A review of dynamics modelling of friction draft gear (1)

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Vehicle System Dynamics
International Journal of Vehicle Mechanics and Mobility
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A review of dynamics modelling of friction draft
gear
Qing Wu, Colin Cole, Shihui Luo & Maksym Spiryagin
To cite this article: Qing Wu, Colin Cole, Shihui Luo & Maksym Spiryagin (2014) A review
of dynamics modelling of friction draft gear, Vehicle System Dynamics, 52:6, 733-758, DOI:
10.1080/00423114.2014.894199
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Vehicle System Dynamics, 2014
Vol. 52, No. 6, 733–758, http://dx.doi.org/10.1080/00423114.2014.894199
A review of dynamics modelling of friction draft gear
Qing Wua∗, Colin Colea, Shihui Luob and Maksym Spiryagina
aCentre for Railway Engineering, Central Queensland University, Rockhampton, Queensland,
Australia; bState Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan,
People’s Republic of China
(Received 12 November 2013; accepted 9 February 2014 )
Longer and heavier trains mean larger in-train forces and more complicated force patterns. Practical
experience indicates that the development of fatigue failure of coupling systems in long heavy trains
may differ from conventional understanding.The friction-type draft gears are the most widely used draft
gears. The ever developing heavy haul transport environment requires further or new understanding
of friction draft gear behaviour and its implications for train dynamics as well as fatigue damage of
rolling stock. However, modelling of friction draft gears is a highly nonlinear question. Especially
the poor predictability, repeatability and the discontinuity of friction make this task more challenging.
This article reviews current techniques in dynamics modelling of friction draft gears to provide a
starting point that can be used to improve existing or develop new models to achieve more accurate
force amplitude and pattern predictions.
Keywords: literature review; friction draft gear; dynamics modelling; train dynamics
1. Introduction
A friction draft gear is an assembly of springs, wedges, plates and other components, which
are enclosed by a housing, as shown in Figure 1. Friction draft gears play two roles in train
operations: transmitting and damping of in-train forces. The very first patent for a friction
draft gear was awarded to George Westinghouse in 1888 and its commercial application
began in 1896.[1,2] The strong points of friction draft gears were widely acknowledged by
the railway industry soon after, meanwhile shortcomings were also revealed.[1–4] Friction
draft gears are of simple construction and low cost because friction components are used. A
negative consequence of using friction components is that they are found to be unstable and
easily worn. Over the past century, numerous studies have been carried out to improve the
performance of friction draft gears; still their shortcomings are not completely fixed. In this
process, new types of draft gear were invented and put into service, including rubber draft
gears, hydraulic draft gears and elastomeric draft gears, etc. Nevertheless, the friction-type
draft gears are still the most widely used and the sheer amount of them now in service suggests
that this situation will last for a considerable period of time.
Simulations of the dynamic performance of draft gears are important for both safety and
commercial concerns. Under heavy haul conditions, coupling systems could be working close
∗Corresponding author: Email: q.wu@cqu.edu.au
© 2014 Taylor & Francis
mailto:q.wu@cqu.edu.au
734 Q. Wu et al.
Figure 1. Schematic of friction draft gear.
to the limits of the available technology. Longer and heavier trains mean larger in-train forces
and more complicated force patterns. Practical experience indicates that the development
of fatigue failure of coupling systems in long heavy trains may differ from conventional
understanding.[5] Therefore, further or even new understanding of draft gear behaviour and
its implications for fatigue damage of rolling stock are required. Most frequently, draft gear
dynamics studies were carried out to improve train safety, with considerable attention being
paid to the accuracy of the maximum forces. Unlike the safety assessment, fatigue assessment
needs to consider all cycles of in-train forces, which means that the fatigue assessment poses
requirements not only to ensure the accuracy of maximum forces but also to that of force
occurrences.
Modelling of draft gears is one of the most difficult and also one of the most important tasks
in longitudinal train dynamics (LTD). It is a highly nonlinear problem complicated by an exten-
sive number of factors: nonlinear stiffness, friction, coupler slack, pre-load and so on. The poor
predictability, repeatability and the discontinuity of friction make this task more challenging.
This article reviews current techniques used in dynamics modelling of friction draft gears.
Efforts are being made to improve existing or develop new models to achieve more accurate
force magnitude and pattern predictions.As the title implies, this article only focuses on dynam-
ics modelling of friction draft gears; modelling of other types of draft gears and non-dynamics
modelling approaches (Finite-Element Analysis, etc.) of friction draft gears are not included.
As linear models and piecewise-linear (PWL) models (without hysteresis) are no longer
taken seriously, they are not discussed. Before the popularisation of computers, modelling
approaches did exist that used electrical analogue and continuum mechanics techniques; these
approaches are dispensable from modern perspective. It is worth mentioning that the friction
wedge suspensions of three-piece bogies have a similar working mechanism to that of friction
draft gears. Modelling and study of suspension wedges are therefore useful for reference.
Considering its complexity, modelling of friction wedge suspensions requires another review.
2. Considerations for draft gear modelling
Figure 2 shows a typical demonstration of the shunting-type (shunting impact or drop hammer
performance) behaviour of a friction draft gear. From this figure, evident differences between
Vehicle System Dynamics 735
Figure 2. Dynamic behaviour (shunting).
loading and unloading processes, i.e., the hysteresis can be seen. Also clear from Figure 2
is the force-peaking behaviour, which means that draft gears become much stiffer whenever
they are approaching the end of loading processes. The force-peaking behaviour is evident
from the literature, such as [6–8]. Cole [7] reported and modelled the differences of draft gear
behaviour between normal train operationsresearch in connected trains. Veh Syst Dyn. 1987;17(s):460–465.
http://www.leconsult.ch/Dokumente/L%E4ngsdynamik_040707.pdf
http://www.leconsult.ch/Dokumente/L%E4ngsdynamik_040707.pdf
Vehicle System Dynamics 757
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Appendix 1. Notation
Notation Description Notation Description
x Draft gear deflection F, Fj Draft gear force or coupler force
xt , xt−�t Draft gear deflections of the current
time-step and previous time-step
respectively
F0 Pre-load
x0 Slack FS Spring force
xmax maximum deflection Fst , Fdy The static and dynamic forces of
draft gear respectively
v, v0 Relative velocity of adjacent
vehicles or hammer velocity
Fe, Fei External force
vt , vt−�t Relative velocity of the current
time-step and previous time-step
respectively
k, k1, k2, k3, k4, k5 Stiffness of draft gear
ve Friction critical speed c, c0, c1 Damping coefficients of draft gear
v1, v2, v3 Relative velocity of internal
components
μ, μ1, μ2, μ3 Friction coefficients
fl Loading characteristics θ , θ1, θ2, θ3 Wedge or plate angles
fu Unloading characteristics h1, h2, h3, h4 Parameters for the expression of
friction coefficient
α, β Dimensionless constants ϕ, ϕ1, ϕ2, ϕ3 Coefficient functions
n Vehicle number
	Introduction
	Considerations for draft gear modelling
	American models
	TOS, TOES and STARCO
	TEDS and ATTIF
	TDA, LEADER and Trip Optimizer
	Hanford engineering development laboratory
	State university of Campinas
	European models
	TRAIN, E-TRAIN and TrainDy
	Polytechnic University of Milan and University of Florence
	Polytechnic University of Bucharest
	Russian and Ukrainian models
	Others
	Chinese models
	Southwest Jiaotong University
	China academy of railway sciences
	Dalian Jiaotong University
	Others
	Australian models
	Western Australia
	Centre for Railway Engineering
	Conclusion
	Acknowledgements
	Referencesand drop hammer tests. Experimental data indicate
that the stiffness of friction draft gears in normal train operations can be much higher than that
revealed from drop hammer tests, especially in small deflection cases.[7] Both force peaking
and stiffness variation phenomena are characterised by changes of stiffness with the change of
velocities; in some publications they are termed as “stiffening”. Hysteresis and stiffening are
both associated with friction, and fundamentally with the directions and values of velocities,
respectively. It is therefore important to incorporate the velocity-dependent friction into draft
gear models.
The nature of friction damping gives draft gears nonlinear hysteresis and results in discon-
tinuities between loading and unloading curves. For most cases, smoothing approximations
or some transitional characteristics are needed to solve this issue mathematically. Otherwise,
switching responses may occur during simulations as reported by Qi et al. [8] and encountered
by Chen [9] (Figure 3). It is evident from Figure 3 that the upper and lower values from 0.4 s to
0.8 s were ‘trimmed’. It can be found by examining the inputs of the model that the upper and
lower values in Figure 3 are values which were prescribed in the input loading and unloading
curves. This phenomenon is actually a kind of numerical instability; without any smoothing
approximation and transitional characteristics, the time history of coupler force was repeat-
edly switching between the loading and unloading curves. This numerical phenomenon is
termed “switching response” in this article and it obviously disagrees with the real behaviour
of draft gears. Note that the absence of transitional characteristics and smoothing approxima-
tions may not always be detected. The authors’ modelling and simulation experience indicates
that the switching response is not so evident in shunting-type simulations in which velocities
change sharply and the transitional phase quickly disappears. But for simulations of normal
train operations, switching responses can be very severe and can even make the simulations
diverge.
For the purpose of train dynamics simulations, draft gear models are usually integrated
with coupling system features (Figure 4). The final model must also consider coupler slack.
Coupling systems are usually simplified and every pair of draft gears is usually modelled in
736 Q. Wu et al.
Figure 3. Switching response.[9]
Figure 4. Integrated draft gear model.
series as a single unit. An integrated draft gear model has to incorporate characteristics that can
simulate the whole coupling system working under both draft and buff conditions. Another
consideration for integrated models is the limiting stiffness after springs are fully compressed
or “bottomed”. The limiting stiffness can be the stiffness of the wagon sill or the overall
stiffness of various wagon components. When installed, draft gears are usually pre-loaded;
for freight cars the pre-load can be as high as 300kN. Pre-load can compensate for the wear of
components to keep the coupling slack from rapidly increasing. It can also improve the train
dynamics performance by limiting the slack actions.
To sum up, a desirable friction draft gear dynamics model should include the above-
discussed elements: velocity-dependent friction, slack, limiting stiffness, pre-load and tran-
sitional characteristics. The first four elements are usually expressed as force–displacement
(F–D) characteristics, so modelling of friction draft gears has two general aspects, F–D and
transitional characteristics.
Vehicle System Dynamics 737
3. American models
3.1. TOS, TOES and STARCO
The well-known Track Train Dynamics (TTD) [10] project (1972–1982) was a milestone of
American railway research. In this project a series of train dynamics models, including lateral,
vertical, longitudinal and interactive dynamics models [11–16] were developed. Regarding
LTD research, the most important results in the TTD project are probably the development
of a detailed LTD model [11] and a software package named Train Operation Simulator
(TOS).[12]
In regard to the draft gear modelling, not enough information was obtained for the authors
to confirm the modelling details in TOS. As for the detailed LTD model,[11] the look-up
table/ intermediate slope (T&S) approach (Figure 4) was used. In the T&S approach, look-up
tables were needed to establish the F–D characteristics and intermediate slopes were used as
transitional characteristics. Generally. two fixed tables are used to match experimental cross-
plots of draft gear (shunting impact or drop hammer performance) characteristics, one named
the loading table (or loading curve) for the loading process and the other named the unloading
table (or unloading curve) for the unloading process (Figure 4). Both curves can be integrated
with coupler slack, fixed pre-load and limiting stiffness. The intermediate slope can “bridge”
the loading and unloading curves whenever the draft gear is switching. Therefore the draft
gear model can be expressed mathematically by Equations (1)–(3).
F(xt , vt) =
{
fl(xt) |xt| ≥ |xt−�| ,
fu(xt) |xt|severity
and the peaking points of forces will change with the impact velocity (Figure 2). During the
TTD project, another model was proposed by Ward et al. [29] to simulate the force-peaking
behaviour. The model was expressed by Equation (4).
F = c1v + k1x + k2 |x| sgn(v) + k3x3, (4)
where F, x and v are the draft gear force, deflection and working velocity, respectively.
In the first term, c1 is the equivalent viscous damping; then in the next two terms, k1 and
k2 simulate the linear hysteresis behaviour; the last term is intended to simulate the force-
peaking behaviour by enlarging the draft gear force mathematically. This model can simulate
the force-peaking behaviour mathematically but is not easy to relate explicitly to draft gear
components.
Along with [29] there are two other serial articles, by Genin et al. [30] and Hsu et al.[31]
While Genin et al. simplified the model of Ward et al. to lower the computational requirement,
Hsu et al. improved it. Hsu et al. found that Equation (4) cannot simulate the actual behaviour
of friction draft gears because the cubic term will give identical loading curves for every
impact speed. The cubic term works on the unloading curve as well, which means that the
unloading simulation cannot follow the actual linear behaviour. Hsu et al. reasoned that the
actual force-peaking behaviour is a result of the transition from kinetic to static friction as the
internal components come to rest in a draft gear. In [31], an improved draft gear model was
presented, as in Equation (5).
F =
{
F0 + k1x + k2x exp
(
−v
ve
)
, v > 0,
F0 + k3x, vshown in Figure 6. The smoothness was determined by the relative velocity
of adjacent vehicles; the higher the velocity, the smoother the transitional curve.
Vehicle System Dynamics 741
Figure 6. Smoothing effect of E-TRAIN.[45]
Figure 7. Smoothing characteristic of TrainDy.[46]
Look-up tables were also applied in TrainDy but this software used a different smoothing
approach. The draft gear model was expressed by Equation (9).
F =
⎧⎪⎨
⎪⎩
fl(x), v vunload,
(9)
where vload and vunload are limiting speeds. When the relative speed is between vload and vunload
the draft gear force is a composition of forces which are calculated from the loading and the
unloading curves separately. The composite coefficient, ϕ(v), is plotted in Figure 7.
4.2. Polytechnic University of Milan and University of Florence
Buffer–hook systems, as shown in or similar to Figure 8, are widely used in Europe. Unlike
the auto-coupler systems, the primary energy dissipating devices in these systems are buffers
742 Q. Wu et al.
Figure 8. Buffer–hook coupling system.
which work only under buff conditions. Laboratory and field tests concerning the characteri-
sation of buffers were reported by Belforte et al. [52] and Cheli et al.[53] In an LTD program
named TrainSet Dynamics Simulator (TsDyn),[52] which was developed by the Polytechnic
University of Milan, buffer models were expressed as in Equation (10).
F = F0 + (k1x + k2x2 + k3x3 + k4x4 + k5x5) + (c1x + c0)v, (10)
where ki(i = 1 − 5) and ci(i = 1 − 2) are stiffness and damping coefficients, respectively;
F0 represents the pre-load; other features of the buffer were described as a nonlinear visco-
elastic element. After some laboratory tests, the model in TsDyn was further detailed by
Cheli et al.[53] Their model consists of five elements: a linear spring which models the pre-
load; a fourth-order nonlinear spring, which is used to model the quasi-static response of
the elastomer; two Maxwell elements which simulate the effects of dynamic loads and high-
frequency excitations; a friction element which models the effect of internal friction. The
friction element was expressed by Equation (11).
Ff = μ1M(1 + μ2x) arctan(μ3v), (11)
where Ff is the friction force; μ1, μ2, μ3 are friction coefficients; M is the moment on the buffer
head. In Equation (11), a smoothing approach was applied, i.e., the arc tangent function. The
arc tangent function can be used as a smoothing approach because it has the property as
shown in Figure 9. The detailed buffer–hook system model has taken various aspects of buffer
responses into consideration; it can give accurate buffer responses under different working
conditions. A challenging point of the detailed model is that nearly 20 parameters have to be
determined before it works.
A novel look-up table approach was developed by Pugi et al.[54] to model the buffer sys-
tems. In this approach, the third dimension, speed, was added to traditional force–displacement
look-up tables (Figure 10). Both slack and limiting stiffness were incorporated. In subsequent
Vehicle System Dynamics 743
Figure 9. Smoothing functions.
Figure 10. 3D look-up table (Source: [54], with permission).
studies by Pugi et al.,[55] the model expressed by Equation (10) was also used. With
different parameters, Equation (10) can also be used to model the hook of buffer–hook
systems.
744 Q. Wu et al.
4.3. Polytechnic University of Bucharest
Extensive studies of buffer–hook coupling systems were conducted at the Polytechnic Uni-
versity of Bucharest, Romania. In these studies, Equation (12) was used by Cruceanu et al.
[56–59] to simulate buffer–hook systems.
F = −[k1 − k2tanh(αv)]x(1 − β)
2
+ [k3 − k4tanh(αv)]x(1 + β)
2
, (12)
where the first part was used to model buffers and the second part to model hooks. Because
hyperbolic tangent functions have the characteristics as shown in Figure 9, they can be used for
the simulation of hysteresis and for smoothing. The coefficient, α, in the hyperbolic tangent
functions allows adjustments of the smoothing range. Note that Equation (12) can only be used
to simulate buff conditions of buffer–hook coupling systems. Under buff conditions, longitu-
dinal forces taken by buffers and hooks are not equal, so the coefficient,β, was incorporated
to tune the distribution of longitudinal forces.
To develop a model that can simulate both buff and draft conditions, Craciun et al. [60] used
Equation (13) to model buffer–hook systems.
F = 1
2 (1 + sgn x)(k1x + k2 |x| sgn v) + 1
2 (1 − sgn x)(k3x + k4 |x| sgn v)
+ 1
2 k5(1 + sgn(x − xmax)k3), (13)
where the first two parts were used to simulate buffers and hooks respectively, and the third part
was used to simulate the limiting stiffness after buffers were fully compressed. It can be seen
from Equations (12) and (13) that both models have linear F–D characteristics. Improvements
in Equation (13) were achieved by the sign functions in the first two parts, which can make
one of the multipliers equals either 0 or 1, further to disable or enable a corresponding part.
In [60], modelling of transitional characteristics was not reported.
A non-smooth draft gear modelling approach was proposed by Oprea et al.[61–63] The
draft gear model was expressed by Equation (14).
F =
⎧⎪⎨
⎪⎩
F0+kx
1+α|v| sgn(v), v �= 0,
−Fe, |Fe| ≤ |F0 + kx| ∩ v = 0,
−sgn(Fe)(F0 + kx) otherwise,
(14)
where F0 is the pre-load; Fe is the external force to which the system is subjected; and k is
draft gear static stiffness. The denominator part of the first case in Equation (14) integrates
the effects of velocities. The coefficient α allows adjustments of the dynamic characteristics
of the model. The sign function allows modelling of hysteresis. For the cases when v = 0,
draft gear forces equal to the counterforce of the sum of external forces, but draft gear forces
cannot exceed the boundary condition, i.e., the static force at the current deflection. The non-
smooth modelling approach of transitional characteristics has advantages in dealing with the
static friction forces. Theoretically, non-smooth models have more accurate responses than
smoothed models, such as Equation (12) which have adopted certain assumptions.
In [63], two smoothing approaches were also introduced; the first approach was to use
the hyperbolic tangent function as in Equation (12). Another approach was expressed by
Equation (15). The difference between these two smoothing approaches is shown as in Figure 9.
F =
{
F0+kx
1+α|v| sgn(v)min(αv, 1), v ≥ 0,
F0+kx
1+α|v| sgn(v)max(αv, −1), vthis step, draft gear forces can be calculated through Equation (18).
F = ϕ(v)Fs(x), (18)
where Fs is the spring force. The spring force function can be nonlinear so as to incorporate
nonlinear components, such as elastomer or polymer springs. Nikolskii also suggested an
expression of friction coefficients (Equation (19)) and its parameters (h1, h2, h3, h4) for
different types of draft gear.
μ(v) = (h1 + h2v) exp(−h3v) + h4. (19)
The detailed modelling approach proposed by Nikolskii is on the basis of practical structures
and working principles of friction draft gears; it is widely used in Russia and Ukraine [65–
71]. In a doctoral programme undertaken by Fatkov,[71] a draft gear analysis software was
developed based on tests of elastomeric elements. In practice, the friction coefficients on
different surfaces in draft gears were generally assumed uniform.
In [64], Nikolskii also presented a simplified friction draft gear model which was expressed
as Equation (20).
F = F1(x)
1 + α |v| sgn(v), (20)
where F1(x) represents the static characteristics of friction draft gears and the denominator
part has the same role as that of the first case in Equation (14). The recommended values of
α for different types of draft gears were documented in [64]. In Refs. [64–69], modelling of
transitional characteristics were not reported; this is probably because they are mainly focused
on the study of impact or static characteristics of friction draft gears and train modelling was not
involved. In Refs. [70,71], where train dynamics was simulated the transitional characteristics
were still not reported. Examination of the results of [70,71] indicates that a transitional
characteristic similar to the intermediate slope was used.
746 Q. Wu et al.
Figure 11. Stick–slip phenomenon by PWL look-up tables.[73]
In another LTD monograph, Grebenyuk [72] used Equations (21)–(23) to simulate friction
draft gears.
F(xt , vt) =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
0, |xt| ≤ x0,
k1(xt − x0), x0of Budapest (now Budapest University of
Technology and Economics) since the 1970s.[83] Linear models were used in [83,84]; non-
linear friction draft gear models along with cylindrical crash element models were presented
in [85] to study LTD during train collisions. The nonlinear friction draft gear models were
similar to the one expressed by Equation (28), differentiated by PWL look-up tables without
coupler slack.
Vehicle System Dynamics 749
5. Chinese models
5.1. Southwest Jiaotong University
From 1986 to 1990, in order to improve the freight transport capacity in China, a government
project named Heavy Train Dynamics (HTD) project was conducted by Southwest Jiaotong
University (SWJTU). Similar to the TTD project of America, in the HTD project a series of
train dynamics models along with a Train Driving Simulator were developed by Rail Vehicle
Institute (now an institute of State Key Laboratory of Traction Power).[86–92] With regard to
the draft gear modelling, most models in this project and its subsequent studies [93,94] are T&S
models. Due to the limited computational capacity at that time, simplified draft gear models
[95] and their corresponding train models [96] were also developed. By introducing equivalent
linear draft gear models and equivalent mass train models the computational efficiency was
significantly improved. Validations of these simplified models were conducted on head-end
unit trains.
A new algorithm named the circular-variable algorithm was designed by Zhang et al. [97,
98] to simulate train dynamics. In these studies look-up tables were used to model F–D
characteristics of draft gears, but the transitional characteristics were not reported. Evident
switching responses of draft gear forces can be seen in [97]. Sun et al. [99] added damping
into the conventional T&S approach. The model proposed by Sun et al. is similar to that in
[72]; both models can overcome the problem caused by the absence of damping in transitional
characteristics as explained in [78]. Wu et al. [100–102] also applied look-up tables to model
draft gear F–D characteristics (Equation (29)). In the model proposed by Wu et al., a limiting
velocity, ve, was used to smooth discontinuities.
F(xt , vt) =
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
fu(xt), |vt| > ve ∩ vtxt ≥ 0,
fl(xt), |vt| > ve ∩ vtxt xt−�t ,
fu(xt) + (F(xt−�t , vt−�t) − fu(xt))exp
(−|xt−xt−�t |
β
)
, xt ≤ xt−�t.
(30)
It used look-up tables and a smoothing approximation. The smoothing approximation was
documented in [105] and originally used to smooth the problem of dry friction in truck leaf
springs.
5.3. Dalian Jiaotong University
Dalian Jiaotong University (DJTU) also developed an LTD simulation software package.[106,
107] In this software, the polynomial fitting approach was used to model friction draft gears.
750 Q. Wu et al.
The model was expressed by Equation (31).
F = k(x, v)x + c(x, v)v, (31)
where the stiffness and damping are functions of deflections and velocities rather than being
given two constant values. The polynomial approach can produce hysteresis loops with differ-
ent sizes, which is desirable to model rubber or elastomeric elements. There are two challenging
points in this model. First, it is not easy to find suitable mathematical expressions and balanced
parameters because of the strong nonlinear nature of friction draft gears. Second, without dry
friction models, some unique features of dry friction cannot be fully revealed.
An intelligent system model of friction draft gears was proposed by Hu et al. [108]; this
model involves the derivation of draft gear forces from measured data using neural network
techniques. The structure of the model includes three layers, these being the input, hidden and
output layers. Inputs in the input layer are velocities and deflections, the outputs in the output
layer are draft gear forces. The advantage of an intelligent model is that the model is shaped
by measured data alone, which can minimise mathematical idealisation. The disadvantage is
that the final model is limited to the range of measured behaviour.
5.4. Others
Researchers from Beijing Jiaotong University used look-up tables to model draft gear F–D
characteristics, but the transitional characteristics were not reported [109–111]. The approach
presented in [104] was also used in [112]. In a vehicle system dynamics monograph authored
by Wang [113], Equation (32) was used to model the F–D characteristics.
F =
(
k1 + k2
2
+ k1 − k2
2
sgn(v)
)
x, (32)
where k1 and k2 are linear stiffness for loading and unloading, respectively. Prior to [104]
being published, Wang [113] proposed to modify the smoothing approach used in [105] to
address the discontinuity of friction draft gears.
A draft gear model was presented by Jin et al. [114] to include the force-peaking behaviour
which was measured from shunting operations. This draft gear model also used the smoothing
approach documented in [105]. Regarding the F–D characteristics, the modelling approach in
[114] was similar to that in [31] but with PWL stiffness. In [114] a suggested value for the
critical velocity constant, ve, was derived from analyses of the Stribeck Friction model. It is
noted that both [31] and [114] suggested that a value of 0.6 m/s for ve was reasonable.
Qi et al. [8] presented a model that incorporated the effects of velocities into the conventional
look-up table approach. The model was expressed as Equation (33).
F =
{(
1 + 0.75v
ve
)
fl(x) loading,
fu(x) unloading,
(33)
where fl and fu are the static loading and unloading characteristics, respectively. This model
may not be completely justified because it will make the dynamic draft gear stiffness stiffer
than that of static cases, which does not agree with the traditional understanding of friction
draft gears. But Qi et al. introduced an important improvement for the discontinuity smoothing
strategy. They pointed out that the traditional discontinuity smoothing approach will cause
considerable error in estimating the maximum coupler forces (Figure 14). Generally, the max-
imum coupler forces occur in the vicinity of the v = 0 part of the loading quadrant. Allocating
Vehicle System Dynamics 751
Figure 14. Smoothing phase.
the smoothing range to the loading quadrant (traditional approach in Figure 14) could result in
the maximum coupler forces being underestimated. This problem can be alleviated by chang-
ing the smoothing range from [-v0, v0] to [-v0, 0] as shown in Figure 14. Note that the improved
approach still retains an error in the unloading quadrant, but it is much less significant.
6. Australian models
6.1. Western Australia
In 1973, sponsored by Mount Newman Mining Co., a train simulator, probably the first train
simulator in Australia, was developed at the University of Western Australia.[115–118] In
this simulator, a T&S draft gear model was used and the F–D characteristics of the draft gear
model were expressed by Equation (34).
F =
{
ϕ1(μ, θ1, θ2)kx loading,
ϕ2(μ, θ1, θ2)kx unloading,
(34)
where ϕ1 and ϕ2 are transfer coefficients similar to Equation (16), and parameters in
Equation (34) are constants. Intermediate stiffness (slope) which had the series stiffness of
two couplers was used to provide a linear transition between the loading and unloading curves.
As the computational ability was limited at that time, two simplifiedtrain models were also
developed. The first of these simplified trains by lumping several vehicles together, and the
second one was a locomotive simulator known as the Train Performance Calculator which was
mainly used to predict the running time and velocity profile.[116] In 1985, the detailed version
of train model [115] was upgraded to a Train Driving Simulator and installed for Queensland
Railways by ACET Ltd. (now MRX Technology).[119]
During some field tests around 1989, the third type of in-train forces was identified, i.e., the
longitudinal vibration forces.[120] Prior to this finding, in-train forces were generally divided
into steady forces and impact forces. Studies discussed in [120] resulted in a new contribution
to friction draft gear modelling, which was the locked stiffness. The locked stiffness was
752 Q. Wu et al.
defined as the longitudinal system stiffness of the whole train when the longitudinal vibration
was observed; and it was integrated into friction draft gear models to express the behaviour
when draft gears were locked. From the transitional characteristics perspective, the locked
stiffness can act as an intermediate slope and has physical meaning rather than being just a
mathematical smoothing strategy.
6.2. Centre for Railway Engineering
In 1994, the Centre for Railway Engineering (CRE) and Queensland Rail started joint research
into train dynamics. Part of the first stage achievement was published in Cole’s doctoral
thesis.[121] Cole introduced three approaches to model F–D characteristics of draft gears,
these being the look-up table approach, stiffness function approach and intelligent system
model. Typical examples of the first two approaches are Equations (1) and (8), respectively.
An example of the third approach can be found in [108].
Various aspects of draft gear behaviour were identified in [121] by examining measured
data. In order to cope with the wide spectrum of draft gear behaviour, a wedge–spring model
shown as Figure 15 was presented. This model is a velocity- and displacement-dependent
model and its F–D characteristics can be expressed as Equation (35).
F = Fs(x)tan(θ)
[tan(θ) ± μ(v)] . (35)
Coincidently, Cole’s wedge–spring model uses essentially the same expression of F–D char-
acteristics as that of Barbosa [43]. The locked stiffness identified in [120] was confirmed again
later [7] and incorporated into Cole’s model so as to widen the draft gear model behaviour
range. The longitudinal vibration force behaviour documented in [120] has been reproduced
in [121–123] by simulation. Cole’s model was reported as being able to match draft gear
responses for both normal train operations and shunting-type impacts. In recent years, the
wedge–spring model has been widely used not only by CRE researchers [122–126] but also
by external researchers.[127–130] Train dynamics simulation software (CRE-LTS) [131] and
intelligent train monitor devices [132,133] have been developed at CRE. Researchers in [127–
129] basically adapted Equation (35) to model F–D characteristics, but they handled the
discontinuity issue differently by smoothing the friction coefficient around the v = 0 point.
Wu et al. [127] used the improved linear smoothing strategy proposed by Qi et al. in [8], while
Nasr and Mohammadi [128,129] used hyperbolic tangent functions shown as Equation (36).
μ(v) =
[
1 +
(
μ1
μ2
− 1
)
exp(−2 |v|)
]
tanh(βv)μ2, (36)
Figure 15. Simplified friction draft gear model.[121]
Vehicle System Dynamics 753
where μ1 and μ2 are static and kinetic friction coefficient constants, respectively, and β is a
parameter to control the smoothing range of the friction coefficient.
7. Conclusion
Accurate maximum forces and force occurrences are required for train dynamics as well as
fatigue assessments. Possible improvements of draft gear modelling can start with a review
and summary of available techniques. This paper attempts to present a comprehensive review
of friction draft gear modelling.
For the purpose of train dynamics simulations, friction draft gear models require the inclu-
sion of velocity-dependent friction, slack, limiting stiffness and transitional characteristics.
The look-up table approach is the most widely used approach to model draft gear F–D charac-
teristics. The fixed tables have usually excluded the implications of velocity values, so some
dynamic characteristics are missed in the simulation results. The polynomial fitting approach
is desirable for modelling of rubber or elastomeric elements, as it can produce hysteresis loops
with different shapes and sizes, but for friction draft gear the dry friction model is necessary.
The intelligent system approach can minimise mathematical idealisations, but the final model
is limited to the range of measured behaviour. With reasonable simplifications, the wedge–
spring models look promising. They are based on the basic working mechanism of friction
draft gears and they can include the elements which relate to the hardware components.
Transitional characteristics (smooth or non-smooth) are necessary to address the issue of
discontinuity.A variety of approaches are available from the literature. The most popular one is
the intermediate slope approach. It would be more rational to combine the intermediate slope
with a certain level of damping. The locked stiffness can act as an intermediate slope, and
it has physical meaning rather than just a mathematical smoothing strategy. The non-smooth
approach has advantages in dealing with the static friction forces; the corresponding results
can be closer to the actual draft gear behaviour. Note that computational efficiency issues may
arise for large-scale whole-train trip simulations.
It is clear that one single paper cannot include all existing friction draft gear models, and it
is safe to conclude that there still remains scope for further model development and validation
of models of friction draft gears.
Acknowledgements
Writing this paper has received support from researchers from eight counties who are too numerous to list. We are
grateful to those who we contacted when seeking information and/or references. The first author is the recipient of an
International Postgraduate Award (IPRA) and University Postgraduate Research Award (UPRA), Central Queensland
University, Australia.
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