Development of a Design Procedure Combining Topological Optimization and a Multibody Environment: Application to a Tram Motor Bogie Frame

Abstract

Nowadays, it is essential to find increasingly rapid and efficient design strategies. This approach becomes crucial in the railway industry, where components must be verified according to multiple reference standards, both structurally and dynamically. In this context, the present research activity aims to develop a fast and effective desin procedure based on European reference standards. The goal was to develop the geometry of a motor bogie frame for a tram vehicle, integrating three fundamental tools for development: finite element simulations and topological structural optimization, a Write Computer Aided Design (CAD) environment, and a multibody environment. Their integration could enhance design accuracy, streamline the traditional design workflow, and support innovation. The optimization process involved the introduction of complex technological constraints, directing the geometry toward production by casting. A tool was developed to automate running dynamics simulations and the output of results for immediate verification of the entire vehicle performance. Finally, the new geometry was tested both structurally and dynamically. The mass was reduced by approximately 7% while ensuring satisfactory mechanical performance. The maximum value of stress was reduced by about 16%. The dynamic performance showed negligible variation, confirming the encouraging outcomes to make this procedure increasingly effective and reliable.

1. Introduction

Nowadays, railway vehicles stand out as one of the optimal solutions for sustainable mobility, aiming to reduce air pollution generated by various types of land and non-land vehicles. The growth of the railway sector necessitates addressing new challenges in both vehicle design and management. A critical aspect of this involves the integration of running dynamics analysis with structural optimization to enhance performance and safety. Running dynamics analysis is crucial for understanding the operational behavior of railway vehicles, leading to improved stability and passenger comfort. When combined with structural optimization, this approach allows for the design of components that are not only lightweight and durable but also capable of maintaining optimal performance under varying operational conditions. This synergy can lead to significant advancements in the efficiency and reliability of railway systems. The focus of this research activity was the innovation of a motor bogie mounted on a tram vehicle. Redesign of this component was achieved through a combination of key tools used in the railway sector for the development of such components: the creation of a high-level Finite Element (FE) model, the addition of a complex structural optimization calculation, the reconstruction of a CAD model, and the development of a multibody model. The ability to effectively and automatically integrate these environments while adhering to all relevant European design standards represents the novelty of this activity, along with the innovative design and validation of the bogie. Structural optimization processes have yet to establish themselves in the railway field, especially as an optimization method. The present research activity has confirmed the potential of this tool in which technological constraints were also included to ensure that the optimized geometry could be produced using a sand-casting process. The results were excellent, as was the impact of a sensitivity analysis, which significantly reduced computation times. Finally, the automation of multibody analyses to study the running dynamics of the entire vehicle, including optimized geometries, further increase the efficiency of the design process. Having a replicable bogie frame model would lead to a significant reduction in production times, both during initial construction and when repairs or component replacements are needed due to damage. In order to reach these objectives, topological structural optimization processes are essential tools that support the design and development phases of the system in question. The literature highlights that topological structural optimization approaches have yet to gain widespread acceptance in the railway field, as well as a design procedure capable of efficiently integrating an FE model, a CAD model, and a multibody model. This aspect is even more accentuated for tram vehicles, which are the focus of the present work. In general, topological and structural optimization processes are effective techniques for designing various types of components. A suspension system adopted in automotive applications has been studied to emphasize a weight reduction [1]. An automotive car body was optimized through material performance indices [2], and a high-performance chassis design of an automotive vehicle body has been innovated using topological optimization [3]. Comparable methods have been applied to enhance the design process for turbomachinery components [4], particularly when integrated with additive manufacturing techniques [5,6,7,8]. Moving to railway applications, the modal behavior of a car body structure was included in a size optimization approach testing an innovative design approach [9]. In addition, composite structures were integrated and optimized, testing their structural and dynamic behavior [10]. Another similar activity regarded an anchor bracket, developed using topological optimization applied on composite materials [11] and a composite wagon [12]. A new generation of freight railway wagons was developed and proposed in [13]. Numerous researchers focused on optimizing car bodies with an emphasis on lightweight design processes [14], using a material selection method [15] or a multidisciplinary approach [16]. A recent optimization application based on a sensitivity analysis for a railway vehicle body is presented in [17]. With reference to manufacturing constraints, the bolster beam and the bogie frame of a freight wagon were optimized to reduce the mass of the system [18]. A topological optimization was conducted using Solid Isotropic Material with Penalization (SIMP) methodology to maximize the stiffness of the structure. Two casted parts were compared, highlighting the impact of manufacturing constraints in the optimization process [19]. Additionally, the existing literature emphasizes the critical importance of accurately predicting the lifespan of railway bogie frames. An FE model and multibody dynamics model were simulated to study the behavior of bogie frames under real operational service loads [20]. A similar approach was adopted to test the crashworthiness performance of a passenger rail vehicle locomotive [21]. The dynamic response acting on a high-speed train was analyzed using a rigid-flexible coupled vehicle dynamic model [22]. The bogie frame of a diesel multiple unit (DMU) was studied using stress and acceleration tests to clarify the reasons for fatigue cracking [23]. With reference to railway dynamics in general, a dynamic model for an asymmetrical vehicle/track system was proposed in [24], while the performance of a pantograph–catenary system was analyzed in [25]. However, all the proposed techniques focused on evaluating vehicle performance, neglecting the aspects of component redesign and innovation. It confirms the importance of the novelty proposed in this research activity, where a complex component like a tramway bogie frame was innovated combining all the main techniques for component development. The process resulted in an enhancement of the traditional procedure, providing a solid foundation for direct integration between structural optimization and the multibody environment, with new applications of a flexible multibody. The structure of this paper is summarized as follows: Section 1 includes an introduction to the research work and a literature review. Section 2 presents the methodology, and the CAD and FE models of the bogie, and the multibody model of the whole vehicle. Section 3 outlines the results obtained from the proposed procedure, focusing on structural, vibrational, and running dynamics aspects. Finally, Section 4 presents the conclusions and potential future developments.

2. Methodology and Benchmark Description

The following section provides a comprehensive overview of the methodology utilized in this study. Along with a brief description of the tram bogie frame, including an FE model, the optimization process and a multibody model are also presented.

2.1. Model Description: The Metro Bogie Frame

The tram bogie frame serves as a critical structural component, engineered to support and integrate various elements within the train undercarriage. In detail, the tram vehicle tested during the activity (P-V category according to EN 12663-1:2010+A2:2023 [26]) had five car bodies, two motor bogies, and a trained one. The motor bogie frame was studied and innovated. This robust frame is designed to deliver the strength, durability, and resilience necessary to withstand the rigorous operational demands of a tram system. Precision welding techniques are employed in the fabrication of the frame, ensuring seamless and uniform connections between steel components. Despite precision in manufacturing, welded joints remain a significant concern, particularly regarding mechanical fatigue acting in these complex systems. The frame comprises two robust longitudinal beams, two cross-members, and other geometric configurations, strategically designed to evenly distribute loads across the structure. This design not only enhances the stability of the bogie but also contributes to the overall safety and efficiency of the tram vehicle system. The original bogie frame was constructed using structural steel and assembled via welding as described. In the middle part of the longitudinal beam are positioned secondary suspension interfaces. At the ends, in opposite positions, supports for the motor and braking system are located. The connection interfaces of the two traction rods are positioned on the two crossbeams. Figure 1 provides a comprehensive view of the complete frame.

2.2. Methodology

The methodology adopted by the authors for the innovation of a tram motor bogie frame is outlined in detail and summed up in Figure 2. Several steps were needed to achieve an innovated configuration that complied with structural and running dynamics requirements. Different environments were involved. FE calculation was fundamental both for static-fatigue structural assessment and for the optimization process. A CAD environment was useful for transitioning from a design concept to a tangible design. Finally, a multibody ambient allowed virtual testing of the complex dynamics of the vehicle in different running conditions. The possibility of combining these tools represents a novelty in the railway field, allowing fast management and checking, which are main design aspects of a component as complex as a railway bogie frame. In addition, the present research work represents the development of an interesting design methodology approach proposed by authors in [27]. It involved the implementation of a robust sensitivity analysis on the loading conditions to which railway bogies are subjected to expedite the complex topological optimization process, which also included manufacturing constraints. Returning to the present research activity, the first step of the procedure involves assessing the original performance of the component to be innovated, both from a structural perspective and in terms of dynamic behavior. Once these are known, the process moves on to the first and only step of topological optimization. This step enables the development of an innovative frame design, based on load conditions proposed by the European standard EN 13749:2021+A1:2023 [28]. This standard includes approximately 20 load conditions, including both static and fatigue-related conditions, summarized in a reference table found in [27]. Information regarding optimization process settings is also included in this reference. Furthermore, the optimization process incorporates manufacturing constraints with the goal of producing a design suitable for casting, thereby eliminating the most critical element of this component: welds. Proceeding to the central block, which includes an automated loop, the first step involves reconstructing the geometry resulting from optimization using commercial CAD software. At this point, a quick modal analysis of the bogie frame in free-free conditions could also be performed to observe its natural vibration frequencies. This is not a mandatory step. The critical frequencies of the bogie from a dynamic behavior perspective are quite low; for example, hunting typically occurs at around 6 Hz. In contrast, bogie frames usually exhibit frequencies well above 40 Hz. However, during the final phase of analysis for the completed frame, a comparison of frequencies should always be conducted to fully conclude the analysis of the innovated component. Therefore, once the initial version of the geometry is completed, through a tool developed by the authors using Python (version 3.12), the data related to the new design automatically update the input file of the multibody software used, specifically Simpack 2021. The model consists solely of rigid bodies, including the bogie frame. Therefore, it is sufficient to update the component data related to moments of inertia, mass, and center of gravity position. Once the new Simpack model is set up, the simulation will automatically start. The final step of the automated cycle is the production of results, where key dynamic performance parameters, such as the vehicle critical speed, axle lateral displacement, derailment coefficient, and normal and tangential pressures, are compared. If the parameters are satisfactory, the dynamic behavior is also confirmed. This allows for moving on to the final testing phase, where static and fatigue mechanical performance are re-evaluated. Is it possible for the new design to exhibit some critical areas based on the material allowable values? Yes, it is possible. However, this will not necessitate a new dynamic test of the entire vehicle for two main reasons. First, the innovative design is the result of an optimization process in which the load conditions have already been considered for strength purposes. Second, as a consequence of the first point, stress concentrations may occur in very localized portions of the material. Therefore, corrective design actions will also be localized, without affecting the dynamic behavior of the entire vehicle system.

2.3. FE Model of the Motor Bogie Frame

Figure 3 illustrates the FE model of the original motor bogie frame. It was a high-level representation of the system, where every part was modeled with different types of elements (0D, 1D, and 3D), according to component requirements. Due to the robust geometry of the frame, characterized by thick sections and three-dimensional development geometries, it was not feasible to represent it using the mathematical formulation of 2D shell elements (QUAD type). All the main components, including longitudinal beams, cross members, and various supports, were modeled with a three-dimensional mesh of second-order TETRA elements (TETRA10 type), adapting the element dimension to the various parts of the bogie. The choice of different types of elements depended on the characteristics of the modeled components. Figure 4 shows a detailed view of the mesh. All the supports were connected to the main frame, using contacts formulation, specifically using freeze/bonded contact. This type of contact configuration imposes the linearity of the model, and the correct redistribution of all the acting stresses (both forces and moments). It enforces zero relative motion on the contact interfaces, and the gap between them remains fixed at the original value fixed by the user. The sliding distance is forced to be zero. This choice allowed accuracy of the results and simplicity of the modeling. Moreover, the model is easier to manage as the various components can be modified independently. It is sufficient to maintain the interface between the components to ensure the connection remains active, expediting potential changes and numerical tests. Depending on the type of loads and the zone of application, forces, pressures, and RBE3 elements were used to load the bogie frame. Using RBE3 elements allowed exclusion of the addition of stiffness to the model, distributing force and moments among the selected nodes. The wheelsets were represented with one-dimensional elements. In detail, a series of connected beams with the proper section dimensions and shape were used. This modeling choice gave the right mass and inertial characteristics to the model, without losing the flexural behavior of the component. The primary suspension (light blue color) mounted between the wheelset and the bogie frame was also modeled using one-dimensional elements, following the same concept described above. Accurately representing this system was critically important as it was vital to precisely replicate this connection point. It was necessary to avoid introducing overly rigid constraints that could create excessively high stresses, leading to unrealistic results. Global constraint conditions applied to the bogie frame; all load cases were represented by an isostatic configuration. The reference points were the four wheels, positioned at the end of the axle.

2.4. Topological Optimization Model and Settings

The topological optimization process applied to the tram motor bogie frame during the present research activity had the objective to propose an innovative design for the frame, which would be suitable for a sand-casting production process. All information about settings is detailed in [27] as explained in Section 2.2. The solver was based on the Gradient based Optimization Method [29]. All simulations were conducted using the same computer that had the following characteristics: Intel(R) Xeon(R) CPU E5-2643 v4 @ 3.40 GHz and RAM 32 GB. The goal of the optimization process was to minimize the weighted compliance evaluated across all load cases. The compliance represents the sum of strain energy of the mechanical model. The higher the value, the more the component is deformed under a load. This essentially establishes an inverse measure of stiffness, enabling the formulation of a more efficient minimization problem. Before proceeding with the optimization process, the model was adapted to the methodology (Figure 5). The volume of modifiable material was expanded, giving the solver greater flexibility (green color), while the interface and connection areas with other vehicle components, the connection areas with various supports, and the load application zones were included in the non-modifiable material volume (red color). The optimization constraint was imposed on the mass fraction, which is the ratio between the final and initial mass of the modifiable material (design space). It was set below 40%, which was the reference value adopted for topological optimization.

2.5. Multibody Model

The multibody model used represents a five-coach tram vehicle, with three coaches supported by bogies and two suspended. The end bogies are motorized, while the central one is a trailing bogie. All components of the vehicle have been modeled as rigid bodies. In addition to the coaches, which are clearly visible in Figure 6a, all the fundamental elements of each bogie have been modeled as shown in Figure 6b. Furthermore, the characteristics of the bogie frame are also included as fundamental parameters: mass, centroidal moments of inertia, and the position of the center of mass. The two wheelsets present on each bogie are also clearly visible. Finally, the rails were of the “grooved” type, which is consistent with the nature of an urban vehicle.

3. Results and Discussions

3.1. Topological Optimizations Result and Innovative Bogie Frame Design

Once the optimization process was completed, an initial material density distribution was obtained, as shown in Figure 7. The red zones represent areas where the material is necessary and cannot be removed to ensure the required performance of the frame in accordance with optimization process settings. Moving towards the lower end of the scale, represented in blue is material that can be removed. As shown in the reference figure, the solver generated large cavities both in the longitudinal beams and crossbeams. This significant result positively directs the component toward the sand-casting production process thanks to the technological constraints introduced in the optimization calculation. Figure 8 shows the trend of the objective function during the optimization process, highlighting the 7% mass savings achieved.

The next step involved reconstructing the geometry of the frame as shown in Figure 9. This new design is highly innovative compared to more conventional geometries, such as the original one. The bodies of the longitudinal beams are hollow as are those of the crossbeams. The crossbeams also feature additional internal reinforcements that ensure the strength and stiffness of the entire frame. Moreover, their presence enhances the casting process by creating a much better flowability condition for the molten material. The two crossbeams have their openings facing downward. This design choice proved to be crucial as it prevents the accumulation of debris during vehicle operation, thereby reducing potential sources of damage. The interface area of the secondary suspensions includes an additional cavity. The supports have retained their geometry at the interface zones, while being globally streamlined and adapted to the new production process, with larger fillet radii and smoother shapes. The large central holes in the longitudinal beams allow for a significant reduction in mass but require the addition of cores to the mold used for production.

3.2. Bogie Frame: Innovative Design and Verification

The mass and inertial properties of the new bogie frame were then automatically incorporated into the vehicle Simpack model. The objective of this step was to observe minimal variations in the dynamic behavior of the entire vehicle, now supported by two newly designed motor bogies with different mechanical and vibrational properties. A modal analysis of the frame under free-free conditions was conducted, confirming the robustness of its vibrational performance. However, the results will be presented later as the test was repeated on the final geometry. Two test conditions were considered: straight track and 100 m curve track. The comparison between the original model and the current one was conducted based on several reference parameters, including the vehicle critical speed, lateral displacement of the wheelsets, Y/Q coefficient (also known as the derailment coefficient), and normal and tangential forces. The critical speed of a railway vehicle is the speed at which the dynamic behavior of the vehicle becomes unstable, potentially leading to unsafe conditions. At this speed, the interaction between the vehicle and the track, particularly the forces transmitted through the wheel–rail interface, can result in excessive vibrations, lateral oscillations, or even derailment. The phenomenon is closely related to the vehicle suspension system, track geometry, and the natural frequencies of the vehicle components. Critical speed is important because it defines the operational limits for the safe and efficient functioning of railway systems. Exceeding this speed can compromise passenger comfort, increase wear and tear on both the vehicle and track, and, in extreme cases, pose serious safety risks. In this case, the critical speed of the vehicle has remained unchanged and is approximately 90 km/h, confirming a positive result. The derailment coefficient (Y/Q) is a crucial parameter in railway dynamics that represents the ratio between the lateral force (Y) and the vertical load (Q) exerted on a railway wheel. Specifically, the lateral force (Y) is the force acting horizontally at the wheel–rail interface, while the vertical load (Q) is the downward force due to the weight of the vehicle. The Y/Q ratio is used to assess the risk of wheel derailment, which occurs when the lateral forces exceed a critical threshold relative to the vertical load. By comparing the two curves obtained for the curved track, no significant differences are observed, confirming only a minimal variation in vehicle dynamic behavior. Additionally, the maximum recorded value of the derailment coefficient remains below the permissible limit of 1.2. This indicates that the vehicle operates safely within acceptable thresholds, with no increased risk of derailment even when navigating curved sections of the track. The derailment coefficient is important because it directly influences the safety of railway operations. A high Y/Q ratio indicates that the lateral forces are approaching a level that could lift the wheel off the rail, potentially leading to derailment. Figure 10 shows the result for the Y/Q coefficient, obtained with a 20 Hz low-pass filter.

For all other parameters, similar results were obtained, with negligible differences compared to the original. Therefore, the representations of these parameters have been omitted. Given the results, the proposed chassis has been deemed suitable.

3.3. Structural Performance Assessment

The bogie frame was tested again for fatigue under load conditions specified by the reference standard. Specifically, seven static conditions and nine fatigue conditions were analyzed. The FE model of the bogie frame was tested by verifying the performance in terms of stress at each node of the structure grid. Starting from the local stress tensor, it was possible to determine the resultant of the principal stress components. The values obtained were compared with the reference allowable limits. Before proceeding with the calculation, the model was subjected to a sensitivity analysis that took into account the effects of mesh size on the maximum stress value. The results are shown in Figure 11, where it can be observed that variation in the normalized maximum stress value is less than 3% after the last two iterations, confirming the stability of the FE model.

Table 1. Fatigue analysis results.

∆σ (Principal Stress) [-]		∆σ (Permissible Stress Range) [MPa]	Utilization Factor (U) [-]
Original	Innovated	-	Original	Innovated
1	0.84	130–150	1.43	1.20
0.81	0.68	130–150	1.16	0.97

summarizes the results obtained from fatigue calculation, exploiting the utilization coefficient of the material, defined as the ratio between the stress calculated value and permissible value. The results were positive, with only one critical area characterized by a utilization coefficient larger than one, identified at the base of a support on the extremity of the longitudinal beam (as indicated by the square in Figure 12). Stress values have been normalized to the maximum observed in the original bogie configuration. The stress permissible value has been expressed as a range.

The maximum allowed value in linear analysis is equal to one. Local exceedances of stress are allowed, taking into account the plastic behavior that characterizes metallic materials. This research activity did not require it, but a non-linear calculation could be exploited to investigate the plastic behavior of the system.

From the perspective of the dynamic behavior of the innovative frame, a comparison with the original frame is presented. This allows us to observe that the first mode of vibration has remained essentially unchanged. As shown in Figure 13, the dark blue areas are more extensive in the new proposed solution, confirming an overall increase in stiffness in the central area of the bogie. The first mode is a flexural mode that locally involves one of the end supports. This analysis was crucial, both to confirm that the first natural frequency of the system exceeds 50 Hz and to identify the area of maximum displacement, which coincides with the peak stress region discussed earlier. Finally, Figure 14 presents a comparison of the first six natural frequencies of the two frames. The match in the first frequency is excellent and nearly identical to that of the original frame, confirming the positive outcome of the analysis.

4. Conclusions and Future Developments

This project involved the combination of several essential tools for the design of a railway bogie. Starting with a high-level FEM model of a tram motor bogie, it was subjected to a structural optimization process. This approach, whose load conditions were based on the relevant European standard for bogie design, enabled the development of an innovative frame design. Additionally, due to the proposed technological constraints, the frame proved to be highly innovative and could be produced through a sand-casting process. In parallel, a multibody model of the vehicle was created to test its running dynamics. The goal of this project was to efficiently integrate these methodologies, effectively and quickly, to develop an innovative bogie frame design that considers both structural and vibrational performance as well as the dynamics of the entire vehicle. The innovative design demonstrated good performance. The mass was reduced by approximately 7% despite the transition to a new manufacturing technique. From both a static and fatigue perspective, it presented only one critical area, which was easily resolved. The first vibration mode maintained both its frequency and modal shape, confirming a first bending mode that locally involved an end support. Finally, the change in mass properties of the new frame did not cause differences in terms of the vehicle critical speed (running stability) or derailment coefficient (Y/Q). Other analyzed parameters also confirmed this positive result. This first phase of the research activity allowed for the development of a satisfactory approach to generating a new design. As a future development, this activity aims to create an automatic loop capable of integrating the FE model of the bogie and the optimization environment directly with the multibody environment, incorporating flexible behavior. In addition, the optimization process itself will need to be improved, with the introduction of new constraint conditions related to mechanical fatigue.