2003 Paper MUBO JPombo JAmbrosio

7/28/2019 2003 Paper MUBO JPombo JAmbrosio

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Multibody System Dynamics 9: 237264, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.

237

General Spatial Curve Joint for Rail GuidedVehicles: Kinematics and Dynamics

JOO POMBO and JORGE A. C. AMBRSIOIDMEC/IST, Av. Rovisco Pais, P-1049-001 Lisboa, Portugal;

E-mail: [email protected]

(Received: 17 December 2001; accepted in revised form: 9 October 2002)

Abstract. In the framework of multibody dynamics for rail-guided vehicle applications, a new kine-matic constraint is proposed, which enforces that a point of a body follows a reference path while thebody maintains a prescribed orientation relative to a Frenet frameassociated to the spatial track curve.Depending on the specific application, the tracks of rail-guided vehicle are described by analyticalline segments or by parametric curves. For railway and light track vehicles, the nominal geometry ofthe track is generally done by putting together straight and circular track segments, interconnected bytransition track segments that ensure the continuity of the first and second derivatives of the railwayin the transition points. For other applications, the definition of the track is done using parametriccurves that interpolate a given number of control points. In both cases, the complete characterizationof the tracks also requires the definition of the cant angle variation, which is done with respectto the osculating plane associated to the curve. The track models for multibody analysis must bein the form of parameterized curves, where the nominal geometry is obtained as a function of aparameter associated to the curve length. The descriptions adopted here ensure, not only that thetype of continuity of the original track definition is maintained, but also that no unwanted deviationsfrom the nominal track geometry are observed, which can be perceived in the dynamic analysis astrack perturbations. In this work different types of track geometric descriptions are discussed. The

application of cubic splines, to interpolate a set of points used to describe the track geometry, leads toundesired oscillations in the model. The parameterization of analytical segments of straight, circularand cubic polynomial track segments does not introduce such oscillations on the track geometry butit is rather complex for the description of railways with large slopes or with vertical curves. Splineswith tension minimize the undesired oscillations of the interpolated curve that describes the railwaytrack nominal geometry, but the curve segment parameters are not proportional to the length of thetrack. It is proposed here that the nominal geometry of the track is described by a discreet numberof points, which are organized in a tabular manner as function of a parameter that is the length ofthe track measured from its origin to a given point. For each entry, the table also includes the vectorsdefining the Frenet frames and the derivatives required by the track constraint. The multibody codeinterpolates such table to obtain all required geometric characteristics of the track. With applicationsto a roller coaster, the suitability of this description is discussed in terms of the choice of originalparametric curves used to construct the table, the size of the length parameter step adopted for thetable and the efficiency of the computer implementation of the formulation.

Key words: railway dynamics, Frenet frame, spatial curve geometry, prescribed motion constraint.


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238 J.POMBO AND J.A.C. AMBROSIO

Figure 1. Two body arrangement to model the track foundation flexibility. The base bodyhas a prescribed motion while the track element has a motion relative to it described by thein-plane degrees-of-freedom, i.e. two translations and one rotation.

1. Introduction

The dynamic analysis of railway [13], roller coaster [4] or any other type of railguided vehicles requires an accurate description of the track geometry. The trackis composed of two rails, which can be viewed as two parallel line defined in aplane that sits in a spatial curve, defined hereafter as the reference path. The basicingredient to define the track is therefore the geometry of the reference path, whichmust include the vertical gradients, lateral curves and cant. Any track irregularitiescan be perceived as deviations from the reference path parallel lines, representingthe rails. Typically, these are modeled by adding to the track perfect geometry smallperturbations that are either measured experimentally or generated numerically.Furthermore, the track flexibility, or the deformability of its foundations, can also

be introduced on the model by allowing that a track body moves with respect toa track reference, as depicted by Figure 1. There, the track reference element hasto move along the reference path and have an orientation compatible with it. Theobjective of this work is to present a description of the spatial geometric features ofthe tracks and its computational implementation in a form suitable to the multibodymethodology adopted for the modeling of railway systems. The introduction of thetrack irregularities and the flexibility of the track foundations are not considered inthis work.

Depending on the specific application, the reference path of the track geometrycan be described by a number of types of parametric curves [5]. For railway andlight track vehicles the description of the nominal geometry of the track is generally

done by putting together straight and circular track segments, interconnected bytransition segments that ensure the continuity of the first and second derivativesof the railway in the transition points [1, 6]. Moreover, these transition elementsare responsible for the smooth variation of the lateral accelerations of the vehicles,when they move from a straight track to a circular track or between two track seg-ments of the same type with different radius or orientations. For other applications,


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GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 239

parametric curves that interpolate a given number of control points are commonlyused to define the track geometry. In any case, the complete characterization of thetracks also requires the definition of the cant angle variation along the referencepath. For flat tracks, the cant angle in a given point of the reference path is mea-sured in the plane perpendicular to the reference path, between a line that seats onboth rails and the horizontal plane. For tracks with a full spatial geometry a newdefinition of this angle is introduced. It is proposed that the osculating plane of thereference track plays the role of the horizontal plane of the flat track in measuringthe cant angle.

In this work two types of track geometric descriptions are discussed in theframework of the multibody models for railway dynamics analysis, i.e., analyti-cal segments [7] and cubic splines [10, 11]. The application of cubic splines tointerpolate a set of control points describing the track geometry leads to undesiredoscillations in the track model [12]. For instance, if a spline interpolation is usedto describe the geometry of the reference path made of a straight segment followed

by a circular segment, the result will not be a perfect straight line but simply acurve that oscillates in turn of the original lines. Another drawback, in the directapplication of this approach, is that the local parameter used in each spline segmentis not linearly related with the length of the segment, e.g., a given point of thereference path for which the local curve parameter is half of the parameter intervalis not necessarily located half way along the curve. Other methodologies usingsplines with tension or Akima splines [11], are alternative techniques for the pa-rameterization of the reference path geometric description in railway applications.Although these alternative forms of interpolation have the potential to minimizethe undesired oscillations of the interpolated curve they are not discussed here.

The reference path parameterization with analytical segments, which use

straight, circular and transition curves, does not introduce unwanted oscillations onthe track geometry. However, this description is rather complex to model railwayswith large slopes or with vertical curves. Some of the commercial codes that adoptthis description impose that the tracks are basically horizontal in order to avoiddifficulties [79].

Regardless of the form in which the reference path geometry is described asuitable kinematic constraint must be defined in order to enforce not only that aparticular point of given body of the multibody systems follows the reference pathbut also that the body orientation does not change with respect to a Frenet-frame as-sociated to the curve. The methodology proposed here for the general spatial curvejoint can use any descriptive form for the curve. The position, the Frenet-framevectors and their derivatives, which are used in the definition of the constraint, are

pre-processed and included in a table as function of the curve length from the originto the actual point position. Therefore, during the dynamic analysis the quantitiesinvolved in the general spatial curve joint are obtained by linear interpolation ofthe tabulated values. The length parameter step is small enough to ensure that forany reasonable speed of the rail guided vehicle not more than once a time step the


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Figure 2. Cant and cant angle.

quantities are obtained by interpolation with the same point limits. The constraint isimplemented in the general purpose multibody computer program DAP-3D [13].The constraint features and its computational efficiency are discussed through the

application to the dynamic analysis of a roller coaster in different tracks.

2. Physical Aspects of Rail Guided Systems

Some physical aspects relevant for the track geometric description of rail guidedsystems are examined here. Special emphasis is put in the description of the meth-ods used to derive the analytic properties of parametric curves required to establishthe general spatial curve kinematic constraint.

2.1. CANT AND CANT ANGLE

When traveling in horizontal curves, rail guided vehicles are influenced by centrifu-gal forces, which act away from the center of the curve and tend to overturn thevehicles. The sum of a vehicle weight and its centrifugal force produces a resultantforce directed toward the outer rail. In order to counteract this force, the outerrail in a curve is raised, which is called cant or superelevation, ht, and is definedaccording to Figure 2 [3]. The base for the definition of cant is the distance 2 b0,between the nominal wheel-rail contact points. Angle t, in Figure 2, called cantangle, is given by:

t = arcsin(ht/2b0). (1)

A curve is designed as being balanced at the equilibrium speed when the travel-

ing vehicles produce a resultant force through the centerline of the track. Under thiscondition, the vertical rail forces are equal, so that maximum utilization of tractioneffort and minimum wear on wheels and rails can be realized [1].

At this point it should be noted that it is assumed that the curve is horizontal and,therefore, the definition of the cant angle uses the horizontal plane as the referenceplane. For a spatial curve it is not clear what the reference plane is, relative to which


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Figure 3. Transition curves and superelevation ramps.

the cant angle should be defined. In this work it is assumed that the cant angle isdefined with respect to the osculating plane, which is presented later together withthe descriptive geometry of the reference path curve.

2.2. TRANSITION CURVES AND SUPERELEVATION RAMPS

When trains are operated at normal speeds, a circular curve with cant cannot befollowed directly by a tangent track, i.e., a straight track segment. A transitionbetween these two types of elements, designated by transition curve, is required inorder to minimize the change of lateral acceleration of the vehicles. Usually, the

radius of a transition curve is changed continuously, decreasing from an infiniteradius at the tangent end to a radius equal to that of the circular curve at theother end. This change of radii provides a smooth transition from tangent to curvesegments, and vice versa, also allowing for the superelevation to change graduallyover its length. Therefore, the cant is also changed continuously leading to the so-called superelevation ramp. In general, the transition curve and the superelevationramp have the same start and end points, i.e., the curvature and the superelevationin transition curves correspond to each other, as illustrated in Figure 3 [1, 3].

2.3. ANALYTIC PROPERTIES OF PARAMETRIC CURVES

The definition of the general spatial curve kinematic constraint requires a ratherelaborate geometric description of the properties of the parametric curves used.The analytic properties of the curves are classified as intrinsic or extrinsic [5]. Theintrinsic properties are local properties that vary from point to point, thus, they areonly computed at specific points. These properties include the principal vectors,designated by tangent, normal and binormal vectors, the principal planes, desig-


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nated by normal, osculating and rectifying planes, the curvature and the torsion.The extrinsic or global properties are those that depend on the over-all character-istics of a geometric element. For a given curve these include the arc length andwhether or not it is a plane curve or a straight line. In this section the focus is onlyon some of the local and global properties, which are important for the definitionof the kinematic constraint. The interested reader can obtain additional informationin [5].

2.3.1. Parametric Curves

A parametric curve consists on a point-bounded collection of points that haveCartesian coordinates given by continuous, one-parameter, single-valued mathe-matical functions in the form:

x = x(u),

y = y(u),

z = z(u), (2)

where u is the parametric variable. The curve is point-bounded because it has twodefinitive end points corresponding to the interval limits of the parametric variableu. The coordinates of any point on a parametric curve are treated as the componentsof the vector g(u) given by

g g(u) = x(u)

y(u)

z(u) . (3)

In order for the curve, represented in Figure 4, to be used in the kinematicconstraint it is required that the moving frame represented by vectors t, n and b isdefined. The unit vectors that characterize the frame, known as Frenet frame, aredefined in the intersection of the different planes represented in Figure 4. For astraight line it is assumed that the osculating plane is either horizontal, if the lineis in the XY plane, or that its orientation is such that its intersection with the XYplane is perpendicular to the straight line.

2.3.2. Unit Tangent Vector

On a parametric curve, the tangent vector at point g is denoted by gu and it is foundby differentiating g(u) with respect to the parametric variable. Thus,

gu(u) =dg(u)

du. (4)


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Figure 4. The moving frame.

Note that when u appears as a superscript, it indicates differentiation with respect tou. It should be also noticed that the relationship between the parametric derivativesand the ordinary derivatives of Cartesian space is:

dy

dx=

dy/du

dx/du. (5)

In many situations, it is necessary to work with the unit tangent vector to the curveat point g(u), which is given by:

t =gu

||gu||. (6)

2.3.3. Principal Unit Normal Vector

The principal normal vector at point g(u) is normal to the curve and consequentlyit must lie in the plane normal to the unit tangent vector [5]. However, amongthe many possible normal vectors, the unit principal normal vector points towardsthe spatial center of curvature of the curve. Given the parametric expression for acurve, the principal normal vector is found by [5]:

k = guu g

uuT

g

u

||gu||2gu, (7)

where guu is the second derivative ofg(u) with respect to the parameter u. Finally,the principal unit normal vector, is obtained as:

n = k/||k||. (8)


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2.3.4. Binormal Vector

In order to define the Frenet frame associated to the reference path another vectornormal to the curve in point g(u) needs to be defined. Using the principal tangent

and normal vectors, given by Equations (6) and (8) respectively, the third vector,denominated by binormal vector, is defined as

b = tn. (9)

The expressions for the three characteristic vectors associated with each pointon a curve have been developed. They are intrinsic properties since they varyfrom point to point. In Figure 4 it is clear that these elements form a local, three-dimensional orthogonal coordinate system consisting of three axis vectors. Thiscoordinate system, known as Frenet frame, is also designated by moving trihedronof the curve in some literature [5].

2.3.5. Arc Length

The arc length is an extrinsic property of the curve since it is a global characteristicthat doesnt vary from point to point. The length of a parametric curve is given by[5]

L =

u2u1

gu

Tgu du, (10)

where u2 > u1 are two arbitrary values of the curve parametric variable.It should be noticed that the parameter u used for the definition of the curve is

not necessarily directly related with the length of the curve from its origin to thecurrent position of the point represented by the parameter. Within the framework

of the application of the parametric description of the spatial curve in the definitionof the general spatial path kinematic constraint a replacement of parameter u bya length representative parameter is necessary. This is discussed together with thecomputer implementation of the kinematic constraint at a later stage.

3. Multibody Systems Methodology

The methodology, developed here, is implemented in the computer program DAP-3D [13], which is suitable for the spatial dynamic analysis of general multibodysystems. The multibody methodology, based on Cartesian coordinates, is brieflydescribed in order to introduce the formulation of the general spatial curve con-

straint. Finally, the new constraint is formulated and its implementation aspects arediscussed.


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3.1. EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS

A multibody system is defined as a collection of rigid and/or flexible bodies con-strained by kinematic joints, which control their relative motion, and eventually

acted upon by a sets of internal and/or external forces. The position and orientationof each body i in the space can be described by a position vector ri and a set ofrotational coordinates pi , which are organized in a vector as [13]

qi = {rT, pT}Ti . (11)

According with this definition, a multibody system with nb bodies is described bya set of coordinates in the form:

q = {qT1 , qT2 , . . . , q

Tnb}

T. (12)

The dependencies among system coordinates, which result from the existenceof mechanical joints interconnecting the several bodies, are defined through theintroduction of kinematic relationships involving the coordinates, which are des-

ignated by kinematic constraints. In order to guide the system during the analysis,driving constraints are also defined to control the system degrees-of-freedom alongthe time. After being joined in a consistent manner, in the global constraints vector,these linear and/or non-linear equations are written in short as [13]

(q, t) = 0, (13)

where q is the generalized coordinates vector, defined in (12), and t is the time vari-able, resulting from the existence of driving constraints. The second time derivativeof Equation (13) with respect to time yields the accelerations equations:

(q, q, q, t) = 0 qq = , (14)

where q

is the Jacobian matrix of the constraints, q is the acceleration vector and is the vector that contains all contributions that depend on the velocities and ontime.

For an unconstrained mechanical system [13], the matricial form of theequations of motion are given by

Mq = f, (15)

where M is the global mass matrix, containing the mass and moments of inertia ofall bodies, and f is the force vector, containing all forces and moments applied onsystem bodies, as well as the gyroscopic forces.

The system kinematic constraints (13) can be added to the equations of mo-tion (15) using the Lagrange multipliers technique [13]. Defining by the vector of

the unknown Lagrange multipliers, the equations of motion for a constrained me-chanical system can be written as a system of differential and algebraic equationsas

M Tq

q 0

q

=

f

. (16)


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The Lagrange multipliers are associated to the kinematic constraints and arephysically related with the reaction forces generated between the bodies intercon-nected by kinematic joints. These reaction forces, due to the kinematic joints, aregiven by [13]

f(c) = Tq . (17)

According to this methodology, the dynamic analysis of multibody systems in-volves the calculation of the vectors f and , for each time step. Equation (16) isthen used to calculate the system accelerations q. These accelerations together withthe velocities q are integrated in order to obtain the new velocities q and positionsq for the time step. This process proceeds until the complete description of thesystem motion is obtained, for the selected time interval. The usual proceduresto handle the integration of sets of differential-algebraic equations must still beapplied in this case in order to eliminate the constraint drift or to maintain it undercontrol [13].

3.2. GENERAL SPATIAL CURVE KINEMATIC CONSTRAINTS

The general spatial curve kinematic constraint equations are derived now and theresulting formulation is implemented in the computer program DAP-3D [13]. Thisconstraint is the basis of the definition of the tracks for the rail guided vehicles byenforcing that a body moves along the railway. When such body travels along thetrack, not only the railway path has to be followed, but also its spatial orientationhas to be prescribed, according to railway characteristics. The formulation usedto implement these kinematic constraints that define the Frenet frame is describednext.

3.2.1. Prescribed Motion Constraint

The objective here is to define the constraint equations that enforce a certain point,of the given rigid body, to follow the reference path. Consider a point R, located ona rigid body i, that has to follow the specified path, as depicted in Figure 5. The pathis defined by a parametric curve g(L), which is controlled by a global parameter L,which represents the length traveled by the point along the curve until the currentlocation of point R. This parameter L should not be confused with parameter uused in Equations (2) through (10). The kinematic constraint is written as

(pmc,3) = 0 rRi g(L) = 0, (18)

where rRi = ri + AisRi represents the coordinates of point R with respect to the

global coordinate system (x,y,z), ri is the vector that defines the location of thebody-fixed coordinate system ( , , )i , Ai is the transformation matrix from thebody i fixed coordinates to the global reference frame and sRi represents the co-ordinates of point R with respect to the body-fixed reference frame. The vector


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Figure 5. Prescribed motion constraint.

g(L) = {x(L), y(L), z(L)}T represents the Cartesian coordinates of the curve

where point R is constrained to move and L is the curve parametric variable. Fornotational purposes (.) means that (.) is expressed in body-fixed coordinates. Theconstraint equations are assigned with a superscript of two indices where the firstdenotes the type of constraint and the second defines the number of independentequations that it involves.

It should be noticed that the constraint requires the introduction of the newcoordinate L in the multibody system, which is the length of the curve traveled bypoint R from the start of the curve up to its current position. Therefore, the velocityand acceleration vectors also include the time derivatives of this parameter.

The velocity equation is obtained as the time derivative of Equation (18) withrespect to time

(pmc,3)

= 0 rRi g(L) = 0

I sRi Ai

dg

dL

rL

= 0, (19)

where the Jacobian matrix is

(pmc,3)q =

I sRi Ai

dg

dL

(20)

in Equation (20), I is a 3 3 identity matrix and sRi = Ai sRi represents the coordi-

nates of point R with respect to the ( , , )i coordinate system, written in globalcoordinates.

The acceleration equation is obtained by the derivative of Equation (19) withrespect to time. The resulting equation is

(pmc,3)

= 0

I sPi Ai

dg

dL

rL

= ii Ai sPi + d2gdL2 L2, (21)


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Figure 6. Local frame alignment constraint.

where ri = {x y z}Ti are the translational accelerations of body i, i =

{ }Ti represents the angular acceleration of the body-fixed coordinate

system ( , , )i, expressed in local coordinates, and L is the second time deriv-

ative of the curve parametric variable. The contribution of the constraint for theright-hand-side of the accelerations Equation is given the 3 1 vector, written as

# = i sRi +

d2g

dL2L2. (22)

Therefore, Equations (18, 20) and (22) represent the quantities that must beimplemented in constraint module of the computer code.

3.2.2. Local Frames Alignment Constraint

The second part of the constraint ensures that the spatial orientation of body i

remains unchanged with respect to the Frenet frame associated to the referencepath curve, as represented in Figure 6.Consider a rigid body i where (u, u, u)i represent the unit vectors associated

to the axis of the body-fixed coordinate system ( , , )i . Consider also that theFrenet frame of the general parametric curve g(L), is defined by the principalunit vectors (t, n, b)L, as depicted in Figure 6. Assume that, at the initial timeof analysis, the relative orientation between the body vectors (u, u, u)i and thecurve local frame (t, n, b)L are such that the following equations hold

(lfac,3) = 0

nT ubT unT u

=

a

b

c

. (23)

The kinematic constraint ensures that this alignment will remain constant through-out the analysis. The transformation matrix from the body i fixed coordinates tothe global coordinate system is written as

Ai = [u u u]i . (24)


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With the purpose of having a more compact notation, let the following unit vectorsbe defined

u1 = {1 0 0}T

; u2 = {0 1 0}T

; u3 = {0 0 1}T

. (25)

Equation (23) is now re-written as

(lfac,3) = 0

nTAiu1bTAiu1nTAiu3

=

a

b

c

. (26)

The velocity equation for this constraint is obtained as the time derivative ofEquation (26), expressed as

(lfac,3)

= 0

0T nTAiu1

dndL

TAiu1

0T bTAiu1

db

dL

TA1u1

0T nTAiu3

dn

dL

TAiu3

r

L

= 0. (27)

The contribution of frames alignment constraint (27) to the Jacobian matrix is thesubmatrix 3 7 given by

(lfac,3)q =

0T

nT

Ai u1 dn

dLT

Ai u1

0T bTAi u1

db

dL

TAi u1

0T nTAi u3

dn

dL

TAi u3

, (28)

where 0T is a 1 3 null vector. The acceleration equation is the time derivative ofEquation (27), and it is written as

(lfac,3)

= 0

0T nTAiu

1 dn

dLT A

iu

1

0T bTAiu1

db

dL

TAi u1

0T nTAiu3

dn

dL

TAi u3

r

L


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=

2L

dn

dL

TAi

i + n

TAii

i + L

2

d2n

dL2

TAi

u1

2L db

dLT

Aii + bTAi

i

i + L2

d2bdL2

TAi

u12L

dn

dL

TAi

i + n

TAii

i + L

2

d2n

dL2

TAi

u3

, (29)

The contribution of the local frames alignment acceleration equation, described byEquation (29) for the right-hand-side of the accelerations equation [13] is the 3 1vector, written as

=

2L

dn

dL

TAi

i + n

TAii

i + L

2

d2n

dL2

TAi

u1

2L

dbdL

TAi

i + b

TAii

i + L

2

d2

bdL2

TAi

u1

2L

dn

dL

TAi

i + n

TAii

i + L

2

d2n

dL2

TAi

u3

. (30)

The complete set of quantities that is necessary to implement computationallyin the general spatial curve constraint is described by Equations (18, 20, 22, 26, 28)and (30) represent the quantities that must be implemented in constraint module ofthe computer code.

4. Pre-Processor for Railway Geometric Description

For multibody analysis, the track models must be defined in the form of parame-terized curves. Here, two different parametric track descriptions, using analyticalfunctions and cubic splines, are presented. A pre-processor program uses theseparametric descriptions in order to define the nominal geometry of a railway us-ing a discrete number of points as function of the curve length parameter. Thisinformation is organized in a database where all quantities, necessary to define thespatial curve constraint, are obtained as a function of the track length, measuredfrom its origin, i.e., from the point where the analysis starts.

4.1. CURVE FOR THE REFERENCE PATH BY ANALYTICAL FUNCTIONS

The pre-processor program, developed to parameterize the reference path geome-try, supports the three types of analytical segments identified before, i.e., tangent,transition and circular segments. These segments are defined analytically and arecharacterized by their length, horizontal/vertical curvature and cant angle. The for-mulation of the analytical segments is similar to the track description adopted by


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the commercial codes, such as ADAMS/Rail [79]. The reference path tangentialanalytical segments are defined by:

x(L) = L,

y(L) = 0,

x(L) = 0. (31)

The transition curves are expressed as

x(L) =2

Khsin

Kb

2L

,

y(L) =2

Kh

1 cos

Kh

2L

,

z(L) = 2Kv

1 cos

Kv

2L

. (32)

While the circular curve segments are depicted by

x(L) = sin(KhL)/Kh,

y(L) = [1 cos(KhL)]/Kh,

z(L) = [1 cos(Kv L)]/Kv, (33)

where L is the distance traveled while Kh and Kv represent, respectively, thehorizontal and vertical track curvatures given by

Kh = R1h ; Kv = R1v (34)

Rh and Rv being the horizontal and vertical track radii, respectively.The analytical expressions for the curve segments are given as function of the

travel distance L for each segment. Therefore, within the framework of Equa-tions (31) through (33), L is a local parameter and the position of a point moving onthe curve, obtained by these equations, must be transformed to global coordinatesby using an appropriate coordinate transformation. Moreover, it should be noticedthat Equations (31, 32) and (33) are based on the simplification that the horizontaltravel distance is approximately equal to the total travel distance. This assumptionis valid as long as the track grade is relatively small [7].

According to the presented formulation, the reference path is obtained byassembling a number of analytical segments. The actual form how these track seg-ments are ordered and their characteristics are left for the user to define. However,in order to ensure the smooth transition between railway segments, the introductionof transition curve segments between a straight and circular curve segments, orbetween segments of the same type, is required if comfort is a concern. This issue,


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Figure 7. Points interpolated by cubic spline segments.

that is application dependent, is left for the user to decide. A detailed descriptionon the track geometries and their reasoning in terms of comfort, vehicle wear andnormalization is out of the scope of this work and it can be found in [6].

4.2. CURVE FOR THE REFERENCE PATH BY CUBIC SPLINES

The reference path curve can also be described using cubic spline curves that in-terpolate a set of control points given by the user. The advantage of these curves isthat the continuity of their first and second derivatives is guaranteed. Furthermore,the position of any point over the curve is defined in terms of a local parameter thatcan be associated to, but it is not, the length traveled over the curve.

A parametric cubic curve is defined as [10]

g(u) = a3u3 + a2u

2 + a1u + a0, (35)

where g(u) is a point on the curve, u is the parametric variable and ai are theunknown algebraic coefficients that must be calculated. Equation (35) can beseparated into the three components ofg(u), such that

x(u) = a3xu3 + a2x u2 + a1x u + a0x ,

y(u) = a3yu3 + a2y u

2 + a1y u + a0y ,

z(u) = a3zu3 + a2zu

2 + a1zu + a0z. (36)

Let a set of points gi , representing the reference path, be defined by their coordi-nates (x,y,z)i as represented in Figure 7. When the cubic spline segments are usedto represent the interpolation curves, the algebraic coefficients ai in Equation (35),are written explicitly in terms of the boundary conditions, i.e., segment end pointsand tangent vectors [10]. In this sense, each spline segment is

g(u) = [u3 u2 u 1]

2 2 1 1

3 3 2 10 0 1 01 0 0 0

g(0)

g(1)g(0)g(1)

, (37)where the spline local parameter u [0, 1] and g(0) and g(1) represent the coor-dinates of the end points of each segment. The spline derivatives, g(0) and g(1),


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at the end points are calculated in order to ensure C2 continuity between splinesegments. Assuming that the second derivatives of the first and last points, of theset to be interpolated, are null, the cubic spline is referred to as natural [10]. In thiscase, the first derivatives in all other control points are obtained as

2 11 4 1

1 4 1. . .

1 4 11 2

g0g1g2...

gn2gn1

=

3(g1 g0)3(g2 g0)3(g3 g1)

...

3(gn1 gn3)3(gn1 gn2

. (38)

Once the gi values are obtained by Equation (38), they are used (Equation (37)) toobtain the points coordinates anywhere in the cubic splines segments.

In order for the cubic splines to be used in the kinematic constraint their geo-metric characteristics must be expressed as function of the reference path lengthL, measured from its origin, and not as a function of the parameter u. The relationbetween L and u is given by

L(u) =

k1n=1

L0n + Lactualk (u), (39)

where k is the number of the spline segment where the point is actually located, uis the spline parametric variable and L0i corresponds to the length of the ith splinesegment that, referring to Equation (10), is given by

L0 =

10

gu

Tgu du. (40)

Note that, according to the cubic splines formulation, the local parametric vari-able u [0, 1]. The parameter Lactual(u) represents the length of the actual splinesegment from its origin to the actual location of the point and it is defined as

Lactual(u) =

u0

gu

Tgu du. (41)

In order to implement the kinematic constraint in the computer code it is nec-essary to find the value of the cubic spline parametric variable u that correspondsto a prescribed segment length L. It is clear from Equations (39) and (41), thatthe relation between these two parameters is not linear. Consider the parametricvariable, uR, corresponding to a point R, located on the kth cubic spline segment,


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Figure 8. Cant angle contribution to the track model.

and to which it is associated a curve length LRk , measured from the kth segmentorigin. In this case, the parameter u is obtained from the parameter L using

uR

0

g

uT

gu

du LR

k = 0. (42)

This non-linear equation is solved, in a pre-processor, using the NewtonRaphsonmethod [11, 13].

4.3. INTRODUCTION OF A PRESCRIBED CANT ANGLE IN THE KINEMATICCONSTRAINT

As referred before, in horizontal curves the outer rail is usually raised in orderto reduce the effects of the centrifugal acceleration on vehicles. In this sense, therailway superelevation has to be taken into account when creating a track model.

The pre-processor program, developed to construct railway databases, automati-cally accounts the contribution of the track cant for the calculation of the curvegeometry. The cant angle is defined here as the angle between vector nR and theosculating plane as measured in the normal plane, all described in Figure 8.

Let the track cant angle, on a point R of the parametric curve g(L), be des-ignated by R. Assume that the reference path moving frame is defined by itsprincipal unit vectors (t, n, b)L, which are defined by Equations (6, 8) and (9).Thus, due to the track cant, the parametric curve reference frame rotates about thet axis by an angle R, as shown in Figure 8. Therefore, it is necessary to calculatethe new components of the principal unit vectors (tR, nR, bR)L of the curve movingframe after the rotation. Such vectors obtained as

tR = ALtR; nR = ALnR; bR = ALbR, (43)

where AL represents the transformation matrix from the parametric curve localframe to the global reference frame (x,y,z) given by

AL = [t n b]L. (44)


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The relationship between the principal unit vectors before and after a rotation Rabout the t axis is

tR = Rt; nR = Rn

; bR = Rb, (45)

where R is the rotation transformation matrix for rotations around t axis given by

R =

1 0 00 cos R sin R

0 sin R cos R

(46)

and t, n, b represent the principal unit vectors expressed in local coordinates,written as

t = {1 0 0}T; n = {0 1 0}T; b = {0 0 1}T. (47)

Notice that, in keeping with the right-hand convention, R is positive in a counter-

clockwise sense when viewed from a point on the positive t axis and toward theorigin [5]. Substituting (45) in (43), and after rearranging, the new components ofthe principal unit vectors, after the cant angle rotation, are expressed as:

tR = t,

nR = n cos(R) + b sin(R),

bR = n sin(R) + b cos(R). (48)

According with this formulation, the user must set the cant angle for the be-ginning and for the end points of every track segment. The values of the cantangle are linearly interpolated between the segment end points, regardless of the

parametric description of the curve [11]. With the complete information available,a pre-processor program uses Equation (48) to calculate the geometric parametersthat define the reference path geometry and store them in the railway database.

4.4. TRACK INFORMATION INCLUDED IN RAILWAY DATABASE

The direct use, in the general spatial curve constraint, of the equations of thereference path, as obtained by any of the parametric descriptions previously pre-sented, is neither practical nor efficient from the computational point of view.As the kinematic constraint is to be used in within the framework of a dynamicanalysis program, where the rail guided vehicles may have a large number of

bodies constrained to move in general spatial curves, the solution of the nonlinearEquations (42) and the sets of Equations (33, 36, 48) and so forth at every timestep would be an heavy burden on the code. An alternative implementation ofthese equations is the construction of a table where all quantities appearing in thedefinition of the kinematic constraint are tabulated as function of the global lengthparameter.


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Figure 9. Structure of reference path geometric information table.

After selecting any of the parametric descriptions of the spatial curve presentedbefore the length parameter step, L adopted for the database construction has tobe chosen. Then, the pre-processor program automatically constructs a table withall parameters necessary to define the geometric characteristics of the referencepath, taking into account the track cant variation. These geometric parameters areorganized in columns as function of the length parameter L of the track, measuredfrom its origin up to the actual point in the track. The multibody program interpo-lates linearly the table in order to obtain all required geometric characteristics of thetrack. If the size of the length parameter step L is set to be similar to the productof the vehicle velocity by the average integration time step used during dynamicanalysis, then only a few number of interpolations, if any, will be performed inbetween two successive lines of the table.

In Figure 9 it is presented the structure of the railway database obtained withthe pre-processor program, where the adopted step size for the track length isL = 0.1 m. As shown, a railway database consists of a table with 37 columnswhere each one corresponds to a railway geometric parameter. The first column ofthe database corresponds to the track length L with a step size L being the corre-sponding Cartesian coordinates (x,y,z) are stored in the following three columns.

Columns 5 through 10 store the first and second derivatives of the Cartesian coor-dinates with respect to L, required for the Jacobian matrix, given by Equation (20)and the right-hand-side of the acceleration equations presented by Equation (22).The next three columns contain the information about the Cartesian components ofthe unit tangent vector t, which is defined in Equation (6). Columns 14 through 19store the first and second derivatives, of the unit tangent vector components, withrespect to L, required by Equations (28). The next three columns of the railwaydatabase contain the Cartesian components of the principal unit normal vector n,which is defined in Equation (8). Columns 23 through 28 store the components ofthe first and second derivatives, of the vector n, with respect to L. The next threecolumns contain the Cartesian components of the binormal vector b, which is de-

fined in Equation (9). Columns 32 through 37 store the first and second derivatives,of the binormal vector components, with respect to L.After the railway database construction, by the pre-processor program, the track

model is completely defined. Therefore, we can assemble with the multibody mod-els of the railway, roller coaster or other types of rail-guided vehicles, in order toperform the dynamic analysis of the whole system. For this purpose, the multibody


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Figure 10. Horizontal track geometry.

program has to interpolate the railway database in order to obtain all requiredgeometric characteristics of the track as described before.

5. Application Examples

The discussion of the methodology proposed for the general spatial curve con-straint is carried here based on some application examples. The first includes ahorizontal track model, with geometric characteristics similar to the ones usedin train and tram railway networks. The second one concern the application to athree-dimensional track model with a geometry analogous to the one used on rollercoasters designs [4].

5.1. HORIZONTAL TRACK MODEL APPLICATION

In this application example, the horizontal track geometry has the characteristicspresented in Figure 10. Since the track has no vertical curves or grade, the limita-tions of using analytical segments are overcome. Therefore, the track model is builtusing either analytical functions or cubic splines. In order to compare the differentdescriptions of the track geometry and their impact in the accuracy of the modelsdeveloped, three reference paths are modeled using analytical segments and splinecurves with different control points increments. As the emphasis of this work is thetrack model and not the vehicle model itself only a single body, representing thecomplete vehicle, is considered for the dynamic analysis performed here. The trackmodels are first pre-processed and the relevant geometric information is included

in the tables format described before.The track models are assembled considering transition curves with lengths of10 m each. The cant angle for the circular curve is 0.2 rad and null for thetangent track. This angle varies linearly n the transition segment. The track cantangle adopted corresponds to the equilibrium cant, i.e., the cant for zero trackplane acceleration at a given curve radius and speed [3, 4]. For the track model


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Table I. Comparative parameters of dynamic simulations performed in the horizontal track models.

Track model Analysis Initial Average CPU Time Constraint Traveled

time velocity time-step violations() distance

Analytical functions 34.5 sec 10 m/s 102 sec 139 seg. 1 102 344.8 m

Cubic splines 34.5 sec 10 m/s 102 sec 139 seg. 2 105 345.0 m

() Maximum

Figure 11. Acceleration of the vehicle center-of-mass in the y direction for tracks describedby cubic splines and by analytical segments.

that uses spline segments the distance between the control points is 1 m. In eithercase the rigid body has a mass of 176 Kg and inertias of I = 144.5 Kg m2,I = 2.2 Kg m2 and I = 144.5 Kg m2. A initial velocity of 10 m s1 is assigned

for the simulations. Several parameters characterizing the simulations with the twotrack models are summarized in Table I.

A first observation is that the integration time-step is not sensitive to the descrip-tion adopted for the track model. This is not surprising because all computationalcosts associated to the interpolation of the railway table are exactly the same re-gardless of the parametric description adopted. At the most, different track modelscould induce more or less oscillations in the curves with consequences in the in-tegration time-steps size, when variable time stepping algorithms are used. In alltrack models used this effect has not been an issue. Another aspect that showsin Table I is that the distance traveled for the 34.5 second of simulation is notexactly the same for both track models. This reflects that the different track models

have slightly different lengths resulting from the different parametric descriptionsadopted.A comparative graphic of the accelerations of the vehicle center of mass in the

y direction obtained for the two track models is presented in Figure 11. As it canbe seen, there is a good agreement between the results obtained with the two formsof parameterization. The discontinuities observed at 10 s and at 26 s suggest that


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Figure 12. Acceleration of the vehicle center of mass in the y direction for track modelsdescribed by cubic splines with different distances between control points.

Figure 13. Roller coaster geometry.

the length of the transition curves is too small in face of the vehicle velocity and

of the track curve radius. The response peaks observed in the results obtained withthe track model described by cubic splines are a direct result of the oscillationsinherent to the interpolation process of the control points.

The influence of the distance between the control points of the splines onthe acceleration response is observed in Figure 12 where the vehicle center ofmass accelerations in y direction are presented for two track models describedby cubic splines, but with distances between control points respectively of 1 mand 10 m. The track with larger distances between control points exhibits asmoother response. However, the acceleration response deviates more clearly fromthe acceleration obtained with the analytic segments.

In Figure 12 it is clear that though larger distance between control points leads

to a smoother track the amplitudes for the acceleration oscillations are also higher.Smaller distances between the control points lead to a perturbation of the dynamicresponse in terms of acceleration that can be perceived in a dynamic analysis of acomplete railway vehicle as perturbations of the railway. Therefore, caution mustbe used in the parametric interpolation curve description selected for the trackmodel.


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Figure 14. Views of the roller coaster as used in the simulations.

Table II. Comparative parameters of dynamic simulations performed in the roller coaster models.

Track model Analysis Initial Average CPU time Constraint Traveled

time velocity time-step violations() distance

Distance = 1 m 49.6 seg. 2 m/s 102 sec 11 m 50 s 9 102 1,010.5 m

Distance = 5 m 49.6 seg. 2 m/s 102 sec 11 m 52 s 9 102 1,009.8 m

() Maximum

5.2. ROLLER COASTER MODEL APPLICATION

The second application example is a three-dimensional track model of a rollercoaster with the geometry illustrated in Figure 13. Since the track has verticalcurves and grade, its model cannot be parameterised with analytical segments and,therefore, the track model is build using only cubic splines. In this roller coasterexample, two track models with distances between control points respectively of1 m and 5 m are build. In Figure 14, two views of the roller coaster are presented. Asingle body vehicle model is assembled with the two track models and the dynamicsimulations of the systems are performed.

The motion resulting from the simulations, observed from two different view-points, is sketched in Figure 15. Table II contains comparative parameters thatcharacterize the dynamic simulations performed with the two roller coaster models.These parameters are similar to the ones observed in Table I, which means that thetime stepping control of the integration algorithms are not sensitive to the trackcomplexity.


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Figure 15. Views roller coaster motion resulting from the simulations.


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Figure 16. Acceleration of the roller coaster vehicle center of mass in the z direction.

In Figure 16, a comparison of the vehicle center of mass acceleration in the z

direction, achieved for the two track models, is presented. There is a good correla-tion between the results obtained with both models. The large peaks of accelerationobserved are direct results of the sudden change of the vertical curvature betweenparts of the track with different geometric characteristics. These sudden changesreflect the fact that no transition curves are used in this roller coaster design. Thesmaller perturbations observed in the results result from the oscillations inherent tothe spline interpolation process and, therefore, do not have a physical meaning.

6. Conclusions

A kinematic constraint representing a general spatial curve joint has been devel-

oped here and its computational implementation has been presented. The strategyadopted for the computer implementation of the joint starts by having the spatialcurve expressed in a parametric form. A moving reference frame is defined in thecurve such a way that the axes are defined in the intersections of the normal, oscu-lating and rectifying planes. The introduction of the cant angle and of its variationalong the curve has also been implemented. The reference plane used in the spatialcurve to define the cant angle is the osculating plane, which is the horizontal planein case of a flat curve. After recognizing that the parameters used in the curvedefinition are not necessarily related to the arc length of the curve a transformationis proposed. Recognizing the fact that the transformation equations are nonlinearand that it is not efficient to calculate the curve vectors and their derivatives during

the dynamic analysis a pre-processor to generate all geometric properties of thecurve is suggested. The result of the use of this pre-processor is a table whereall quantities involved in the constraint are tabulated as function of the arc lengthtraveled by the constrained point of a system body in the curve.

This methodology has the advantage of making the time required for the dy-namic simulation of the rail-guided vehicle completely independent of the track


27/28


complexity and of the type of parametric curve used. Any descriptive form ofparametric curves is dealt with in the pre-processor while the dynamic analysisprogram only has to proceed with linear interpolations of the railway table. Byensuring that the arc-length step is small enough the linear interpolation proceduredoes not introduce any significant error in the geometric description of the curve.

The results presented show that the parameterization with cubic splines canbe used either to describe the track geometry of horizontal tracks or fully three-dimensional roller coasters. The major drawback of this formulation is its de-pendence on the distance between the control points used to describe the trackgeometry. The use of cubic splines also leads to undesired oscillations in the trackmodel, which can be perceived as track irregularities during dynamic analysis.The parameterization with analytical functions does not produce the oscillationsobserved in the cubic splines formulation but it is limited to tracks with horizon-tal geometry. Other parametric descriptions of the general spatial curve, such assplines with tension and Akima splines, can be implemented in the pre-processor

program as alternative techniques for railway parameterization. These alternativeforms of interpolation are expected to minimize the undesired oscillations of theinterpolated curve.

The general spatial curve kinematic now described serves as the basis for theconstruction of the railway as it provides a moving frame, associated to the curvewhere the osculating plane is defined with respect to which the cant angle can bedefined. The actual geometry of the tracks can now be described with respect to thismoving frame providing one of the basic ingredients for the use of the multibodycode in the context of railway dynamics applications.

Acknowledgements

The support of Fundao para a Cincia e Tcnologia (FCT) through the PRAXISXXI Project, with the reference BD/18180/98, on Mtodos Avanados para Apli-cao Dinmica Ferroviria (Advanced Methods for Railway Dynamics) madethis work possible and is gratefully acknowledged.

References

1. Dukkipati, R.V. and Amyot, J.R., Computer-Aided Simulation in Railway Dynamics, MarcelDekker, New York, 1988.

2. Garg, V.K. and Dukkipati, R.V., Dynamics of Railway Vehicle Systems, Academic Press,Toronto, 1984.

3. Andersson, E., Berg, M. and Stichel, S., Rail Vehicle Dynamics, Fundamentals and Guidelines,Royal Institute of Technology (KTH), Stockholm, 1998.

4. Wayne, T., Roller Coaster Physics An Educational Guide to Roller Coaster Design andAnalysis for Teachers and Students, Charlottesville, 1998.

5. Mortenson, M. E., Geometric Modeling, John Wiley & Sons, New York, 1985.


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6. Pinto, A. R. V., Bases Tcnicas dos Traados do Metroplitano de Lisboa, (Track Geome-try Technical Bases of Lisbon Underground Company), Revista da Ordem dos Engenheiros,Portugal, 1978 [in Portuguese].

7. MDI Mechanical Dynamics, ADAMS/Rail 9.1 Technical Manual, 1995.

8. MDI Mechanical Dynamics, ADAMS/Rail 9.1.1 ADtranz Milestone I Release Notes, 1999.9. MDI Mechanical Dynamics, ADAMS/View 9.0 Training Documentation, 1998.10. Anand, V., Computer Graphics and Geometric Modeling for Engineering, John Wiley and

Sons, New York, 1994.11. Pina, H. L. G., Mtodos Numricos (Numerical Methods), McGraw-Hill, Lisboa, Portugal,

1995 [in Portuguese].12. Ambrsio, J., Trainset kinematic: A planar analysis program for the study of the gangway

insertion points, Technical Report IDMEC/CPM/97/005, Lisbon, Portugal, 1997.13. Nikravesh, P. E., Computer-Aided Analysis of Mechanical Systems, Prentice Hall, Englewood

Cliffs, NJ, 1988.14. Ambrsio, J., Implementation of typical track geometries, Technical Report PEDIP

No. 25/00379, Lisbon, Portugal, 2000.15. Jalon, J. de and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems, Springer-

Verlag, Heidelberg, 1993.

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