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SENSITIVITY ANALYSIS OF THE CRITICAL SPEED IN RAILWAY
Daniele Bigoni, Hans True, Allan P. Engsig-Karup
Department of Applied Mathematics and Computer Science, Technical University of Denmark
Matematiktorvet, building 303B
DK-2800 Kgs. Lyngby, Denmark
We present an approach to global sensitivity analysis aiming at the reduction of its computational cost without
compromising the results. The method is based on sampling methods, cubature rules, High-Dimensional Model
Representation and Total Sensitivity Indices. The approach has a general applicability in many engineering fields
and does not require the knowledge of the particular solver of the dynamical system. This analysis can be used as
part of the virtual homologation procedure and to help engineers during the design phase of complex systems.
The method is applied to a half car with a two-axle Cooperrider bogie, in order to study the sensitivity of the
critical speed with respect to suspension parameters. The importance of a certain suspension component is
expressed by the variance in critical speed that is ascribable to it. This proves to be useful in the identification of
parameters for which the exactness of their values is critically important.
The past couple of decades have seen the advent of computer simulations for the study of deterministic dynamical
systems arising from any field of engineering. The reasons behind this trend are both the enhanced design
capabilities during production and the possibility of understanding dangerous phenomena. However, deterministic
dynamical systems fall short in the task of giving a complete picture of reality: several sources of uncertainty can
be present when the system is designed and thus obtained results refer to single realizations, that in a probabilistic
sense have measure zero, i.e. they never happen in reality. The usefulness of these simulations is however proved
by the achievements in Computer-Aided Design (CAD).
The studies of stochastic dynamical systems allow for a wider analysis of phenomena: deterministic systems can
be extended with prior knowledge on uncertainties with which the systems are described. This enables an
enhanced analysis and can be used for risk assessment subject to such uncertainties and is useful for decision
making in the design phase.
In the railway industry, stochastic dynamical systems are being considered in order to include their analysis as a
part of the virtual homologation procedure , by means of the framework for global parametric uncertainty
analysis proposed by the OpenTURNS consortium. This framework splits the uncertainty analysis task in four
A. Deterministic modeling and identification of Quantities of Interest (QoI) and source of uncertainties
B. Quantification of uncertainty sources by means of probability distributions
C. Uncertainty propagation through the system
D. Sensitivity analysis
Railway vehicle dynamics can include a wide range of uncertainty sources. Suspension characteristics are only
known within a certain tolerance when they exit the manufacturing factory and are subject to wear over time that
can be described stochastically. Other quantities that are subject to uncertainties are the mass and inertia of the
bodies, e.g. we don’t know exactly how the wagon will be loaded, the wheel and track geometries, that are subject
to wear over time, and also external loadings like wind gusts.
In this work the QoI will be the critical speed of a fixed half-wagon with respect to uncertain suspension
components (step A). The deterministic and stochastic models will be presented in section 2. Step B requires
measurements of the input uncertainty that are not available to the authors, so the probability distribution of the
suspension components will be assumed to be Gaussian, without losing the generality of application of the
methods used in C and D.
Techniques for Uncertainty Quantification (UQ) will be presented in section 3.1. They have already been applied
in  and  to perform an analysis of Uncertainty propagation (step C). They will turn useful also in section 3.2
and 3.3 for the sensitivity analysis technique to be presented (step D). This is based on Total Sensitivity Indices
(TSI) obtained from the ANOVA expansion of the function associated to the QoI . Section 4 will contain the
results of such analysis.
2. THE VEHICLE MODEL
In this work we will consider a fixed half wagon equipped with a Cooperrider bogie, running on tangent track with
wheel profile S1002 and rail UIC60. The position of the suspension components is shown in Fig. 1. In  a
framework for the simulation of the dynamics of complete wagons running on straight and curved tracks has been
implemented and tested based on the Newton-Euler formulation of the dynamical system:
F and i
M are respectively the forces and the torques acting on the bodies, m and
J are the mass and
inertia of the bodies, a
are the acceleration and the angular acceleration of the bodies.
In this work the wagon will be fixed in order to alleviate the lateral oscillations during the hunting motion that
would, in some cases, break the computations. The mathematical analysis and the generality of the methods
proposed are not weakened by this assumption, even if the results may change for different settings. The wheel-rail
interaction is modeled using tabulated values generated with the routine RSGEO  for the static penetration at the
contact points. These values are then updated using Kalker’s work  for the additional penetrations. The creep
forces are approximated using Shen-Hedrick-Elkins nonlinear theory . The complete deterministic system
is nonlinear, non-smooth, and it has 28 degrees of freedom.
2.1 Nonlinear dynamics of the deterministic model
The deterministic dynamics of the complete wagon with a couple of Cooperrider bogies were analyzed in . The
stability of the half-wagon model considered in this work is characterized by a subcritical Hopf-bifurcation
at smvL/114, as it is shown in Fig. 2a, and a critical speed smvNL /47.50
. The critical speed is found
using a continuation method from the periodic limit cycle detected at a speed greater than the Hopf-bifurcation
v. In order to save computational time, we try to detect the periodic limit cycle at speeds lower than L
perturbing the system as described in . This is the approach that we will take during all the computations of
critical speeds in the next sections. The criterion used in order to detect the value of the critical speed is based on
the power of the lateral oscillations in a s1 sliding window of the computed solution. Fig. 2b shows how this
criterion is applied.
Fig. 1 The half-wagon equipped with the Cooperrider bogie.
2.2 The stochastic model
In the following we will assume that the suspension characteristics are not deterministically known. Rather, they
are described by probability distributions stemming from the manufacturing uncertainty or the wear.
If experimental information is available, then some standard distributions can be assumed and an optimization
problem can be solved in order to determine the statistical parameters of such distributions (e.g. mean, variance,
etc.). Alternatively the probability density function of the probability distribution can be estimated by Kernel
Due to the lack of data to the authors, in this work the probability distributions associated with the suspension
components will be assumed to be Gaussian around their nominal value, with a standard deviation of 5%. We
define Zto be the d-dimensional vector of random variables
iii Nz 1
describing the distributions of
the suspension components, where d is called the co-dimension of the system. The stochastic dynamical system is
then described by
3. SENSITIVITY ANALYSIS
Sensitivity analysis is used to describe how the model output depends on the input parameters. Such analysis
enables the user to identify the most important parameters for the model output. Sensitivity analysis can be viewed
as the search for the direction in the parameter space with the fastest growing perturbation from the nominal
One approach of sensitivity analysis is to investigate the partial derivatives of the output function with respect to
the parameters in the vicinity of the nominal output. This approach goes by the name of local sensitivity analysis,
stressing the fact that it works only for small perturbations of the system.
When statistical information regarding the parameters is known, it can be embedded in the global sensitivity
analysis, which is not restricted to small perturbations of the system, but can handle bigger variability in the
parameter space. This is the focus of this work and will be described in the following sections.
3.1 Uncertainty Quantification
The solution of (3) is
Zu ,t, varying in the parameter space. In uncertainty quantification we are interested in
computing the density function of the solution and/or its first moments, e.g. mean and variance:
ddFttt zzuZuE Zu Z
2zzuZuV Zuu ZdFtttt d
(a) bifurcation diagram (b) Critical speed detection criteria
Fig. 2 Left: complete bifurcation diagram where the folding point is detected by continuation (ramping) method
from the periodic limit cycle. Right: criterion for the determination of the critical speed based on the power of the
lateral oscillations in a sliding window. LB, LLW and LTW stand for the bogie frame, the leading wheel set and the
trailing wheel set respectively.
F are the probability density function (PDF) and the cumulative distribution function
(CDF) respectively. Several techniques are available to approximate these high-dimensional integrals. In the
following we present the two main classes of these methods.
Sampling based methods
The most known sampling method is the Monte Carlo (MC) method, which is based on the law of large numbers.
Its estimates are:
tt uuu Zu
Zare realizations sampled randomly within the probability distribution of Z. The MC method has
a probabilistic error of
MO /1 , thus it suffers from the work effort required to compute accurate estimates.
However the MC method is very robust because this convergence rate is independent of the co-dimensionality of
the problem, so it’s useful to get approximate estimates of very high-dimensional integrals.
Sampling methods with improved convergence rates have been developed, such as Latin Hypercube sampling and
Quasi-MC methods. However, the improved convergence rate comes at the expense of several drawbacks, e.g., the
convergence of Quasi-MC methods is dependent of the co-dimensionality of the problem and Latin Hypercube
cannot be used for incremental sampling.
The integrals in (4) can also be computed using cubature rules. These rules are based on a polynomial
approximation of the target function, i.e. the function describing the relation between parameters and QoI, so they
have superlinear convergence rate on the set of smooth functions. Their applicability is however limited to
low-co-dimensional problems because cubature rules based on a tensor grid suffer the curse of dimensionality, i.e.
if mis the number of points used in the one dimensional rule and dthe dimension of the integral, the number of
points at which to evaluate the function grow as )( d
mO . They will however be presented here because they
represent a fundamental tool for the creation of high-dimensional model representations that will be presented in
the next section.
Let Z be a vector of independent random variables in the probability space ),,( Z
BD F, where d
RD , Bis
the Borel set constructed on D and Z
F is a probability measure (i.e. the CDF of Z). For this probability
measure we can construct orthogonal polynomials
for di 1
, that form a basis for each
independent dimension of D . The tensor product of such a basis forms a basis for D. From these orthogonal
polynomials, the Gauss quadrature points and weights
,,,, ,wz can be derived using the
Golub-Welsch algorithm , obtaining approximations for (4):
Gauss quadrature rules of order Nare accurate for polynomials of order up to degree 12 N. This high accuracy
comes at the expense of the curse of dimensionality due to the use of tensor products in high-dimensional
integration. This effect can be alleviated by the use of Sparse Grids techniques proposed by Smolyak  that use
an incomplete version of the tensor product. However, in the following section we will see that we can often avoid
working in very high-dimensional spaces.
3.2 High-dimensional model representations
High-dimensional models are very common in practical applications, where a number of parameters influence the
dynamical behaviors of a system. These models are very difficult to handle, in particular if we consider them as
black-boxes where we are only allowed to change parameters. One method to circumvent these difficulties is the
HDMR expansion , where the high-dimensional function RD :f,n
RD is represented by a function
decomposed with lower order interactions: .),,,(),()()( 21,,2,10
iii xxxfxxfxffxf (7)
This expansion is exact and exists for any integrable and measurable function
, but it is not unique. There is a
rich variety of such expansions depending on the projection operator used to construct them. The most used in
statistics is the ANOVA-HDMR where the low dimensional functions are defined by
,)()()()()( 0xfPxdxfxfPxf A
1is the hypercube excluding indices l
is the product measure
iii xx )()(
. This expansion can be used to express the total variance of
, by noting that
However, the high-dimensional integrals in the ANOVA-HDMR expansion are computationally expensive to
An alternative expansion is the cut-HDMR, that is built by superposition of hyperplanes passing through the cut
,)()()( 00 yfxfPxf CC
,)()()()( 0xfPxfxfPxf C
where ),,( 1
is the function )(xf with all the remaining variables set to y. This expansion
requires the evaluation of the function
on lines, planes and hyperplanes passing through the cut center.
If cut-HDMR is a good approximation of
, i.e. considering up to
-terms interactions in (7), such
expansion can be used for the computation of ANOVA-HDMR in place of the original function. This reduces the
computational cost dramatically: let nbe the number of parameters and sthe number of samples taken along each
direction (being them MC samples or cubature points), then the cost of constructing cut-HDMR is
iin n (11)
3.3 Total Sensitivity Index
The main task of Sensitivity Analysis is to quantify the sensitivity of the output with respect to the input. In
particular it’s important to know how much of this sensitivity is accountable to a particular parameter. With the
focus on global sensitivity analysis, the sensitivity of the system to a particular parameter can be expressed by the
variance of the output associated to that particular input.
One approach to this question is to consider each parameter separately and to apply one of the UQ techniques
introduced in section 3.1. This approach goes by the name of one-at-a-time analysis. This technique is useful to get
a first overview of the system. However, this technique lacks an analysis of the interaction between input
parameters, which in many cases is important.
A better analysis can be achieved using the method of Sobol’ . Here single sensitivity measures are given by
for ,1 1nii l
where Dand l
1are defined according to (9). These express the amount of total variance that is accountable to
a particular combination l
1of parameters. The Total Sensitivity Index (TSI) is the total contribution of a
particular parameter to the total variance, including interactions with other parameters. It can be expressed by
Sis the sum of all l
1that does not involve parameter i.
These total sensitivity indices can be approximated using sampling based methods in order to evaluate the integrals
involved in (9). Alternatively,  suggests to use cut-HDMR and cubature rules in the following manner:
1. Compute the cut-HDMR expansion on cubature nodes for the input distributions.
2. Derive the approximated ANOVA-HDMR expansion from the cut-HDMR.
3. Compute the Total Sensitivity Indices from the ANOVA-HDMR.
This approach gives the freedom of selecting the level of accuracy for the HDMR expansion depending on the
level of interaction between parameters. The truncation order
of the ANOVA-HDMR can be selected and the
accuracy of such expansion can be assessed using the concept of “effective dimension” of the system: for 1
the effective dimension of the integrand
is an integer
where tis a multi-index l
1and tis the cardinality of such multi-index. The parameter q is chosen based
on a compromise between accuracy and computational cost.
4. SENSITIVITY ANALYSIS ON RAILWAY VEHICLE DYNAMICS
The study of uncertainty propagation and sensitivity analysis through dynamical systems is a computationally
expensive task. In this analysis we adopt a collocation approach, where we study the behaviors of ensembles of
realizations. From the algorithmic point of view, the quality of a method is measured in the number of realizations
needed in order to infer the same statistics. Each realization is the result of an Initial Value Problem (IVP)
computed using the program DYnamics Train SImulation (DYTSI) developed in , where the model presented in
section 2 has been set up and the IVP has been solved using the Explicit Runge–Kutta–Fehlberg method ERKF34
. An explicit solver has been used in light of the analysis performed in , where it was found that the hunting
motion could be missed by implicit solvers, used with relaxed tolerances, due to numerical damping. In particular
implicit solvers are frequently used for stiff problems, like the one treated here, because their step-size is bounded
Fig. 3: Histogram of the Critical Speed obtained using
Latin Hypercube sampling and the estimated density
function (KDE) obtained using Kernel Smoothing.
Fig. 4: Pie plot of the Total Sensitivity Indices on the
reduced stochastic model, where only the most
influential components are analyzed. (See Table 1 for an
explanation of the notation used)
by accuracy constraints instead of stability. However, the detection of hunting motion requires the selection of
strict tolerances, reducing the allowable step-sizes and making the implicit methods more expensive than the
explicit ones. Since the collocation approach for UQ involves the computation of completely independent
realizations, this allows for a straightforward parallelization of the computations on clusters. Thus, 25 nodes of the
DTU cluster have been used to speed up the following analysis.
The first step in the analysis of a stochastic system is the characterization of the probability distribution of the QoI.
Since the complete model has co-dimension 24, a traditional sampling method, among the ones presented in
section 3.1, is the most suited for the task of approximating the integrals in eq. (4). Fig. 3 shows the histogram of
the computed critical speeds with respect to the uncertainty in the suspension components. In order to speed up the
convergence, we used 200 samples generated with the Latin Hyper Cube method . Kernel smoothing  has
been used to estimate the density function according to this histogram. The estimated mean and variance are
and 22 /07.4 sm
Suspension One-at-a-time ANOVA ANOVA - Refined
Tot. Sensitivity v
PSLL_LEFT_K1 0.00 0.03 0.01
PSLL_LEFT_K2 0.06 0.18 0.06 0.18 0.09
PSLL_LEFT_K3 0.02 0.13 0.04 0.14 0.07
PSLL_RIGHT_K1 0.00 0.05 0.02
PSLL_RIGHT_K2 0.06 0.17 0.06 0.22 0.11
PSLL_RIGHT_K3 0.03 0.17 0.06 0.10 0.05
PSLT_LEFT_K1 0.00 0.02 0.01
PSLT_LEFT_K2 0.54 1.71 0.56 1.29 0.63
PSLT_LEFT_K3 0.14 0.20 0.07 0.11 0.05
PSLT_RIGHT_K1 0.00 0.05 0.02
PSLT_RIGHT_K2 0.55 1.73 0.56 1.22 0.59
PSLT_RIGHT_K3 0.03 0.13 0.04 0.17 0.08
SSL_LEFT_K4 0.00 0.01 0.00
SSL_LEFT_K5 0.00 0.01 0.00
SSL_LEFT_K6 0.00 0.02 0.01
SSL_LEFT_D1 0.00 0.02 0.01
SSL_LEFT_D2 0.02 0.04 0.01
SSL_LEFT_D6 0.00 0.02 0.01
SSL_RIGHT_K4 0.00 0.01 0.00
SSL_RIGHT_K5 0.00 0.00 0.00
SSL_RIGHT_K6 0.00 0.02 0.01
SSL_RIGHT_D1 0.00 0.03 0.01
SSL_RIGHT_D2 0.00 0.04 0.01
SSL_RIGHT_D3 0.00 0.02 0.01
Table 1: Variances and Total Sensitivity Indices obtained using the One-at-a-time analysis, the ANOVA expansion
of the complete model and the more accurate ANOVA expansion of the reduced model. The naming convention
used for the suspensions works as follows. PSL and SSL stand for primary and secondary suspension of the leading
bogie respectively. The following L and T in the primary suspension stand for leading and trailing wheel sets. The
last part of the nomenclature refers to the particular suspension components as shown in Fig. 1.
4.1 One-at-a-time analysis
When each suspension component is considered independently from the others, the estimation problem in (4) is
reduced to the calculation of a 1-dimensional integral. This task can be readily achieved by quadrature rules that
have proven to be computationally more efficient on problems of this dimensionality than sampling methods .
Fourth order quadrature rules have been used to approximate the variances due to the single components. The
convergence of this method enables a check of accuracy through the decay of the expansion coefficients of the
target function .
The second column in Table 1 lists the results of such analysis. The amount of variance described by this analysis
is given by the sum of all the variances: 22
OAT /47.1 sm
. This quantity is far from representing the total
variance of the stochastic system, suggesting that interactions between suspension components are important.
Anyway the method is useful to make a first guess about which components are the most important: the critical
speed of the railway vehicle model analyzed in this work shows a strong sensitivity related to the longitudinal
springs (K2) in the trailing wheel set.
4.2 Total Sensitivity Analysis
The technique outlined in section 3.3 can fulfill three important tasks: taking into account parameter interactions,
performing the analysis with a limited number of realizations and enabling an error control in the approximation.
In a first stage we consider the full stochastic model and we construct a cut-HDMR expansion which takes into
account 2nd order interactions and describes the target function through 2nd order polynomials, computing the
realizations for up to 2-dimensional cubature rules. The ANOVA-HDMR expansion of the cut-HDMR expansion
can be quickly computed, due to the low dimensionality of the single terms in (10). At this point, the l
in (9) can be obtained and the “effective dimensionality” of the target function, given by (14) for 95.0q, is
found to be 2L. This confirms that the 1st and 2nd order interactions are sufficient to describe most of the
variance. The third and fourth column of Table 1 list the total variances induced by each parameter, including
interactions with other parameters, and the Sobol’ total sensitivity indices.
Once the first approximation of the sensitivities is obtained, the parameters with the lowest sensitivity indices can
be fixed to their nominal values and we can perform a more accurate analysis of the remaining stochastic system.
Longitudinal and vertical springs (K2 and K3) in the primary suspensions have shown to be very influential for the
critical speed of the analyzed model, thus a new cut-HDMR expansion, with 2nd order interactions and 4th order
polynomial approximation is constructed. The resulting total variances and total sensitivity indices are listed in the
fifth and sixth column of Table 1. A visual representation of the sensitivity indices is shown in the pie chart in Fig.
The results obtained by the one-at-a-time analysis are confirmed here by the total sensitivity analysis, but we stress
that the latter provide a higher reliability because they describe a bigger part of the total variance of the complete
4.3 Discussion of the obtained results
Even if the results obtained are formally correct, the interpretation of such results can raise some questions. A
railway engineer might wonder why the yaw dampers D2 are not listed among the most important by the
sensitivity analysis. The yaw dampers in the secondary suspension are known to provide stability to the vehicle
ride, helping to increase its critical speed. This result is true also with the vehicle model considered here, in fact
low values of D2 cause a drastic worsening of the ride stability. However, the total sensitivity indices embed the
probability distributions of the uncertain parameters in the global sensitivity analysis: the impact of a component is
weighted according to these distributions. Thus we say that the yaw damper has little influence on the riding
stability with respect to the distributions chosen. A change in the distributions can dramatically change these
results, thus particular care should be taken with the quantification of the source of uncertainty.
Finally, observe that, even if they are not as important as the primary suspension components, the yaw dampers
seem to be the most important components among the secondary suspensions.
4.4 Remarks on sensitivity analysis on non-linear dynamics
Uncertainty quantification and sensitivity analysis require a rigorous preliminary formulation of the stochastic
system, its sources of uncertainty and the Quantities of Interest. We already mentioned in section 2.2 that in this
work the characterization of the sources of uncertainty was bypassed by assuming Gaussian distributions for all the
parameters, without loss of generality for the methods presented. The selection of the QoI, however, merits some
more discussion. In section 2.1 the continuation method used to estimate the critical speed was presented and the
threshold used to determine the end of the hunting motion was chosen in a conservative way, as it is shown in Fig.
2b. However, the value of the computed critical speed will depend also on the deceleration chosen for the
continuation method, i.e. the computed critical speed will be exact in the limit when the deceleration goes to zero.
Of course, the exact computation of the critical speed is not computationally feasible. With the limited
computational resources available, we then chose a fixed deceleration coefficient for the continuation method, and
thus we introduced numerical uncertainty in the computations. Therefore, the variance expressed from the analysis
is given both by the variance due to the stochastic system and the variance introduced by the computation of the
QoI. This is, however, a conservative consequence, meaning that a decision taken on the basis of the computed
results is at least as safe as a decision taken using the “exact results”.
Sensitivity analysis is of critical importance on a wide range of engineering applications. The traditional approach
of local sensitivity analysis is useful in order to characterize the behavior of a dynamical system in the vicinity of
the nominal values of its parameters, but it fails in describing wider ranges of variations, e.g., caused by long-term
wear. The global sensitivity analysis aims at representing these bigger variations and at the same time it embeds the
probability distributions of the parameters in the analysis. This enables the engineer to take decisions based on the
risk of a certain event to happen.
Wrongly approached, global sensitivity analysis can turn to be a computationally expensive or even prohibitive
task. In this work a collection of techniques are used in order to accelerate such analysis for a high-co-dimensional
problem. Each of the techniques used allows for a control of the accuracy, e.g., in terms of convergence rate for the
cubature rules in section 3.1 and the “effective dimension” in section 3.3. This makes the framework flexible and
easy to be adapted to problems with more diversified distributions and target functions.
The analysis performed on the half wagon equipped with a Cooperrider bogie shows a high importance of the
longitudinal primary suspensions, and this reflects the connection between hunting and yaw motion. Furthermore,
the importance of the yaw damper in the secondary suspensions is confirmed, even if its influence is little
compared to the primary suspensions.
It is important to notice that the same settings for global sensitivity analysis can be used for the investigation of
different Quantities of Interests, such as wear in curved tracks, angle of attack etc., once they have been properly
defined. Furthermore, the “non-intrusive” approach taken allows the engineer to use closed software for the
computations. The machinery for sensitivity analysis needs only to be wrapped around it, without additional
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