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A discontinuous model simulation for train start-up dynamics.

1. Introduction

During the last decade, the increase of the trains' operating speed led to an enhancement of the rail vehicles' start-up analysis. The main aims of this research are the reduction of the damage risks and derailment probability. Both matters require the study of the train longitudinal dynamics.

The International Union of Railways and railway researchers have been developing numerical studies and computer applications for the analysis of the train start-up longitudinal dynamics. Related European Rail Research Institute (ERRI) reports are also available. Common approaches involve nonlinear ordinary differential equations (ODEs) integrated by standard solvers, and a few works proposed the use of commercial multibody system simulation software. Train crashworthiness also employs finite element analysis. A description of the methods may be found in Cole et al. (2017); a common feature is the high integration cost.

The main fact which encumbers the system resolution is the presence of the dry friction in the wagon connection elements (Iwnicki 2006), i.e. the buffers and the draw gears. This feature suggests that a non-smooth model would best feature the studied system (Grabner and Kecskemethy 2003; Kulkarni 2006; Wu, Spiryagin, and Cole 2015). However, only physical models, using smoothened laws for the constraints, have been used until now. As a consequence, the models imply high-resolution costs and yield poor results (Thomsen and True 2010; Baraff 1989).

The longitudinal train dynamics encounters alternative stick--slip phases in the couplers, wherein the two surfaces in contact stick, respectively slip, over each other and the displacement-time evolution has a saw-tooth profile, owing to the friction force. The laws of the dry friction are usually different for static friction between surfaces which are not moving one in respect to the other and for kinetic friction (sometimes called sliding friction or dynamic friction) between surfaces with relative displacement, see for instance Andersson, Soderberg, and Bjorklund (2007).

As a consequence that the stick--slip motion phases involve different mechanisms, the differential equations which describe it are discontinuous. Such equations may be solved by means of a regularisation (sometimes called penalty) method which replaces the discontinuous equations by a smooth adjoint system. This solution is easy to follow as it consists of a common ODE system. In reverse, the equations are stiff, determining high integration costs and inexact values and solution behaviour. For instance, the occurrence of stick--slip may not be observed.

The alternative method to solve the problem is to deal with the nonlinear characteristic of the dry friction force. This choice is not only more suitable but also more difficult to apply. The non-smooth approach faces disjoint subintervals of the computation time. On each subinterval, the system features a certain configuration of stick or slip phases between the bodies in contact which shall be further referred as system mode, while the phase's switches will be referred as events.

In this study, a numerical method which allows the computation of the system states values for each mode is presented. The simulation is fast, accurate and may be done with any standard ODE solver. Employing generalised matrix inverses (GI), an intuitive and effective computation method of the static friction forces is deduced, considering Gauss' least constraint principle. The integration uses an event-driven method which detects phase's switches and distributes the computation interval accordingly. On each subinterval, the system model is a set of ordinary non-stiff differential equations which can be computed by any standard ODE solver. Static friction force computation is accomplished by means of an algorithm compatible with the GI method which is developed.

2. The non-smooth model of the train start-up

The train collision model comprises the point masses of the vehicles; the vehicles are connected by the couplers and actuated by the applied forces, Figure 1; the vertical and lateral degrees of freedom are neglected. The classical Newton--Euler equations which describe unconstrained motion of a multibody system with n degrees of freedom may be stated as:

[mathematical expression not reproducible] (1)

The multipliers [lambda], usually forces or torques, are defined by set-valued laws and the constraint vectors w [member of] [[reals].sup.n] specify how they are applied. M [member of] [[reals].sup.nxn] is the inertia matrix, q is the generalised coordinates and h is the sum of the forces described by constitutive laws, including viscous damping and elasticity. The most important force which influences the longitudinal start-up dynamics of the train is the locomotive traction force, F. The equation of the longitudinal dynamics may be written as:

M[??] - h(t, x, [??]) - WT = 0 (2)

where M = diag([m.sub.1],..., [m.sub.n]) is a diagonal matrix containing the wagons masses and the generalised coordinates q are substituted by the longitudinal displacements x = [[x.sub.1],..., [x.sub.n].sup.T]. In the start-up phase, the effective constraint forces are the friction forces [T.sub.i] present in the couplers, i [member of] {1,..., n - 1}. The constraints are imposed for the relative motion in the tangential contact points, i.e. between coupled wagons or between wagons and rails; the relative displacements in the buffers are [xi] = [[x.sub.1] - [x.sub.2],..., [x.sub.n-1] - [x.sub.n].sup.T].

The column vector T [member of] [[reals].sup.m] will contain the friction forces. Their values may be either kinetic or static, depending on the system mode, as described by the index sets [I.sub.k] and [I.sub.s], Equation (6).

The action of the constraint forces over the systems component masses is defined by the constraint matrix W [member of] [[reals].sup.nxm] ([W.sup.T] = J, the Jacobian) which may be obtained by assembling the constraint vectors w.

[mathematical expression not reproducible] (3)

The columns of the constraint matrix specify the action of the coupling forces, and they are assembled in matrix with values + 1 on the first diagonal and - 1 on the diagonal below. The total number of columns is n - 1. In accordance with the previous notations, the contact space coordinates are [xi] = [[x.sub.1] - [x.sub.2],..., [x.sub.n-1] - [x.sub.n].sup.T] = [W.sup.T]x, i [member of] {1,..., n}, and Equation (2) may be expressed as

M[??] - [W.sub.1][DW.sub.1.sup.T][??] - [W.sub.1][CW.sub.1.sup.T]x - f(t,x,[??]) - WT = 0 (4)

where [c.sub.i], i [member of] {1,..., n - 1} are couplers damping coefficients and D = diag([c.sub.i]) defines the viscous damping matrix, [k.sub.i] are couplers stiffness and C = diag([k.sub.i]) defines the stiffness matrix and f(t, x, [??]) is the vector of the externally applied forces, including the traction force F.

2.1. Dry friction forces

Dry friction model is different for static friction and for kinetic friction (Olsson et al. i998; Leine, Van Campen, and De Kraker 1998). The stick force opposes to the resultant of all the other exerted forces (including inertia force, if applicable) and takes any value from zero up to the maximum stick force, [T.sub.sMax]. It follows that the set-valued expression of the force law may be written as

[mathematical expression not reproducible] (5)

The indexes k and s are used for kinetic and static values, respectively. The following index sets may be used to define the system mode:

[I.sub.c] = {1, 2,..., m} [I.sub.k] = {k [member of] [I.sub.k]|[[??].sub.k][not equal to]0} [I.sub.s] = {s [member of] [I.sub.s]|[[??].sub.s] = 0} (6)

The set [I.sub.c] contains the indexes of all the m contact points, while the complementary subsets [I.sub.k] and [I.sub.s] contain the indexes of the sliding contacts and those of the potentially sticking contacts, respectively.

In the case of the kinetic friction, the slip force opposes the relative movement between the contact points (which may differ from the relative movement between the bodies). There have been proposed many laws for its value--see for instance Oprea, Cruceanu, and Spiroiu. (2013) or Oprea (2012)--most of them velocity dependent.

The static friction force is a function of the applied forces and it exactly cancels them; their sum will be denoted by [SIGMA]F. When the other applied forces overcome this threshold value, the motion would commence. At this very moment, the value of the friction force is [T.sub.SMax], and its orientation opposes [SIGMA]F. Hence, a general description of friction may be of the form (Equation (7)):

[mathematical expression not reproducible] (7)

The slip force, [T.sub.K], is an arbitrary function of the relative velocity which best fits the studied system; the constant value slip force is used in models known as the Coulomb friction law. The main characteristic of the models, which takes into account this law, is the discontinuity arisen at zero relative velocity.

The static value of the friction force [T.sub.s] equals the external and inertia forces resultant if this is smaller than the limit value [T.sub.sMax]; otherwise, its limit value opposes to the external forces, but a slip phase will begin. Such a friction law does not explicitly specify the friction force at zero velocity; the stick force counteracts the external resultant below its maximum value and thus keeps objects in contact not to move relative to each other.

While transitions from stick to slip may not occur until [T.sub.sMax] is reached, during slip to stick transitions, when [??] vanishes, the force value may jump from [T.sub.sMax] anywhere in the set [- [T.sub.sMax] [T.sub.sMax]]. This feature reveals discontinuity and hysteretic behaviour shrunk to one point, [??] = 0. In the present work, we shall use a Coulomb model where the slip force does not depend on the relative velocity, and it is equal to the limit value [T.sub.sMax].

3. An event-driven algorithm for the numerical integration of a non-smooth system

General considerations about the numerical integration of non-smooth systems are revised in the sequel. The algorithm used for the demonstrative application included in this work is further presented. An extended treating of the subject may be found in Oprea, Cruceanu, and Spiroiu. (2013) and Oprea (2012), the main sources of the following two chapters.

The event-driven method considers the simulation interval as a union of disjoint subintervals on which the mode, stick or slip between the bodies in contact remains unchanged. Until any of the contact phases switches, the system may be described by a set of ordinary equations; when at least one of the contact phase changes, from sticking to relative motion or vice-versa, the corresponding friction force law and, possibly, its value become different. Even more, stick phases lead to degree of freedom (DOF) collapses. Consequently, on the following subinterval, the system might be described by a new set of equations as is the case of the switching diagrams, (Acary and Brogliato 2008; Leine and Nijmeijer 2004). Such strategy may not be pursued unless there are only a few contacts, as the logical complexity of the model exponentially increases with the number of contacts. The alternative is to preserve the full slip dynamics system and to assign the proper values of the constraints. This solving manner involves special techniques to determine the moments of the switches, the system modes and the values of the static forces.

The algorithm includes the following stages (Acary and Brogliato 2008):

(1) Determine the next smooth mode, initialise the system and update the equations.

(2) Integrate the smooth state vector with any ODE solver while constraints are not violated.

(3) Detect within imposed tolerance the moment of the next event.

In the implementation of this algorithm, two issues have to be solved: event detection and static friction computation.

3.1. Karnopp method event detection

In the case of systems with Coulomb friction models, transition from slip to stick may occur when relative velocity [??] vanishes in a contact point. The mode change is confirmed if the contact force lies strictly inside the friction cone |T| [less than or equal to] [T.sub.sMax].

A review of the accurate methods to locate zero-crossings may be found in Acary and Brogliato (2008), but, as a general feature, they are very expensive in terms of the required computational power. A solution is given by the Karnopp (1985) model which considers that the stick occurs when the relative velocity is 'small enough'; this condition is formulated as [??] < [eta] (instead of [??] = 0), where [eta] should be much smaller than the average speed values of the system elements. The slip mode is defined by the complementary relative velocities which lie outside the narrow stick band, [??] < [eta] (instead of [??] = 0). It results a discontinuous system, non-stiff in the stick interval.

This method overcomes the problems of the zero velocity detection and allows efficient simulations; the stick band may be quite coarse, but the stick and slip periods are nevertheless distinguished. On the other hand, because the relative acceleration is put to zero in the stick phase, the constant offset of the relative velocity causes a drift-off effect for large intervals and can cause numerical instability of the ODE integrator (Acary and Brogliato 2008). However, because of its desirable properties, this technique is used to obtain all simulation results presented in this paper.

In the stick mode, transitions may occur when the contact force reaches the boundary of the friction cone. This is accomplished when the static force equals [T.sub.sMax] as expressed in Equation (7). Testing such an event requires a root-finding procedure.

4. Matrix GI solution of the constrained system

When events take place, the ensuing system configuration has to be identified. Relative velocities and friction forces values grant the proper mode selection. In fact, event detection, mode selection and integration step depend on each other and may be simultaneously resolved. The core of the problem is to determine the constraints values (i.e. the static friction forces).

The alternative approach to the evaluation of the static friction forces is based on the Gauss' principle of least constraints and on the notion of generalised inverse (or pseudoinverse) of matrices. This formulation allows a convenient computer implementation, taking into account that mathematical packages contain GI computation subroutines. But its most remarkable features are that redundant sets of equations may be reduced to systems of ODE, DOF changes do not involve any additional programming in the solution formulation and rheonomic and scleronomic constraints or forward and inverse dynamics are handled in a unified manner. Some improvements regarding computational efficiency are also mentioned, e.g. De Falco, Pennestri, and Vita (2005).

In order to introduce the formulation deduced by Udwadia and Kalaba (2002), we shall denote by

a = [M.sup.-1]h(t, q, [??]) (9)

The unconstrained acceleration, which results from the equation of motion

M[??] - h(t, q, [??]) = 0 (10)

Motion constraints may be expressed in a general form as

[W.sup.T](t, q, [??])[??] = b(t, q, [??]) (11)

where W [member of] [[reals].sup.nxm] is a generalised constrained matrix, the constraints expressionsb [member of] [[reals].sup.m], and they may be ensured if additional generalised constraint forces W[lambda] are applied. The constrained equation of motion has been previously enounced (Equation (1)).

Taking into account Equations (1), (9), (10), and (11), the constrained system may be described by the following differential algebraic equations. Because the equations use more coordinates than the underlying system, 12 is known as the redundant coordinate system.

[mathematical expression not reproducible] (12)

According to Gauss' principle, the system's accelerations must fulfilsufficient optimality conditions for the 'acceleration energy' which is a convex function and grants existence and uniqueness of the solution. The principle asserts that the values of the accelerations of a system subjected to constraints are the closest possible to the accelerations of the unconstrained system, and, consequently, the constraints take the minimum possible values. Therefore, the resulting acceleration [??] minimises the function G over the set which satisfies the constraint Equation (11).

minG([??]) = 1/2 [([??] - a).sub.T] M([??] - a) = 1/2[??] - [a.sub.M.sup.2] (13)

A similar minimum condition may be imposed for the constraints [lambda]. This development is described by Udwadia and Kalaba (2002). The expressions of [??] and [lambda] may be obtained formally inverting the matrix

[mathematical expression not reproducible] (14)

which yields the generalised constraint accelerations and forces as

[??] = a + [M.sup.-1]W[([W.sup.T][M.sup.-1]W).sup.-1](b - [W.sup.T]a) (15)

[lambda] = [([W.sup.T][M.sup.-1]W).sup.-1](b - [W.sup.T]a) (16)

Considering the substitution [J.sub.M.sup.T] = [M.sup.-1/2]W, Equations (15) and (16) may be written as

[??] = a + [M.sup.-1/2 ([m.sup.-1/2]W) [([W.sup.T][M.sup.-1/2][M.sup.-1/2]W).sup.-1] (b - [W.sup.T]a) = a + [M.sup.-1/2][J.sub.M.sup.T][([J.sub.M][J.sub.M.sup.T]).sup.-1] (b - [W.sup.T]a) (17)

W[lambda] = [M.sup.1/2]([M.sup.-1/2]w) [([W.sup.T][M.sup.-1/2][M.sup.-1/2W]).sup.-1] (b - [W.sup.T]a) = [M.sup.-1/2][J.sub.M.sup.T][([J.sub.M][J.sub.M.sup.T]).sup.-1] (b - [W.sup.T]a) (18)

Even though the matrices involved in the above relations may not be invertible, the minimum condition expressed by the Gauss' principle defines a unique solution. Yet the GI may be used for least square problems or for minimum norm solutions of l(inear) systems. Therefore, employing the GI [J.sub.M.sup.+] = [J.sub.M.sup.T][([J.sub.M][J.sub.M.sup.T]).sup.-1], the constraint acceleration and force which satisfy Gauss' principle are explicitly given in Equations (19) and (20). Formulation simplicity and concision are the main arguments for this approach.

[??] = a + [M-1/2][J.sub.M.sup.+](b - [W.sup.T]a) (19)

W[lambda] = [M.sup.1/2][J.sub.M.sup.+](b - [W.sup.T]a) (20)

4.1. Mode selection

In the following, an algorithm to compute the constraint forces involved in the longitudinal dynamics is developed. The underlying idea is that the GI may be used for least square problems or for minimum norm solutions of linear systems. The aim of the model is to determine the static friction forces as the others, including kinetic friction forces, are functions of the system states or time and they are known. Therefore, the optimisation problem may be formulated only for the static friction which occurs only in sticking contacts, i.e. when the relative velocity and acceleration vanishes.

If any of the computed forces are outside the friction cone, a switch from stick to slip occurs at the given contact. The force value is limited to the maximum stick force (but the force direction does not change). The mode configuration and, correspondingly, W and h matrices are modified. Constraints are computed afresh until all of them lie inside the friction cone. The numerical scheme can be summarised by the following steps.

Compute explicit forces h = D[??] + Kx + f(t)

Provisional index sets assignment

For [member of] [I.sub.c], If [??] [greater than or equal to] [eta], Then i [member of] [I.sub.k]

Kinetic friction forces vector

[T.sub.k]([I.sub.k]) = [T.sub.k](t,[xi],[xi]), [T.sub.k]([I.sub.s]) = 0

Add kinetic friction forces to h

[??] = h + W[T.sub.k]

Static friction forces computation

[I.sub.s] = [I.sub.c]\[I.sub.k]

[??] = Remove [I.sub.k] columns in W

[T.sub.s] = - [??]+[??] Index sets reassignment

For [member of] [I.sub.s], If |[T.sub.s](s)| > [T.sub.SMax], Then s [member of] [I.sub.k] and [T.sub.k](s) = [-T.sub.sMax]sign([SIGMA]F)

If [I.sub.k] changed, Then Go To Add kinetic friction forces to h

Else Update state vector

The above procedure should be performed at each time step before the state vector is updated. Some particular tasks must be accomplished. The computing of the static forces may necessitate iteration. The decision to proceed the computation anew may be taken by analysing the [I.sub.k] index set. Another specific task is to adjust the constraint matrix [??] and the known forces vector [??] in accordance to the provisional sticking contacts.

As deduced by Oprea (2012), static friction force computation may be alternatively formulated as T = -[([W.sup.T]W).sup. +] [W.sup.T]h. In this case, both lines and columns corresponding to sliding contacts [I.sub.k] are removed and also the elements [I.sub.k] from the vector [W.sup.T]h. Even if mathematically equivalent, formulations and numerical techniques employed proved to be different from the computational perspective.

5. Demonstrative application

The demonstrative application depicts a six-vehicle train set in motion by the first one. The simulation starts when the train leading vehicle (virtually, the train engine) applies the traction force. The longitudinal dynamics initially exhibits a transient phase which lasts for about 5 s.

The simulation is intended to illustrate the appropriateness of the multibody approach to the train collision dynamics. Only relevant forces and features are considered. Parameter values are plausible for railway vehicle description. The same value is used for the vehicle masses, and, likewise, all couplers have the same stiffness and friction characteristics. Viscous damping will not be considered ([c.sub.i] =0,i [member of] {1,..., 6}). The kinetic friction forces are constant, and their magnitudes are equal to the threshold values, [T.sub.k] = [T.sub.SMax] (the classical Coulomb model). The traction force is constant during the entire simulation.

Parameter values used in the application are given in Table 1. Masses and couplers are numbered beginning with the first vehicle, as in Figure 1; the train engine has index 1. The zero for both displacements and for velocities.

Figure 2 depicts the order of the train vehicles. The green vehicle at right applies the traction force (as in Figure 1). Figures 3 and 4 describe relative displacements and velocities between each pair of two consecutive vehicles in the train. Each plot identifies a pair of vehicles using the colour of the vehicle at right in each pair (e.g. for vehicles 5-6, the color is magenta).

Relative velocities are plotted in Figure 3. Null values correspond to stick phases; otherwise, slip occurs. The relative velocity between vehicles 1 and 2 grows from the beginning of the simulation due to the constant force applied by the leading vehicle. While the forces in the couplers increase, the following pairs of vehicles exhibit relative movement. The relative movement between vehicles changes the sign a few times until the stationary phase when relative rest of the train is achieved. Relative rest periods are also encountered during the transient phase.

Relative rest may also be observed in the relative displacements plot, Figure 4,defined by constant values. The stationary phase, beginning after the fifth second, also exhibits constant displacements between the vehicles. These latter displacements have the greatest value for the first pair of vehicles (green plot), where the coupler force has to be the largest, and they decrease towards the last coupler (blue plot).

The fourth coupler forces are plotted in Figure 5. The force is null until the fourth vehicle begins to move. A distinctive characteristic is given by the friction force change of sign which brings on gaps in the coupler force values. The value remains constant in the stationary phase.

6. Conclusions

A method for the event-driven integration of a non-smooth train start longitudinal dynamics is developed in the present work. Redundant systems and DOF changes may be provided by the rank-deficient constraint matrix. Based on Gauss' least constraint principle, the matrix GI is employed to compute constraint forces. Algorithms which determine matrices GIs use robust, use iterative schemes and do not have specific limitations for the input size; therefore, they will always provide a solution. Mathematical packages include routines for the pseudo-inverses calculus, so they may be easily implemented in ODE solvers. At last, the GI algorithm is simple, gives an intuitive description of the constrained system and may be employed without detailing the optimisation methods. Hence, the method may be used for models with a large number of degrees of freedom and contact points.

A demonstrative application illustrates the capabilities and benefits of the non-smooth approach to the train start dynamics. Distinctive non-smooth dynamics phenomena are revealed. The model is able to handle any number of vehicles and contact points.

Notes on contributor

Razvan Andrei Oprea graduated and started academic career in 1993. The scope of his research includes railway vehicles vibrations and dynamics. During the last years he was mainly interested in non-smooth models and their numerical simulation

References

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De Falco, D., E. Pennestri, and L. Vita. 2005. "The Udwadia-Kalaba Formulation: A Report on Its Numerical Efficiency in Multibody Dynamics Simulations and on Its Teaching Effectiveness." Multibody Dynamics (2005): 21-24.

Grabner, G. A., and A. Kecskemethy. 2003. "Dynamic Simulation of Colliding Trains for Buffer Stop Dimensioning." Proceedings in Applied Mathematics and Mechanics 3: 156-157. doi:10.1002/pamm.200310353.

Iwnicki, S. 2006. Handbook of Railway Vehicle Dynamics. Boca Raton: Taylor & Francis.

Karnopp, D. 1985. "Computer Simulation of Slip-Stick Friction in Mechanical Dynamic Systems." Journal of Dynamic Systems, Measurement, and Control 107: 100-103. doi:10.1115/1.3140698.

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Razvan Andrei Oprea

Railway Vehicles Department, University Politechnica of Bucharest, Bucharest, Romania

ARTICLE HISTORY

Received 11 August 2017

Accepted 8 December 2017

CONTACT Razvan Andrei Oprea [??] razvan.oprea@upb.ro

https://doi.org/10.1080/14484846.2018.1485297
Table 1. Parameter values for the train collision dynamics model.

Parameter      Value                 Description

[M.sub.i]        25   Vehicle i mass, i [member of] {1,..., 6}
[k.sub.i]       100   Coupler i stiffness, i [member of] {1,..., 6}
[T.sub.sMaxi]     5   Coupler i friction, i [member of] {1,..., 6}
F                10   Traction force

Parameter      Units

[M.sub.i]        t
[k.sub.i]      kN/m
[T.sub.sMaxi]   kN
F               kN
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Publication:Australian Journal of Mechanical Engineering
Date:Jun 1, 2018
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