An influence of track stiffness discontinuity on pantograph base vibrations and catenary–pantograph dynamic interaction

The simulation method used in this article assumes the decomposition of the catenary–train–track system into two main subsystems: (A) catenary and pantographs and (B) track and train. The physical contact between them is the pantograph base fixed rigidly to the vehicle roof. Each of the highlighted main subsystems (A) and (B), in accordance with its name, consists of two internal subsystems, and these, in turn, consist of many elements which are shown in Fig. 1.

Figure 1

It is assumed that the contact between main subsystems (A) and (B) is modeled by the use of one-way coupling method which is often applied in the fluid–structure computational mechanics, for instance [20]. For one-way coupling calculations of the fluid–structure interaction, the fluid affects the structure motion but the motion of structure does not affect fluid properties. Similarly, it is assumed for the considered catenary–train–track system that the pantograph base motion affects the catenary–pantograph subsystem, constituting a kinematic excitation of the pantograph vibrations, but this does not occur reversely. Vibrations of the catenary–pantograph subsystem are not transferred to the train–track subsystem because they do not significantly affect a heavy railway vehicle. This approach, that is, the use of one-way coupled simulations is consistent with the physical behavior of the considered system and gives a possibility for reducing the computational effort.

In each main subsystem: (A) catenary and pantographs or (B) track and train, two internal subsystems are coupled in a two-way manner [20], excluding the moments when the loss of contact occurs. The contact loss is manifested by detaching the collector head from the contact wire (in (A)) or the wheel from the rail (in (B)). Hence, for a moment, there is no transmission of vibrations between internal subsystems in the given main subsystem. Loss of contact is the source of specific geometrical non-linearity in the main subsystems. A similar type of non-linearity occurs in the main subsystem (A) due to droppers’ behavior, because droppers do not transfer a compression. This problem is described in detail [4, 14, 15]. In the presented simulation method, the geometrical non-linearities due to the loss of contact between internal subsystems are neglected, similarly as in many publications by other authors [4, 5, 7], while the non-linearity caused by droppers’ behavior is accounted for.

The most significant simplifications of the assumed computational model are as follows: (1) the model is a flat one (two-dimensional) and (2) the complex internal structure of the railway track is replaced by the Euler beam resting on the homogeneous viscoelastic foundation. The first simplification is acceptable because the vertical motion is the dominant vibration component of the considered catenary–train–track system; hence, a flat model is sufficient enough to fulfill the objective of the article which is to assess the impact of a train passage through the track stiffness discontinuity on catenary vibrations. The second simplification which consists of omitting the internal structure of the track is also justified because, from the research purpose viewpoint, the very fact of an occurrence of track stiffness discontinuity is important – not the nature of it.

The assumed dynamic model of the main subsystem (A), shown schematically in Fig. 2, is described in detail in earlier published papers [13, 14]. Compared with the solutions presented in the literature, the model stands out by the consistent application of the theory of cable vibrations in order to formulate partial differential equations governing the motion of messenger and contact wires. An original feature of this model is also the method for considering the non-linear behavior of droppers.

Figure 2

The model of the main subsystem (A) is referred to as the catenary which consists of one contact wire and one messenger wire. The catenary section being analyzed is composed of a given number of spans. The messenger wire, supported slidingly on rigid supports, is represented by a multispan flexible cable with a parabolic route within each span (in an unloaded state). The contact wire is treated as a straight cable (string) which is suspended to messenger wire by flexible droppers modeled by nonlinear elastic constraints (springs). Messenger wire and contact wire are tensed by different constant static forces.

Pantographs are modeled by two-degree-of-freedom dynamic systems. Contact between the pantograph collector head and the contact wire is represented by a linear contact spring that does not reflect a real behavior of the structure because the effects due to loss of contact are neglected in that way. However, such simplification is often applied in the literature and is allowed by the European standard EN 50318 [16] which gives guidelines for vibration simulation methods of the pantograph–catenary system. In the presented model, the loss of contact is identified by a negative value of the contact force.

Equations of motion of the catenary–pantographs subsystem have been derived based on the Lagrange equations, using the Ritz approximation method for the catenary, as shown in Bryja and Prokopowicz [13]. They may be written in matrix notation: (1)B(A)q¨(A)(t)+C(A)q˙(A)(t)+[Kconst+K^const−(1−κ)K^dc(q(A))+K˜(t)](A)q(A)(t)=f(A)(t)\matrix{ \matrix{{{{\bf{B}}_{({\rm A})}}{{{\bf{\ddot q}}}_{({\rm A})}}(t) + {{\bf{C}}_{({\rm A})}}{{{\bf{\dot q}}}_{({\rm A})}}(t) + } \cr {{{[{{\bf{K}}_{{\rm{const}}}} + {{{\bf{\hat K}}}_{{\rm{const}}}} - (1 - \kappa ){{{\bf{\hat K}}}_{{\rm{dc}}}}({{\bf{q}}_{({\rm A})}}) + {\bf{\tilde K}}(t)]}_{({\rm A})}}{{\bf{q}}_{({\rm A})}}(t) = {{\bf{f}}_{({\rm A})}}(t)}}} where an explicit form of the inertia, damping, and stiffness matrices, as well as the vector of excitation forces, can be found in Bryja and Prokopowicz [13] and Bryja and Hyliński [14]. The excitation forces gathered in the vector f_(A)(t) depend on the static contact forces of pantographs, moving along the catenary at the constant speed. Dynamic increments in the moving contact forces, being a measure of dynamic interaction between pantographs and contact wire, are included in the time-dependent stiffness matrix component: [K˜(t)](A)q(A)(t){[ {{\bf{\tilde K}}(t)} ]_{({\rm A})}}{{\bf{q}}_{({\rm A})}}(t) . In this article, the vector f_(A)(t) depends also on the dynamic displacements WJp(t)W_J^p(t) and their velocities W˙Jp(t)\dot W_J^p(t) at contact points between the pantograph and contact wire, where J denotes the pantograph number. These functions are computed by the solver of track–train subsystem.

The equations of motion (1) are non-linear due to the non-linear stiffness characteristic of droppers [14]. It is assumed that the droppers are characterized by the tensile stiffness k_t=k and the residual compressive stiffness k_c=κk which value is a given small percentage of a tensile stiffness. Note, that the assumption κ=0 leads to droppers which do not carry any compression (have zero compressive stiffness). In order to take the nonlinear droppers’ stiffness into account, two components related exclusively to droppers have been isolated in the general stiffness matrix: [K^const](A){[ {{{{\bf{\hat K}}}_{{\rm{const}}}}} ]_{({\rm A})}} and [K^dc(q(A))](A){[ {{{{\bf{\hat K}}}_{{\rm{dc}}}}({{\bf{q}}_{({\rm A})}})} ]_{({\rm A})}} . Both components have the same structure and satisfy the assumption that k_c= k_t=k. The first component covers all droppers of the catenary, while the second applies only to droppers detected as being under compression at time t (therefore, it depends on q_(A)). Expression [K^const−(1−κ)K^dc(q(A))](A){[ {{{{\bf{\hat K}}}_{{\rm{const}}}} - (1 - \kappa )\;{{{\bf{\hat K}}}_{{\rm{dc}}}}({{\bf{q}}_{({\rm A})}})} ]_{({\rm A})}} reduces the stiffness of droppers subjected to compressive forces, into the value k_c=κk. The component [K_const]_(A) is related only to the elastic properties of messenger and contact wires.

To solve equation (1), the unconditionally stable variant of the Newmark numerical integration method is applied with the use of an iterative loop for calculating the stiffness matrix component: K^dc(q(A)){{\bf{\hat K}}_{{\rm{dc}}}}({{\bf{q}}_{({\rm A})}})) , at each time step of numerical integration. The method of solving the equations of motion is described in detail in Bryja and Hyliński [14, 15]. At each time step of the simulation, the contact wire displacements at indicated points, dynamic contact force, etc, are generated based on general coordinates q_(A) (t_i) determined at time points t_i. Discrete records of resulting time histories can be further processed.

The considered section of the track is straight and horizontal, as shown in Fig. 3. Its total length L₀ is divided into two segments: L₁ and L₂, due to an occurrence of the track discontinuity having the form of a sudden step change in the track stiffness. Such track stiffness discontinuities are usually of structural nature and exist along the track at such places as transitions from ballasted track to slab track (see Fig. 1) or bridge approaches, culverts, turnouts, ends of tunnels, etc. A train passage through the stiffness discontinuity of the track induces significant vibrations of vehicles and, thus, rail vibrations [21].

Figure 3

In the presented vibration simulation method, the track stiffness discontinuity is considered leaving aside its nature. The applied dynamic model of the train–track main subsystem is shown in Fig. 4. It is assumed that the railway track is the Euler–Bernoulli beam resting on the viscoelastic foundation of the Winkler type. To transform the generally known, partial differential equation of the beam vibrations, into equations of motion in a time domain, the GFEM has been adopted. This approach allows to easily account for the track stiffness discontinuity, by setting different stiffness and damping parameters of the Winkler foundation over two track segments (L₁ and L₂). To satisfy the GFEM requirements, both ends of the whole track section L₀ have been assumed as rigidly fixed, which is shown in Fig. 3. The entire GFEM procedure adopted in this article has been explained by Bryja et al. [17, 22, 23].

Figure 4

It is assumed that the train is an electric trainset composed of a given number of repetitive units (vehicles), each based on two 2-axle bogies, as shown in Fig. 4. The trainset is equipped with a maximum of two pantographs mounted on the roof of one selected unit or two units. The pantograph base is rigidly connected with the vehicle roof, moves at constant speed together with the train, and vibrates only in a vertical direction. The assumed vehicle model of 10 degrees of freedom, presented in Fig. 4, allows to calculate vertical vibrations of the vehicle body at the pantograph attachment point. The model consists of seven rigid mass elements representing: vehicle body, two rigid bogies, and four wheelsets, connected by sets of linear springs and viscous dampers which constitute two-stage vehicle suspension. It is assumed that the wheelsets remain in full contact with rails during the vibrations. As a further stage of the research, it is planned to extend the model by introducing the Hertz contact spring between wheelsets and rails that will allow accounting for the loss of contact.

The equations of motion of the described train–track subsystem derived in Bryja et al. [17, 22] on the basis of equations in Bryja et al. [23], can be written in the following general form: (2)B(B)(t)q¨(B)(t)+C(B)(t)q˙(B)(t)+K(B)(t)q(B)(t)=f(B)(t){{\bf{B}}_{({\rm B})}}(t){{\bf{\ddot q}}_{({\rm B})}}(t) + {{\bf{C}}_{({\rm B})}}(t){{\bf{\dot q}}_{({\rm B})}}(t) + {{\bf{K}}_{({\rm B})}}(t){{\bf{q}}_{({\rm B})}}(t) = {{\bf{f}}_{({\rm B})}}(t) In the simulation method, the solutions of equation (2), that is, general coordinates q_(B)(t), their velocities q˙(B)(t){{\bf{\dot q}}_{({\rm B})}}(t) , and accelerations q¨(B)(t){{\bf{\ddot q}}_{({\rm B})}}(t) are obtained by the use of the Newmark method, and on that basis, time histories of displacements WJp(t)W_J^p(t) , velocities W˙Jp(t)\dot W_J^p(t) , and accelerations W¨Jp(t)\ddot W_J^p(t) of vehicle body vibrations at the pantograph base attachment are calculated. These functions are transferred to the catenary–pantographs solver.

The analysis is conducted for a hypothetical trainset consisting of eight repetitive units (vehicles) which include mass parameters, primary and secondary suspension parameters, and dimensions, which are gathered in Table 1. These parameters are taken from Bryja et al. [23]; they reflect the typical unit of Shinkansen high-speed train. The trainset is equipped with a single pantograph mounted on the first unit. A theoretical point of the pantograph base attachment lies on the vertical axis of the leading bogie. In order to find a scenario where the vehicle vibrations influence the catenary, a hypothetical vehicle equipped with a primary suspension alone is additionally considered. In the computational model, such a hypothetical vehicle is defined by setting a very high stiffness of the spring and zero damping parameter in the secondary vehicle suspension.

The calculations have been carried out for two train speeds, 60 m/s and 100 m/s, assuming that the length of the track section is equal to 300 m and the track stiffness discontinuity occurs in the middle of this section. Flexural stiffness and unit mass of the Euler beam representing two rails are 1.2831×10⁷ Nm² and 1.21×10² kg/m, respectively. Damping of the rail material is taken into account assuming the retardation time of 2.1×10⁻⁵ s.

Based on the experience from previous research described by Bryja and Popiołek [19], two types of track stiffness discontinuity are considered. The type 1 is a sudden 50-fold decrease in the stiffness of the elastic Winkler foundation: from the value of k_f to k_f/50, while type 2 means the reverse transition: from the value of k_f/50 to k_f, where k_f=1.1×10⁸ N/m² is the assumed basic value of the Winkler foundation stiffness, specific for the track of a very good quality (see Esveld [21]). The damping coefficient characterizing the track foundation amounts to 2.8667×10⁵ Ns/m².

Simulation results of vehicle body vibrations at the point of pantograph base attachment are shown in Fig. 5 (for the track stiffness discontinuity of type 1) and Fig. 6 (for the track stiffness discontinuity of type 2). Time histories of displacement, velocity, and acceleration are presented for two cases of vehicle suspension (two-stage and single-stage suspension). The graphs on the left side relate to the passage speed of 60 m/s, those on the right side – 100 m/s. These graphs are prepared on the same vertical scale for both speeds, in order to facilitate a comparison of the obtained results. Similarly, time axes are equally scaled for both speeds, so that the time nature of vibrations can be compared. A vertical dotted green line denotes the moment of passing through the track stiffness discontinuity.

Figure 5

Figure 6

It should be recalled that both ends of the considered track section are fixed in the computational model, and the vehicle before entering the track is in a static equilibrium state (i.e., its dynamic displacements are zero). Entering on a deformable track induces vehicle body vibrations in the form of decaying oscillations which are visible in the first phase of vehicle motion, in all graphs presented in Figs. 5 and 6. This initial effect is much more intense when the vehicle enters the track resting on a weak substrate (k_f/50) – Fig. 6. Local amplitudes of displacement, velocity, and acceleration are then compared with those that appear after the passage through the track stiffness discontinuity of type 1 (change from k_f to k_f/50) – Fig. 5. This indicates that the track substrate with the elasticity coefficient k_f = 1.1×10⁸ N/m² does not differ much from a rigid foundation, as evidenced also by a weak initial effect that appears when the vehicle enters the track resting on a strong substrate (k_f) – Fig. 5.

In the case of time histories obtained for the speed of 60 m/s, displacements of the vehicle without secondary suspension are lower than for a vehicle with two-stage suspension but velocities and accelerations are higher, which is consistent with practice. In particular, the maximum absolute value of vibration acceleration is more than twice higher in relation to the train with two-stage suspension. This effect is even greater at a speed of 100 m/s. It shows the intensity of the impulse load induced by a high-speed passage through the track stiffness discontinuity and also confirms the need to use a two-stage suspension in high-speed trains. The obtained results demonstrate that the secondary suspension effectively reduces accelerations of vehicle body vertical vibrations and, thus, improves passenger comfort as well as operational conditions of the pantograph mounted on vehicle roof.

An interesting phenomenon is observed when comparing pantograph base displacements calculated for the train speed of 60 m/s and 100 m/s. At the speed of 100 m/s, local amplitudes of vibrations of the vehicle with two-stage suspension are smaller than for one-stage suspension, contrary to the case of 60 m/s. This is caused by relatively long retardation time of the reaction of two-stage suspension, due to additional dampers. Comparing the results obtained at the speed of 60 m/s and 100 m/s, for the same two-stage suspension of the vehicle, one can conclude that this long retardation time results in decreasing the vehicle body vibrations as well as their velocities and accelerations – all local amplitudes are a bit smaller. Therefore, it is seen that a two-stage suspension efficiency increases with the train speed. The opposite effect appears for a single-stage suspension, especially in the case of vibration accelerations. The intensity of vibrations caused by a train passage through the track stiffness discontinuity is definitely higher at the speed of 100 m/s than at 60 m/s because exclusion of secondary suspension results in a much lower delay in the dynamic response of the vehicle body. This is in line with an engineers’ intuition. All these conclusions refer to both considered types of track stiffness discontinuity.

It should be noted that the assumed track length preceding the track stiffness discontinuity (i.e., 150 m) is too short as the initial vehicle body oscillation is not completely decayed, especially in the case shown in Fig. 6. Hence, the dynamic effect caused by a train passage through the track stiffness discontinuity is amplified because of overlapping the initial vibrations induced by entering the track. However, this amplification is advisable; it may help to identify such a train passage scenario in which the vibrations of a railway vehicle considerably influence the pantograph–catenary dynamic interaction. For this purpose, all solutions presented in Figs. 5 and 6 will be examined in the second step of numerical analysis.

The second step of numerical analysis is devoted to an evaluation of the impact of pantograph base vertical oscillations, induced by train passages through the track stiffness discontinuity, on catenary vibrations. According to the assumed research methodology, vehicle body vibrations at the pantograph base attachment point, obtained in the first step of numerical analysis, have been transferred to the solver of pantograph–catenary subsystem, constituting the kinematical excitation of the pantograph vibrations. While simulating the pantograph–catenary vibrations, the time step of numerical integration of equations of motion has been taken the same as the time step of recording the simulation results obtained from the train–track solver.

It is assumed that the test section of the catenary consists of five identical spans with a length of 60 m each, which gives a total length of 300 m being equal to the length of the track section considered. The material and geometric characteristics of the catenary structural elements and pantograph parameters are summarized in Table 2.

To analyze the catenary vibrations, the contact wire uplift at the right steady arm of the middle catenary span is examined, due to the localization directly behind the place where the stiffness discontinuity appears in the track. In turn, the dynamic interaction between the pantograph and catenary is analyzed based on time histories of the pantograph contact force. The simulation results for different scenarios of a train passage are presented in Figs. 7 and 8, following the pantograph position that changes in time. Fig. 7 shows the time histories generated for the track stiffness discontinuity of type 1, while Fig. 8 is related to discontinuity of type 2. As previously, the location of the track stiffness discontinuity is indicated by a dotted vertical green line. As a reference, the corresponding time histories obtained without taking the vehicle body vibrations into account are shown. All graphs illustrate the same fragment of time histories, where the pantograph moves along the third, fourth, and half of the fifth span.

Figure 7

Figure 8

The obtained results demonstrate that the vehicle body vibrations induced by crossing the track stiffness discontinuity do not considerably influence the catenary vibrations. Differences between time histories of the contact wire uplift are almost unnoticeable. In case of the dynamic contact force, the influence of vehicle body vibrations is more noticeable, but it is relevant only in one-load scenario, in which the train without the secondary suspension passes the track stiffness discontinuity of type 1 at the speed of 100 m/s (Fig. 8). Under these conditions, an additional peak of the contact force oscillation is observed in the last phase of the pantograph run (from 240 m to 270 m), which leads to a strong local decrease in the contact force and thus to loss of contact (the loss of contact occurs when the contact force is zero or negative). It means that the railway vehicle vibrations might increase the probability of loss of contact between pantograph head and contact wire, but the two-stage suspension used in high-speed trains effectively prevents such incidents.