Introduction

Longitudinal train dynamics (LTD) is considered one of the important aspects in the design of trains. In common parlance, it is the movement of the locomotive and coaches of a train and the incidental in-train forces of the coaches with respect to the track. LTD depends on many factors such as braking dynamics (train brake lag time), coupler dynamic specifications, load distribution of coaches, operational parameters, and the train velocity. The study of train longitudinal dynamics and stipulating out an optimal set of design parameters is a complex investigation. Any wrong design with respect to the above-mentioned factors will lead to a significant impact on the running stability, loading integrity, safety of the vehicle’s structure and passengers’ comfort [1, 2]. Due to this reason, in the preceding two decades, studies on LTD were conducted in the direction to reduce the longitudinal in-train forces and to augment the dynamic performance of trains [1, 3, 4].

When brakes are initiated, the train body is first exposed to an accentuated compression, trailed by bounce back and then with a run-over of oscillatory movements that determine a mechanical wave which traverses through the train numerous times. Subsequently, sundry tensile in-train forces and longitudinal compression forces develop, which generate stresses in couplers, affecting passengers’ comfort [5]. Therefore, the study of braking dynamics contributed significantly to predict the behaviour of train system and led to improvements in comfort and running safety, optimisation of train structure, capacity growth, lower maintenance costs and lower damage risks.

Apart from braking dynamics, the in-train forces or coupler forces also play an important role in LTD. The in-train forces are the corollary of the sudden contrasts, i.e., tensile or compressive, of the longitudinal forces occurring between coaches. Immediate contrasts among forces that act on every coach determine the assembly of complex longitudinal responses, acting and being transmitted through the shock and traction apparatus. Wu et al. [6] presented a detailed review of modern dynamic models of draft gears. They concluded that draft gears modelling is one of the most important and difficult tasks in LTD. The ride quality and comfort of a vehicle depend on the velocity, displacement, acceleration, jerk of the vehicle along with other environmental factors, i.e., dust, noise, temperature and humidity [7, 8].

As mentioned before, braking dynamics and coupling between coaches are affecting the ride quality and comfort. Therefore, these need to be considered while designing the trains. There is no literature available on the study of LTD considering braking dynamics and coupling between the coaches for Indian Railway (IR) systems to the best knowledge of the authors. Therefore, in the present article, LTD for a passenger train was investigated with braking dynamics and coupling dynamics.

Longitudinal Train Dynamics Modelling

Mathematical models were developed to represent the dynamics of various structures of train, i.e., train modelling, modelling of coupler or in-train forces, modelling of braking phenomena and rolling resistance.

Train Modelling

A typical train model for simulation of longitudinal dynamics, described in many papers is used in the presented research [5,6,7,8]. The train model is a damped elastic system that consists of n number of rigid bodies of masses, \( m_{1} ,m_{2} \ldots m_{n} \) representing the train coaches as exhibited in Fig. 1. Only a longitudinal degree of freedom, connected by coupling devices along with the forces acting on each coach is considered. Lateral and vertical dynamics were not considered [13]. The various forces acting on each coach in the train body are inertia forces, \( F_{i,1} ,F_{i,2} , \ldots F_{i,n} \); braking forces, \( F_{{{\text{brake}},1}} \left( t \right),F_{{{\text{brake}},2}} \left( t \right), \ldots F_{{{\text{brake}},n}} \left( t \right) \); forces in the coupling devices, \( F_{1} \left( {\Delta x_{1} ,\Delta \dot{x}_{1} } \right), F_{2} \left( {\Delta x_{2} ,\Delta \dot{x}_{2} } \right), \) \( \ldots F_{n - 1} \left( {\Delta x_{n - 1} ,\Delta \dot{x}_{n - 1} } \right) \); rolling resistances, \( F_{{{\text{rr}},1}} ,F_{{{\text{rr}},2}} , \ldots F_{{{\text{rr}},n}} \) and traction/braking effort.

Fig. 1
figure 1

Forces acting in a train

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The system of differential equations is shown in Eqs. (1–3):

$$ m_{1} \ddot{x}_{1} - F\left( {\dot{y}_{1} ,y_{1} ,\dot{y}_{2} ,y_{2} } \right) = F_{{t/{\text{bd}}}} - F_{{{\text{brake}},1}} (t) - F_{{{\text{rr}},1}} . $$
(1)

For the ith coupling device (coach number i and i + 1), one can write it as

$$ m_{i} \ddot{x}_{i} - F\left( {\dot{y}_{i} ,y_{i} ,\dot{y}_{i - 1} ,y_{i - 1} } \right) = F_{\text{ti/bd}} - F_{{{\text{brake}},i}} (t) - F_{{{\text{rr}},i}} . $$
(2)

For the (n − 1)th, the last coupling device (coach number n − 1 and n), one can write it as

$$ m_{n} \ddot{x}_{n} - F\left( {\dot{y}_{n} ,y_{n} ,\dot{y}_{n - 1} ,y_{n - 1} } \right) = F_{\text{tn/bd}} - F_{{{\text{brake}},n}} (t) - F_{{{\text{rr}},n}} . $$
(3)

The solution to the above-given equations is further obscured by the need to compute the system’s input forces, i.e., Fn, Fbrake, and Frr. Methodologies to model coach connections with every single forcing inputs are included and discussed in the subsequent subsections.

Coaches Connection System Modelling

In a train, the adjacent vehicle is connected mechanically using a device known as coupler which plays an important role in the analysis of LTD [9,10,11,12,13]. This technique is easier to apply than the AAR and has no significant error consequence unless used for very sharp curves (less than 0.1% at R = 100 m). In this study, AAR type H tight lock coupler (Fig. 2), with a low preload draft gear has been considered [14, 15].

Fig. 2
figure 2

AAR type H tight lock coupler [15]

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The coupler system is a connection between two adjacent coaches in a train. It consists of coupler head with drawbar and its guide and draw and buffing gear in draft gear. The coupler system is usually shortened into single-element models, so every two draft gears are exhibited in series as a single unit. A unit model of coupler system has to incorporate characteristics that can simulate under both draft and buff conditions. The final model must also consider coupler slack as well as the limiting stiffness that appears after springs are fully compressed. When installed, draft gears are usually preloaded, which should also be incorporated.

To sum up, a desirable coupler system model should include the above-discussed elements: slack, limiting stiffness, preload, velocity dependent friction, and transitional characteristics. In Eq. (4), ka is an elastic constant and kfr is a frictional spring constant [15, 16]. The coupler system consists of preload for holding the adjacent coupler head which is here denoted by P. Where, y is the relative displacement of draft gear stroke and \( \dot{y} \) is its relative speed. The conditions of coupler assemble in buff mode, preload tensile and draw gear are y < 0, y = 0, and y > 0. For performing simulations, the main parameters used for the buffers in buff mode are: P = 25 kN, kab= 11,785 kN/m, kfrb= 5457 kN/m; and kad= 9430 kN/m and kfrd= 4365 kN/m [16] for the draw gear. Thus, coupler force was evaluated by the following equation and shown in Fig. 3.

Fig. 3
figure 3

Force-relative displacement characteristics of the coupler at different initial velocities for 0.5, 1.0 and 2 m/s

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$$ F_{i} (\dot{y},y) = \left\{ \begin{aligned} & k_{\text{ab}} y_{i} + k_{\text{frb}} \left| {y_{i} } \right|\tanh \left( {u \cdot \dot{y}} \right) + P\quad & {\text{for }}y < 0, \\ & P\quad & {\text{for }}y = 0, \\ & k_{\text{ad}} y_{i} + k_{\text{frd}} \left| {y_{i} } \right|\tanh \left( {u \cdot \dot{y}} \right) + P\quad {\text{for }}y > 0 \\ \end{aligned} \right.. $$
(4)

Air Brake System Modelling

The passenger train under study is fitted with automatic air brake system. The peculiar air brake system begins with the reduction of air pressure in the brake hose. This is achieved through brake control valve. Simultaneously, the distributor valve of every compartment comes into operation, due to which air flows from auxiliary air reservoir to the brake cylinder. This ultimately causes the cylinder thrust to push into moving wheels of each individual compartment. The ratio of the transmission is a function of parameters such as the size of the cylinder, dynamic efficiency of rigging and cylinder, rigging ratio, friction coefficient of disc brake along with other operational conditions. Hence, to calculate the braking force Eq. (5) is used [16, 17]. Table 1 gives the value of brake modelling parameter:

$$ F_{{{\text{brake}}(n)}} = \left( {\left( {\frac{{\pi d_{\text{bc}}^{2} }}{4}} \right)\eta_{\text{cyl}} p_{\hbox{max} (n)} - F_{\text{sR}} } \right)i_{t} n_{\text{bc}} \frac{{ 2r_{m} }}{{D_{0} }}\mu_{d} \eta_{\text{br}} . $$
(5)
Table 1 Brake modelling parameters
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Rolling Resistance Modelling

The sum of air resistance and rolling resistance is known as propulsion resistance of the vehicle. Its effect is largely dependent upon the design and shape of the vehicle and aerodynamic drag complexity [18,19,20]. The reckoning of the summation process is still a function of the empirical formulae. In the study, a relation similar to that of Davis relation was used for predicting propulsion resistance [21] as given in (6):

$$ \begin{aligned} & {\text{Locomotive}}\left( {\text{WAP5}} \right) = 1.34819 + .02153v + 0.00008358v^{2} \\ & {\text{Passenger coaches}}\left( {\text{LHB}} \right) = 0.69873 + .021533v + 0.0000835v^{2} . \\ \end{aligned} $$
(6)

Running resistance is given by an empirical formula, \( F_{\text{rr}} = a + bv + cv^{2} , \) where a, b and c are dependency constants, v is speed in km/h and Frr is train resistance in kg/ton.

Case Study and Software Modelling

Case Study Information

In the case study, the India’s fastest passenger train, i.e., Rajdhani Express is considered which runs at 40 m/s. The arrangement of various types of coaches is shown in Fig. 4. The various types of coaches and their respective weight for tare and the gross condition are shown in Fig. 5. The length of railway track considered from Palwal to Mathura is 87 km.

Fig. 4
figure 4

Arrangement of coaches in Rajdhani express (LHB) Rake

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Fig. 5
figure 5

Types of coaches and their weight [15]

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Locomotive Design Aspects

The WAP-5 locomotive is considered for the study. For the typical configuration, the design characteristics and specifications are given in Table 2.

Table 2 Characteristics and specifications of WAP5 locomotive [22, 23]
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These attributes and specifications were utilized to model locomotive performance curves for each standard unit for their implementation in the train dynamics studies, so as to describe the capability of the locomotive and dynamic braking. Manufacturers measure tractive effort (TE) and braking effort (BE) as a function of speed [24, 25]. The TE and BE of WAP-5 electric locomotive are shown in Fig. 6. These graphs were obtained experimentally by RDSO [25, 26]. In the graphs, the maximum tractive effort is 258 kN and braking effort is 160 kN [26,27,28].

Fig. 6
figure 6

Tractive effort and braking effort performance curve for WAP-5

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Train Model in Universal Mechanism Software

A study on heavy haul train and car dumper model were carried out using universal mechanism (UM) to analyze train longitudinal dynamics during dumping by Kovalev et al. [13]. Hence, for the analysis of LTD, the 3D dynamic train model was developed in UM software using the information provided in the previous sections. The train model is shown in Fig. 7.

Fig. 7
figure 7

Model of the train in universal mechanism software

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Condition for Simulation

The train was moving at a speed of 40 m/s. The braking process that began in different modes at different times in the simulation are given in Fig. 8. It has been assumed that the speed of propagation of air in the brake line (the so-called wave speed braking) is 220 m/s. The final pressure in brake cylinder of locomotive is 53,200 Pa.

Fig. 8
figure 8

Different modes at different times in the simulation

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Results and Discussion

The various braking phases considered for analysis have been shown in Fig. 8. In this analysis, application of the dynamic brake system was applied solely to the locomotive. However, these brake phases lead to change in the acceleration and in-train forces, which could be observed in Figs. 9 and 10.

Fig. 9
figure 9

Coaches acceleration due to braking phases

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Fig. 10
figure 10

In-train forces due to braking phases

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Phase 1: Auxiliary Brake (Service Braking)

Phase 1 (0 s ≤ t ≥ 25 s) was a type of auxiliary braking in service braking mode. In this phase, the initial velocity drops from 40 to 39.6 m/s and acceleration lies between + 0.54 to − 1.375 m/s2. The maximum in-train force occurred in the last three coaches. The maximum in-train force was about 63, 62 and 68 tons, respectively.

Phase 2: Brake Pipe (Service Braking)

In this phase (25 s ≤ t ≥ 80 s), the service brake was applied to the train which caused more reduction in the train velocity up to 38.3 m/s and acceleration varied between + 0.0098 to − 0.11 m/s2. The brake application time was observed as 20.41 s for full-service application on the last coach. That resulted in a sequential brake lag time of 0.97 s among the adjoining cars. In contrast to the preceding phase, at the preliminary stage of this phase, a high magnitude in-train compressive force occurred, at first three coaches of train, i.e., 24.47, 24.21 and 24.37 tons for first three coaches, respectively.

Phase 3: Auxiliary Brake (Running Repeater)

The running repeater mode braking was applied in Phase 3 (80 s ≤ t ≥ 155 s) of braking operation. In this phase, the velocity of the train was initially decreased from 38.3 to 32.94 m/s and then it started to increase eventually to reach to 28.5 m/s at the end of the phase. This variation of reduction and increment in the velocity was due to the track elevation as the train goes on uphill and downhill at the end of this phase. The average acceleration was − 0.11 m/s2. Moreover, the train experiences buff mode on the uphill and the maximum in-train compressive force was of considerable magnitude. In the later stage of braking, the train gets stretched as it starts going downhill, and the amount of in-train tension forces increased rapidly. The maximum in-train force was about − 124, − 127 and − 141 ton for the last three coaches, respectively, as shown in Fig. 10.

Phase 4: Brake Pipe (Release)

In this phase, the release of service brake was considered (155 s ≤ t ≥ 210 s). The train velocity increased and reached to 35 m/s and acceleration lied between 0.44 and 0.025 m/s2. In contrast to the previous phases, the extent of in-train tension force among the coaches was reduced. Moreover, the train was making up and the train dynamic motion appeared to be very complex with multi-mode behaviour. The maximum in-train tension force value lied between 27.17 and − 17.8 tons for the middle coaches, i.e., between 10th and 18th coach.

Phase 5: Brake Pipe (Emergency Braking)

In this phase, the emergency brake was applied to stop the train. The train velocity changed from 35 to 0 m/s and the time lag, which was defined as the time duration in which the pressure signal initiated from the locomotive reached the end of the train was measured to be 5.3 s. That amounted to a sequential brake lag time among the succeeding cars of 0.12 s. The maximum in-train forces lied between 143 and − 120 ton.

Conclusions

The results of different braking phases on the train longitudinal dynamics are thoroughly studied using multibody dynamics analysis. The train consists of a locomotive and 20 coaches fitted with AAR type H tight lock coupler, with a low preload draft gear running at maximum speed. A mathematical model is formulated to analyse the coupler characteristics and air brake dynamics. The brake delay times are considered to be 0.12 and 0.97 s for emergency and service brakes. This period of time paved the way for inter coaches’ states to slowly move from bunched to stretch or vice versa, avoiding large in-train forces. The simulation results suggested that for all braking phases considered the maximum compression force increased when the forward velocity of the train is reduced. It is observed that compression with maximum magnitude is noticed at the third quarter of the train. The maximum compression force observed is around 143 tons.