution modules that compose the presented system. Furthermore, they can be used to integrate engineering models and thus organise them in a function-oriented way and reuse them in product development. For instance, testing the fatigue of the solution element “lubricated mechanical line-rolling contact (rolling effect)” within a tooth contact can be conducted with the engineering models of DIN 3990‑2 [37]. For testing the temperature within the contact, the engineering models of DIN 3990‑4 [38] can be allocated and used, whereas the power loss analysis for the same solution element can be executed by the engineering models of Winter/Niemann [39]. The stated solution elements can be now stored in SysML-model libraries as functional modules that can be reused to develop new physical systems.

6 Conclusion

In this paper an approach is presented, to identify and derive solution elements by a classification procedure. Using the proposed classification procedure, solution elements can be defined. It shows that only few solution elements are sufficient to describe a complex system such as the electro-mechanical drive train, as these solution elements are often repeated in a complex system. Building a library from the solution elements (e.g. a SysML-model library) is expected to amortise quickly since these few solution elements are used repeatedly. New systems are composed through different arrangements.

Engineering models can be next linked to the solution element based on an engineering model classification as stated in Sect. 2. This allows engineering models to be organised and reused in a function-oriented way.

However, a few open points remain. In the case of the identified spring, for example, the function fulfilment (save mechanical energy) does not take place in the contact, but between the two active surfaces (force transmission points) of the spring. In order to be able to map functions of this type, further description concepts are required. The concept of an “active volume” that e.g. conducts energy offers a possible solution. How well this concept can be integrated into the MBSE motego method is the subject of future research.

Furthermore, this publication only addresses mechanical contacts. In the next step, the scope will be extended to other fields such as fluidic, thermal or magnetic interactions. Here, the concept of the “active volume” is also a potential solution approach to be able to functionally describe the interactions of e.g. magnetic fields in solution elements.