numerical study of the noninertial systems: application to train coupler systems

Nonlinear Dyn (2012) 68:215–233DOI 10.1007/s11071-011-0220-2

O R I G I NA L PA P E R

Numerical study of the noninertial systems: applicationto train coupler systems

Alberto Massa · Luca Stronati ·Ahmed K. Aboubakr · Ahmed A. Shabana ·Nicola Bosso

Received: 31 March 2011 / Accepted: 13 September 2011 / Published online: 11 October 2011© Springer Science+Business Media B.V. 2011

Abstract Car coupler forces have a significant effecton the longitudinal train dynamics and stability. Be-cause the coupler inertia is relatively small in compar-ison with the car inertia; the high stiffness associatedwith the coupler components can lead to high frequen-cies that adversely impact the computational efficiencyof train models. The objective of this investigation isto study the effect of the coupler inertia on the traindynamics and on the computational efficiency as mea-sured by the simulation time. To this end, two differentmodels are developed for the car couplers; one model,called the inertial coupler model, includes the effect ofthe coupler inertia, while in the other model, called thenoninertial model, the effect of the coupler inertia isneglected. Both inertial and noninertial coupler mod-els used in this investigation are assumed to have thesame coupler kinematic degrees of freedom that cap-ture geometric nonlinearities and allow for the relativetranslation of the draft gears and end of car cushioning(EOC) devices as well as the relative rotation of thecoupler shank. In both models, the coupler kinematicequations are expressed in terms of the car body and

A. Massa · L. Stronati · N. BossoDipartimento di Meccanica, Politecnico di Torino, C. soDuca Degli Abruzzi 24, 10129, Torino, Italy

A.K. Aboubakr · A.A. Shabana (�)Department of Mechanical and Industrial Engineering,University of Illinois at Chicago, 842 West Taylor Street,Chicago, IL 60607, USAe-mail: [email protected]

coupler coordinates. Both the inertial and noninertialmodels used in this study lead to a system of differen-tial and algebraic equations that are solved simultane-ously in order to determine the coordinates of the carsand couplers. In the case of the inertial model, the cou-pler kinematics is described using the absolute Carte-sian coordinates, and the algebraic equations describethe kinematic constraints imposed on the motion ofthe system. In this case of the inertial model, the con-straint equations are satisfied at the position, velocity,and acceleration levels. In the case of the noninertialmodel, the equations of motion are developed usingthe relative joint coordinates, thereby eliminating sys-tematically the algebraic equations that represent thekinematic constraints. A quasistatic force analysis isused to determine a set of coupler nonlinear force al-gebraic equations for a given car configuration. Thesenonlinear force algebraic equations are solved itera-tively to determine the coupler noninertial coordinateswhich enter into the formulation of the equations ofmotion of the train cars. The results obtained in thisstudy showed that the neglect of the coupler inertiaeliminates high frequency oscillations that can nega-tively impact the computational efficiency. The effectof these high frequencies that are attributed to the cou-pler inertia on the simulation time is examined usingfrequency and eigenvalue analyses. While the neglectof the coupler inertia leads, as demonstrated in this in-vestigation, to a much more efficient model, the re-sults obtained using the inertial and noninertial cou-pler models show good agreement, demonstrating that

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216 A. Massa et al.

the coupler inertia can be neglected without having anadverse effect on the accuracy of the solution.

Keywords Longitudinal train forces · Couplergeometric nonlinearities · Railroad vehicle dynamics ·Multibody systems

1 Introduction

Accurate prediction of coupler forces is necessary inthe analysis of train longitudinal dynamics and sta-bility. In order to achieve higher degree of computa-tional efficiency, longitudinal train dynamics (LTD) al-gorithms tend to be simpler as compared to the moregeneral multibody system (MBS) algorithms which re-quire the solution of a system of differential/algebraicequations (DAE’s). The algebraic equations that relateredundant coordinates in MBS algorithms must be sat-isfied at the position, velocity, and acceleration levels.The use of redundant coordinates in MBS algorithmsleads to a sparse matrix structure that can be exploitedin order to efficiently solve the dynamic equations ofmotion. Most LTD algorithms, on the other hand, donot employ a DAE’s solver, and they use indepen-dent coordinates instead of redundant coordinates. Theequations of motion are expressed in terms of the sys-tem degrees of freedom by systematically eliminatingdependent variables. Nonetheless, the implementationof efficient force elements may necessitate introduc-ing algebraic equations in LTD algorithms as will bediscussed in this investigation.

Train car couplers, which are designed to absorbshock forces resulting from sudden brake applicationsand traction, include elastic, viscoelastic, and/or fric-tion elements. While in many investigations on trainlongitudinal dynamics, the couplers are represented bymassless discrete spring-damper elements, MBS algo-rithms allow for the development of detailed modelsthat capture the coupler geometric nonlinearities re-sulting from the relative rotations of the coupler com-ponents. In order to develop such detailed MBS cou-pler and train models, most MBS algorithms take intoaccount the effect of the coupler inertia. However, be-cause couplers have high stiffness and small inertia incomparison with the train car inertia, including the ef-fect of the coupler inertia in the dynamic model canlead to high frequency oscillations that can adverselyaffect the computational efficiency. One example of

a coupler that has such a high stiffness coupling el-ement is the knuckle-type automatic coupler, whichconnects firmly when one car is bumped against an-other. This coupler was introduced in the 1880s andis still in use despite the fact that improved designsare available [5]. The use of the automatic coupler inNorth America became standard in 1917 with the in-troduction of the knuckle coupler, also called buckeyecoupler or Janney coupler [8] that replaced the MillerHook which was never widely used in freight vehi-cles to replace the link and pin coupling system [5].The knuckle coupler, shown schematically in Fig. 1,consists of a knuckle attached to the end of a shankfastened to a housing mechanism on the car. The as-sembly is designed to allow for some lateral play inorder to avoid derailments during curve negotiations.Cars are automatically coupled by engaging the openknuckle on one car to a closed knuckle on the othercar. A conventional auto-coupler package is shown inFig. 2 [4]. The coupler element consists of a head andshank that can be connected to a flexible unit such as adraft gear or an end-of-car-cushioning (EOC) devicethat is attached to the car body. The draft gear andEOC device are energy dissipating elements; the draftgear employs dry friction using friction wedges, whilethe EOC device includes oil-based dashpots (dampers)that produce damping force that depends on the rela-tive velocity [4, 6]. It is known that longitudinal trainforces during braking, traction, and curve negotiationsheavily depend on the design of the couplers, their de-grees of freedom, and their ability to absorb impactforces and dissipate energy. For this reason, differ-ent coupler designs, with the improvements made overmany years, are studied extensively in order to exam-ine their performance and their effect on railroad vehi-cle system dynamics and stability [10, 17].

2 Scope and objective of this investigation

As previously mentioned, in most railroad vehiclecomputer formulations, the coupler is modeled as aspring-damper element with no kinematic degrees offreedom [11]. The force in this spring-damper elementis function of the relative displacements between thetwo cars connected by this coupler. This simplified ap-proach does not take into account the effect of the cou-pler degrees of freedom and fails to capture the cou-pler geometric nonlinearities that can influence the car

Numerical study of the noninertial systems: application to train coupler systems 217

Fig. 1 Automatic coupler

Fig. 2 Conventionalauto-coupling package

motion. Developing a more accurate coupler model re-quires including coupler kinematic degrees of freedomthat account for the geometric nonlinearities resultingfrom the motion of the coupler components. For sucha detailed coupler model, two fundamentally differentapproaches that require the use of two different solu-tion procedures can be used. In the first approach, theinertia of the coupler is taken into account. In sucha model, called inertial coupler model, the algebraicequations, if they are present in the model, describeconnectivity constraint equations which must be sat-isfied at the position, velocity, and acceleration lev-els since the coupler accelerations appear in the equa-tions of motion. The coupler degrees of freedom havegeneralized inertia forces associated with them and,therefore, their second derivatives appear in the sys-tem equations of motion. This leads to an increase inthe number of the system state equations that must beintegrated numerically.

In the second approach, on the other hand, the in-ertia of the coupler, assumed small compared to thecar body inertia, is neglected. In this model, calledthe noninertial coupler model, no generalized inertiaforces are associated with the coupler degrees of free-dom, and as a consequence, the second derivatives ofthese coordinates do not appear in the final form ofthe equations of motion. The use of such noninertialcoupler model can lead to significant reduction in the

number of state equations and can also contribute toeliminating the high frequency oscillations that mightresult from the high coupler stiffness and its relativelysmall inertia. A set of quasistatic coupler equilibriumconditions can be developed and used to define a set ofnonlinear algebraic equations that can be solved iter-atively for the coupler noninertial coordinates. Thesecoupler noninertial coordinates enter into the formu-lation of the equations of motion of the train cars[16, 18].

It is, therefore, the objective of this investigationto examine the effect of the coupler inertia on the dy-namics and computational efficiency of train models.The results obtained using the two models, inertial andnoninertial coupler models, are compared in order toexamine the assumption of neglecting the coupler in-ertia. In the case of the noninertial coupler model, thequasi-static coupler condition used in this investiga-tion do not require having velocity dependent termsas compared to methods previously published in theliterature [1]. Only algebraic equations are requiredin order to be able to use the procedure employed inthis study. Nonetheless, velocity dependent forces canstill be included in the noninertial coupler formulationused in this paper. A frequency domain analysis is alsoperformed in order to identify the frequencies in thesolution associated with the coupler inertia.

218 A. Massa et al.

3 Inertial and noninertial coordinates

The concept of the inertial and non-inertial coordi-nates is discussed in this section. Inertial coordinateshave generalized inertia forces associated with them,while the noninertial coordinates have no generalizedinertia forces. In order to avoid having a singular iner-tia matrix and/or high frequency oscillations, the sec-ond derivatives of the noninertial coordinates are notused when formulating the system equations of mo-tion in this study. In this case, the system coordinatesare partitioned into two distinct sets; inertial and non-inertial coordinates, leading to a formulation similarto the one used in the case of nongeneralized coor-dinates [14]. The use of the principle of virtual workleads to a coupled system of differential and algebraicequations expressed in terms of the inertial and nonin-ertial coordinates. The differential equations are usedto determine the inertial accelerations which can be in-tegrated to determine the inertial coordinates and ve-locities. The noninertial coordinates are determined byusing an iterative Newton–Raphson algorithm to solvea set of nonlinear algebraic force equations obtainedusing quasistatic equilibrium conditions. The nonin-ertial velocities are determined by solving these alge-braic force equations at the velocity level. The non-inertial coordinates and velocities enter into the for-mulation of the generalized forces associated with theinertial coordinates.

3.1 Inertial system

If all the coordinates of the bodies in a MBS are treatedas inertial coordinates, the equations of motion of abody i can be written as (Roberson and Schertassek[9]; Shabana et al. [15])

Mi qi = Qie + Qi

c + Qiυ (1)

where Mi is the mass matrix of the body, qi =[ RiT θ

iT ]T is the vector of the accelerations of the

body with Ri defining the body translation and θ i

defining the body orientation, Qie is the vector of ex-

ternal forces, Qic is the vector of the constraint forces

which can be written in terms of Lagrange multipliersλ as Qi

c = −CTqi λ,Cqi is the constraint Jacobian ma-

trix associated with the coordinates of body i, and Qiυ

is the vector of the inertia forces that absorb terms thatare quadratic in the velocities [13]. The nonlinear al-gebraic kinematic constraint equations can be written

in the vector form C(q, t) = 0, where q is the vectorof the system generalized coordinates, and t is time.The constraint equations at the acceleration level canbe written as Cqq = Qd , where Qd is a vector thatabsorbs first derivatives of the coordinates. Using (1)with the constraint equations at the acceleration level,one obtains[

M CTq

Cq 0

][qλ

]=

[Qe + Qυ

Qd

](2)

This matrix equation, which ensures that the constraintequations are satisfied at the acceleration level, can besolved for the accelerations and Lagrange multipliers.In order to ensure that the algebraic kinematic con-straint equations are satisfied at the position and ve-locity levels, the independent inertial accelerations qi

are identified and integrated forward in time in orderto determine the independent velocities qi and inde-pendent coordinates qi . Knowing the independent co-ordinates from the numerical integration, the depen-dent coordinates qd can be determined from the non-linear constraint equations using an iterative Newton–Raphson algorithm that requires the solution of thesystem Cqd

�qd = −C, where �qd is the vector ofNewton differences, and Cqd

is the constraint Jaco-bian matrix associated with the dependent coordinates.Knowing the system coordinates and the independentvelocities, the dependent velocities qd can be deter-mined by solving a linear system of algebraic equa-tions that represents the constraint equations at thevelocity level. This linear system of equations in thevelocities can be written as Cqd

qd = −Cqiqi − Ct ;

where Cqiis the constraint Jacobian matrix associated

with the independent coordinates, and Ct = ∂C/∂t isthe partial derivative of the constraint functions withrespect to time.

Lagrange multipliers, on the other hand, can beused to determine the constraint forces. For a givenjoint k, the generalized constraint forces acting onbody i, connected by this joint, can be obtained fromthe equation

(Qic)k = −(Ck)

Tqi λk = [

FiTk TiT

k

]T (3)

where Fik and Ti

k are the generalized joint forces as-sociated, respectively, with the translation and orienta-tion coordinates of body i. Using the results of (3), thereaction forces at the joint definition point can be de-termined using the concept of the equipollent systemof forces.


3.2 Noninertial coordinates

In developing the augmented form of (2), it is assumedthat all coordinates have generalized inertia forces as-sociated with them. In some MBS applications, theinertia associated with some coordinates can be rel-atively small. This can lead to high frequency oscil-lations that have negligible effect on the solution butcause significant deterioration of the computational ef-ficiency. For this reason, it is important to develop acomputational procedure that allows for eliminatingthese high frequency oscillations by neglecting the ef-fect of the small inertia. If the small inertia can be ne-glected, the virtual work of the inertia forces δWi andthe virtual work of the applied forces δWe can be writ-ten, respectively, as

δWi = QTi δq, δWe = QT

e δq + QTniδqni (4)

In this equation, q is the vector of the system iner-tial coordinates that have inertia forces associated withthem, qni is the vector of the system non-inertial co-ordinates which have no inertia forces associated withthem, Qi is the vector of generalized inertia forces,Qe is the vector of generalized applied forces associ-ated with the inertial coordinates, and Qni is the vec-tor of generalized applied forces associated with thenoninertial coordinates. Using (4) and the principle ofvirtual work, which states that δWi = δWe, one has

QTieδq − QT

niδqni = 0 (5)

In this equation, Qie = Qi − Qe. If the system is sub-jected to kinematic constraints that describe specifiedmotion trajectories and mechanical joints, the vectorof nonlinear algebraic constraint functions can be writ-ten as

C(q,qni, t) = 0 (6)

Considering a virtual change in the coordinates, oneobtains Cqδq + Cqni

δqni = 0. Multiplying this equa-tion by Lagrange multipliers λ , adding the results to(5), and using the generalized coordinate partitioningas inertial and noninertial, one obtains

Qie + CTq λ = 0, Qni + CT

qniλ = 0 (7)

Differentiating (6) twice with respect to time, one ob-tains the constraint equations at the acceleration level,

Cqq + Cqniqni = Qd , where Qd is the quadratic ve-

locity vector that arises from the differentiation of thekinematic constraint equations twice with respect totime. By combining the constraint equations at the ac-celeration level with (7), one obtains⎡⎣M 0 CT

q0 0 CT

qni

Cq Cqni0

⎤⎦

⎡⎣ q

qni

λ

⎤⎦ =

⎡⎣Qe

Qni

Qd

⎤⎦ (8)

In the special case in which the noninertial coordinatesare not subjected to constraints, the preceding equationleads to[

M CTq

Cq 0

][qλ

]=

[Qe

Qd

],

(9)Qni(q,qni, q, qni, t) = 0

Furthermore, if an embedding technique is used tosystematically eliminate the dependent inertial coor-dinates such that all the components of the vector qare independent, the preceding equations reduce to

Mq = Qe, Qni(q,qni, q, qni, t) = 0 (10)

The second equation of (10) depends on the coordi-nates and velocities. The velocity terms appear in thisstudy when coupler damping forces are considered.The procedure developed in this investigation is basedon developing a quasistatic equilibrium force condi-tions that depend on the noninertial coordinates. Thisleads to a system of nonlinear algebraic equations, rep-resented by the second equation of (10) that can be it-eratively solved, using a Newton–Raphson algorithm,for the noninertial coordinates for given set of iner-tial coordinates. The non-inertial velocities are deter-mined by differentiating the quasistatic algebraic forceequations. This procedure is fundamentally differentfrom the procedure presented by Arnold et al. [1] andBurgermeister et al. [3] in which the force equationsmust include velocity dependent terms and these forceequations are treated as first-order ordinary differen-tial equations that are integrated with the equations ofmotion.

4 Coupler kinematics using redundantcoordinates

As previously mentioned, in most coupler models re-ported in the literature, the coupler is represented us-ing a discrete massless spring-damper element which

220 A. Massa et al.

has no kinematic degrees of freedom. This simplemodel does not capture geometric nonlinearities re-sulting from the relative motion between the couplercomponents as well as the relative motion of the cou-pler with respect to the car body. In this investigation,the coupler kinematic degrees of freedom are consid-ered. There are two methods for the description ofthe coupler kinematics. In the first method, the cou-pler nonlinear kinematic equations are formulated interms of redundant coordinates that are related by al-gebraic constraint equations. This approach is used inmost general purpose MBS computer algorithms andit leads to a sparse matrix structure of the equationsof motion. Nonetheless, the redundant coordinate ap-proach requires the solution of a system of differentialand algebraic equations and the use of the technique ofLagrange multipliers that can be used to determine thegeneralized constraint forces. In the second approach,the kinematic equations of the coupler are expressedin terms of the coupler degrees of freedom. This ap-proach is more suited for the implementation in lon-gitudinal train force dynamics algorithms which tendto be simpler as compared to general MBS algorithmsand do not in general allow for the solution of cou-pled sets of differential and algebraic equations. Sucha coupler model will be discussed in later sections ofthis study.

In this section, the kinematics using the redundantcoordinate approach is discussed. As shown in Fig. 1,the coupler consists of a shank and a draft gear or anEnd-of-car cushioning (EOC) device unit. The draftgear or EOC device is connected to the car body us-ing a prismatic joint, while the shank is connected tothe draft gear or EOC device using a pin joint that al-lows for the relative rotation of the shank with respectto the car body. The draft gear produces dry frictionforce, while the EOC device produces viscous damp-ing force; both oppose the sliding motion of the cou-pler with respect to the car body. In order to have ameaningful comparison between the effects of the twotypes of forces, the stiffness of the coupler is assumedto be the same for both coupler types. As previouslymentioned, in the coupler model considered in thisstudy, the draft gear or end of car cushioning are as-sumed to be connected to the car body using a pris-matic (translational) joint that allows for relative trans-lation along the joint axis. In general MBS algorithms,the nonlinear algebraic equations that define the pris-matic joint are expressed in terms of the absolute co-ordinates of the two bodies i and j connected by the

joint. The five algebraic constraint equations that elim-inate five degrees of freedom can be written in termsof the absolute Cartesian coordinates of the two bodiesas [13].

C(qi ,qj )

= [viT

1 vj viT2 vj viT

1 rijP viT

2 rijP hiT hj

]T = 0

(11)

where hi and hj are two orthogonal vectors drawn per-pendicular to the joint axis, defined respectively onbodies i and j ; vi and vj are two vectors definedalong the joint axis on bodies i and j , respectively;vi ,viT

1 ,viT2 form an orthogonal triad defined on body

i; and

rijP = ri

P − rjP = Ri + Ai ui

P − Rj − Aj ujP (12)

In this equation, Ai and Aj are the transformation ma-trices that define the orientation of bodies i and j , re-spectively; and ui

P and ujP are the local position vec-

tors of points P i and P j with respect to bodies i andj , respectively. Points P i and P j are defined on theaxis of the prismatic joint on bodies i and j , respec-tively. One can show that the Jacobian matrix of theprismatic joint constraints is defined as

Cq = [Cqi Cqj ]

=

⎡⎢⎢⎢⎢⎢⎣

vjT Hi1 viT

1 Hj

vjT Hi2 viT

2 Hj

rijTP Hi

1 + viT1 Hi

P −viT1 Hj

P

rijTP Hi

2 + viT2 Hi

P −viT2 Hj

P

hjT Hih hiT Hj

h

⎤⎥⎥⎥⎥⎥⎦

(13)

where Cqi and Cqj are the constraint Jacobian matricesassociated with the coordinates of bodies i and j , re-spectively; and other vectors and matrices that appearin the preceding equation are

HiP = [

I Ai ˜uiT

P Gi], Hj

P = [I Aj ˜ujT

P Gj],

Hi1 = ∂vi

1

∂qi= ∂

∂qi

(Ai vi

1

), Hi

2 = ∂vi2

∂qi= ∂

∂qi

(Ai vi

2

),

Hj = ∂vj

∂qj= ∂

∂qj

(Aj vj

), Hi

h = [0 Ai ˜hiT

Gi],

and

Hjh = [

0 Aj ˜hjT

Gj]


In these equations, ˜h is the skew symmetric matrix as-sociated with the vector h.

The shank of the coupler is assumed to be con-nected to the draft gear or EOC device using a pinjoint. In this model, only one relative rotation degreeof freedom about the Z axis is considered [5]. In thecase of the pin joint, the algebraic constraint equa-tions in terms of the absolute Cartesian coordinatesare the same as the equations of the prismatic jointgiven by (11) except of replacing the last equation bythe constraint equation rijT

P rijP − kθ = 0, where kθ is a

constant [13], and rijP = ri

P − rjP . Using the algebraic

constraint equations of the pin joint, one can show thatthe Jacobian matrix of the revolute joint constraintscan be written as

Cq = [Cqi Cqj ]

=

⎡⎢⎢⎢⎢⎢⎣

vjT Hi1 viT

1 Hj

vjT Hi2 viT

2 Hj

rijTP Hi

1 + viT1 Hi

P −viT1 Hj

P

rijTP Hi

2 + viT2 Hi

P −viT2 Hj

P

2rijTP Hi

P −2rijTP Hj

P

⎤⎥⎥⎥⎥⎥⎦

(14)

where the vectors and matrices that appear in thisequation are the same as previously defined in this sec-tion.

5 Coupler force equations and algorithm

In this section, the formulations of the forces resultingfrom the connection between the car body and shankwith the draft gear or EOC device; and the forces re-sulting from the coupler knuckle connection are pre-sented. The generalized forces associated with the ab-solute Cartesian coordinates that are used in generalMBS algorithms are obtained. The computational al-gorithm used with the redundant coordinate formula-tion is also summarized before concluding this sec-tion in order to shed light on the fundamental differ-ences between general MBS algorithms used in thisinvestigation for developing the inertial coupler modeland the simpler LTD algorithms used for developingthe noninertial coupler model. While linear stiffnessand damping coefficients are used in this investiga-tion, nonlinear coefficients obtained using measure-ments can also be considered using a spline functionrepresentation.

5.1 Draft gear/EOC connection

The car body is assumed to be connected to the draftgear or EOC device using a compliant force elementthat has stiffness kB and damping coefficient cB . Theforce due to this compliant force element can be writ-ten as

FB = kB(l − l0) + cB l (15)

where l is the current spring length, l0 is the initialspring length, and l is the time rate of deformation ofthe spring. Using the virtual work, δW = −FBδl, and

the definition of l as l =√

rijTB rij

B , where i and j re-fer to the two components connected by the spring,one can write the virtual change in the spring length interms of the virtual changes in the absolute Cartesiancoordinates as

δl = (rijTB rij

B

)−1/2rijTB δrij

B = 1

lrijTB δrij

B

= rijTB

l

[δRi − ui

BGiδθ i − δRj + ujBGj δθ j

](16)

where rijB = ri

B − rjB defines the position vector of the

spring attachment point Bi on the draft gear or EOCdevice with respect to the spring attachment point Bj

on the car body; Gi and Gj are the matrices that re-late the angular velocity vector to the derivatives ofthe orientation parameters; ui

B and ujB are the skew

symmetric matrices associated with the vectors Ai uiB

and Aj ujB that define the locations of the spring at-

tachment points with respect to the origin of the coor-dinate systems of bodies i and j , respectively; and Ai

and Aj are the transformation matrices that define theorientation of the two bodies. Defining the direction ofthe force FB by the unit vector rij

B = rijB / l, the virtual

work can be written as

δW = −FB rijTB

[δRi − ui

BGiδθ i − δRj + ujBGj δθ j

]= QiT

R δRi + QiTθ δθ i + QjT

R δRj + QjTθ δθ j (17)

which can be rewritten in a compact form as δW =QiT

B δqi + QjTB δqj , where qi and qj are the general-

ized coordinates of bodies i and j , and

QiB =

[Qi

R

Qiθ

]=

[−FB rij

B

FBGiT uiTB rij

B

],

222 A. Massa et al.

QjB =

[Qj

R

Qjθ

]=

[FB rij

B

−FBGjT ujTB rij

B

](18)

This equation defines the generalized forces associ-ated with the absolute Cartesian coordinates due to theconnection of the draft gear or EOC device with thecar body.

5.2 Shank connection

In this investigation, the shank is assumed to be con-nected to the draft gear or EOC device by a pin jointwhich allows yaw rotation. The restoring torque re-sulting from the relative shank rotation is introducedusing a rotational spring-damper element. This torqueis assumed to take the following form:

T ij = krθij + cr θ

ij (19)

where θij represents the relative rotation of the shank.The virtual change in this rotation can be defined as

δθij = hijT(Giδθ i − Gj δθ j

)(20)

where Gi and Gj are as defined before, and hij is aunit vector along the joint axis. Using the virtual work,one has

δW = −T ij δθ ij = QiTr δqi + QjT

r δqj (21)

where Qir = [0 −(T ij GiT hij )T ]T , and Qj

r =[0 (T ij GjT hij )T ]T .

These generalized forces are nonlinear functions ofthe relative shank rotation.

5.3 Knuckle force model

The connection between the couplers of two cars isrepresented by a compliant element that define theknuckle force model. Following a procedure similar tothe one used previously in this section, one can definethe following vectors of generalized forces associatedwith the absolute Cartesian coordinates of the shanksof the two couplers that connect two cars:

Qirr =

[Qi

R

Qiθ

]=

[−FQrij

Q

FQGiT uiTQ rij

Q

],

Qirr =

[Qj

R

Qjθ

]=

[FQrij

Q

−FQGjT ujTQ rij

Q

](22)

where the force is defined as

FQ = krr lrr + crr lrr (23)

and krr and crr are, respectively, the stiffness anddamping coefficients of the knuckle; lrr is the relativedisplacement between the two shanks; and lrr is therate of the relative displacement.

5.4 Solution algorithm

The inertial coupler model is developed using a gen-eral MBS algorithm based on the redundant coordi-nates. The dynamic equations of motion of the trainsystem are augmented with the nonlinear algebraicequations that describe mechanical joints and speci-fied motion trajectories. These algebraic equations aresatisfied at the position, velocity, and acceleration lev-els. As previously mentioned, LTD algorithms tend tobe much simpler than general MBS algorithms that re-quire the use of a procedure for the solution of differ-ential and algebraic equations. The simpler LTD algo-rithms will be discussed in later sections of this paper.

General MBS algorithms employ the technique ofLagrange multipliers and they are, for the most part,based on (2). These algorithms also exploit sparse ma-trix techniques in order to obtain efficient solution forthe nonlinear dynamic equations of motion. Using thedevelopments presented in Sect. 2, the following al-gorithm is used for developing the inertial couplermodel:

1. An estimate of the initial conditions that define theinitial configuration of the MBS is made. The ini-tial conditions that represent the initial coordinatesand velocities must be a good approximation of theexact initial configuration.

2. Using the initial coordinates, the constraint Jaco-bian matrix Cq can be constructed, and based onthe numerical structure of this matrix an LU fac-torization algorithm can be used to identify a set ofindependent coordinates.

3. Using the values of the independent coordinates,the constraint equations C(q, t) = 0 can be consid-ered as a nonlinear system of algebraic equationsin the dependent coordinates. This system can besolved iteratively using a Newton–Raphson algo-rithm and sparse matrix techniques.

4. Assuming that the independent coordinates and ve-locities are known from the numerical integration


and the dependent coordinates are determined fromthe previous step, one can construct the constraintequations at the velocity level, Cqq = −Ct , whereCt is the partial derivative of the constraint func-tions with respect to time. This linear system of al-gebraic equations has a number of scalar equationsequal to the number of dependent velocities.

5. Having determined the coordinates and velocities,(2) can be constructed and solved for the accel-erations and Lagrange multipliers. The vector ofLagrange multipliers can be used to determine thegeneralized reaction forces.

6. The independent accelerations can be identifiedand used to define the state space equations whichcan be integrated forward in time using a direct nu-merical integration method. The numerical solutionof the state equations defines the independent coor-dinates and velocities, which can be used to deter-mine the dependent coordinates and velocities asdiscussed in steps 3 and 4.

7. This process continues until the desired end of thesimulation time is reached.

The solution algorithm used for developing thenoninertial coupler model differs significantly fromthe solution algorithm discussed in this section as willbe explained in later sections of this paper. Nonethe-less, both algorithms require the solution of differen-tial and algebraic equations. The algebraic equationsin the case of the noninertial coupler model, however,are obtained from force equations instead of kinematicconstraint equations.

6 Longitudinal train dynamics (LTD)

Specialized LTD algorithms tend to be simpler as com-pared to general MBS algorithms. LTD algorithmsdo not in general require the solution of DAE’s sys-tem and are based on formulations that employ inde-pendent coordinates. They are designed to solve effi-ciently long train consists for the purpose of analyz-ing the longitudinal forces acting on the consist cars.These longitudinal forces can include tractive, brak-ing, resistance, and coupler forces. Efficient imple-mentation of some of these force elements such as cou-plers may require modifying LTD algorithms to allowfor solving algebraic equations simultaneously withthe system equations of motion. As previously men-tioned, the implementation of efficient coupler models

may require the neglect of the coupler inertia leadingto noninertial coordinates that can be determined bysolving a system of algebraic equations obtained us-ing a quasistatic force analysis [16]. In this section,the kinematic equations of the coupler are developedin terms of independent relative coordinates in orderto allow for a systematic implementation in LTD algo-rithms.

6.1 Draft gear/EOC connection

The slider block Bi , shown in Fig. 3, can represent adraft gear or EOC device. The global position vectorof the center of this block, which slides with respect tothe car body i, can be written as

riB = Ri + Ai

(ui

P o + uid

) −((

Ri − Rio

)T diB

)di

B (24)

In this equation, Ri is the global position vector of theorigin of the coordinate system of car body i,Ai isthe transformation matrix that defines the orientationof the coordinate system of car body i, ui

P o is the lo-

cal position vector of point P i, uid = (lio + di) ˆdi

B, ˆdi

B

is a unit vector that defines the block sliding axis, liois the initial distance between point P i and the center

of the block Bi, diB = Ai ˆdi

B,Rio is the global position

of the origin of the car body coordinate system beforedisplacement, and di is the distance that measures the

location of the block along ˆdi

B . Due to the fact that di

in (24) is defined as di = did + (Ri − Ri

o)T di

B , wheredid is the relative displacement of the sliding block

with respect to the car body i, the last term in (24)is introduced to eliminate the effect of the translationof the car body in the direction of di

B on the displace-ment of the slider block. The virtual displacement ofthe position vector of (24) can then be written as

δriB = δRi − Ai

( ˜ui

o + ˜ui

d

)Giδθ i + Ai ˆdi

Bδdi

+ LiRdδqi

= (Li

B + LiRd

)δqi + Li

Bdδdi (25)

where

LiB = [

I −Ai( ˜ui

P 0 + ˜ui

d

)Gi

],

LiBd = Ai ˆdi

B,

LiRd = [−(

diB ⊗ di

B

)2((

Ri − Rio

)T diB

)Ai

˜d

i

BGi]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(26)

224 A. Massa et al.

Fig. 3 Two-car system

The kinematic equations of the slider block will beused to define the draft gear and EOC kinematics.

6.2 Shank rotation

As shown in Fig. 3, the shank is subjected to a basemotion defined by the sliding di

d and is allowed to ro-tate by the angle αi with respect to the car body. Thelocation of the end point Qi of the shank with respectto the origin of the car body coordinate system is de-fined by the vector ui

Q = uiP 0 + ui

d + uir , where as

shown in Fig. 3, uir is a vector defined along the rotat-

ing arm which is assumed to have length lir . It followsthat

uir = lir

[cosαi sinαi 0

]T (27)

One can then write the global position vector of pointQi shown in Fig. 3 as

riQ = Ri + Ai ui

Q −((

Ri − Rio

)T diB

)di

B (28)

The virtual change in this position vector can be writ-ten as

δriQ = (

LiQ + Li

Rd

)δqi + Li

Bdδdi + Lir δα

i (29)

where

LiQ = [

I −Ai ˜ui

QGi], Li

r = Ai(ui

r

)αi (30)

In this equation (uir )αi = lir [− sinαi cosαi 0 ]T .

7 LTD noninertial generalized forces

By neglecting the effect of the inertia of the couplercomponents, one can develop a quasistatic equilib-rium conditions that define a set of nonlinear algebraicequations that can be solved for the noninertial couplercoordinates. These nonlinear equations can be solvediteratively using a Newton–Raphson algorithm to de-termine the noninertial coordinates di

d and αi intro-duced in the preceding section.

7.1 Coupler forces

The compliant force element that connects the carbody i and the draft gear or EOC device is denotedas fiB . Similarly, the force associated with car body j

is denoted as fjB . The forces fiB and fjB can be writtenas

fiB = (kiBdi

d + ciB di

d

)di

B,

fjB = (kjBd

jd + c

jBd

jd

)dj

B

}(31)


In this equation, kiB , k

jB , ci

B , and cjB are the stiff-

ness and damping coefficients associated with the draftgear or EOC of car bodies i and j , respectively. Notethat the spring deformations di

d and djd can be ex-

pressed in terms of di and dj . The virtual work ofthe forces given in the preceding equation can be writ-ten as δW

ijB = fiTB δri

P − fiTB δriB + fjT

B δrjP − fjT

B δrjB .

Using this equation and the expressions of the virtualdisplacements, one obtains

δWijB = QiT

B δqi + QjTB δqj + (

QijB

)T

niδqij

ni (32)

In this equation, qijni is the vector of the coupler non-

inertial coordinates defined as

qijni = [

did αi d

jd αj

]T(33)

and

QiB = (

LiP − Li

B − LiRd

)T fiB,

QjB = (

LjP − Lj

B − LjRd

)T fjB,(Qij

B

)ni

= [−LiTBd fiB 0 −LjT

Bd fjB 0]T

⎫⎪⎬⎪⎭ (34)

In order to account for the stiffness at the joint betweenthe coupler shank and the draft gear/EOC of each car,a torsional spring with stiffness ki

r and damping cir is

introduced at this joint. Let dir and dj

r be unit vectorsalong the shank and draft gear/EOC joint axis of carbodies i and j , respectively. It follows that the general-ized forces associated with the noninertial coordinatesαi and αj can be written, respectively, as

Qir = −ki

r

(αi − αi

o

) − cir α

i ,

Qjr = −k

jr

(αj − α

jo

) − cjr α

j

}(35)

This leads to the following generalized force vectorassociated with the noninertial coupler coordinates

(Qi

r

)ni

= [0 Qi

r 0 Qjr

]T (36)

The force at the knuckle due to the interaction be-tween the two couplers that connect the two cars isdenoted as fijrr . This force vector can be written as

fijrr = (kijrr

(lijr − l

ijro

) + cijrr l

ijr

)dij

rr (37)

In this equation, kijrr and c

ijrr are the spring stiffness and

damping coefficients, lijr is the current spring length,

lijro is the un-deformed length of the spring, and dij

rr

is a unit vector along the line connecting points Qi

and Qj , that is, dijrr = (ri

Q − rjQ)/|ri

Q − rjQ|. The vir-

tual work of the force fijrr can be written as δWijrr =

−fijTrr δri

Q + fijTrr δrj

Q. This equation upon using thekinematic equations previously developed in this studycan be written as

δWijrr = QiT

rr δqi + QjTrr δqj + (

Qijrr

)T

niδqij

ni (38)

In this equation,

Qirr = −(

LiQ + Li

Rd

)T fijrr ,

Qjrr = (

LjQ + Lj

Rd

)T fijrr ,(Qij

rr

)ni

= [−LiTBd fijrr − LiT

r fijrr

LjTBd fijrr LjT

r fijrr]T

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(39)

7.2 Generalized inertial and noninertial forces

Using the expressions of the forces developed in thissection, the generalized coupler forces associated withthe inertial coordinates of the car bodies i and j canbe written as

Qic = Qi

B + Qirr = (

LiP − Li

B − LiRd

)T fiB− (

LiQ + Li

Rd

)T fijrr ,

Qjc = Qj

B + Qjrr = (

LjP − Lj

B − LjRd

)T fjB+ (

LjQ + Lj

Rd

)T fijrr

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(40)

The forces associated with the noninertial coupler co-ordinates are

(Qi

c

)ni

= (Qi

B

)ni

+ (Qi

rr

)ni

+ (Qi

r

)ni

=

⎡⎢⎢⎢⎢⎣

−LiTBd(fiB + fijrr )

−LiTr fijrr + Qi

r

−LjTBd(fjB − fijrr )

LjTr fijrr + Q

jr

⎤⎥⎥⎥⎥⎦ (41)

The generalized forces associated with the car bodyinertial coordinates of (40) can be introduced tothe dynamic equations of motion, while the four-dimensional force vector associated with the couplernoninertial coordinates can be used to solve for thenoninertial coordinates di

d , αi, djd , and αj using an

iterative procedure described in the following sec-tion.

226 A. Massa et al.

8 LTD noninertial coupler solution algorithm

As previously mentioned, most LTD algorithms tendto be simpler and do not include a procedure forsolving DAE systems. If the inertia of the coupleris neglected, however, (41) can be used to definequasistatic equilibrium conditions of the coupler as(Qi

c)ni = 0. This vector equation includes, for eachcoupler connection, four nonlinear scalar equationsthat can be solved for the noninertial coupler coor-dinates di

d , αi, djd , and αj , assuming that the inertial

coordinates and velocities are known from the nu-merical integration of the train dynamic equations ofmotion. Therefore, when noninertial coupler modelsare considered in LTD studies, the resulting differen-tial and algebraic equations must be solved simultane-ously.

The LTD algorithm used in this investigation,which is implemented in the computer code ATTIF,is based on the trajectory coordinate formulation. Thenonlinear train equations of motion are automaticallygenerated in ATTIF and are solved using direct nu-merical integration method in order to determine theinertial coordinates and velocities. Knowing the iner-tial coordinates and velocities, the coupler quasistaticequilibrium conditions can be solved iteratively usinga Newton–Raphson algorithm in order to determinethe noninertial coordinates that enter into the formula-tion of the generalized forces associated with the traininertial coordinates. In developing the quasistatic al-gebraic equations used in this investigation, the effectof the noninertial velocities is neglected, leading toa nonlinear system of algebraic equations expressedin terms of the noninertial coordinates. The noniner-tial velocities are determined by solving the algebraicequations at the velocity level. The steps of the nu-merical algorithm used in this investigation to solvethe resulting DAE system can be summarized as fol-lows:

1. For a given set of initial conditions q0 and q0

associated with the train inertial coordinates,the quasistatic equilibrium conditions of each cou-pler element are used to define (41). This equa-tion is used to define the train system quasistaticcoupler conditions which can be written asQni(q,qni, q, qni, t) = 0, where qni is the vectorof system noninertial coordinates. If this system de-pends on the noninertial velocities, the effect of thevelocities is neglected, thereby defining the system

of algebraic equations Qni(q,qni, t) = 0, where qand q are assumed to be known from the initial con-ditions or the numerical integration of the equationsof motion.

2. Given the inertial coordinates and velocities q andq, the system Qni(q,qni, t) = 0 can be solved forqni using a Newton–Raphson algorithm that itera-tively solves the following system of equations:

∂Qni

∂qni

�qni = −Qni(q,qni, t) (42)

In this equation, �qni is the vector of Newton dif-ferences. Convergence is achieved if the norm of�qni or the norm of Qni is smaller than a specifiedtolerance.

3. If the vector Qe is function of the noninertialvelocities, that is, Qe = Qe(q,qni, q, qni, t); anapproximate solution for the noninertial veloci-ties can be obtained by differentiating the systemQni(q,qni, t) = 0 to define the following linear sys-tem in the noninertial velocities:

∂Qni

∂qni

qni = −∂Qni

∂qq − ∂Qni

∂t(43)

The solution of this system can be used to determinethe noninertial velocities qni . In obtaining (43), anassumption is made that the inertial coordinates andvelocities are assumed to be known.

4. Knowing the inertial coordinates and velocities andthe noninertial coordinates and velocities from theprevious two steps, the first equation of (10), Mq =Qe(q,qni, q, qni, t), can be constructed and solvedfor the inertial accelerations q.

5. The inertial accelerations q can be integrated for-ward in time using direct numerical integrationmethod in order to determine the inertial coordi-nates q and the inertial velocities q.

6. The simulation stops if the end time is reached;otherwise the procedure is repeated starting withStep 2.

In order to solve for the noninertial coordinates,the coefficient matrix in (42) must be evaluated. Theevaluation of this matrix requires the following deriva-


tives:

∂Lir

∂qijni

=[0 −uir 0 0 ], ∂Lj

r

∂qijni

=[0 0 0 −ujr ],

∂dijrr

∂qijni

=[ diB Li

r −djB −Lj

r ], ∂lijr

∂qijni

= 1lijr

dijTrr

∂dijrr

∂qijni

,

∂dijrr

∂qijni

= lijr

(∂dij

rr /∂qijni

)−dijrr

(∂l

ijr /∂qij

ni

)(lijr

)2 ,

∂fiB∂qij

ni

=[kiB di

B 0 0 0 ],∂fjB∂qij

ni

=[0 0 kjB dj

B 0 ], ∂fijrr∂qij

ni

= kijrr

(dij

rr∂l

ijr

∂qijni

+ (lijr − l

ijro)

∂dijrr

∂qijni

),

∂Qir

∂qijni

=[0 −kir 0 0 ], ∂Q

jr

∂qijni

=[0 0 0 −kjr ]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(44)

In this equation, ukr = Akuk

r , k = i, j . Using the def-initions of (44), one can show that the 4 × 4 matrixassociated with the coupler ij that can be used to con-struct the coefficient matrix in (42) is defined as

∂Qijni

∂qijni

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−LiTBd(

∂fiB∂qij

ni

+ ∂fijrr∂qij

ni

)

−LiTr

∂fijrr∂qij

ni

− fijTrr

∂Lir

∂qijni

+ ∂Qir

∂qijni

−LjTBd(

∂fjB∂qij

ni

− ∂fijrr∂qij

ni

)

LjTr

∂fijrr∂qij

ni

+ fijTrr

∂Ljr

∂qijni

+ ∂Qjr

∂qijni

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(45)

This matrix is nonsingular in the case of nonzero stiff-ness coefficients ki

B, kjB , and k

ijrr . It is also impor-

tant to point out, as previously mentioned in Sect. 3that since di and dj are considered as independentnoninertial coordinates, these coordinates are pre-dicted when evaluating the right-hand side of (42) andthe force vector associated with the inertial coordi-

nates using the equation (ukd − ((Rk − Rk

o)T dk

B) ˆdk

B −lko

ˆdk

B)T ˆdk

B, k = i, j .

9 Two-car model

The main goal of this study is to examine the effectof the coupler inertia on the solution accuracy andcomputational efficiency using simple train models.To this end, two different models are developed. Thefirst model is developed using a general MBS algo-rithm that employs redundant coordinates and allowsfor the solution of DAE systems. This model, which

Fig. 4 Forward position ( Rear car loaded, Leadingcar loaded, Rear car empty, Leading car empty)

Table 1 Model parameters

Coupler parameter Value Coupler parameter Value

l0 (m) 0.289 cir , c

jr (N·m·s) 1.0 × 105

kiB , k

jB (N/m) 3 × 107 k

ijrr (N/m) 5.0 × 108

ciB , c

jB (N·s/m) 2.5 × 105 c

ijrr (N·s/m) 5.0 × 105

kir , k

jr (N·m) 3 × 107 lr0 (m) 0.204

accounts for the coupler inertia, is called the inertialcoupler model. The second model is developed usingthe LTD algorithms implemented in ATTIF code. Thisalgorithm is modified to allow for the solution of thequasistatic coupler algebraic equations with the dif-ferential equations of motion. In this second model,which is referred to as the noninertial coupler model,the effect of the coupler inertia is neglected.

The type of coupler considered in this numericalinvestigation is called long shank E-type, which hasproperties reported in the literature [5]. The simpletwo-car train model, shown in Fig. 3, is used to ob-tain the numerical results reported in this numericalinvestigation. The coupler stiffness and damping co-efficients are assumed as kB = ki

B = kjB, cB = ci

B =cjB, kr = ki

r = kjr , cr = ci

r = cjr , krr = k

ijrr , and crr =

cijrr , with the values shown in Table 1. The masses and

mass moments of inertia of the coupler componentsare assumed as mshank = 140 kg, Izz,shank = 8 kg·m2,and mEOC = 50 kg. In order to examine the effect ofthe coupler inertia, two different simulation scenariosare considered in this investigation; the first is withempty cars, while the second is with full loaded cars.

228 A. Massa et al.

Table 2 Car parameters

Parameter Full loading Empty

m (kg) 93800 23800

Ixx (kg·m2) 117000 20900

Iyy (kg·m2) 1370000 237000

Izz (kg·m2) 1380000 243000

Ixy , Ixz, Iyz (kg·m2) 0 0

Fig. 5 EOC and shank knuckle relative displacements (loadedcar scenario) ( di

d inertial, did noninertial, d

ijrr

inertial, dijrr noninertial)

In both scenarios, the traction force is applied to therear car with a magnitude of 3 × 105 N. For both sce-narios, the initial velocities and angles are assumed tobe zero. The initial coordinates of the center of massof the two cars along the longitudinal global X axis areassumed to be 0 m for the rear car and 13.976 m for theleading car; while the vertical Z coordinate of the cen-ter of mass is assumed to be 1.85 m for the full loadingcondition and 1.085 m for the empty condition. The in-ertial properties of the cars in the empty and full casesare reported in Table 2 [16]. In case of full loaded car,the ratio of the mass of the coupler to the mass of thecar is 0.00205; while in the case of empty car, this ra-tio is 0.00798. Note that this ratio is less than 1%, evenin the case of the empty car. The coupler model con-sidered in this investigation is assumed to have EOCunit.

9.1 Simulation results

Figure 4 shows the forward position of the two cars,for the two scenarios of empty and loaded cars. Their

Fig. 6 EOC and shank knuckle relative velocities (loaded carscenario) ( d i

d inertial, d id noninertial, d

ijrr in-

ertial, dijrr noninertial)

Fig. 7 EOC and shank knuckle relative displacements (emptycar scenario) ( di


ijrr

inertial, dijrr noninertial)

velocities increase linearly due to the application ofthe constant force. Figure 5 shows the EOC and shankknuckle relative displacements di

d and dijrr , respec-

tively, for the full loaded car scenario using both theMBS and LTD algorithms, while Fig. 6 shows the timederivatives of these relative displacements. Figure 7shows the EOC and shank knuckle relative displace-ments di

d and dijrr , respectively, for the empty car sce-

nario using both the MBS and LTD algorithms, whileFig. 8 shows the time derivatives of these relative dis-placements. Each of these figures shows the effect ofneglecting the coupler inertia.


The results presented in this section demonstratethat neglecting the coupler inertia does not have asignificant effect on the accuracy of the solution.Nonetheless, the neglect of the coupler inertia leadsto significant improvement in the computational effi-ciency. The LTD model that does not take into accountthe effect of the coupler inertia was found to be morethan 85 times faster than the MBS model that takesinto account the effect of the coupler inertia. The sameexplicit numerical integration method is used to ob-tain the results of the inertial and non-inertial couplermodel.

Fig. 8 EOC and shank knuckle relative velocities (empty carscenario) ( d i


ijrr in-

ertial, dijrr noninertial)

9.2 Eigen value and frequency domain analysis

Figure 9 shows the mode shapes and frequencies ob-tained by solving the eigenvalue problem of the in-ertial coupler model [7, 12], for the full loaded carmodel. The dashed line in this figure shows the ref-erence configuration, while the solid line shows thedisplaced configuration for the given mode. Using theresults presented in this figure, it is possible to rec-ognize some frequencies that are associated with thesmall inertia of the coupler components, including themodes that describe the rotation of the shanks or rela-tive displacement between the slider and the car. Thefirst mode with a low frequency of 2.8 Hz is the onlymode that is present in both the inertial and noniner-tial coupler models. Other modes presented in Fig. 9are mainly due to the coupler inertia. It is clear thatthe second mode, for example, which has frequency ofapproximately 63.26 Hz corresponds to the longitudi-nal motion of the coupler with respect to the car body.This frequency can be easily calculated by using theinertia of the coupler and the stiffness coefficient kB .It is also important to point out that the highest fre-quency is associated with the rotational motion of theshanks. The mode associated with this frequency de-pends on the coupler inertia and, therefore, this modedoes not appear in the noninertial system.

In order to have a better understanding of the effectof these frequencies, a Fast Fourier Transform (FFT)of the acceleration is performed [2]. The accelerationsare directly related to the coupler forces and can beused as good FFT measure. Figure 10 shows the fre-quency domain analysis of the acceleration of the rear

Fig. 9 Mode shapes

230 A. Massa et al.

Fig. 10 Frequency domainanalysis of the acceleration

coupler EOC unit which has the same frequency con-tent as the shank acceleration. The results presentedin this figure are obtained for different values of thedamping coefficients cB of the rear coupler. Figure 11shows the frequency domain results of Lagrange mul-tiplier associated with longitudinal generalized reac-tion of the revolute joint constraints of the inertial cou-pler. The results presented in this figure show clearlythe effect of the first mode (2.8 Hz) and there is nearlyno effect from the other higher modes.

10 Ten-car model

In order to check the robustness of the LTD cou-pler model implemented in ATTIF (Analysis of Train/Track Interaction Forces), a 10-car model is devel-oped. This train model, shown in Fig. 12, consistsof ten fully loaded cars; the cars and EOC deviceshave the same specifications and properties as thefully loaded cars and EOC devices used in the 2-carmodel discussed in the preceding section, except forthe EOC damping coefficient which is assumed to becB = ci

B = cjB = 5 × 105 N·s/rad. The value of cB

has been chosen in order to reach steady state in areasonable time. The tractive force is assumed to be3 × 106 N, and the simulation is performed for 20 sec-onds. Figures 13 and 14 show the relative displace-ments and velocities of the 5th coupler using MBSand LTD algorithm. The results presented in these fig-ures show that the LTD and MBS solutions are in agood agreement. Nonetheless, the LTD model is morethan 300 times faster than the MBS model, indicatingthat as the number of cars increases, the LTD model

Fig. 11 Lagrange multiplier frequencies

will become more efficient as compared to the MBDmodel.

11 Curved track simulation

In the examples considered in the preceding two sec-tions, a tangent track was used, and therefore, αi andαj remain nearly constant. In order to examine theeffect of the geometric nonlinearities resulting fromthe coupler kinematic degrees of freedom αi and αj ;the full loaded two-car model is considered and itis assumed to negotiate the curved track shown inFig. 15. This track is assumed to consist of 60.96 mtangent segment, 121.92 m spiral segment, 3-degreeconstant curve of length 91.44 m, spiral segment oflength 91.44 m, tangent segment of length 121.92 m,spiral segment of length 91.44, −3-degree curve of


Fig. 12 Ten-car model (ATTIF interface)

Fig. 13 EOC and shank knuckle relative displacements (5thcoupler) ( di


ijrr iner-

tial, dijrr noninertial)

length of 91.44, spiral segment of length 121.92 m,and tangent segment of length 60.96. A 76.2 mmsuper-elevation is assumed for the first two spiral seg-ments, and −76.2 mm for the last two spiral segments.In this example, car j is considered as the locomo-tive that pulls car i which represents a freight car. Thelength of the shank knuckle lr0 is assumed to be 0 m inthe simulation scenario considered in this section. Theinitial forward velocity of each car is assumed to be20 m/s, and the simulation time is increased to 40 s in

Fig. 14 EOC and shank knuckle relative velocities (5th cou-pler) ( d i


ijrr inertial,

dijrr noninertial)

order to capture the nonlinear behavior of the coupleron different track sections. Figure 16 shows the for-ward position during the first 5 seconds of car i, sliderblocks i and j that represent EOC devices, and car j

as the result of the application of constant forward ve-locity 20 m/s using both the MBS and LTD algorithms.Figure 17 shows the rotational deformation of the tor-sional spring of car i for both MBS and LTD mod-els. The results presented in these figures show thatthe LTD and MBS solutions are in a good agreement.

232 A. Massa et al.

Fig. 15 Curved track

Fig. 16 Forward position ( Rear car inertial, Sliderblock i inertial, Slider block j inertial, Leading carinertial, Rear car noninertial, Slider block i nonin-ertial, Slider block j noninertial, Leading car non-inertial )

Nonetheless, the LTD model is more than 200 timesfaster than the MBS model, indicating that the LTDmodel is accurate and more efficient as compared tothe MBD model in the case of curved track simula-tions.

12 Summary and conclusions

LTD algorithms tend to be simpler as compared toMBS algorithms that require the use of DAE solver.LTD are designed for the efficient solution and anal-

Fig. 17 Torsional spring deformation ( Inertial,Noninertial)

ysis of longitudinal train forces of long consists. Carcoupler forces have a significant effect on the behav-ior of trains in response to braking, tractive, and resis-tance forces. For this reason, it is important to developefficient coupler models that can be implemented inLTD algorithms. The development of such an efficientcoupler model requires modifying existing LTD al-gorithms to include a procedure for solving differen-tial and algebraic equations simultaneously as demon-strated in this study.

This investigation is focused on studying the ef-fect of neglecting the coupler inertia which can leadto significant deterioration of the computational effi-ciency. The relatively small inertia of the coupler canlead to high frequency oscillations in the solution; re-quiring the use of smaller integration time steps andsignificantly increasing the CPU time. In order to ad-dress this problem, a noninertial coupler model is de-veloped by replacing the coupler differential equationswith quasistatic nonlinear algebraic force equations.These algebraic equations are iteratively solved usinga Newton–Raphson method in order to determine thenoninertial coordinates. The noninertial velocities aredetermined by solving these quasistatic equations atthe velocity level. The noninertial coordinates and ve-locities are then used in formulation of the generalizedcoupler forces associated with the coordinates of thecar bodies. In order to examine the effect of neglectingthe coupler inertia on the accuracy of the solution andthe computational efficiency, an inertial coupler modelis developed using MBS algorithms. The results ob-tained using the inertial and noninertial coupler mod-


els are compared. The numerical results obtained inthis study showed that the neglect of the coupler inertiadoes not have a significant effect on the accuracy of thesolution. On the other hand, the neglect of this inertialeads to significant improvement in the computationalefficiency. The results obtained showed that the LTDimplementation that neglects the effect of the couplerinertia becomes more efficient as the number of carsincreases. An eigenvalue analysis and FFT are usedto identify the frequencies associated with the couplerinertia. As discussed in this study, these high frequen-cies do not appear when the noninertial coupler modelis used.

Acknowledgements This research was supported by the Fed-eral Railroad Administration, Washington, DC.

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numerical study of the noninertial systems: application to train coupler systems

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