Vehicle System Dynamics: International Journal of Vehicle Mechanics ...

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Modelling of suspension components in a rail vehicle dynamics context a

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Stefano Bruni , Jordi Vinolas , Mats Berg , Oldrich Polach & Sebastian Stichel

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Dipartimento di Meccanica , Politecnico di Milano , Via La Masa 1, Milano, 20156, Italy b

Applied Mechanics , CEIT and Tecnun (University of Navarra) , P. Manuel Lardizábal, 15, San Sebastian, 20.018, Spain c

Department of Aeronautical and Vehicle Engineering , KTH (Royal Institute of Technology) , Teknikringen 8, Aeronautical and Vehicle Engineering/Rail Vehicles, Stockholm, SE-1004, Sweden d

Bombardier Transportation , Zürcherstrasse 39, Winterthur, 8401, Switzerland Published online: 24 Jun 2011.

To cite this article: Stefano Bruni , Jordi Vinolas , Mats Berg , Oldrich Polach & Sebastian Stichel (2011) Modelling of suspension components in a rail vehicle dynamics context, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 49:7, 1021-1072, DOI: 10.1080/00423114.2011.586430 To link to this article: http://dx.doi.org/10.1080/00423114.2011.586430

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Vehicle System Dynamics Vol. 49, No. 7, July 2011, 1021–1072

Modelling of suspension components in a rail vehicle dynamics context Downloaded by [Monash University Library] at 01:15 04 December 2014

Stefano Brunia *, Jordi Vinolasb , Mats Bergc , Oldrich Polachd and Sebastian Stichelc di Meccanica, Politecnico di Milano, Via La Masa 1, Milano 20156, Italy; b Applied Mechanics, CEIT and Tecnun (University of Navarra), P. Manuel Lardizábal, 15, San Sebastian 20.018, Spain; c Department of Aeronautical and Vehicle Engineering, KTH (Royal Institute of Technology), Teknikringen 8, Aeronautical and Vehicle Engineering/Rail Vehicles, Stockholm SE-1004, Sweden; d Bombardier Transportation, Zürcherstrasse 39, Winterthur 8401, Switzerland a Dipartimento

(Received 23 February 2011; final version received 27 April 2011 ) Suspension components play key roles in the running behaviour of rail vehicles, and therefore, mathematical models of suspension components are essential ingredients of railway vehicle multi-body models. The aims of this paper are to review existing models for railway vehicle suspension components and their use for railway vehicle dynamics multi-body simulations, to describe how model parameters can be defined and to discuss the required level of detail of component models in view of the accuracy expected from the overall simulation model. This paper also addresses track models in use for railway vehicle dynamics simulations, recognising their relevance as an indispensable component of the system simulation model. Finally, this paper reviews methods presently in use for the checking and validation of the simulation model. Keywords: suspension components; rail vehicle modelling; model validation; vehicle acceptance; running safety; ride quality

1.

Introduction

For modern railway vehicles, multi-body vehicle dynamics simulation has become a major design instrument, allowing the assessment and optimisation of vehicle performance from the early stage of the design process, before a prototype is built. Typical applications of multi-body simulation include the verification of running stability and safety, the evaluation of ride quality, the analysis of wear and other damage effects at the wheel–rail interface, the verification of dynamic gauging and the numerical estimation of design loads to be used for durability analysis [1]. Furthermore, multi-body simulation is increasingly being used to complement and partially replace physical testing to demonstrate running safety and acceptability in view of the admission of a new or modified rail vehicle into service [2], a topic sometimes referred to as ‘virtual homologation’ [3]. *Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online © 2011 Taylor & Francis DOI: 10.1080/00423114.2011.586430 http://www.informaworld.com


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Quite obviously, the above-mentioned applications call for a high degree of reliability and trustworthiness of the models used, since any significant deviation of the models from the real vehicle actual behaviour would imply the need for troubleshooting at a late stage of the design process or even during the acceptance process, especially considering that some of the issues addressed are safety critical. Therefore, one important challenge regarding modelling and simulation of railway vehicle dynamics is to define accurate and reliable procedures for building multi-body vehicle models. A second, strictly related problem is concerned with the validation of multi-body models, considering that, in a ‘virtual homologation’ perspective, validation shall not rely on comparison with line tests performed on the same vehicle but could instead make use of results from line tests performed on similar vehicles and/or of results of tests performed on single suspension components. The accuracy of rail vehicle multi-body models is mainly affected by the model of wheel– rail contact and by the models of vehicle suspension components. As far as the model of wheel–rail contact is concerned, a very large research effort has been spent in the past to define modelling approaches ensuring a satisfactory level of detail and, at the same time, requiring an affordable computational effort, and recent surveys are available on this topic, such as the one done by Chollet and Piotrowski [4], describing the wheel–rail contact models most widely used in the state of the art of rail vehicle dynamics simulation. On the other hand, rail vehicle suspension component models have also been the subject of extensive research work, especially in recent years, but few updated reviews of the work done are available, with the two most noteworthy being the survey paper written in 1995 by Eickhoff, Evans and Minnis [5] and the chapter on simulation in the Handbook of Railway Vehicle Dynamics [6]. The aims of this paper, which is partly based on the work being undertaken in WP5 of the European research project DynoTrain, are to provide a comprehensive and updated description of railway vehicle suspension component models in use for railway vehicle dynamics multibody simulations, to describe how model parameter data can be defined and to discuss the required level of detail of component models in view of the accuracy required for the overall model of the complete vehicle or train, also considering the specific scope of the multi-body simulation being performed. Mathematical models of railway vehicle suspension components may range from relatively simple linear ones for coil springs to more sophisticated load-sensitive models of friction elements and nonlinear and frequency-dependent models of rubber springs and bushes and may also include multi-physics models for example, in the case of air springs and active suspension components. Even in the case of apparently ‘simple’ suspension components, there are issues that need to be carefully considered in view of the accuracy of the overall vehicle model, such as secondary effects related to gravitational loads, uncertainty in the parameters, geometrical imperfections and deviations from the nominal behaviour. This paper also addresses track models in use for railway vehicle dynamics simulations: although it is not intended here to cover track dynamics and train–track interaction models which are specific to the study of track vibration and track damage problems (such as those surveyed in [7]), it is recognised that the modelling of track dynamics bears a relevant influence on the simulation of vehicle dynamics, deserving a presentation of viable approaches and a discussion of their respective advantages and disadvantages. Another notable issue which will be discussed in this paper is the effect of model uncertainties on simulation results and how the complete vehicle model shall be checked and validated before it is used to simulate vehicle dynamics: this subject is particularly relevant to the issue of ‘virtual homologation’, a task often requiring that model accuracy is assessed based on a limited amount of line test measurements, possibly referring to a slightly different vehicle or to different service scenarios than the ones being considered for acceptance.

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Before concluding this paper, a critical review of the challenges and future trends in the modelling of railway vehicle suspension components is also provided.


2.

Rail vehicle models and suspension component models

As indicated above, the physical/practical rail vehicle appearance needs to be represented in a mathematical/theoretical model, imitating the mechanical behaviour of the vehicle running along the track. The level of detail in the vehicle modelling and the accuracy of the input data depend on the purpose of the analysis at hand and how accurate the output quantities need to be. Based on the review [8], Figure 1 presents the main types of analyses and associated calculation methods. From a safety perspective, the key issues in the figure are derailment, track shift forces and running stability. These issues are emphasised in vehicle acceptance standards such as [9,10]. In Figure 1, it is also obvious that the calculation methods required are dominated by dynamics simulation in the time domain. A recent review of multi-body simulation in railway vehicle design is given in [11], including the software packages available. In the dynamics simulation, the choice of vehicle model is also dependent on the frequency range of interest; the higher the frequency, the more detailed the model. Traditionally, the main range of interest is 0–20 Hz. This is certainly sufficient for issues such as car body sway, gauging, wear and usually also for ride comfort. For wheelset accelerations, wheel–rail forces, and wheel and rail fatigue issues, this is less obvious. For the dynamics simulations, the vehicle–track system model is essentially represented by a number of first-order differential equations including nonlinearities, mainly originating from the wheel–rail interfaces and many of the suspension components. The unknowns in these equations are mainly translations and rotations, and the corresponding velocities, of the

Figure 1.

Common analysis types and calculation methods applied in rail vehicle engineering. After Polach [8].


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car body, bogie frames and wheelsets of the vehicle. These bodies essentially also host the inertial properties of the vehicle, including masses of suspension components and equipment ‘lumped’ to these bodies. On the other hand, relative motions between vehicle bodies are mainly made possible through the flexibility of the suspension components. As pointed out in the introduction, the modelling of the suspension components is decisive for the accuracy of the vehicle model and the simulation results. This is due to the fact that the suspension components play key roles for the vehicle dynamic behaviour, mainly in terms of vibration isolation and limiting wheel–rail forces. Suspension spring components also carry static forces due to gravitational effects and quasi-static ones in vehicle curving where centrifugal effects are also present. Some suspension components are also engaged in vehicle traction and braking. The most common suspension components are coil springs, friction-based components such as leaf springs, rubber springs, air springs and hydraulic dampers, but components such as traction links, anti-roll bars and bump stops should also be mentioned in this context. All these components are passive components meaning that for given component properties and possible pre-load, the forces (and moments) that they give at their interfaces to the vehicle bodies in question only depend on the displacement and velocities at these interfaces. This stands in contrast to active suspension components, including semi-active ones, for which other factors also affect the forces. Modelling of active suspensions is also briefly described in this review paper.

3.

Models and data of suspension components

3.1. Principles of component modelling The art of suspension component modelling involves several steps and requires engineering experience and judgement to be successful. Too simple suspension models and/or inaccurate model parameter data may produce misleading results from vehicle–track simulations in terms of wheel–rail forces, ride stability, ride comfort, etc. The mechanical behaviour of suspension components originates from the fields of solid mechanics, fluid mechanics and tribology. The solid materials dealt with are usually steel or rubber, whereas the fluids are almost exclusively oil and pressurised air. The tribology here is often associated with steel-to-steel or rubber-to-steel sliding motion involving friction and wear issues. In many cases, the suspension components consist of combinations of these parts, for instance, steel and rubber forming a layered spring, air and rubber for an air spring, and oil and rubber bushings for a hydraulic damper. A suspension component may be modelled in detail by describing its geometry and the material and/or fluid properties as well as possible pre-load and friction interfaces. However, in many cases, such models would be too complicated and time consuming to be used in the vehicle–track simulations. Instead, spring elements, dashpots and friction elements are used and combined in representing suspension components in such simulations. The spring elements and dashpots often also have linear characteristics. In this way, the key properties of suspension components are often stiffness, viscous damping, pre-load and friction break-out force. Still, the more advanced suspension models might be useful in pre-processors, defining stiffnesses, etc. of the simpler models from the physical appearance of the suspension component in question. In some cases, the suspension model parameters might also be obtained from measurements on the component in question or at least on a similar component. In this process, it is often convenient to use harmonic displacement excitations and evaluate the corresponding force



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responses. Generally, the response will depend on the excitation frequency, the displacement amplitude and a possible pre-load. Non-elliptic force–displacement graphs imply that the component has a nonlinear behaviour. The component stiffness, often defined as the force amplitude divided by the displacement amplitude, might be illustrated as a function of excitation frequency, displacement amplitude and/or pre-load. The component damping is sometimes expressed by the force–displacement phase angle, but for nonlinear component behaviour, this does not fully represent the hysteresis. The viscous damping of hydraulic dampers may be seen as the impedance, that is, the force amplitude divided by the velocity amplitude. In Europe, rail vehicle suspension measurements should be carried out according to EN standards. In, for instance, spring standards, ‘stiffness’ is often split into a static one and a dynamic one. Then, great care has to be taken regarding the definition of ‘dynamic stiffness’. As an example, a single (vertical) dynamic stiffness value is not sufficient in modelling most air spring systems (Section 3.5). For air springs and most coil springs, significant horizontal effects take place, meaning that one-dimensional vertical models are not sufficient. Thus, horizontal forces and bending moments are introduced at the interfaces with the bodies connected, and the destabilising effect of the compressive pre-load needs to be considered. For some metal-to-metal interfaces, two-dimensional friction sliding is possible. In UIC link suspensions, found in some freight wagons, complicated rolling–sliding phenomena also need to be represented in the modelling. Spring–dashpot–friction suspension models sometimes resemble the component physical appearance, but they can also be a pure mathematical representation where such a direct coupling cannot be identified. For instance, a rubber model may include several parallel sets of dashpots in series with springs to represent the rubber dynamics over a wide frequency range. Often, the suspension models include internal degrees of freedom (DoF) (or first-order differential equations), either to imitate, say, the emergency spring of an air spring system or to allow for models such as the rubber one just mentioned. In what follows, the modelling of different suspension components is reviewed. The review starts with metal (steel) components such as coil springs and friction-based components such as leaf springs. Then, rubber springs and bushes are covered, including internal friction-like effects, followed by air springs. Thereafter, the fluid of air is replaced by oil to review models of hydraulic dampers. Briefly, also modelling of semi-active and active suspension components is surveyed. Last but not least, a number of components, strictly not called suspension components, are reviewed. Examples of such components are traction links, anti-roll bars and bump stops.

3.2. Coil springs Modelling coil springs is more difficult than initially expected depending on the deformations that they are subjected to. In many cases, one has to take into account more than just the axial stiffness, considering the coupling between different directions and the reaction moments due to the non-axial displacement of the vertical force. Additionally, the compressive pre-load provokes destabilising effects in the transverse plane, and consequently, some cases require a particular analysis of buckling limit (critical axial load or critical axial deflection) which depends on how the coil spring is able to deform and how its ends are fixed. The simplest way to model a coil spring is by a single, linear stiffness (which corresponds to the axial direction). Three-dimensional models include three perpendicular springs. Nonlinearities associated with bump stops or vertical nonlinearity, which commonly appears in the primary suspension due to inner coil springs with vertical clearance, have to be included. However, these simple models frequently need an upgrade that includes transverse effects to


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Figure 2. Analytical (Lee–Thompson) and numerical (Abaqus) solutions for the axial dynamic stiffness of a typical rail vehicle coil spring.

represent adequately real coil spring suspensions. Transverse and bending stiffnesses play an important role (e.g. coil springs located in the secondary suspension of a railway vehicle) as transverse shear deformation may compromise vehicle dynamics. The contribution of coil springs to damping can be neglected as load/unloading curves coincide showing nearly no energy dissipation/hysteresis. Additionally, the dynamic behaviour of coil springs used in railway applications has nearly no dependency on frequency in the range of 0–20 Hz. For higher frequencies, some analytical formulae might be used to find the dynamic response of the spring [12,13], although finite element models can also be used. Figure 2 compares analytical and numerical solutions for a typical rail vehicle spring and shows that for frequencies lower than 20 Hz, the coil spring can be modelled using its static properties. 3.2.1.

Stiffness calculations

The European standard EN 13906-1:2008 [14] provides different methods for the calculation and design of cylindrical helical springs made from round wire and bar. It includes transverse, buckling and impact loading, stress correction factors over the cross section of the wire, material property values for the calculation of the spring and a review of the formulae needed in the design process. Axial stiffness, spring rate R, is estimated using a simple expression that depends on geometric data of the coil spring. An equation is also proposed for the transverse spring rate, RQ , although the standard notes that the ‘transverse spring rate is only constant for short transverse spring deflections, sQ , for a given length L under compression’. Based on Timoshenko’s work [15–19], the concept for the Haringx model [16] is the division of the spring into small elements consisting of ordinary, linear springs; it assumes a small helix angle. By integration, the relationship between axial and shear stiffnesses is obtained. For the coil spring shown in Figure 3 to be in equilibrium when compressed between two non-parallel plates, the equations for axial and lateral reactions at the seats are as follows: F=

Gd 4 (L0 − L) 8nD3

(1)


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F M1 FQ

L

1


N Q

FQ M2

2

F

Figure 3.

Spring subjected to axial, shear forces and moments at both ends.

F(ψ1 + ψ2 ) (4G/E + 2)(L0 − L)/(qD2 tan (qL/2)) − 2 EGqLd 4 FQ 4G L0 − L M1 = ψ1 + 1− +2 16(2G + E)(nD)tan(qL/2) F E qD2 sin qL FQ =

(2)

(3)

M2 = M1 + FQ L, where D is the mean coil diameter; n is the number of active coils; d is the nominal diameter of the wire; E is the Young modulus; G is the shear modulus; L0 is the unloaded spring length; ψ1 and ψ2 are the angles of the ends of the spring or seat angles when the spring is seated and q is the so-called buckling factor. Buckling refers to the loss of stability of a component and is usually independent of material strength. In this case, it can be calculated as 2 2 1 2G L 2G E L 0 0 q=

. + −1 + −1 D E G L E L

(4)

Krettek and Sobczak [17] worked on the same problem. They estimated different correction factors following extensive tests which relate the axial and shear deformations for a given loading factor, as they found differences between experiments and previously proposed formulae. These correction factors, ai , depend on the relative axial deformation of the spring and are foreseen for slenderness ratios between 1.5 and 3: • a1 = 1.9619ξ + 0.6740 Correction factor for lateral stiffness • a2 = 1.3822ξ + 0.6513 Correction factor for mixed term (lateral/bending coupling) • a3 = 0.8945ξ + 0.6690 Correction factor for bending stiffness ξ=

s L0 − L = . L0 L0

(5)

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They also provided the following formulae for lateral and bending stiffness (Ky∗ and Kψ∗ ):

F = a1 Ky = a1 · 2/c(1 + F/RQ ) · tan(cL/2) − L

Rψ cL F ∗ Kψ = a3 Kψ = a3 . Ky . 1+ − F RQ tan(cL)


Ky∗

lateral stiffness (6a) bending stiffness.

The contribution of bending angle to lateral force and the contribution of lateral displacement to bending moment can be included using a ‘mixed term’, with it being cRψ cL ∗ , (6b) = a2 Kyψ = a2 · Ky . tan Kyψ 2 F where c is a non-dimensional factor which relates the axial load with the bending and shear properties of the spring through the equation:

F F 2 (7) c = 1+ Rψ RQ and Rψ =

Ed 4 L 32nD(ν + 2)

Gd 4 L (ν + 1) 4nD3 Gd 4 L R = RL = 8nD3

RQ =

(8)

Figure 4 compares the lateral stiffness for a particular coil spring using the different approaches. This coil spring has a slenderness ratio of 1.63; in the case of higher values of λ, the results of the Timoshenko formulae would give less accurate results. The influence of the correction factors proposed by Krettek–Sobczak is also appreciated, which justifies why in applications where the transverse stability of the spring is an important operational factor, the calculated values should be verified by practical tests as suggested by EN 13906-1:2008. 3.2.2.

Modelling

There are two main options to model spring components, in general, in a multi-body vehicle model [20]. The first one is the so-called point-to-point force element (PtP), which exerts only an axial force along the line of action. The second one is the compact force element (Cmp), which enables axial and shear forces and reaction moments. In both kinds of elements, linear and nonlinear characteristics can be modelled, if necessary pre-compression can also be accounted for. All the force elements have in common being mass-less. The mass and inertia of the component are very small compared with those of other vehicle bodies. The options are to neglect the masses or to share them among the bodies connected to retain the mass of the actual vehicle. PtP elements act along the connecting line of their coupling markers (Mi , Mj ) with all their outputs (forces/torques) applied in this direction. An example is shown in Figure 5. At t = 0, Mi and Mj are the coupled markers defining the line (Mi − Mj ) in which the forces are acting.



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Figure 4. Lateral stiffness (transverse spring rate) of a coil spring vs. axial load, F (coil spring data L0 = 0.32 m; d = 0.036 m; D = 0.196 m and n = 4.5).

Mi Mi’

Bi

Bi

Mj Bj

Figure 5.

Point-to-point force element diagram.

At t = t1 , Bi moves towards the final position defining the final position of marker Mi , which is Mi ; Bj does not change its position. Consequently, at t1 , a different line of action is defined (Mi − Mj ), that is, the direction of the acting force has changed, as well as its magnitude, because of the new distance between these markers (|Mi − Mj |). Compact spring elements allow the user to take into consideration the three main directions: X, Y and Z. In addition, the stiffness curve of each direction can be different. In this case,

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1 ·(r x F ) 2


-F

F

(r x F )

F

-F

1 ·(r x F ) 2

Figure 6. Compact spring element connecting two bodies. Left, reaction moment only at one end. Right, reaction moment distributed between both ends.

the reaction moments are not neglected; therefore, moments are generated from the offset, as indicated in Figure 6. The reaction moments can be distributed between the connected bodies. This element is recommended when bending and reaction moments are important. Frequently, the coil spring has rubber seats at both ends in order to reduce vibration transmission and improve its seating. Not only does the geometric configuration of the spring and seats affect the force line, but it has also been observed experimentally that the spring seat material has an effect [18].

3.3. Friction-based suspension components In many suspension components in railway vehicles, friction damping is used, especially in freight wagons or deflated air springs. In some components, such as rubber and (inflated) air springs, friction damping is only a side effect. These components are described in other sections of this paper. This section will concentrate on components where damping in the system is generated by two – in many cases metallic – surfaces that slide on each other. One advantage of such friction components is that they are relatively cheap and almost maintenance free. Another major advantage is that the amount of damping in many arrangements is more or less proportional to the axle load. This is an important property, especially in freight wagons where a loaded wagon can have up to five times higher axle load than an empty one. A disadvantage is that the efficiency of friction dampers depends on parameters such as the friction coefficient in the contacting surfaces and the flexibility in the damper arrangement. The friction coefficient can vary quite significantly during operation of a vehicle, which can change the running behaviour of a freight wagon. The uncertainty regarding the parameters also makes it a challenge to simulate the running behaviour of vehicles, which, to a great extent, rely on friction damping [21].


Fx

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Fx

x

x


Figure 7. Force–displacement curve of Coulomb friction model (right) and Coulomb model with spring in series like those shown in Figure 8 (left).

N m

F

F

ks Figure 8.

Friction element with spring in series.

3.3.1. Modelling of friction In most cases, friction in rail vehicle suspensions is modelled as dry Coulomb friction. In this model, the friction force is proportional to the normal load. Another possibility is to model the friction considering two friction coefficients, static and kinetic. The static friction coefficient during sticking is higher than the kinetic one during sliding. The disadvantage of the Coulomb model is that it is non-smooth, multi-valued and non-differentiable, which causes numerical problems in simulations, see the force-displacement curve in Figure 7 (right). Therefore, most authors presenting work on simulation of dry friction dampers apply regularisation to avoid the difficulties mentioned above, e.g. [22,23]. One possibility to avoid the problem of a multivalued function is using a linear spring in series with a friction slider (cf. Figure 8). The resulting force-displacement characteristic is shown in Figure 7 (left). Since most structures containing friction have a finite flexibility, such models could also be regarded as more realistic. Another possibility for regularisation can be found in the model proposed by Bosso et al. [24]. The friction force is calculated according to the following expression: F=

v·X 1 − (v · X/μ · N)2

,

(9)

where F is the friction force and v is the relative velocity. The X parameter represents the angle between the velocity axis and the friction curve around the origin, μ is the kinetic friction coefficient and N is the force normal to the friction surfaces. The force–velocity curve of the element is shown in Figure 9. A disadvantage of this model is the missing stick state. When the velocity is very less, the body interface slips too. For dynamic analysis, the model is a good representation for simulation of friction, but even for very slow force variations, there is always a slip between the bodies, which does not reflect the real behaviour. The impact of the regularisation on simulation results depends on the regularisation function used. Piotrowski [25,26] has developed a non-smooth rheological model that does not resort to the regularisation. Instead, it employs the notion of the differential succession involving a contingent derivative of the non-smooth, multi-valued characteristics of Coulomb friction. In his model, the action of the friction slider in series with spring ks is replaced by the friction force T (cf. Figure 10). In order to derive the differential equation for the friction force, the

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Friction force Continuous function Relative velocity


If-else function

Figure 9.

Continuous friction model.

Figure 10. The slider and the spring replaced by the friction force T [25].

continuity condition for the slider and spring in series is considered. It has the form T˙ + vs = ς˙ − Y˙ , ks

(10)

where ς˙ is the velocity of the base of the element, Y˙ is the velocity of the top and vs is the sliding velocity. Tan and Rogers [27] proposed equivalent viscous damping models to avoid the numerical problems of Coulomb friction. They claim that this substitution works very well for cases where sliding motions predominate. The friction model can be one dimensional or two dimensional. In a two-dimensional model, sliding in two directions is possible. A two-dimensional Coulomb friction model can be found, for example, in [28]. The model is shown in Figure 11. The components of the friction force must satisfy the relation 2 + F 2 ≤ μN. Fμ = Fxμ (11) yμ The sliding velocity which determines the direction of the kinematic force reads |v| = x˙ 2 + y˙ 2 .

(12)

Another two-dimensional model for anisotropic friction was derived in [26]. It is based on the one-dimensional model of Piotrowski described above and is shown in Figure 12. 3.3.2. The influence of dither in friction damping on the running behaviour of freight wagon Piotrowski [26] also described how dither (i.e. high-frequency vibrations superimposed on low-frequency vibrations), generated, for example, in the wheel–rail contact, influences the



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Figure 11. A two-dimensional friction oscillator [28].

Figure 12. Two-dimensional friction model for anisotropic friction proposed by Piotrowski [26].

running behaviour of freight wagons. With the help of experiments, it is shown that a friction damper that is influenced by dither, either perpendicular or co-linear to the sliding velocity of the body, more or less acts like a viscous damper. In a simulation example, it is shown that mid-frequency dither in some operational conditions could remove the instability of a two-axle freight wagon. Therefore, this effect has to be included to achieve relevant simulation results. 3.3.3. Friction wedges in three-piece bogies The three-piece bogie is probably the most common type of freight wagon bogie in the world. Therefore, a large number of authors have developed simulation models of this type of bogie, see e.g. [29–33]. The most critical components to model in a three-piece bogie are the friction wedges (cf. Figure 13). The friction damping can either be independent of the load (b) or be dependent on the load (a). The friction between bolster and wedge is usually modelled in one

Figure 13.

Friction damping in three-piece bogie with load-dependent damping (a) and constant damping (b).


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Figure 14. (a) Suspension of the standard UIC Y25 bogie. 1, bogie frame; 2, inner spring; 3, outer spring; 4, friction piston; 5, sliding surface; 6, Lenoir link; 7, axle box and 8, pivot. (b) A photograph of the suspension [26].

dimension, while the friction between wedge and side frame is modelled in two dimensions, because both vertical sliding and lateral sliding are possible. A typical value of around μ = 0.2 for the friction between bolster and wedge and a value of ca. μ = 0.3 for that between wedge and side frame could be found in, for example, [34,35]. The difference in the friction is due to the inclined surfaces between bolster and wedge usually being greased. These values, however, will probably vary significantly during operation. 3.3.4. Lenoir link in Y25 bogie The Lenoir link in theY25 bogie shown in Figure 14 is another example of a friction component in a freight wagon running gear. Models of the Y25 bogie have been developed for example, by Eickhoff et al. [5], Evans and Rogers [36], Jendel [37], Keudel [38] and Stichel [39]. The Lenoir link transfers a normal force onto the friction surfaces, which is needed to obtain the damping. As shown in Figure 14, the outer coil spring is connected to this link via a spring holder. When the bogie frame moves downwards, the link will pull the spring holder down and via the friction piston apply a normal load to the friction surfaces. A higher load on the wagon will thus result in a larger force in the link. The longitudinal component of the link force is the initial normal force on the friction surface, that is, when the wheelsets are not yawed relative to the bogie. The Lenoir link also affects the stiffness in the primary suspension. Since the link is inclined, there will be a coupling stiffness in the vertical and longitudinal directions. According to Jendel [37], the stiffness matrix is written as

Fx kxx kxz x kz tan2 α kz tan α x = = . (13) kz tan α z z Fz kzx kzz kz Primary damping in the Y25 bogie is provided by two-dimensional friction damping acting in the Y –Z plane. There are friction surfaces on both sides (front/rear) of the axle boxes, as can be seen in Figure 15. The normal force on these surfaces is the longitudinal component of the link force plus the force from the metallic stop between the friction piston and the bogie frame if the 4-mm clearance is overcome. When the wheelset begins to yaw, one of the friction surfaces out from bogie centre, left or right side of the bogie, will lose its contact, and the damping will be zero. This behaviour can be modelled using a nonlinear stiffness property, having the value of the longitudinal component of the pre-loaded link force at zero yaw (x = 0) and a zero force if the contact is lost (x > 0).



Figure 15.

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Friction surfaces in primary suspension of Y25 bogie.

Friction damping will also occur in the lateral bump stops. Damping will be provided in the vertical and longitudinal directions when a 10 mm play has been overcome. The stiffness in the normal direction will be the result of a ‘metallic’ contact though and thus is very high. 3.3.5. Leaf springs Leaf springs and link suspensions – either single or double links – are still the most common suspension components in two-axle freight wagons in Europe. They are also frequently used in four-axle bogie freight wagons. Leaf springs and links provide both stiffness and damping in one component. In these components, the suspension characteristics also adapt to the axle load, that is, they are more or less constant when normalised with the axle box load. A typical leaf spring and double-link arrangement of a two-axle freight wagon is shown in Figure 16. 3.3.6. Typical model of leaf spring and link suspension Leaf springs are used as vertical suspensions. The horizontal flexibility is usually provided by other suspension components, for example, double links (cf. Figure 16). The leaf spring in these cases is regarded as rigid in both the longitudinal and lateral directions. In Figure 17 (left), typical force–displacement curves of a single-stage and a two-stage (progressive) leaf spring for large displacements are shown. For dynamic displacements around a static equilibrium position, leaf springs are characterised by a relatively high stiffness for small displacements and a significantly lower stiffness for larger displacement (Figure 17, right). Leaf springs are described in the ORE reports [40,41].

Figure 16.

Suspension arrangement of two-axle freight wagon.


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Figure 17. Typical force–displacement diagram of leaf spring/link suspension. Example of a curve for large displacements of leaf spring (left). Example of a curve for small displacements around a static equilibrium (right).

Figure 18.

Model for leaf spring or link suspension as used, for example, by KTH [39]. See also Figure 19.

Since link suspensions show very similar characteristics, they are often modelled in a way that is similar to the way leaf springs are modelled, at least for the lateral link behaviour. The initial higher stiffness k1 is caused by friction, for example, the leaves of a leaf spring stick together for small displacements and start to slide on each other for larger displacements. In the same way, the link rolls in the end bearing as long as there is no sliding in the contact area. The lower stiffness k2 is the value for sliding in the leaf spring or the so-called pendulum stiffness of a link. The force Fd determines the amount of damping in the hysteresis. It should be taken into account that the characteristic of leaf springs strongly varies due to run-in, deterioration or lubrication state. A commonly used method to represent the two different stiffness values with the hysteresis is to use a linear spring and a friction element in series, in parallel with another linear spring (cf. Figure 18). The three parameters in the model described above can be derived from measurements of the components. Measurement results and more detailed descriptions of link suspensions can be found in [42–51]. This model, however, is simplified, since the shape of the hysteresis curve is usually rounded (cf. Figure 17). 3.3.7. Model for longitudinal link stiffness For lateral displacements of a double link, all four joints are assumed to start to slide at the same time. Therefore, the model shown in Figure 18 is sufficient. In the longitudinal direction, however, it is more likely that the joints start to slide at different displacements as shown, for example, by Piotrowski [25]. He used a set of four sliders and spring elements with different



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Figure 19. Model for leaf spring or link suspension with several friction elements in parallel as used, for example, by Stiepel and Zeipel [51].

break-out forces in parallel to describe these characteristics. Also, in a model used by Alstom, several elements in parallel are used, as can be seen in Figure 19. 3.3.8. Models using an exponential expression To give a better representation of the rounded shape of the hysteresis curves, Fancher developed a model for truck leaf springs [52,53] using exponential expressions. The equations given below were suggested by Jönsson et al. [48] and are based on Fancher’s model. The total force over the suspension component is separated into piecewise elastic and friction forces. The model is used for both leaf springs and double links: F = F e + F f = F0 + K e · δ + F f

(14)

where F0 is the static pre-load, Ke is the stiffness and δ is the deformation over the coupling. The friction force is given by kfA · (1 − eαA ·(δ1 −δ) ), αA kfB Ff = Ff 2 + · (1 − eαB ·(δ−δ2 ) ), αB

F f = Ff 1 +

δ˙ ≥ 0, δ˙ < 0,

kfA . FfA − Ff1 kfB αB = . FfB − Ff2

αA =

(15a) (15b)

The difference compared to Fancher’s approach is that the force gradient, ∂F/∂δ, is assumed to be constant at every point when the direction of loading is changed: ∂Ff ∂Ff = kfA , = kfB . (16) ∂δ 1 ∂δ 2 The characteristics described above are shown in Figure 20. If the direction of loading is changed at point 1, the response from both models, α1 and β1 , forms a closed loop and hence energy is dissipated. At point 2, the response from Fancher’s model, β2 , does not form a closed loop. Another possibility to describe a hysteresis with rounded shape for link suspensions is using rolling contact theory. This was proposed by Piotrowski [54]. Based on the slip velocity, the creepage in the contact is calculated. With the help of the heuristic formula of Shen et al. [55], the creep force and the angle of the pendulum are calculated, which gives a transition


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Figure 20.

Force–displacement characteristics for the friction force model proposed in [48].

between the state of pure rolling and the state of pure sliding. Piotrowski concludes, however, that ‘neglecting the creep in the joints leads to marginal discrepancies in the parameters, as compared to the parameters resulting from the model accounting for creep’.

3.3.9.

Buffers

Side buffers are design components that – if considered – are usually modelled with friction elements. Side buffers, in general, stabilise a train. Therefore, to achieve a realistic running behaviour, it is important that several vehicles be modelled with buffers in between. Especially, the so-called low-frequency instability, a kind of lateral snaking mode, usually at a frequency of 1–2 Hz, can be influenced significantly by taking into account the side buffers between vehicles. In derailment studies, in the case of push operation of a passenger train or in the case of braking of a long freight train, detailed modelling of the buffers is also important. While dealing with freight wagons, however, one has to take into account that the authorities usually require a wagon to fulfil the running behaviour requirements without help from other wagons. This means that the draw coupling between wagons is not tightened so that the buffers are not in contact or are in contact without pre-load. A buffer is usually an energy-absorbing component at least if the buffer force in the longitudinal direction exceeds a certain level. Most designs rely on friction damping. The simplest model for the longitudinal dynamics of side buffers is a nonlinear spring in parallel with a dashpot. Also, models with Coulomb friction as those shown in Figures 8 and 18 could be used depending on the buffer design. The friction that arises in the lateral and vertical directions between two pre-loaded buffers is also significant. The two-dimensional friction could be modelled as shown in Figure 11. The pre-load of the buffer is given by the longitudinal springs in the buffers. The quasi-static load is produced by the draw coupling at the centre of the vehicle ends. In reality, the buffer heads are not planar, but at least the buffer on one vehicle has a curved shape. Especially when studying the risk of derailment, it is necessary to take the real shape of the buffer into consideration to get realistic results. Belforte et al. [56] and Cheli and Melzi [57,58] developed a very detailed model of the longitudinal dynamics of side buffers taking into account also misaligned loading of the buffer in the case of lateral displacement. The model, shown in Figure 21, is based on laboratory measurements on a side buffer under quasi-static and dynamic loading conditions. The parameters are derived by least-square approximation with time series of measured buffer forces on a freight train. The experiments reveal that an increase in the excitation frequency increases



Figure 21.

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Rheological model of side buffer (from [58]).

the stiffness and lowers the dissipation force. Internal friction in the buffer increases when misaligned loads or tangential contact forces are applied. The model consists of a linear spring kp to reproduce the presence of a pull rod that generates a pre-compression of the elastomer rings. A nonlinear spring k allows modelling of the quasistatic response to the elastomer. A first nonlinear Maxwell element [59] is introduced to reproduce the change in stiffness and damping properties of the elastomer associated with dynamic loads. A second Maxwell element is used to fit the changes in stiffness and damping properties of the elastomer at higher frequencies. Finally, a friction element allows the effect of internal friction to be included, which is emphasised by the application of misaligned load or tangential contact forces. 3.4.

Rubber springs and bushes

Rubber springs found in rail vehicles serve as part of the primary and secondary suspension systems [60]. An example of a rubber spring is the layered rubber-metal spring (chevron) used in the primary suspension (Figure 22). Sometimes, these components can provide flexibility in all three directions, whereas they only give flexibility in the horizontal plane as shown in Figure 22 (right). Rubber in the secondary suspension mainly shows up as the bellow of air springs, hosting the pressurised air but also affecting the lateral air spring characteristics. Air springs are discussed in Section 3.5. Rubber is also used as bushes for hydraulic dampers, traction rods, etc.

Figure 22. (right).

Example of layered rubber-metal components: single component (left) and part of primary suspension

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Figure 23. One-dimensional model often used to represent rubber (springs). Parallel stiffness k, viscous damping c and series stiffness ks .

In contrast to the traditional coil springs, rubber springs provide damping as well as increased stiffness with increased excitation frequency and decreased amplitude. Rubber spring modelling in the present context mainly focuses on the frequency range of 0–20 Hz and on displacement (deformation) amplitudes typically found in primary and secondary suspensions. 3.4.1. One-dimensional models In many cases, the very simple model of a spring and dashpot in parallel is used. This might be acceptable if only a very limited frequency range is studied, but the increase in stiffness with frequency is significant and often not representative for rubber. A common rubber spring model is instead the one shown in Figure 23 [61,62]. Here, a series spring has been added to the dashpot (Maxwell element), which means that the model stiffness will stay in the range of k to k + ks and monotonically increase with increasing excitation frequency. This better matches the results of rubber spring measurements. The characteristic frequency ks /c will determine where the main transition between the two stiffness levels will take place. To improve the frequency dependency, additional Maxwell elements may be added in parallel to the model components shown in Figure 23 [63,64]. But such, and more advanced, linear models, in principle, need additional measurements to be justified and more input parameters. The models described above imply that for harmonic excitation, the hysteresis effect will tend to zero when the excitation frequency tends to zero. But such behaviour is generally not supported by measurements. Instead, a certain hysteresis effect remains even for very low frequencies. This can be related to internal rubber friction associated with the introduction of carbon black in the rubber manufacturing. Some rubber springs also allow for friction sliding between rubber and metal parts. In such cases, the hysteresis can be very significant, also for low frequencies. The internal and external rubber friction will also increase the rubber spring stiffness, in particular, for low-amplitude motions. The trend of higher rubber stiffness for higher frequencies is also due to the lower displacement amplitudes associated with highfrequency motions. To extend the models described above to account for the phenomena described, friction needs to be represented. In this way, the models will become nonlinear. One way of introducing the friction is to add another parallel element to the model shown in Figure 23. This approach is shown in Figure 24, where the elastic and viscous forces may, for instance, originate from, respectively, the k and c and ks components shown in Figure 23. The friction part of the model shown in Figure 24 may be represented by a Coulomb friction element with a series spring. As for the viscous part, several such friction components may be added in parallel to imitate a successive stick/slip, see, for instance, Figure 25 referring to [65]. Models used for leaf springs may also come into question (Section 3.3). Other examples are given in [66–71]. In [66–70], two-parameter models have been used to represent ‘smooth


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Figure 24. One-dimensional model principle to represent rubber (springs). The total force is the sum of the elastic, friction and viscous force contributions.

Figure 25.

Example of one-dimensional model for elastomeric components: generalised Zener model from [65].

friction’. In [66,67], a logarithmic function was used, whereas a fractional expression was used in the work of Berg [68–70]. In [71], the friction was included in a nonlinear viscous part by a velocity-dependent friction model. Also, a nonlinear elastic part was used through a three-parameter polynomial expression. Also, see models for lateral behaviour of air springs (Section 3.5). In principle, the parameters of rubber spring models need to be determined by component measurements. Measurements on rubber specimens can also support this process. In the component measurements, the dynamic behaviour at different excitation frequencies and displacement amplitudes need to be evaluated, often also different pre-loads. Traditionally, rubber (spring) damping is expressed as a loss angle, or the tangent of this angle, describing the phase angle by which displacement lags force at harmonic excitation. For strong frictional rubber behaviour, cf. above, alternative damping definitions may be considered, since at low-frequency, large-amplitude excitation, the frictional hysteresis can be significant but still the phase angle can be very small.

3.4.2. Multi-dimensional models For two- or three-dimensional rubber spring models, one-dimensional ones are often superimposed. This may also be the case for nonlinear one-dimensional models, although this is not theoretically correct. Aspects such as reference points/levels and possible resisting moments must be considered as described for coil springs in Section 3.2.

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3.5. Air springs Air springs are often used in the secondary suspension of passenger railway vehicles, and their modelling has important implications for the accuracy of quasi-static and dynamic multi-body simulations. The overall behaviour of this suspension element can be described in terms of vertical and horizontal behaviour, generally with a weak interaction between the two, although the vertical pre-load has an important influence on the lateral behaviour of the suspension. In the vertical direction, air spring suspensions show behaviour highly dependent on the preload and on the amplitude and frequency of dynamic displacements. Hence, specific models have been defined, which are reviewed in this section. In the horizontal plane, air springs represent a particular case of shear springs, and the relationship between shear and rotational deformations and the shear forces and moments reacted by the spring is often not negligible, requiring a three-dimensional modelling approach. When deflated, air springs sit down on a rubber emergency spring and can be modelled using the models for rubber elements and models for friction elements in two dimensions, discussed in Sections 3.4 and 3.3, respectively. Models of the vertical air spring behaviour can be classified into ‘equivalent mechanical models’ and ‘thermodynamic models’. Equivalent mechanical models are based on the use of lumped parameter springs, dashpots and masses. These allow a relatively simple mathematical description of the suspension, but they generally do not account for the levelling system behaviour and do not provide an estimate of air consumption. Furthermore, these models may not be well suited to consider non-conventional suspension configurations (e.g. cross piping of the bellows) or active/semi-active suspension control. Thermodynamic models instead aim at representing the actual mechanical and thermodynamic processes occurring in the air spring suspension, and hence, all parameters in such models have a clear physical meaning. Despite this, tuning may be needed to define the values of model parameters describing the concentrated and distributed losses in the pneumatic circuit. More details on equivalent mechanical and thermodynamic models are provided in the next two subsections, while it is not the intention of this paper to cover in detail all existing air spring models, it focuses on some representative ones. In a subsequent subsection, models of the air spring suspension in the horizontal plane are reviewed. The final subsection deals with the definition of model data. 3.5.1. Equivalent mechanical models The simplest model of the air spring suspension in the vertical direction consists of a spring with a viscous dashpot in parallel. This model, however, only reproduces the quasi-static stiffness of the suspension, and it is difficult to define a correct value for the damper parameter, because the actual dissipative effects in the suspension are far from linear. A different model, appropriate in a wider frequency range and known as a ‘Nishimura model’ [72], is shown in Figure 26(a). It consists of a spring K1 representing the bellows, in series with the parallel combination of a spring K2 representing the compressibility of air in the surge reservoir and a viscous damper C accounting for dissipations in the surge pipe and, in suspension systems with air damping to replace hydraulic dampers, also the dissipation from the orifice between the bellow and the reservoir. Optionally, a spring K3 may be added in parallel to represent the additional stiffness effect due to the change in the effective area with the suspension height [73]. A similar model is the ‘linear air-spring element FE83’, implemented in Simpack [73], see Figure 26(b): this model also includes an additional damper in parallel to spring K3 and a spring-damper element K4 and C4 in series to the rest of the suspension, representing the emergency spring.



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Figure 26.

(a) The Oda–Nishimura model (from [73]). (b) The Simpack FE83 model [73].

Figure 27. [75]).

(a) The Vampire model in vertical direction (from [1]). (b) The Berg model in vertical direction (from

The simplest Nishimura model with K3 = 0 is sufficient to define a frequency-dependent behaviour of the air spring: when the frequency of deformation is low, the force in the damper C is negligible, and the model behaves as the series of springs K1 and K2 , but the damper force increases with frequency until the deformations of the damper and of spring K2 become negligible and the complete model approaches the behaviour of the single spring K1 . Hence, the Nishimura model allows the reproduction of the transition between the ‘lowfrequency’ and ‘high-frequency’ stiffness of the pneumatic suspension, but it does not include internal suspension resonances which may occur on account of inertial effects in the air mass contained in the surge pipe. As pointed out in [5], this is a relatively small mass, but it is accelerated to high velocities when air is exchanged between the bellows and the reservoir, and therefore, the equivalent inertia can be important. Two models allowing consideration of this effect are the ‘Vampire model’ [5,74] and the ‘Berg model’ [75]. The Vampire vertical model is shown in Figure 27(a): it consists of a Nishimura model to which a lumped mass M and a series stack stiffness K4 are added. The K4 spring represents the emergency spring and may be taken to be infinite. The lumped mass M represents the inertia of the air mass in the surge pipe. It is worth remarking that in the Vampire model, the damping term can follow a square law, which is considered to be more accurate than linear damping. Berg [75] defined a three-dimensional model, of which, only the vertical part is described here (Figure 27(b)). This is similar to the Vampire vertical model, but it also includes a friction force element and a velocity exponent parameter (β).

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14

x 10

K Nishimura VAMPIRE

Dynamic Stiffness (N/n)


12

10

8

6

4

2

0 0

5

10

15

20 Freq (Hz)

25

30

35

40

Figure 28. Comparison of the vertical dynamic stiffness for an air spring system, Nishimura and Vampire models (K1 = 7.6E5 N/m, K2 = 5.4E5 N/m, C = 1.29E5 Ns/m, M = 5.4E-3 kg, κ = 209) from [84].

When compared with the Nishimura model, both the Vampire and Berg models are capable of accounting for an internal resonance of the air spring suspension, leading to a maximum value of the dynamic stiffnesses in an intermediate frequency range, with the quasi-static and high-frequency stiffness approaching the same values as those for the Nishimura model, see Figure 28 for the Vampire model and Figure 29 for the Berg model. 3.5.2. Thermodynamic models Thermodynamic models of an air spring suspension [76–81] include the following main elements (Figure 30): • a model of the bellows and surge reservoir; • a model of the surge pipe connecting the bellows with the reservoir. Depending on the scope of the air spring suspension model, a model of the levelling system can also be introduced, as done in [80,81]. The bellows and reservoir are modelled, respectively, as variable size and constant size air volumes, whose thermodynamic states are varying on account of the boundary conditions applied and of fluid exchange between the volumes. The volume of the bellows is expressed as a function of the air spring height: Quaglia and Sorli [76], Nieto et al. [78] and Docquier [81] reported examples of measured volume vs. height functions for different types of bellows, and they all observed that the actual nonlinear relationship between these quantities is well approximated by a linearised expression. The vertical force F generated by the air spring is expressed as F = prel Ae ,

(17)

where prel is the relative pressure of the air in the bellows and Ae is a geometric parameter called ‘effective area’. This can be defined either based on the geometry of the bellows or from measurements. Examples of the relationship between the effective area and the air spring height are also reported in [76,78,81], where it was observed that the effective area can be



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Figure 29. Dynamic stiffness and damping for different air spring models (from [77]). Model 1: oscillating air mass. Model 2: incompressible differential. Model 3: incompressible algebraic. Model 4: ISO 6358. Model 5: Berg’s equivalent mechanical model.

well approximated using either a constant value or a linear expression as a function of the air spring height. The thermodynamic state of the air volumes in the bellows and surge reservoir can be described using a formulation based on mass and energy balances. Instead of the energy conservation equation, one can consider a polytropic law of the type:

V p M

k = p0

V0 M0

k ,

(18)

where V is the volume of the bellows or tank, M is the air mass in the bellows or tank and p is the (absolute) air pressure in the bellows or tank, while p0 , V0 and M0 are the given reference conditions and k is the polytropic exponent whose value depends on assumptions made on the energy balance of the system, k = 1 describing an isothermal transformation and k = γ (the specific heat ratio) representing the case of an adiabatic transformation. It shall be pointed out that the suspension behaviour is significantly affected by the exponent k used in the polytropic law (25): this issue is specifically addressed in [81,82], where a sensitivity analysis on the effect of heat transfer was performed, including the two extreme cases of zero and infinite heat exchange capacity, corresponding, respectively, to the adiabatic and isothermal transformations. The case of a metro railway car is considered, showing that different assumptions on the heat transfer coefficient lead to significant differences in wheel unloading while the vehicle negotiates a track twist and also heavily affect the estimate of air consumption associated with passenger loading/unloading; note, however, that both the conditions considered are happening on a time scale in the range of 102 s, much larger than the

Figure 30. Thermodynamic model of an air spring suspension (from [77]).


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time scales typically associated with vehicle dynamics. Vehicle dynamics on a shorter time scale were dealt in [76,77,83] under the assumption of heat exchange being negligible in a fast manoeuvre and hence using the adiabatic exponent; in [83], this assumption was corroborated by a good fit with full-scale measurements. On the other hand, in [78], it was reported that measurements performed on an air spring suspension in working condition supported the assumption of an isothermal transformation, but no information was given on the time scale of the experiments performed. The second main component of air spring thermodynamic models is the model of the surge pipe connecting the bellows with the reservoir. This can be defined at different levels of complexity: the simplest model, used, for example, in [76], consists of a simple fluidic resistance, defined according to ISO 6358 and accounting for concentrated and distributed losses, and may include the effect of an orifice introduced in the duct to increase the damping of the suspension. A more detailed model, used, for example, in [77,81,83], is the ‘incompressible differential’ model, which accounts for the inertial effects associated with the oscillation of the fluid in the pipe, assuming a one-dimensional incompressible flow; in this case, a first-order differential equation is written for the pipe, and one additional state is introduced for each pipe. A similar approach is implemented in the Simpack FE82 element [79], a thermodynamic model in which the mass of the air pipe is considered as a rigid mass oscillating between the bellows and the reservoir, subject to dissipative forces representing losses in the pipe. Docquier et al. [77] and Docquier [81] also proposed an ‘incompressible algebraic’ pipe model, which is derived from the incompressible differential one neglecting the inertial effects related to the air contained in the pipe. Like the case of the model based on ISO 6358 formulae, this model results in an algebraic equation relating the mass flow rate in the pipe to the pressure drop between the reservoir and the bellows. The different pipe models were compared in [77,81], see Figure 29, showing that the incompressible differential model (or similar ones) is able to account for an internal resonance of the air spring in the same way as the Vampire and Berg equivalent mechanical models. Docquier [81] in his PhD thesis also proposed the use of different kinds of compressible flow models to describe fluid motion in the piping and showed that by considering fluid flexibility, a second internal resonance of the suspension can be described, which, depending on the length of the pipes, may fall below 20 Hz.

3.5.3. Air spring models in the horizontal plane The behaviour of the air spring in the horizontal plane is determined by the structural stiffness of the bellows and by the effect of the vertical load, which combined with shear and rotational deformation of the bellows give rise to shear forces and moments on the end mountings. Friction effects at the rubber–metal interface at the bellow upper perimeter may also contribute. In the Vampire model (Figure 31(a)), the horizontal air spring behaviour is modelled using an elastic force element having either a linear or a nonlinear characteristic, in parallel with a model of rubber hysteresis and a visco-elastic element consisting of a damper with series stiffness [73,74]. Additionally, a balancing moment is introduced to satisfy the static equilibrium of the air spring, given the pre-load, shear force and deformation of the spring. The user is allowed to define a non-uniform share of the balancing moment on the upper and lower ends of the air spring. The Berg [75] model in the horizontal plane consists of three components: a set of elastic forces, a frictional contribution and a viscous one. The elastic component consists of two shear forces, a roll moment and a pitch moment applied at the air spring upper end, defined as linear functions of the shear deformation in longitudinal and lateral directions and of the roll



Figure 31. [75]).

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(a) The Vampire model in lateral direction (from [73]). (b) The Berg model in lateral direction (from

and pitch rotations of the end mountings. The forces and moments at the lower end mounting are obtained from the static equilibrium of the air spring. The friction and viscous forces are instead defined only in the lateral and longitudinal directions. The complete Berg model for the lateral direction, shown in Figure 31(b), is hence similar to the Vampire model in the lateral direction, with the difference being the linearity of the elastic term and the different function for the ‘smooth’ friction. Despite the fact that the elastic component of the Berg model accounts for the coupling of shear deformation and rotations of the end mountings and for the effect of pre-load, in [75], only the direct shear stiffness term was identified, and comparisons with measurements were limited to the shear force generated under pure shear deformation. Facchinetti et al. [83] proposed a quasi-static, linear elastic model for the air spring suspension in the horizontal plane, similar to the elastic component of the Berg model but limited to lateral and roll movements. All stiffness parameters were identified from full-scale measurements performed by applying combinations of shear and roll deformation. In the same paper, vehicle dynamics simulations were performed using the proposed model and a simplified one neglecting the direct roll stiffness term and the shear–roll coupling term, and the results were compared in terms of wheel–rail contact forces, showing that the stiffness terms under examination have a non-negligible effect on the load transfer effects when the vehicle is negotiating a curve or is subject to the effect of crosswinds.

3.5.4. Definition of air spring model parameters Presthus [73] collected formulae to define the parameters of various equivalent mechanical models, including the Nishimura, Simpack FE83 and Berg models, based on the physical properties of the air spring suspension (effective area, air volumes, concentrated and distributed loss coefficients), and similar expressions are available for the Vampire model [74]. Alonso et al. [84] reported a comparison between the measured and simulated dynamic stiffnesses of an air spring suspension, showing a good agreement between the two sets of data for different cases of concentrated losses in the surge pipe. For thermodynamic models, the definition of the input parameters is more straightforward, since these are directly represented by the physical parameters of the system. However, the accuracy of both equivalent mechanical and thermodynamic air spring models may be affected by uncertainties in some physical parameters: in particular, the variation of the effective area with the air spring height can hardly be defined other than by a direct measurement and the range of validity of semi-empirical formulae defining the loss coefficients based on the geometry of

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the surge pipe needs to be carefully considered, especially in the case of complicated geometry of the pipes and orifices. Based on the examination of the state of the art, the direct measurement remains at present the most frequently used way to define some critical air spring geometric parameters, namely the relationships between the effective area and the volume of the bellows and the height of the air spring, although attempts have been made, for example, by Qing and Shi [85], to compute some of these parameters based on the analysis of the air spring geometry.


3.6. Hydraulic dampers The damping mechanism in hydraulic dampers comes from oil passing through restricted orifices. It is the most common suspension component for many types of vehicles. Different technical and manufacturing solutions exist depending on the application (bikes, automobiles, trucks, buses, rail vehicles, etc.) which defines displacements, piston speed, maximum force allowed, temperature, sealing requirements, impacts, etc. The fluid inside the damper is generally oil as gas dampers are not used in the railway sector, although they are common in other types of vehicles. Hydraulic dampers, also called shock absorbers, behave in a nonlinear and time-variant manner, and consequently, modelling their dynamic behaviour is not straightforward. Damper designers need to introduce nonlinear characteristics in their components in order to satisfy the conflicting requirements of comfort, stability and reduced train–track interaction forces, among other criteria. Force–velocity diagrams are the straightforward way to represent damper dynamics. However, on the track or road, the behaviour of the vehicle depends not only on some absolute values of these diagrams but also on the shape of the curves, which are influenced by the excitation frequency and amplitude of the imposed motion and by temperature. This leads to the use of force–displacement diagrams and force history (force–time) plots if good characterisation and understanding of the damper are the goals. Hydraulic dampers are used in vertical and lateral secondary suspensions of rail vehicles and often also in primary vertical suspensions. Yaw dampers need a special approach and accurate characterisation. They are critical for the vehicle running dynamics as they assure bogie stability with respect to hunting. Therefore, suitable models are required to replicate the actual operating conditions of the component which, during vehicle motion, undergoes displacements with different amplitudes: small ones 1–3 mm, as far as hunting motion is concerned, higher ones if curve negotiation is considered, and at various frequencies, the frequency varies from relatively low values during curve negotiation to high values (e.g. 4–8 Hz) when hunting motion occurs. An introduction to hydraulic damper modelling can be found in [5], and also in Duym et al. [86] and Carrera et al. [87], related to automotive applications. Most detailed damper models are those that relate their parameters directly to the physical properties of the different damper parts. These mechanistic models help to improve the understanding of the damper’s behaviour. An example of a detailed physical model of a shock absorber is that of Lang [88]. It is based on the processes of the oil/gas flow through the various internal chambers and describes the behaviour of the damper in a broad range of operating conditions. The results of the model with 87 parameters show a very good agreement with experiments; however, it is rather uneconomical in terms of computational effort and modelling complexity. Other simpler physical models are possible. Alonso and Giménez [89] recently defined a detailed physical model of a hydraulic damper of a rail vehicle, based on differential equations reproducing the dynamics of oil fluid. The results were compared with measurements on a real damper, showing a satisfactory agreement. Furthermore, a simplified model suitable for being introduced in a multi-body model of a complete rail vehicle was developed in the same


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Figure 32. The ‘dashpot with series stiffness’ model of hydraulic dampers (top). The flexibility and damping of the rubber bushings can be included (bottom). If Cs = 0, then 1/K = 2/Ks + 1/Ka and C = Ca .

paper, consisting of one first-order differential equation, having two different mathematical expressions to separately represent the compression and extension phases. The simplified model was validated by comparing it to the more detailed one, showing very good agreement. Van Kasteel et al. [90] defined a damper model based on the physics of the component, a ‘white box’ model, which allows for insight into the damper physics and includes all nonlinearities (e.g. nonlinear effect associated with leakage) and dynamics within a frequency range up to 30 Hz. In the paper, the issue of modelling the damper component was developed, providing information on the valve and pressure modelling. The alternative to ‘physical models’ is the use of elementary models constructed with combinations of spring and ideal viscous damping elements whose values are fitted from experiments or from component specifications. The model consisting of a dashpot with series stiffness is the most widely used (Figure 32). The dashpot can have a linear/piecewise linear/fully nonlinear characteristic curve, relating the velocity to the force generated, whereas the series stiffness is mostly assumed to be linear, with the notable exception of [91], as discussed below. The ‘dashpot with series stiffness’ model is also one of the modelling options for hydraulic dampers available in multi-body simulation software packages. Additionally, the model should take into account the effect of blow-off valves by limiting the maximum force of the damper model. In the ‘dashpot with series stiffness’ model, the stiffness is used to represent the effect of internal damper flexibility, the deformation of the end bushings and other possible effects (e.g the flexibility of the damper supports). The combination of series stiffness and dashpot acts effectively as a pure damper at low frequencies and a pure stiffness at high frequencies. The cut-off frequency fc depends on the values of damping C and series stiffness K, f = K/(2π C). Low series stiffness reduces the energy transmission at high frequencies, so the use of softer damper bushes can help to reduce vibration levels and improve running comfort. However, it is important to ensure that the cut-off frequency of the damper is sufficiently higher than the frequency intended to be damped. When flexibility effects are neglected, the model reduces to a single dashpot with either a linear or a nonlinear behaviour. However, the use of a correctly tuned series stiffness parameter can be important to represent the damper’s behaviour at high frequency of deformation, with relevant implications on both the transmission of high-frequency vibrations through the damper and the effectiveness of yaw dampers to control the hunting oscillation of the bogie. Reasons for modelling flexibility effects in yaw dampers are detailed by Wrang [92] and Bruni et al. [93], where the force generated by the damper is shown to be the sum of an elastic term and a dissipative one, and the importance of the two force components is discussed regarding the frequency and the amplitude of the excitation. Furthermore, Conde Mellado et al. [91] compared the results of the ‘dashpot with series stiffness’ model with the results of experiments performed on a yaw damper designed for a high-speed vehicle and found some important deviations from the actual behaviour of the real

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Figure 33. The ‘pinlink element’ model of hydraulic dampers.

component. The paper shows that the yaw damper model could be improved substantially by introducing two modifications: • replacement of the linear series stiffness with a bilinear, asymmetric stiffness element to reproduce a decrease in stiffness close to 35% in compression and 25% in extension when exposed to small displacements. • use of an asymmetric force–velocity characteristic to reproduce the different behaviour of the component in compression and extension. In the same paper, the influence of the yaw damper model on the numerical estimation of the vehicle critical speed was examined, showing a variation of the critical speed from 480 km/h when the original yaw damper model was used to 400 km/h using the modified yaw damper model. The sensitivity of the critical speed calculations to the yaw damper model was also addressed by Alfi et al. [94]. In this paper, three different yaw damper models were compared: a dashpot without series stiffness, a dashpot with series stiffness and a dashpot with stiffness in parallel. All the three models are linear, and their parameters are tuned based on laboratory tests performed on a yaw damper for high-speed vehicles. By means of running dynamics calculations performed using a nonlinear wheel–rail contact model, critical speed values in the range between 300 and 355 km/h are obtained depending on the model used for the yaw damper: this shows a sensitivity to the yaw damper model in the range of 18%, which is comparable to the corresponding results reported in [91]. A different model of hydraulic dampers including yaw dampers is the ‘pinlink’ element available in Vampire [74], shown in Figure 33. With respect to the ‘dashpot with series stiffness’ model described previously, this element additionally includes a nonlinear stiffness and a friction element in parallel with the dashpot. The nonlinear stiffness K and friction element F can be disabled, obtaining a model similar to the ‘dashpot with series stiffness’ one (called ‘damper element’ in Vampire), with the notable difference that the line of action of the force generated by the damper element remains fixed irrespective of the relative movements of its ends in Vampire, whereas in the pinlink element, the line of action changes orientation to keep the direction parallel to the relative position of the two ends of the element. When the friction element is activated, a finite value of the series stiffness Ks shall be prescribed, and a series damping Cs is calculated automatically, to avoid the occurrence of numerical oscillations produced by stick–slip effects. In summary, many options are available for hydraulic damper models. The simplest option is just a linear or nonlinear viscous damper, which can be upgraded using a linear or nonlinear viscous damping element, to which a friction damping element could also be added. For yaw dampers, in most cases, the choice is a model based on a nonlinear viscous element with series


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stiffness. For other dampers, the preferred model varies as the selection of a particular damper model is related not only to the type and properties of the damper itself but also to the kind of suspension the damper is part of. In general, due to the combined effect of rubber end bushes and oil/structural flexibility, the modelling of the internal damper flexibility is recognised as a fundamental effect which cannot be disregarded.


3.7. Active and semi-active suspensions The use of semi-active and active devices is being increasingly considered in passenger rail vehicle suspensions, mostly at the secondary suspension level, as a means to improve ride comfort or to provide the same level of comfort for increased service speed, and a number of railway vehicles in commercial service are nowadays being equipped with active or semi-active secondary suspensions. On a more research-related level, active primary suspensions have also been proposed, particularly to remove the design conflict between curving and stability which is typical for the passive primary suspension in a railway vehicle [95]. Appropriate models for active and semi-active suspensions are hence being developed to assess the system performance, verify the implications of active control on system stability, estimate energy consumption, define actuator requirements and analyse the behaviour of the system in fault conditions. Active and semi-active suspensions can be categorised based on the function that they perform in the vehicle, for example, primary vs. secondary or vertical vs. lateral actuation, or otherwise based on the principle of actuation, with pneumatic, hydraulic, electro-mechanical and magneto-rheological being the most widely used. The choice in this section is to categorise the models based on their principle of actuation, as this profoundly affects the modelling.

3.7.1. Pneumatic semi-active and active suspensions Active and semi-active pneumatic suspensions are used in the secondary suspension stage of passenger railway vehicles and can act either in the vertical direction (active/semi-active air springs) or in the lateral direction (lateral secondary suspension). Mathematical models of active and semi-active vertical suspensions are derived from both the ‘equivalent mechanic’ and ‘thermodynamic’ air spring suspension models presented in Section 3.5. Tang [96] proposed an equivalent mechanical model for a semi-active air spring with a controlled variable size orifice in the surge pipe: the model consists of a modified Nishimura model in which the effect of the variable size orifice is represented by changing the damping rate C of the viscous damper. The same modelling approach was followed by Sugahara et al. [97], where a model for the vertical vibration of a complete rail vehicle equipped with semi-active air springs and adaptive primary dampers was also defined. A thermodynamic model of an active air spring suspension was defined by Alfi et al. [98] as a modification of the model shown in Figure 30, by modelling one additional constant air volume representing an additional supply reservoir and the servo-valve connecting the additional reservoir with the surge reservoir. The model of the active suspension was compared with experiments performed on a full-scale prototype, showing a good match of numerical simulations with the measurements. The model of the active suspension was then used in combination with a multi-body model of the entire vehicle to investigate the use of the active suspension to improve ride comfort in curves and to increase the vehicle’s resistance to overturning in the presence of crosswind.

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As far as pneumatic active lateral suspensions are concerned, a thermodynamic model was proposed by Sorli et al. [99], which was integrated into a multi-physics simulator of the lateral dynamics of a Pendolino vehicle; numerical results from the model were compared with line measurements, showing good agreement. Conde et al. [100] proposed a model of an active lateral suspension interconnected to the vertical pneumatic secondary suspension, and the model was used to compare the performances of the proposed active suspension to a conventional passive one.


3.7.2. Active and semi-active hydraulic suspensions Active and semi-active hydraulic devices are being proposed in primary and secondary rail vehicle suspensions. Hydraulic devices allow a larger pass-band than pneumatic actuators and can be used to improve the performance of the active or semi-active suspension, also allowing the use of these devices in the primary suspension [97]. However, hydraulic suspensions imply a higher transmissibility at high frequency and, therefore, their use needs to be carefully considered in view of high-frequency vibration transmitted to the car body. Furthermore, hydraulic actuation requires that a dedicated hydraulic circuit be introduced in the vehicle, whereas pneumatic actuators may use the same circuit feeding brakes, doors and other pneumatic devices in the train. Sugahara et al. [97] proposed a model for a semi-active vertical damper for use in the primary suspension of Shinkansen trains. The model consists of an ideal orifice in parallel to a controlled blow-off valve. The ideal orifice produces a quadratic relationship between the speed of deformation and the force generated by the damper, which is saturated by the blow-off valve to a linear characteristic. The value of the damping force at which the blow-off valve intervenes is controlled by an input signal representing the current fed into the valve’s solenoid. The time lag between the current signal and the opening of the blow-off valve is reproduced by a first-order state equation, whose time constant is calibrated based on experiments. Cheli et al. [101] used a third-order linear model of hydraulic actuators, in combination with a four-bar linkage model of the tilting bolster to model body tilt actuation in a Pendolino vehicle. Results of the model were compared with those obtained from the experiments performed on a test rig which allows the reproduction of curve negotiation on a single railway vehicle by imposing independent roll rotations to each wheelset and with line tests. A similar model of a servo-hydraulic actuator for use in the active secondary suspension of a railway vehicle was proposed by Foo and Goodall [102]. 3.7.3. Electro-mechanical active suspensions The use of electro-mechanical actuators has been proposed for rail vehicle active suspensions, on account of their good dynamic properties and high bandwidth. Modelling an electro-mechanical actuator requires a multi-physics approach, by which the dynamics of both the electrical motor and the mechanical gear are accounted for. Examples of DC motor models applied in active suspensions for rail vehicles can be found in Pearson et al. [103] and Pacchioni et al. [104]. A model of an AC synchronous motor and mechanical actuator controlling the yaw motion of the bogie was defined by Resta and Bruni [105], and numerical results were compared with those obtained from the laboratory tests. The same model was applied to define the expected performance of the actuator as a means to increase vehicle stability and curving performance and the results obtained were compared with line test measurements, partly performed with the actively controlled vehicle [106].


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3.7.4. Magneto-rheological dampers Magneto-rheological dampers are semi-active devices exploiting a physical property of controllable fluids capable of changing their viscosity properties when exposed to a magnetic field. A number of applications have been proposed for magneto-rheological dampers in the automotive field as semi-active suspension components, and a limited number of references can be traced where the same technology has been proposed for rail vehicle suspensions. In particular, Lau and Liao [107] and Wang and Liao [108] proposed a magneto-rheological semi-active lateral secondary suspension in a railway vehicle, introduced two slightly different equivalent mechanical models of the suspension and compared the numerically predicted force–displacement hysteresis loop with laboratory measurements. The model of the magnetorheological suspension was then implemented in Simulink and integrated into a Vampire model of the complete vehicle to assess the semi-active suspension performance. 3.8.

Linkages and other vehicle components

The vehicle may possess other components to be represented in the model, for example, traction connections between car body and bogies, anti-roll devices, bump stops, metallic stops, links, car body tilting mechanism, wheelset guides, steering linkages, drive system, brake system, articulation joints or coupling devices between car bodies, inter-car dampers, buffers and couplers between vehicles. The modelling of these components is very rarely a topic of publications, but the review papers [1,5] mention some of them. Simulation tools usually do not contain special elements for modelling of these components. Besides the functional stiffness of the component such as the longitudinal stiffness of a traction link or the roll stiffness of an anti-roll bar, several components inevitably provide stiffness in other directions too. The term ‘parasitic stiffness’ is used to characterise the stiffening of the suspension which was not intended during the design. The typical sources of such stiffening are the traction link, anti-roll device, brake piping, traction motor cables and hydraulic dampers (due to parasitic friction in series with the end stiffness). It is important to include the parasitic stiffness in the vehicle model unless it is proven that this effect is negligible. Basically, there are two ways to model complex components: • simplified representation by equivalent stiffness and damping parameters derived from the parameters of the bushes and other connecting parts and • detailed modelling with rigid bodies representing each single body (link, rod, etc.) and coupling elements representing each physical component (rubber parts, slide bushes, etc.). Both ways will be explained in the following two subsections using examples of a traction link and an anti-roll bar. 3.8.1. Traction connection between car body and bogie The function of the traction connection is to transfer the longitudinal traction and braking forces between the bogie frame and car body. Many different solutions exist, for example, • • • •

traction rod, centre pivot and Watts (lemniscate) linkage, centre pivot with play and rubber bump stop and pre-stressed layered rubber-to-metal components.

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k

k

1

2

kx1

kx2 kz1

kz2 b


Figure 34.

Horizontal traction link and its stiffness parameters (dashpots representing the damping are not shown).

We have explained the different ways to model such a component using an example of a traction link consisting of a horizontal steel rod with spherical rubber joints on both ends (Figure 34). Considering infinite stiffness of the steel rod, the equivalent longitudinal stiffness kx of this component yields kx =

1 kx1 kx2 = , 1/kx1 + 1/kx2 kx1 + kx2

(19)

with kx1 , kx2 – stiffness of spherical joints in the longitudinal direction. This is the intended design stiffness, while the stiffness of this component in the vertical and lateral directions should be as low as possible. The presented traction link design results in parasitic stiffness in the vertical direction: kz =

1 1/(1/(1/kz1 + 1/kz2 )) +

1 (1/b2 )(kφ1 +kφ2 )

=

kz1 kz2 (kφ1 + kφ2 ) , (kφ1 + kφ2 )(kz1 + kz2 ) + b2 kz1 kz2

(20)

with kz1 , kz2 – stiffness of spherical joints in the vertical direction and kφ1 , kφ2 – torsional stiffness of spherical joints around the horizontal lateral axis. The parasitic stiffness in the lateral direction can be calculated similarly by replacing the stiffness parameters by the values in the horizontal plane. A model of this traction link design can consist of • longitudinal spring and damper to model the equivalent stiffness and damping between the bogie and car body, while the parasitic stiffness is considered in other suspension elements, • a rigid body, representing the rod, and two bushes, each representing the stiffness of the actual rubber bush. Another design solution for the transfer of forces between the bogie and the car body is a combination of traction connection with the car body support to a flat or a spherical centre bowl as typically used in freight wagons. The paper by Fergusson et al. [109] presents the modelling of the rotational friction at the centre plate of a freight wagon used in simulations related to the optimisation of self-steering bogies regarding curving. 3.8.2. Anti-roll bars To achieve soft ride characteristics in the vertical direction and, at the same time, operation of the vehicle within the limits of the operating gauge, passenger vehicles are often equipped with anti-roll bars. They provide an additional rotational stiffness between bogie and car body around a longitudinal axis. A complete anti-roll assembly comprises a torsion bar, levers, links and support bearings (Figure 35). The anti-roll bar may be a single-piece bent bar or a three-piece subassembly, where the separate levers are shrunk onto a straight torsion bar. The spherical joints and support bearings can be either of rubber or a sliding type.


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Longitudinal axis Vertical links

Bent torsion bar


Spherical joints

Support bearings Figure 35.

Lever

Design of anti-roll device.

Although the anti-roll bars primarily provide rotational stiffness, they will, in practice, also generate parasitic stiffness in the lateral and vertical directions that should be considered and included in the vehicle model. Similarly to the previous example of the traction link, the model of an anti-roll bar can be realised either by the equivalent rotational stiffness of the complete anti-roll bar while the parasitic stiffness is considered in other suspension elements or by a detailed model, consisting of rigid bodies and connecting elements, respectively. Both these ways have advantages and disadvantages. The first method leads to a less complex model and is well suited for numerical integration. However, it requires the calculation of equivalent parameters representing the complete component, which can be a source of errors. Furthermore, the forces acting on bodies from the anti-roll bar are not represented at the correct positions, so that the model may be insufficient for simulations considering a flexible car body or for calculation of loads on the bogie frame and car body. The second method leads to a rather complicated model with bodies of small masses, connected by stiff elements, which is disadvantageous for numerical simulations. The parameters of coupling components, however, can be directly derived from data sheet or measurements, and the forces act on the bodies in the correct locations. In the questionnaire carried out in the project DynoTrain within 15 partners (rolling stock suppliers, operators, universities, research institutes and a safety body), 70% responded to be using the roll stiffness only to model the anti-roll bar. This confirms that the simplified modelling of the anti-roll bars can be considered as the state of the art. This simplified model, however, may be insufficient for certain anti-roll bar designs. For example, the anti-roll device with inclined vertical links (Figure 36) leads to a kinematic effect similar to a pendulum suspension, which would not be considered by using only a torsional spring between car body and bogie. 3.8.3.

Bump stops and metallic stops

Bump stops and metallic stops are modelled as a single-sided or double-sided nonlinear stiffness with a zero force inside the clearance. It is important to consider the change of the play of vertical stops and bump stops when varying the loading conditions. Even for the nominal conditions, these plays can change if there is any transient movement during the calculation


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Figure 36.

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Kinematics of an anti-roll bar with vertical (left) and inclined links (right).

of nominal forces after the model assembly. In such a case, the vehicle static equilibrium has to be checked and the plays in stops have to be approved before using the model. Stops or bump stops with horizontal play dependent on curve radius are sometimes used to allow a widening of the car body by reducing the maximum lateral displacement between bogie and car body in very small-radius curves (typically smaller than 250 m). The coulisse used in this component reduces the play in the lateral direction depending on the yaw rotation of the bogie frame against the car body in curves. This specific design of stops either can be modelled by a two-dimensional elastic contact element (if available) or requires a specific set of parameters for the simulation of each single curve radius, respectively. 3.8.4. Wheelset guidance The function of wheelset guidance can be integrated in the primary suspension using rubber springs or it can be performed by a separate component, respectively. Possible wheelset guidance design solutions are • • • • •

sliding wheelset guide with play, pivot sliding guide, guidance by links (rods): longitudinal link, triangular link and lemniscate link, swing arm link and guidance by layered rubber-to-metal springs.

The model of the wheelset guide can sometimes be included in the model of primary suspension even if this functionality is not integrated in the real vehicle. This simplified modelling can be used if the effect of the guide does not influence the direction of forces and moments. This is valid if the guide acts horizontally in the wheelset axis as, for example, sliding guide or lemniscate link. For other wheelset guide designs, this simplification can lead to significant differences in the results, see the analysis in [5,110] for a swing arm. 3.8.5. Steering linkages Steering linkages are used for a cross-coupling of wheelsets or a coupling of the wheelset yaw angle with the yaw angle between the car body and bogie or between two car bodies, respectively. These or other linkage mechanisms can be modelled either simply by introducing the links between DoF or in more detail by rigid bodies coupled through elastic components. Though the simplified modelling is better suited for numerical integration, it neglects the parameters which influence the wheelset yawing due to the longitudinal elasticity. Polach


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analysed the equivalent axle guidance stiffness of a self-steering wheelset cross-coupling mechanism consisting of a lateral torsional shaft fixed through rubber bearings on the bogie frame in [111]. The paper investigated the sensitivity of this self-steering mechanism to the influence of operating conditions such as curve radius, rail inclination, rail wear, tractive force and wheel flange lubrication. 3.8.6. Drive system


The drive system can be unsuspended or suspended (partly or fully) by primary suspension or by both primary and secondary suspensions. The drive components can be fixed • in the car body (inside or below the floor of the coach), • on the bogie frame and • around the wheelset axle or independent wheel. While the last option is very rare, the drive system suspended in the bogie frame represents the design most widely used in electric locomotives as well as in other traction units. The drive system model can be simplified for the running dynamics assessment, if the traction motor is rigidly fixed to the bogie frame, car body or other parts of the vehicle. The mass of drive components must, however, be included in the mass and other inertia parameters. A traction motor and/or other parts of the drive system suspended in the bogie frame may have a significant influence on the stability assessment and the dynamics of the bogie, see [112]. They should be modelled by separate rigid bodies and relevant connecting elements. Simulation of running dynamic performance is usually carried out without considering the tractive effort. For heavy traction vehicles, there can be an important impact of the tractive effort on track loading, wear and damage of wheels and rails. The effect of traction effort on the wheelset radial steering was investigated by Polach [113]. For this kind of assessment, the traction torque can be modelled as a moment on the wheelset. Investigations of interaction between the running dynamics, traction chain dynamics and traction control, however, require not only a detailed modelling of the torsional drive system but also an extension of the creep force model with the negative slope at large creepages as described and illustrated in [114]. 3.8.7. Brake system Similarly as for tractive effort, simulations of running dynamics are usually carried out without considering any braking forces. However, the impact of braking can be important for long trains due to the delay of brake onset and for the assessment of the derailment risk and track loading forces when braking in curves. Because of a rather small mass of brake equipment, the mass of brake components can be added to the relevant body masses and moments of inertia and the centres of gravity (CoG) replaced by the actual values considering the mass of the brake equipment. Heavier brake components, however, may require to be modelled as a separate rigid body. This can be the case, for example, for a magnetic brake suspended in the bogie frame or a low hanging magnetic brake fixed to the axle boxes as used in trams. Braking usually does not influence the running performance. In specific cases, the brakes can have relevant influence on running performance and should be considered when modelling the vehicle as illustrated in the recently published investigation by Kovtun et al. [115]. They

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investigated the influence of braking on the running performance of a freight wagon due to the interaction of the conventional brake rig with the wheelsets and frame of a three-piece bogie. The results obtained show that the brake blocks are excited to virtually undamped longitudinal oscillations by the up and down motions of the bogie bolster which worsen the wagon dynamics.


3.8.8.

Coupling between vehicles and between car bodies of articulated vehicles

Vehicles in a train are connected by coupler and buffers (Section 3.3). Car bodies of a traction unit are usually coupled by a short coupler instead of by a standard coupler and buffers. Such compositions as well as articulated vehicles require a model consisting of the whole composition or at least its sequence with more than one car body. The modelling of the inter-car joints can be carried out either using a constraint reducing one or more DoF or using a bush element considering all six stiffness parameters. Besides the articulation joint connecting the neighbouring car bodies, the vehicle can also possess bump stops and/or stops, inter-car dampers and possibly also linkages between the car bodies. The modelling options of these components are the same as those described in the previous sections about the suspension components.

4. Track models for use in rail vehicle dynamics simulation Track modelling is taken to include the definition of • • • •

Track design geometry. Rail profile. Track irregularities. Track dynamics.

The track design geometry definition includes the track layout (curve radius, curve transition, track cants and track gauge), which can be either design or measured data. The rail profile can be obtained from the standard EN 13674-1 [116] or other standards and include the nominal rail inclination. It might be important to consider worn rail profiles too. An additional effect which modifies the wheel–rail contact conditions is the effect of rail roll due to lateral forces [117]. If available, measured data should preferably be used for track irregularities. It is important to use irregularity data which contain wavelengths covering all investigated frequencies of the vehicle under study for the required speed range. The track irregularity data should contain sufficiently long wavelengths to excite the lowest car body modes at the maximum speed and also sufficiently short wavelengths to excite the highest frequency at low speeds. This requirement is often difficult to fulfil, especially for long wavelengths due to the limitations present in the measuring vehicles. When classifying the irregularity data, the difference between stochastic and periodic irregularities should be mentioned first and then the different irregularity types in each group, for example, sleepers or rail joints as a periodic irregularity. Track irregularities could also be classified according to their wavelength [118]. The stochastic track irregularities can also be randomly generated based on the specified power spectra density (PSD). PSDs based on measured irregularity data ‘Low Level’ and ‘High Level’ according to ORE B176 [119] are often used in specifications for simulations in continental Europe.


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Other interesting disturbances are


• • • •

voided sleepers, irregular sleeper pitches, varying rail support stiffness (e.g. failure of rail fastening) and track transitions before and after bridges and tunnels.

Track dynamics is a very wide field, and in the last 150 years, a great effort has been made to characterise the dynamic behaviour of the track. The level of complexity of the track model varies depending on the phenomena to be studied. It is not intended to cover here track dynamics and train–track interaction models which are specific to the study of track vibration problems [118,120,121]. This section focuses on answering two basic questions: how is track flexibility included in MBS, and when is it important to consider track flexibility in a vehicle dynamic study. 4.1. How is the track flexibility included in MBS? The flexibility of the track might be included in MBS software in different ways. Usually, a track model with moving equivalent mass is used in multi-body simulations. This moving equivalent mass represents the sleeper and a part of the track oscillating together with the wheelset. This simple model of moving elastic track contains stiffness and damping (linear or nonlinear) in the vertical and lateral directions. Default values are often used, for example, the ones suggested for ballasted track in [119,122] (Table 1). This approach is available in several MBS codes. Figure 37 shows the Discrete Track Model option in SIMPACK. In the default model, each wheelset rests on a rigid body. This rigid body represents the equivalent mass and inertia of the track section under each wheelset (for ballasted track, the sum of sleeper mass and ballast mass). Roll, vertical and lateral displacements of the track rigid body are allowed. The track is connected to the ground by spring-damper elements acting in the vertical and lateral directions. This track model could be easily modified in order to represent additional elastic layers of the track and also include springs and dampers connecting masses under each wheel or wheelset. A critical issue is then to find the correct values of these parameters to represent adequately the actual track behaviour. This simple track model is adequate for simulating vehicle–track interaction and vehicle dynamics for frequencies up to about 20 Hz, but it is not suited to study other effects such as forces in individual track components, effects of wheel flats [123], local variations in track support and broken rail [124]. A more detailed modelling option for the track consists of adding to the vehicle model, one by one, each rail and possibly sleepers as flexible bodies. The flexibility of these bodies is given by a previous modal analysis. This might be useful for investigations considering Table 1. Measured values of a ballasted track for a simple model of moving elastic track containing stiffness and damping, published in [119].

Lateral stiffness (per rail) Vertical stiffness (per rail) Relative lateral damping (per rail) Relative vertical damping (per rail)

Unit

Value

MN/m MN/m – –

20 75 0.3 (0.1) · · · 0.3


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Figure 37.

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SIMPACK discrete track model (default configuration).

higher frequencies. However, as the frequency increases, with no doubt, it is wise to simplify the vehicle model (just the unsprung mass) as the dynamic behaviour is decoupled by the soft secondary suspension. Consequently, as frequency increases, approaches other than MBS models are more efficient. The question then is obviously which track model has to be used when studying a particular problem. Simple models are preferred, as they avoid numerical complications and limit the number of parameters which, in fact, are either not known or difficult to obtain. A moving equivalent mass track model with parameters obtained usually from the literature is a widely used option. Due to lack of track dynamic properties knowledge, a simple rigid track model is also used for analyses in the frequency range up to 20 Hz. Németh and Schleinzer [125] compared typical track models used in MBS tools and concluded that there is no difference between wheel displacements or wheel–rail forces among the chosen track models up to 40 Hz. Alfi et al. [126] confirmed that the spectrum of wheel–rail contact forces, up to 20 Hz, is not affected by the introduction of track flexibility. However, they reported changes in the vehicle stability calculations with the three different track models used (a rigid track, a 3 DoF moving track model and a FE model of the track): use of a rigid track gives an overestimation up to more than 10% of the vehicle critical speed and an overestimation of 20% in the hunting frequency. Table 2 recalls three typical deformation shapes and their occurrence frequencies (F1, F2 and F3) of a ballasted track on sleepers. This is the result of calculating frequency response functions or measuring frequency response functions of the track. Clearly, if the phenomena to be studied are below 20 Hz, it is unlikely that the flexibility of the track will affect the results. Additionally, as F2 is much higher than F1, it is more important to include the flexibility of the ballast/sleeper system than the stiffness of the rail pad itself which will affect the phenomena occurring at higher frequencies. An interesting comparison of moving track models, and correlation with experiments, was reported by Chaar and Berg [127] (Figure 38). The model A does not consider the track flexibility under the sleepers in contrast to models B and C. Model B does not consider the vertical flexibility at the rail. Up to 80 Hz, the magnitude of the vertical receptance from model A agrees rather well, and models B and C agree with the measurements up to 120 Hz. Above


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Table 2. Typical deformation shapes and their occurrence frequencies (F1, F2 and F3) of a ballasted track on sleepers. F1: around 100 Hz. Rail and sleepers on phase vibration over ballast

F2: around 300–500 Hz. The rail is vibrating over rail pads


F3: pinned–pinned frequency, 700–1100 Hz. The rail is vibrating with a wavelength which value is twice the sleeper spacing

Figure 38.

Moving track models A, B and C (from [127]).

120 Hz, model C agrees with the measurements far better than models A and B. The calculated lateral receptances agree very well with measurements up to nearly 200 Hz.

5. Assembly, checking and validation of vehicle models 5.1.

Introduction

An adequate representation of a real vehicle–track system through a multi-body model requires an experienced specialist, a proven computer tool able to represent the phenomena under

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investigation and an application of appropriate modelling technique. The work has to be carried out according to quality requirements to avoid mistakes and errors. Nevertheless, every model needs to be checked and tested before being used for the assessment of vehicle performance. The plausibility check can help to identify errors or mistakes at an early project stage; however, it does not provide satisfactory approval of a vehicle model. The validation of the final model by comparison with measurements should demonstrate that the model is a sufficient representation of the investigated system for the intended range of simulations and that it is ‘fit for purpose’.


5.2. Model assembly and checking During the assembly of a vehicle model, the substructures, which can be used with different models, are compiled into a single model and the model parameters are adapted to the specific case under consideration (e.g. empty or loaded vehicle). The testing and checking of the assembled model should be carried out with all model versions, including different vehicle loading variants. The change of vehicle payload does not only lead to another car body mass and moment of inertia. The suspension displacements due to loading may lead to variation of stiffness and damping parameters in coupling elements. The suspension displacements due to loading lead to changed positions of CoG and of the end points of various components. While many of the parameter changes can possibly be neglected, if they are very small and do not change the direction of the suspension component action, some of them are very important, for example, the consideration of changed plays in the bump stops or pre-loads in the vertical suspensions. Other effects such as stiffening of rubber elements or air springs can be either included in the modelling or represented by new parameter values, which means another model. The calculation of static equilibrium, that is, the computation of nominal forces, either can be repeated with a new parameter set or the model can be ‘set down’ by numerical simulation to a new position of bodies and CoGs. The second variant allows the changes of all position parameters to be omitted; however, it is not correct, if the stiffening of components such as rubber springs and changes of play in the bump stops are not implemented in the component models used. Model checking can be carried out by • plausibility check and • comparison with simulation results of similar vehicles. The comparisons with results of similar vehicles can help to identify and eliminate errors and mistakes. However, they could support incorrect results or raise doubts about correct results if the differences between the compared similar models are underestimated. A set of a few simulations representing all possible applications (eigenvalue calculation, runs on an ideal straight track, straight track with irregularities, ideal curved track, curved track with irregularities, etc.) can be used to test the vehicle model. The plausibility of simulation results can be assessed by checks used to prove the signals measured during the tests on a real vehicle. 5.3. Model validation The experience with validation of railway vehicle multi-body models differs significantly between countries, institutions and dynamics specialists. The publications on validation of railway vehicle models usually deal with a particular simulation tool, vehicle type and application case without assessing and generalising the validation process. The survey done by



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Cooperrider and Law [128] in 1978 deals with testing for validation and theoretical rail vehicle dynamic analyses in the advent of modern computer simulation techniques. Meanwhile, computer simulations are now widely being used in the design of railway rolling stock and in research studies, but the progress of validation methodologies is rather limited. From recent publications on this topic, [2,129] can be mentioned. The review paper by Evans and Berg [1] dealing with simulation of rail vehicle dynamics provides some input regarding the model validation too. In the present paper, the validation procedure is related mainly to the application of simulations for the acceptance of running characteristics of railway vehicles according to EN 14363 [10]. Significant experience with validation of railway vehicle models suited for the assessment of running characteristics has been gained in the past in the UK, where there is a long history of simulation as part of the acceptance process. The model validation is described in Appendix G of the draft Railway Group Standard [130]. Although this draft has never been issued, the principles described in this document are used by dynamics specialists in the UK. The validation of a vehicle dynamics simulation model is mentioned in the draft of the standard prEN 15827 [131], which is related to bogies and running gear. The recently issued revision of UIC 518 [9] specifies conditions for application of simulations for vehicle acceptance instead of tests. The revision of EN 14363, currently in preparation, should introduce the possibility of using simulations of railway vehicle dynamics as a part of the vehicle acceptance process, whereby the starting point is the document UIC 518 [9]. The validation of vehicle models is commonly carried out by comparisons with measurements. The measurement error, scatter and the deviation of repeated measurements, however, lead to many questions. The best agreement with a particular measurement does not mean that the model correctly represents the statistical behaviour of the investigated system. Generally, there are two validation approaches; the validation can be realised either by checking the correctness of the physical relationships in each model component or a signal-based verification can be used by comparing the input and output data (i.e. using a ‘black box’ model), respectively. The verification of suspension and coupling component modelling is a suitable part of the validation process; however, the validation of component modelling does not mean that the complete model is validated. Contrarily, the use of the black box model can lead to the risk of a good agreement for the complete model by chance. This kind of validation may be insufficient for railway vehicles because of a limited number of measurements and a large number of unknown or uncertain parameters. The best practice of model validation in railway applications is a combination of physical and signal-based verification. Each particular model is validated by a synthesis of stationary tests, low-speed tests and on-track measurements. The comparisons between simulations and measurements should be carried out for various conditions (static, quasi-static, dynamic) and for several signals (wheel–rail forces, accelerations, displacements, etc.). If simulations are to be used for a vehicle in different conditions (e.g. tare, laden and inflated and deflated air suspension), separate models will need to be validated for each condition. In the UIC 518 revision from 2009 [9] as well as in the revision of EN 14363, now under preparation, the validation of the model relies upon the static, slow-speed and on-track tests as specified in EN 14363 for testing of running characteristics. The static and slow-speed tests are intended to be used to validate different aspects of the vehicle model, namely • • • •

wheel loads and load distribution, behaviour on twisted track, yaw moment during bogie rotation and sway or roll coefficient.

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Simulations of the static and slow-speed tests should be carried out and the output values should be compared with the test results. For example, for the measurements of wheel loads and load distribution, the following values should be compared:


• • • •

load on each individual wheel, load on each axle (sum of two wheels), load on each bogie (sum of wheels) and load on each side of the vehicle (sum of wheels on that side).

For the main part of the proposed assessment comparisons, however, there is no limit required for a successful validation. Consequently, the validation may mature to a very subjective assessment. Even the few criteria specified in [9] do not necessarily ensure a well-validated model. For example, the limit prescribed for the comparison of individual wheel loads related to the appropriated test results is 15%. The acceptance of such large deviations between the simulation and measurement was introduced because of the wheel load variation between successive measurements of the same vehicle, particularly, for vehicles with friction damping like freight wagons. Fulfilment of this modest criterion, however, cannot guarantee a satisfactory representation of the actual real vehicle. Another stationary test suited for model validation but not mentioned in EN 14363 and UIC 518 could be the modal analysis. A simple identification of the eigenfrequencies and eigenmodes of the vehicle car body is carried out by analysing the vehicle reaction on a single excitation in a so-called wedge test, rolling with a walking speed over 15–25-mm-high wedges. This test can provide a useful validation input for modes with rather low damping. It is well suited for conventional vehicles but more difficult to apply to articulated vehicles. An advanced modal analysis is only rarely used as it requires a special test device. Other kinds of tests could be useful too, for example, a test on a special track section with extreme track irregularities to allow an extrapolation of the vehicle assessment. Such tests would, however, need an adaptation of existing test centres or building of new test centres. The questionnaire carried out in the frame of the project DynoTrain identified the following stationary tests as the most useful for model validation: • Measurement of static wheel loads. • Wheel unloading test on a test rig or twisted track. • Measurement of guiding force in a flat curve with a radius of 150 m as specified in the test of safety against derailment according to Method 2 in EN 14363 [10]. The on-track tests represent another relevant part of model validation exercises. The dynamic behaviour can differ from the static and slow-speed behaviour due to the effect of dynamic stiffening and due to the contact geometry in the coupling between wheel and rail. Simulation of on-track test and comparison with measured values, therefore, represent an indispensable part of model validation. The revision of UIC 518 from 2009 [9] as well as the revision of EN 14363 (in preparation) proposes the validation of vehicle models using the on-track tests specified for testing of running characteristics in the same documents. The approval of running characteristics of railway vehicles according to EN 14363 [10] yields the maximum estimated values of measured quantities in each of the four test zones (straight, curves with large radius, curves with small radius and curves with very small radius). These maximum estimated values are based on the statistical assessment (either one- or two-dimensional) of the measurement results in the specified number of sections from the relevant zone, considering normal (Gaussian) distribution of measured quantities. The maximum estimated values together with the results of statistical analysis as provided in the reports for vehicle acceptance according to EN 14363 are, however,



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not well suited for validation. For comparison between simulation and measurement, the data from single sections are required together with a detailed knowledge of boundary conditions such as track irregularity, track gauge, rail profiles and inclination and weather conditions (friction coefficient between wheel and rail). The results of statistical analysis can be used for model validation too. The two-dimensional analysis as a function of the lateral acceleration at track level as used today is less suited for validation, because the effect of lateral acceleration on the measurement results is rather weak and ambiguous. The dependency of measured quantities on the curve radius is more significant, at least for the curves with very small radius, and should preferably be used for the two-dimensional statistical analysis intended for model validation. The validation of the simulation model by comparison of simulations with the on-track test results as described in UIC 518 [9] and in the latest draft of the revision of EN 14363 includes the following parameters: (1) assessment quantities according to UIC 518 or EN 14363, respectively (section values, mean, standard deviation and estimated maximum as appropriate); (2) power spectral densities (PSDs) and key frequencies of the following measurement quantities: • vehicle body lateral and vertical accelerations at each end, • vehicle body bounce and pitch accelerations (derived from in- and out-of-phase values of body end vertical accelerations), • calculated vehicle body lateral and yaw accelerations (derived from in- and out-of-phase values of body end lateral accelerations), • bogie lateral and yaw accelerations, • bogie vertical and pitch accelerations and • sum of guiding forces (key frequencies); (3) distribution plot of values for lateral and vertical wheel–rail forces as function of curve radius or cant deficiency, respectively; (4) sample time histories over straight and curved track sections for all the measurement quantities. Unfortunately, there is no specification of allowable differences between simulation and test results for a well-validated model, so that the comparison of time histories results in a subjective assessment. The experience with model validation using the acceptance tests according to EN 14363 is rather limited. Two trials were initiated by UIC during the preparation of the revision of UIC 518. The experience with one of them is published in [2]. As the measured rail profiles were not available, the validation started with the theoretical, design rail profile. However, no satisfactory agreement could be achieved. An application of a worn rail profile measured previously at another location than at the investigated test line improved the agreement between the simulations and measurements and confirmed the importance of the boundary condition knowledge for a successful validation. The actual track irregularities and rail profile data along the line during the on-track tests, which represent important input data, are very often missing. The track irregularity data are nowadays being recorded regularly; however, the data are often either not available outside of the companies handling the infrastructure maintenance or measured using a chord system which is not well suited for simulations. Even more problematic is the knowledge of rail profile shapes. A continuous measurement of rail profile shapes has recently been introduced in some countries; however, the measurements are only used for calculation of equivalent conicity or of the radial steering index according to UIC 518 [9], which both represent

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insufficient information regarding the validation of multi-body railway vehicle models. The availability and accessibility of measured track irregularities and measured rail profiles are crucial for successful model validation and for the use of simulations partly replacing physical testing.

6.

Challenges and future trends


Based on the state-of-the-art examination proposed above, some challenges and trends for future research are foreseen. Modelling of rubber suspension components: the examination of the state of the art shows that there are not many references dealing with the modelling of rubber for railway suspension components, although rubber springs and bushes are widely used in rail vehicles and their behaviour is critical to vehicle dynamics. It is believed that this is a field where more research is needed in the future, to define models able to correctly capture the hysteretic behaviour of these materials, considering, for example, frequency- and amplitude-dependent behaviour at the same time and possibly also the influence of temperature and other parameters. Also, the modelling of rubber-to-steel contact and the effect of friction and wear is a subject that deserves further research. Multi-physics modelling of suspension components: passenger train suspensions increasingly tend to be composed of several interacting subsystems, including mechanical, pneumatic/hydraulic and electronic ones. Active and semi-active suspensions are an obvious example of this trend, but the increased popularity of pneumatic suspensions represents another example of the same tendency. This calls for multi-physics approaches, incorporating a mathematical description of all the different physical phenomena influencing the force–deformation characteristics in the suspension. At the same time, the trend in vehicle modelling is increasingly tending towards the definition of one single numerical model able to handle different types of analyses such as running dynamics, comfort and gauging rather than using different ‘specialised’ models for each task. This means that using simple, equivalent mechanical suspension models is likely to become less attractive than in the past, since these models might turn out to be not suited to represent the multifaceted suspension behaviour in a variety of operating conditions. Hence, increased focus towards multi-physics models of suspension components is expected in the future, posing new challenges to consider, for example, the effect of high-frequency dynamics in the control and actuation system and the specificities of the validation process for an active vehicle model. Interaction between testing and modelling of components: as discussed in Section 3 of this paper, many suspension component models rely on the identification of some key model parameters from specific tests performed on the single component. For instance, air springs require the definition of the bellows volume and effective area as a function of height, which is often measured to ensure the required accuracy. Similarly, the parameters of an oil damper model can often be defined in a more accurate and reliable way through the direct measurement of the force–deformation curves rather than from a detailed physical model of the damper. However, the definition of the complete vehicle model shall not entail a large experimental work for component testing to take the best advantage from the use of simulation instead of vehicle testing. Hence, a best deal has to be sought between the proven accuracy of the vehicle model and the effort and time required to set up the model. This also entails the relationship between the validation of single-component models and the validation of the whole-vehicle


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model, as treated in Section 5 of this paper, and how the accuracy of a vehicle model can be reliably quantified. Effect of suspension parameter dispersion: one of the most important perceived advantages of extending the scope of vehicle dynamics simulations, especially in the framework of the vehicle homologation process, is the possibility of considering the effect of non-deterministic dispersion of parameters between nominally identical suspension components, which can be expected on account of the variability implied by the component-manufacturing process. Hence, not only the ‘average’ behaviour of a vehicle but also the maximum deviation from the average shall be quantified to ensure that all vehicles in the fleet will meet performance, safety and comfort requirements. This issue can hardly be handled by means of physical testing, since for obvious reasons, only a very limited number of vehicles will be tested, but it can be solved by the proper use of numerical simulation. Indeed, by the use of appropriate mathematical and statistical tools, it is possible to quantify the dispersion of typical vehicle running dynamics indicators such as safety indexes and comfort indexes based on the assumed scatter of suspension parameters in the fleet described in statistical terms. To this end, the use of Monte Carlo methods [3], FORM/SORM methods and the generalised polynomial chaos theory [132,133] has been proposed, but a well-established procedure is not available yet, and more research work and alternative approaches to treat the problem are likely to appear in the near future. Payload modelling: to date, the effect of payload in rail vehicle models has been considered as mainly affecting the mass properties of the car body. Recent work has shown, however, that the passengers possess damping properties, which shall be considered in dynamic analysis, especially when ride comfort is at stake. For freight vehicles, the modelling of liquid in a tank wagon is also a challenging problem, which has been studied quite extensively in road vehicles and has also been approached for railway vehicles, but surely deserves a more comprehensive investigation.

7.

Conclusions

This paper has reviewed existing models for various types of railway vehicle suspension components, showing the key role that they play in the building of the overall multi-body vehicle model. Considering the increased accuracy of wheel–rail contact models which has been achieved after decades of model refinements, it is likely that future research shall also look at improved models of suspension components to allow for further improvements in the accuracy of rail vehicle models. Although reliable models are now well established for nearly all suspension components in a rail vehicle, the availability of modelling approaches having different levels of complexity, accuracy and ease of use calls for an increased awareness of the implications that choices made at the suspension component modelling level have on the accuracy and trustworthiness of the overall rail vehicle model. Furthermore, no matter how accurate the suspension component model is, it will produce correct results only when fed by an appropriate set of parameter values. Hence, the development of new suspension component models shall be considered together with the definition of appropriate (theoretical and experimental) methods to derive the parameter data for use in the simulations. For most components considered in this paper, the experimental approach based on laboratory tests presently represents the preferred option, but in the future, extended use of refined modelling and simulation techniques can be anticipated to reduce the experimental effort and provide improved theoretical background to the data-definition process.


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This paper has also recognised the track flexibility model as an essential component of the overall vehicle model. Various track models are available, with different levels of complexity from simple lumped parameter ones to a combination of multi-body and finite elements. The usefulness of each approach depends on the frequency range of interest. Simple track models work well for frequencies up to, at least, 20 Hz, but to extend the frequency range of the analysis, a more detailed track model is needed. Finally, model checking and validation represent key steps of the overall model-building process. To date, no clear specification of a validation process is available in the standards in force: although there is probably a wide consensus on the good practices required to build and verify a railway vehicle model, a formal definition of the model verification process is still lacking, as well as the quantitative definition of the expected matching error of the model with respect to line test measurements, so that model validation remains a matter of subjective assessment. It is expected, however, that the work being performed in the DynoTrain project and in working groups attending to the revision of railway standards will in the future achieve a clearer and more objective definition of the model validation process.

Acknowledgements This paper describes the work partly undertaken in the context of the DynoTrain project, Railway Vehicle Dynamics and Track Interactions: Total Regulatory Acceptance for the Interoperable Network (www.triotrain.eu). DynoTrain is a collaborative project – medium-scale focused research project supported by the European 7th Framework Programme, contract number: 234079.

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