A finite element methodology to model flexible tracks with arbitrary geometry for railway dynamics applications
Introduction
Vehicle-track studies involve a contact model that establishes the interaction between two subsystems: the vehicle and the track [1], [2], [3], [4], [5], [6]. Although several methodologies are available to represent these subsystems in vehicle-track interaction problems, the multibody dynamic analysis and the finite element method became the most used approaches. Studies concerning the vehicle subsystem favor a multibody dynamics approach and focus on describing the mechanical components of the vehicle and their interactions [7], [8], [9], [10], [11], [12], [13]. On the other hand, studies concerning the track subsystem tend to favor the finite element method and focus on describing the structural and material behavior of the track and its components [14], [15], [16], [17], [18], [19], [20], [21], [22].
Most commercial and in-house codes for railway dynamics applications use multibody dynamic analysis to study vehicle curving behavior, passenger comfort, and stability. The track is usually considered as a semi-rigid or rigid structure, and the representation of the rails is purely geometrical [1], [2], [23]. This modeling approach is reasonable in the low-frequency domain as the suspension of the vehicles filters most of the high-frequency excitations, and the track behaves as a stiff spring [24], [25], [26]. However, recent studies show that wheelset and track flexibility influence the dominant frequency of hunting motion [26], the critical speed of the vehicle [26], [27], and the wheel-rail contact forces [26], [27], [28], and thus should not be overlooked in stability studies. Furthermore, the flexibility of other track components becomes important in the mid and high-frequency range [29], [30], [31], [32], [33], [34], [35]. In the mid-frequency range to study the effects of unsupported sleepers [36], [37], [38], rail corrugation and wheel wear [10], [25], [28], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], and ground-borne vibration [49], [50], and in the high-frequency range to study noise and squeal [24], [51]. The finite element method is well established in the literature as a suitable approach to model track flexibility [14], [15], [52], [53].
The multibody dynamic analysis and the finite element method have different integration requirements and satisfying both within a single integration scheme can be challenging. Moreover, binding the two methodologies together increases the complexity of the problem and decreases its numerical efficiency. Therefore, it is common to find in the literature studies where a multibody model of a railway vehicle runs on a rigid track or, conversely, studies where the behavior of the vehicle is represented in a simplified manner on a finite element model of a railway track. The flexible multibody dynamic analysis provides a reasonable approach that represents track flexibility without compromising the complexity of the vehicle model, and unifies the multibody dynamic analysis and the finite element method under one integration scheme [54]. However, this approach involves modal synthesis [55], [56] and additional considerations [57] to keep the computational cost of the procedure under control. Alternatively, co-simulation procedures provide a general approach to simulate coupled systems [37], [58], [59], [60], [61], [62]. Instead of using a single formulation and unified integration, co-simulation allows the subsystems to have independent formulations and integration schemes.
In vehicle-track co-simulation, the coupling between the two subsystems is associated with the wheel-rail contact forces, which requires the assessment of contact conditions between the wheel and rail surfaces. The contact conditions depend on the relative motion and velocity between the wheel and the rail [2], [63], [64]. Therefore, when track flexibility is considered, the evaluation of the contact forces must also include the deformation of the rail. When modeling track flexibility, the rails are usually modeled as finite elements, and their position is evaluated using the corresponding shape functions. In the literature, the details regarding this implementation often refer to classic finite element references, i.e., unidimensional straight beam elements. This modeling approach is suitable if the track comprises only straight sections or when the wheel-rail interaction is replaced by simplified loading conditions. In curved sections, describing the rails with unidimensional straight elements results in high-frequency oscillations without physical meaning in the contact forces. This behavior is due to the wheel changing direction by rolling over straight elements with different orientations, instead of rolling over a smooth curve. These numerical difficulties are mitigated by using different interpolation schemes for the geometry and deformation of the rail [55], [56], or by improving the resolution of the rail geometry, i.e., increasing the number of straight beam elements used to represent the curve. However, increasing rail mesh density also increases the computational cost and only mitigates the undesired effects without solving them. Hence, making this approach less appealing for engineering applications.
This work proposes a new approach to model the rails as unidimensional curved Timoshenko beam elements. This approach improves the representation of rail geometry, with a direct impact on the prediction of the wheel-rail forces, and reduces the computational cost of dynamic studies involving vehicle-track interaction with flexible tracks. The impact of the finite element chosen to represent the rails is demonstrated with a case study of a vehicle running on a flexible track with curved geometry.
Section snippets
Track modeling methodology
The structure of the ballasted railway track is commonly divided into two subsystems: the superstructure and the substructure. The superstructure comprises the rails, rail pads, fastening systems, sleepers, ballast, and sub-ballast, while the substructure comprises the formation layer and the ground. The track model used in this work considers only the superstructure flexibility between the rail and the ballast, as shown in Fig. 1. The remaining components are disregarded because they do not
Railway vehicle modeling
A multibody model of a railway vehicle is characterized by a set of rigid and flexible bodies interconnected by joints and force elements. The equations of motion representing a multibody model of a railway vehicle are written as [73]where M is a matrix comprising the mass and inertia properties of the rigid bodies, Φq is the Jacobian matrix comprising the partial derivatives of the constraints equations with respect to the coordinates, is the vector with the accelerations of
Vehicle-track interaction
In the vehicle-track co-simulation procedure, the wheel-rail contact establishes the constitutive relation between the vehicle and track subsystems [1], [2]. Both subsystems exchange information to assess contact conditions between the wheel and the rail surfaces, after which the wheel-rail forces are evaluated with a suitable contact model. Finally, the contact forces are transferred from the interacting surfaces to specific points in the models, i.e., to the centers of mass of the bodies on
Case study
A case study of a vehicle running on a flexible track demonstrates the implications of different rail modeling options, such as element choice and mesh refinement. Two different elements based on the Timoshenko beam theory are used to represent the rails: a quasi-exact two-node straight element [67], [68], and a three-node curved element [67]. The coupled dynamics of the railway vehicle and the flexible track are captured with a co-simulation procedure that ensures the synchronization,
Conclusions
In applications involving vehicle-track interaction and curved flexible tracks, modeling the rails using straight beam elements might compromise its geometric representation, which is fundamental in the evaluation of the wheel-rail contact forces. This work proposes the use of curved beam elements to describe the rails when dealing with vehicle-track interaction and flexible track models with arbitrary geometry. A case study demonstrates the benefits of the new approach over the typical
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The first author expresses his gratitude to the Portuguese Foundation for Science and Technology (Fundação para a Ciência e a Tecnologia) and the Luso-American Development Foundation (Fundação Luso-Americana para o Desenvolvimento) through the grants PD/BD/128138/2016 and Project – 140/2019, respectively. The third author expresses his gratitude to the Portuguese Foundation for Science and Technology through the PhD grant SFRH/BD/96695/2013. This work was supported by FCT, through IDMEC, under
References (96)
- et al.
A study on the dynamic train-track interaction over cut-fill transitions on buried culverts
Comput Struct
(2017) - et al.
Short soil–binder columns in railway track reinforcement: three–dimensional numerical studies considering the train–track interaction
Comput Geotech
(2018) - et al.
Methodology for the comprehensive analysis of railway transition zones
Comput Geotech
(2018) - et al.
Corrective countermeasure for track transition zones in railways: Adjustable fastener
Eng Struct
(2018) - et al.
Numerical simulations to improve the use of under sleeper pads at transition zones to railway bridges
Eng Struct
(2018) - et al.
Railway critical velocity - Analytical prediction and analysis
Transp Geotech
(2016) - et al.
The influence of track modelling options on the simulation of rail vehicle dynamics
J Sound Vib
(2012) - et al.
Sensitivity analysis of free vibration characteristics of an in situ railway concrete sleeper to variations of rail pad parameters
J Sound Vib
(2006) - et al.
The use of elastic elements in railway tracks: A state of the art review
Constr Build Mater
(2015) - et al.
Experimental and finite element fatigue assessment of the spring clip of the SKL-1 railway fastening system
Eng Struct
(2019)