Influence of the friction modelling decisions on the acceptance of the running behaviour of a friction-damped locomotive

Multibody simulations are widely used for the dynamic analysis of railway vehicles, including the acceptance of vehicle running behaviour. Among these, the freight vehicles have, generally, friction damped suspensions for which numerical modelling is particularly challenging due to the contacting geometries, the intermittent contact, and the frequent stick-slip transitions. The impact of modelling decisions of friction-damped suspensions on the acceptance quantities for vehicle behaviour is the focus of this work. The selection of friction models and tuning of their parameters is challenging. In particular, the selection of friction coefficients is not straight forward, as friction conditions vary during operation and over time. This work aims to address both problems. A model of a friction-damped freight locomotive is used to simulate the dynamic response of the vehicle under realistic operation conditions and assess the impact of the friction modelling decisions. The results show that the use of different friction models provides similar results if the parameters are appropriately tuned. In contrast, the selection of friction coefficients significantly impacts the acceptance quantities, with emphasis on the wheel-rail forces at small radius curve transitions. The results recommend that the standard should include variations of the friction coefficients used for the friction-damped suspensions.


Introduction
The railway vehicle acceptance process can be supported by multibody simulations to extend the range of test conditions, approve vehicles following modification, and investigate fault modes [1].However, vehicle acceptance requires not only validated numerical models but also validated computational tools [2][3][4].Moreover, the modelling effort requires in general a close attention on modelling the vehicle suspension mechanisms [5], the track [6][7][8], and the wheel-rail contact [9][10][11][12][13].In regards to the former, frictiondamped suspensions, generally used on freight vehicles, are particularly challenging to model due to the contacting geometries, the intermittent contact, and the frequent stickslip transitions [14].The results from the European project DYNOTrain [2,15] reflect the difficulty on validating freight vehicle models equipped with friction damping.
Modelling of friction damped suspensions in multibody simulations can be divided in three main parts: representation of the contacting geometries, including clearances; selection and tuning of the normal and tangential contact models; and selection of friction coefficients that represent the condition of the friction surfaces.The modelling of friction mechanisms requires using suitable modelling elements that represent the relative degrees of freedom between the bodies.Although one-dimensional force elements are commonly used [16,17], they imply an over simplification of the contacting geometries by using several one-dimensional spring elements to approximate the real geometry.On the other hand, clearance joints [18][19][20][21][22][23], provide a more accurate description of the contact conditions by considering the three-dimensional geometry of the bodies and all their relative degrees of freedom.
The normal contact forces developed on the contacting surfaces of friction damped suspensions are often calculated as a function of indentation and penetration speed between contacting bodies.There are several normal contact models in the literature that can be used to calculate the normal contact forces [24], requiring the selection of their stiffness and damping coefficients [14].The tangential contact forces, or friction forces, are often calculated using the Amonton-Coulomb friction model, where the transition between positive and negative sliding velocities is smoothened to avoid the numerical non convergence and instability [14,16].The Amonton-Coulomb friction model requires the tuning of a small number of parameters and provides a simplistic representation of the stick-slip transition [25].A number of alternative tangential contact models are found in the literature [25] and provide alternative modelling possibilities that are better suited for particular operation conditions.Opala [26] evaluates the effect of using different friction models in the centre bowl of a bogie, concluding that in the presence of track irregularities the wheel-rail contact forces are insensitive to different friction models.However, the impact of different friction models on the vehicle accelerations is not addressed.In addition, the effect of using different friction models on other suspension elements, such as horn guides, is not studied.The use of the different normal and tangential contact models requires selecting a different number of parameters that are required for their numerical implementation and that are crucial for the quality of the results obtained.In general, the more detailed the contact models the larger the number of parameters required to be tuned.Marques et al. [25] provide a study on how the parameters of several friction models affect the dynamics of simple mechanisms.However, the effect on more complex systems, such as friction damped railway vehicles, remains unclear.A step to clarify this issue is given in the work by Millan et al. [27] in which the use of different friction models for railway friction damped suspensions is tested, allowing to conclude that different friction models influence the dynamics of railway vehicles only in particular operation conditions, such as in transition curves.
The friction coefficients represent the state of the friction surfaces and are independent of the friction models employed [25].These can vary during the operation and over time, due to temperature, wear, humidity, among others.Therefore, the selection of friction coefficients is also a challenging task.Nonetheless, computational studies which consider vehicles with friction damped suspensions require a selection of friction coefficients.These can affect the response of the suspension mechanisms, as studied by Lyu in [28], and ultimately affect the dynamic behaviour of a vehicle.Götz, in [29], studies the friction coefficient of a Lenoir damper in the validation results of a vehicle model, to find that there are significant changes in the validation quantities.However, the impact of the friction coefficients in the dynamic response quantities used for vehicle acceptance, which require a different post-processing than that for model validation, is still unknown.It is also unclear how to account for the uncertainty in the friction coefficient values in acceptance analysis studies.
This work intends to shed some light on the uncertainties associated with the modelling of friction damped freight vehicle models by investigating the impact of different friction models and the selection of their respective parameters on the dynamic behaviour of a friction damped locomotive.The selection of a friction model and its required parameters is proposed based on realistic computational simulations.In addition, the impact of the friction coefficients on the acceptance quantities as described in the standard EN14363 [1] is analysed and recommendations provided for their more robust modelling.The present work not only covers some topics discussed in the master thesis of the first author [30] and a previously published article [27] but also provides novel insights into the friction modelling aspects and their implication on the vehicle dynamics, including validation.

Contact models
When contact between two surfaces is detected contact forces develop.The contact force components are decomposed into normal and friction forces, each requiring specific models, which are applied in contact points in each one of the contacting surfaces.

Normal contact force models
Normal contact force models evaluate the normal force developed between two contacting surfaces, often calculated as a function of the indentation and indentation speed.The modified Kelvin-Voigt model is employed in this work because it is a simple model that includes the dissipative effect of the normal contact.It was developed to represent the nonlinearity of the force characteristic of the Hertz model while preserving the viscous-elastic effect of the original Kelvin-Voigt model [24].Its numerical implementation includes a smoothed transition from the loading phase to the unloading phase of the contact force, as depicted in Figure 1, to deal with the normal force discontinuity caused by the change in the penetration velocity.The modified Kelvin-Voigt contact model is expressed as: where K and n are, respectively, the penalty constant and the nonlinear power exponent associated with the material and geometric properties of the contacting bodies, δ is the indentation, δ is the penetration velocity, e is the restitution coefficient, υ 0 is a tolerance velocity that defines the transition between the restitution phase ( δ < 0) and compression phase of the contact ( δ > υ 0 ), and υ n is the vector of the relative velocity of the bodies, taken as perpendicular to the surfaces of the contacting bodies surfaces in the point of contact.
It must be noted that in the Kelvin-Voigt normal contact model the Hertzian elastic contact conditions are fulfilled.Alternatively, there are other normal contact force models that can be used but are not discussed here [24,[31][32][33].

Tangential contact force models
Tangential contact force models replicate the friction forces of two contacting bodies sliding relative to each other.These are generally classified into static and dynamic models, depending on the details of the phenomena considered.The static tangential contact models describe friction as a function of the normal force and tangential velocity, while the dynamic friction models additionally consider the local deformation at contact dynamics and, consequently, the time lag between the kinematic conditions and the force development.This work discusses, alternatively, two static models, that are often available in commercial software, and one dynamic model that provides a more detailed description of the friction phenomena but that is more complex to implement, and has a greater number of parameters to select.Thus, a broad range of friction contact models available for friction damped freight vehicle suspensions modelling is discussed here.

Amontons-Coulomb friction model with Threlfall Smoothing
The Amontons-Coulomb friction model is the a static friction model that assumes that the friction force f t , which opposes the relative movement between two contacting surfaces, is related with the normal contact force f n through the friction coefficient μ k [34].The model is defined by: where: in which f C is the Coulomb friction force, f n is the normal contact force, f E is the resultant of the external forces on the bodies in the tangential direction and υ t is the relative velocity of the bodies along the tangent plane to the contacting bodies surfaces in the pairs of contact.This simple friction model has poor computational features because of the friction force discontinuity in the vicinity of null sliding υ t = 0, thus, being unsuitable to represent the slide-stiction transition.Threlfall [35] proposes a solution to overcome this problem by smoothing the transition between positive and negative sliding velocities as: in which υ 0 is a tolerance velocity that defines a region of smooth transition between positive and negative sliding velocities, guaranteeing the continuity of the friction force model close to null velocities.The Amontons-Coulomb model with Threlfall smoothing, represented graphically in Figure 2(a), presents a good solution from the numerical point of view but fails to describe the stiction-sliding physics.

Bengisu-Akay friction model
Bengisu and Akay [36] formulate a static friction model that includes a representation of the stiction behaviour of friction, or Stribeck effect.In this work, a modified version of the Bengisu and Akay model, according to Marques et al. [25], is used and represented graphically in Figure 2(b), being defined as: where f S is the static friction force, due to the Stribeck effect, defined as: while f C is the Coulomb friction force expressed by Equation ( 3), μ k is the dynamic friction coefficient, μ s is the static friction coefficient, υ 0 is the Stribeck velocity and ξ is a positive parameter related to the slope of the sliding state associated with the Stribeck effect.Note that the Bengisu-Akay friction model is not a dynamic friction model because the elastic deformation of the contact surfaces in the tangential direction is not represented.However, the transition from stiction to sliding is represented approximately.

Gonthier friction model
The Gonthier model [37] is a dynamic friction model that considers friction as a result of the interaction of surface bristles that represent a mechanical structure or the contacting surfaces.When a tangential force is applied, the bristles bend elastically in the sticking regime.When the applied force is high enough the bristles break away and the contacting bodies slip.The force due to the deformation of the bristles is represented by: where σ 0 is the bristle stiffness coefficient, σ 1 is the damping coefficient and z is a state variable that represents the bristle deflection.The transition between the stick and slip phases is smoothed by the parameter s, defined by: where υ t is the tangential relative velocity of the contacting surfaces and υ 0 is the Stribeck velocity.The deformation rate is governed by: where f C is the Coulomb friction force vector, defined as: where dir(υ t, υ ε ) returns the unit vector with velocity direction, smoothing the vector oscillations for velocities under a certain limit, υ ε = 0.01υ 0 , and is expressed by: The effect of the dwell time dependence is modelled by an additional state variable, defined as: where τ dw is the dwell-time dynamics time constant and τ br = σ 1 /σ 0 is the bristle dynamics time-constant.The maximum friction force is computed according to: Table 1.Technical characteristics of the locomotive FGC254 (adapted from [30]).where f S is the static friction force.The total friction force is given by: where σ 2 is a viscous damping coefficient.
As for any other dynamic friction model, the Gonthier friction model requires the selection of several parameters which are σ 0 , σ 1 , σ 2 , τ dw , besides μ k , μ s and υ 0 , all explained when first appeared in this work.It must be noted that most of these parameters are difficult to determine and are highly dependent on the normal load and contact conditions.For this reason, it is common to use benchmark values in the first stages of using any dynamic friction models [18].The use of dynamic force models requires the solution of 1st order differential equations (ODE) associated with the local state of deformation of the contact surfaces, represented by Equations ( 8) and (12).Therefore, it is necessary to integrate in time as many extra local ODE as the number of contact points considered in the system model.

Vehicle model description and operation conditions
The vehicle considered in this work is a FGC254 freight locomotive, depicted in Figure 3. General technical characteristics of this vehicle are listed in Table 1.The locomotive suspensions are equipped with several friction mechanisms, or with clearance joints.In particular, the horn guides and the centreplate are contacting pairs, better modelled with clearance joints, which play a significant role in the suspension damping and consequently the vehicle dynamic behaviour.
The multibody model of this locomotive is developed in the computer code MUBO-Dyn [38], which allows to represent the wheel-rail contact [9] and include a number of modelling elements to represent the suspension mechanisms [39].The vehicle body, bogie frames, bolsters, wheelsets, and electric motors are assumed as rigid bodies, connected by its respective suspension elements.Therefore, the influence of the power train in the vehicle forward dynamics is disregarded here.Springs and dampers are modelled with linear force elements while the friction damped suspensions are represented by clearance joints  [27]).[18,39].The clearance joints require the selection of normal and tangential contact models and allow the three-dimensional geometry of the friction contact surfaces to be considered.For a detailed description of the locomotive multibody model the reader is referred to [27].The proposed vehicle model has been validated against experimental data obtained in the framework of Project LOCATE [40], following an adapted post-processing procedure defined in the standard from vehicle acceptance and numerical model validation EN 14363 [1].
The simulation scenarios consider the locomotive running in tracks with realistic irregularities.The track geometry used in the following section, detailed in Figure 4, consists of a 250 m radius curve set in between two straight sections with corresponding linear transitions.In that case, the locomotive speed is kept constant, at 70 km/h.In what regards the acceptance procedure, simulations are performed considering the locomotive running in a variety of track layouts, according to the requirements in standard EN14363 [1].The quantities obtained through simulations are sampled at 200 Hz, although the variable time step numerical integrator used by MUBODyn uses a maximum time step of 10 −4 s.

Selection of friction models and parameters
This section overviews the tuning of the most relevant parameters inherent to the three friction models considered in this work and the impact of using the different friction models in the locomotive dynamics.For that purpose, the vehicle behaviour and contact forces developed at the suspensions elements are studied for the vehicle running in a specific track section, as specified in the previous chapter.

Effect of transition/Stribeck velocity
The Stribeck velocity, on the Bengisu-Akay friction model, is the velocity at which the force between two sliding bodies equals the static friction force, as depicted in Figure 2(b).Above this velocity the friction force is equal to the dynamic friction force.The Bengisu-Akay model includes a smooth transition between the static and dynamic friction force.When the sliding velocity surpasses the Stribeck velocity, the force asymptotically decreases towards the dynamic friction force.In the case of the Amonton-Coulomb model with Threlfall smoothing, the transition velocity is used instead that is the velocity at which the dynamic friction force is reached.On the Gonthier model, the Stribeck effect is dependent on both sliding velocity and bristle deformation.Hence it is more difficult to discern the contribution of the Stribeck velocity on the forces developed at the stick-slip transitions.
Figure 5 shows the friction force as a function of the sliding velocity between two contacting bodies when using two values of the Stribeck velocity, υ 0 , for the Bengisu-Akay friction model.For the higher value of the Stribeck velocity, the friction force developed, f 2 , is below the expected dynamic friction force, f 1 .Therefore, the Stribeck velocity must be lower than the absolute value of sliding velocity between the contacting bodies.However, if it is too small it might lead tounnecessary higher computational effort resulting from the need for the time integration used for multibody dynamics to decrease the time step, in order to adjust to the fast varying conditions experienced by the system state variables.Considering that the behaviour of mechanical systems under realistic operation conditions do not have a constant sliding velocity, the median of the sliding velocity between the bodies is proposed here as an indicator to define the maximum value for the Stribeck velocity, υ 0 .
Figure 6(a) shows the sliding velocity at a contact point in the centreplate base of the freight locomotive when using two values of Stribeck velocity, 10 −2 m/s and 10 −3 m/s, when travelling in a zone of the track that includes a straight section followed by a transition and a curve.The sliding velocity is similar for the two Stribeck velocities.The dashed lines that represent the median of the sliding velocities at each zone of the track, are also similar in the two cases.The median of the sliding velocities is below 10 −2 m/s for all track sections but above 10 −3 m/s.The median values suggests that a Stribeck velocity of 10 −3 m/s is suitable to model the friction forces developed.In fact, Figure 6(b) shows that the friction moment developed at the centreplate base differs significantly between using 10 −2 m/s or 10 −3 m/s but is relatively constant for values below 10 −3 m/s.Therefore, the use of the  median of the sliding velocity at the friction surfaces as a maximum threshold value for the selection of the Stribeck velocity proves suitable.Finally, for the Amonton-Coulomb with Threlfall smoothing and the Gonthier friction models the same approach is used being the same value of the transition/Stribeck velocity, 10 −3 m/s, obtained.The respective plots with the various values of the transition velocity are omitted for the sake of brevity.

Effect of Dwell-time constant on Gonthier friction model
The Gonthier dynamic friction model provides a more detailed representation of the stickslip transition, when compared to the static models, by considering the friction force at break-out as a function of the tangential velocity and the local deflection of bristles.In fact, this model considers the local dynamics on the contact pair and represents the time lag between the relative displacement at the contacting surfaces and the development of the friction forces.However, it requires the selection of five parameters, besides the static and dynamic friction coefficients, that are essential for the accuracy of the model.Gonthier [37] and Marques [36] identify the dwell-time constant as an important parameter for the maximum friction force developed during the stick-slip transition when using the Gonthier friction model.To understand the impact of the dwell-time constant on the maximum force developed at the friction surfaces in the locomotive, the friction coefficient is estimated by dividing the tangential contact force by the normal contact force at the contact points.
Figure 7 shows the friction coefficients estimated as a function of the track length for two contact points in the centreplate, one in the centreplate base and one in the centreplate wall.The track length domains are selected to capture the most significant phenomena and enhance the discussion on the effect of the dwell-time constant.The selection of a reference value of τ dw = 2s, based in [25], does not allow to capture the static friction force at the friction surfaces in the centreplate base or walls, as the estimated friction coefficient, shown in Figure 7, does not exceed the dynamic friction coefficient, μ k .Marques [36] shows that the decrease of the dwell-time constant allows to increase the maximum friction force developed at break-out.Therefore, the dwell-time constant is incrementally reduced to reach the static friction coefficient at break-out.The reduction of the dwell-time constant to 0.0002 s, as shown in Figure 7(a), allows to capture the Stribeck effect at the centreplate wall as the estimated friction coefficient reaches the static friction coefficient, μ s .However, for the contact at the centreplate base, the dwell-time constant is incrementally reduced until 0.0002 s without allowing to capture the Stribeck effect for any of the experimented values, as shows in Figure 7(b).Values below 0.0002 s are not used since it is the lowest acceptable value to avoid exceeding the programme numerical precision.The other parameters of the Gonthier friction model were also varied to understand if they may increase, or not, the maximum friction force at break-out.However, the test for these quantities proved inconclusive.In addition, the variation of the other parameters showed a small impact on the friction forces developed at the friction suspensions.Therefore, the values are selected according to Marques [25].The results obtained in this section suggest that, even though the Gonthier model allows a detailed representation of the behaviour at stick-slip transition, it is unable to capture the Stribeck effect for the type of contact presented in the centreplate base for the parameter values experimented.Nevertheless, the impact of the Stribeck effect on the vehicle dynamics is proven negligible as discussed in the following section.

Comparison of friction models
The present section studies the effect of using the different friction models in the friction forces developed at the centreplate and in the lateral acceleration measured at the vehicle body.The analysis provided in sections 4.1 and 4.2, for the selection of the parameters inherent to each of the friction models, leads to the selected parameters described in Table 2.
Figure 8 shows the friction moment at the centreplate base and the lateral acceleration measured at the vehicle body when using different friction models for all the friction surfaces that compose the friction damped suspensions on the locomotive.The results show a strong agreement between the three models.This suggests that, if friction parameters are carefully selected any of the three models can be used.It is also worth to note that despite the limitations described in the previous sections, the Gonthier dynamic friction model does not imply any significant differences in the friction forces or accelerations obtained.Therefore, the selection of the friction model can be set on the basis of availability or of easiness of implementation and the computational efficiency.Finally, the simulations using the models and parameters, presented in Table 2, show that the two static models require similar computation times while the Gonthier model requires half of the computation time for the static models.This shows an impact of the smoother variation of the friction forces on the stability of the time integration algorithm, with variable time stepping, used for the multibody dynamic solution of railway vehicles.

Effect of friction coefficient on vehicle acceptance
This section studies the impact of the friction-coefficient of the friction surfaces on the quantities for the acceptance of the running behaviour of railway vehicles, according to the standard EN14363 [1].The friction coefficients are often difficult to quantify, varying during operation and over time, due to wear, temperature, humidity, among others.Therefore, different friction coefficient values are tested here to understand their impact in the vehicle acceptance parameters.
The following two sections focus on the acceptance quantities for straight tracks and curves with constant radius and cant.These quantities are obtained through a statistical post-processing and filtering of measurements from different track sections with the vehicle running at different operation conditions.The last subsection focusses on the entry and exiting of transitions curves.In curve transitions, the statistical processing is not applicable, according to the EN14363 standard [1], and the maximum quantities are obtained for each   Q m,qst i = 1,2 above the leading/trailing wheelset of the leading bogie j = 1,2 above the leading/trailing bogie k = 1,2,3 leading/middle/trailing wheelset of the leading bogie m = 1,2,3 outer wheel of the leading/middle/trailing wheelset transition section alone.The acceptance quantities considered in this work are described in Table 3 and include the normal and simplified measuring methods.The normal method allows to evaluate safety and track loading through the measurement of the wheel-rail contact forces while the simplified method only allows to evaluate the safety through the bogie and vehicle body accelerations.
In what follows, the results obtained correspond to simulations performed employing the Bengisu-Akay friction model in all suspension elements.This choice is set as the Bengisu-Akay model accounts for the Stribeck effect, allowing the representation of the transition between sliding and stiction, and is an easier model to implement and tune when compared to the Gonthier friction model.The friction coefficient is the same for all the friction surfaces that compose the locomotive.The simulations are performed and the dynamic response obtained with a sampling rate of 200 Hz.The track irregularities are set according to the requirements described in the standard [1].The number of track sections and the length requirements set by the standard are equally respected.

Track zone 1 -tangent tracks and very large radius curves
Track zone 1 considers the vehicle running under a cant deficiency below 40 mm at speeds between 100 and 110 km/h.This occurs in tangent tracks and very large radius curves.The impact of different friction coefficients in the safety and track loading acceptance quantities at track zone 1 are shown in Figure 9, which presents the values normalised by the limit values for each quantity.The quantities are all below the acceptance limits.However, the decrease of the friction coefficient leads to a small increase on all the lateral quantities, including the accelerations and wheel-rail contact forces.

Track zone 4 -very small radius curves
Track zone 4 considers the vehicle running under high cant deficiencies in tracks with curve radius within 250 and 400 m, considered very small radius curves.The detailed operation conditions can be found in the standard [1].Figure 10 shows the impact of the friction coefficients on the safety and track loading acceptance quantities at track zone 4. Using the simplified method, the lateral accelerations at the vehicle body exceed the acceptance limit values regardless of the friction coefficient.However, on the two lowest friction coefficients, all quantities in the normal method remain below the limits.This suggests that the limits for the simplified method are more conservative than for the ones on the normal method.In addition, the friction coefficients do not play a significant role in the acceptance results,  except in the sum of the lateral wheel-rail contact forces where the increase of the friction coefficient to 0.6 exceeds the acceptance limit.

Curve transitions
Curve transitions require a different data processing of the quantities when compared to the track zones with constant radius.In the case of transitions, the quantities obtained for each of the transitions in the track and the maximum values are evaluated considering a portion of the preceding and following tangent and curved tracks.The acceptance quantities at transitions focus only on safety related quantities [1].
Figure 11 shows the acceptance quantities for the entry and exit transition sections of a 250 m radius curve and 105 mm cant, with the vehicle travelling at 85 km/h.Both entry and exiting transitions are 50 m long.The 20 m before and after the transitions are also considered as specified on standard EN14363 [1].The results obtained at the entry transition, depicted in Figure 11(a), show a significant impact of the friction coefficient in the sum of the lateral wheel-rail contact forces and Y/Q at the leading wheelset.In contrast, the impact of the friction coefficient on the simplified quantities is much less important.The results presented in Figure 11(b) permit to observe that the vehicle behaviour is less affected by the friction coefficients at the exit of transition curves.In this case, the results permit to conclude that the limits of the simplified method are more conservative than those of the normal method.

Conclusions
This work studies how the modelling decisions of friction damped suspensions affects the dynamic behaviour of a friction damped locomotive.A detailed multibody model of the locomotive is considered, in which clearance joints are employed to represent the friction damped suspensions.Three different friction models are tested and the most relevant parameters inherent to each of the models are discussed.Two static friction models and one dynamic friction model are considered.Finally, the effect of the friction coefficient in the acceptance quantities according to standard EN14363 is assessed, including the analysis of the vehicle accelerations and the wheel-rail contact forces.
The analysis of the results obtained stresses the importance on adequately tuning the sliding-stiction transition, represented by the transition/Stribeck velocity for all the different friction models.The selection of this parameter significantly affects the friction forces developed on the friction damped suspensions and, consequently, the dynamic behaviour of the vehicle.The Gonthier dynamic friction model requires the selection of a larger amount of modelling parameters when compared with the two static models considered.Therefore, it is more challenging to adequality tune its parameters.In particular, the dwelltime parameter poses a significant challenge to the Stribeck effect, requiring a careful selection.Nonetheless, the adequate tuning of the different parameters for each of the models leads to the development of similar friction forces and dynamic behaviour of the vehicle regardless of the friction model applied.This complements the prior work of the authors, by showing that the results for using the different models converge, if the parameters are carefully selected.Therefore, in the context of railway vehicle dynamic analyses, the selection of a friction model for friction damped suspensions can be considered on a basis of computational efficiency and easiness of implementation or use.
This work also covers a series of investigative numerical dynamic analysis on the impact of the friction coefficient in the acceptance quantities for vehicle running behaviour.The results demonstrate that the friction coefficient plays a significant role in the acceptance quantities, particularly in the lateral wheel-rail contact forces in curve transitions and small radius curves.In contrast, the quantities measured in tangent track are less sensitive to the friction coefficients.The results suggest that the acceptance procedure for friction damped vehicles, when based on numerical analysis, should include different friction coefficients to surpass the difficulty of quantifying the friction coefficients and to cover a wider range of operation conditions.Finally, the comparison of the normal and simplified measuring methods, as presented in EN14363 standard, suggests that the simplified method is more conservative.However, the correlation between the simplified and normal methods should be further investigated to understand if the use of the different methods can result in different validation judgements.

Figure 1 .
Figure 1.Normal force as a function of penetration depth and velocity according to the modified Kelvin-Voigt model [27].

Figure 4 .
Figure 4. Schematic representation of the track layout used in the simulations [27].

Figure 5 .
Figure 5. Schematic representation of the friction force developed for the Bengisu-Akay friction model for two alternative Stribeck velocities.

Figure 6 .
Figure 6.Impact of Stribeck velocity when using the Bengisu-Akay model: (a) sliding velocity at the surfaces of the centreplate; and (b) friction moment developed at the centreplate.

Figure 7 .
Figure 7. Effect of dwell-time constant in the estimated friction coefficient, f t /f n , for the contact at (a) centreplate wall and (b) centreplate base, for the locomotive travelling in a track with irregularities.

Figure 8 .
Figure 8.Effect of friction models on (a) the friction moment developed at the centreplate base and (b) lateral acceleration at the vehicle body above leading bogie frame, for the locomotive travelling in a track with irregularities.

Figure 9 .
Figure 9.Effect of varying friction coefficients of all friction surfaces on the acceptance quantities for track zone 1 -straight tracks and very small radius curves.

Figure 10 .
Figure 10.Effect of varying friction coefficients of all friction surfaces on the acceptance quantities for track zone 4 -very small radius curves.

Figure 11 .
Figure 11.Effect of varying friction coefficients of all friction surfaces on the acceptance quantities for (a) entry and (b) exit transition curve for a very small radius curve.

Table 2 .
Parameters for the tangential contact models.