2007 Paper IJVD JPombo JAmbrosio

7/28/2019 2007 Paper IJVD JPombo JAmbrosio

1/31

470 Int. J. Vehicle Design, Vol. 45, No. 4, 2007

Copyright 2007 Inderscience Enterprises Ltd.

Modelling tracks for roller coaster dynamics

Joo Pombo*

Escola Superior de Tecnologia de Abrantes,Instituto Politcnico de Tomar,Rua 17 de Agosto de 1808, 2200-370 Abrantes, PortugalE-mail: [email protected]*Corresponding author

Jorge Ambrsio

IDMEC, Instituto Superior Tcnico,Av. Rovisco Pais, 1049-001 Lisboa, PortugalE-mail: [email protected]

Abstract: A methodology for the accurate description of three dimensionaltrack geometries is proposed. Cubic, Akima and shape preserving splines areproposed for track parameterisation. The track description uses Frenet framesthat provide the track referential at every point. During dynamic analysis apre-calculated database is linearly interpolated to form the kinematicconstraints that enforce the wheelsets of the vehicle to move along the trackwith a prescribed angular orientation. This approach allows calculating thereaction forces that develop between the vehicle and the track. The formulationis discussed emphasizing the influence of the parameterisation methods toconstruct the track geometry.

Keywords: railway parameterisation; Frenet frame; spatial curve geometry;prescribed motion constraint; Akima splines; shape-preserving splines; trackreaction forces.

Reference to this paper should be made as follows: Pombo, J. and Ambrsio, J.(2007) Modelling tracks for roller coaster dynamics, Int. J. Vehicle Design,Vol. 45, No. 4, pp.470500.

Biographical notes: Joo Pombo graduated in Mechanical Engineering fromthe Technical University of Lisbon in 1998 and got his PhD in MechanicalEngineering, also from the Technical University of Lisbon, in 2004. He is theauthor of several papers in international journals, international conferences.During his research activities, he received two innovation prizes related to thestudy of the dynamic behaviour of the trainset ML95 from the Lisbon SubwayCompany. He is a Post Doctorate Researcher at IDMEC/Instituto SuperiorTcnico and invited Professor at Escola Superior de Tecnologia de Abrantes,Instituto Politcnico de Tomar.

Jorge Ambrsio is an Associate Professor with aggregation in the Departmentof Mechanical Engineering of Instituto Superior Tcnico of the TechnicalUniversity of Lisbon. His areas of activity are multibody dynamics, structuraldynamics, vehicle dynamics and biomechanics. He graduated in MechanicalEngineering from the Technical University of Lisbon in 1981, got his MastersDegree in Mechanical Engineering also from the Technical Universityof Lisbon in 1984, and his PhD in Mechanical Engineering from the University


2/31

Modelling tracks for roller coaster dynamics 471

of Arizona in 1991. He is the author of several books in the area of multibody

dynamics and more than 200 papers in international journals and internationalconferences. He is the co-editor of the international journal Multibody SystemDynamics and belongs to the editorial board of several international journals.

1 Introduction

The dynamic analysis of any type of rail-guided vehicle requires the accurate descriptionof its track geometry. The definition of railway (Andersson et al., 1998; British-Rail,1997; Dukkipati and Amyot, 1988; Garg and Dukkipati, 1984) or roller coaster(Wayne, 1998) tracks have in common the geometric description of their centrelines,

usually done with some curve-interpolation scheme, and the reference plane wherethe lines that define the rails are seated. The track irregularities can also betreated if considered as deviations from the parallel lines that represent the rails.Typically, these are modelled by adding, to the track perfect geometry, smallperturbations that are measured experimentally or generated numerically (Pombo, 2004;Pombo and Ambrsio, 2003b, 2003c). For the sake of focus on the track nominalgeometry, the track irregularities are not considered in this work.

In general, the nominal geometry of the roller coaster track consists of straight andcircular segments interconnected by transition curves that ensure the continuity of thefirst and second derivatives of the track in the transition points. This assemblageguarantees curvature continuity, making the ride smoother and survivable for the riders(Wayne, 1998). Another important issue in the description of three-dimensional trackgeometries is the existence of horizontal curves, which are curves that do not rise or fall.

In such curves, the roller coaster cars are influenced by centrifugal forces that act in adirection away from the centre of the curve and tend to overturn them. When travelling ina horizontal flat curve, the rider has the sensation of being thrown sideways, and if thecars velocity is high enough and/or the curve radius small enough, the lateralaccelerations can be tremendous. In order to avoid such problems, horizontal bankedcurves are used. This is done by raising the outer rail of the track in order to tilt the carwhen travelling in the curve. The difference of heights between the two rails in ahorizontal banked curve is called cant (Wayne, 1998). The ideal banked curve is the onewhere no outside forces are needed to keep the car on the track. In other words, if thebanked curve were covered with ice (no friction) and the coaster did not have a steeringmechanism, the car would stay on the track.

In this work, the development and implementation of a methodology for the accurate

description of fully three-dimensional track geometries are presented. It is proposed touse parametric curves that interpolate a given number of control points that arerepresentative of the track centreline. Among the methods available in the literature,three different parameterisation schemes, using cubic (Anand, 1994; de Boor, 1978;Pina, 1995), Akima (Akima, 1970; de Boor, 1978; Visual Numerics, 1997) andshape-preserving splines (Irvine et al., 1986; Micchelli et al., 1985; Visual Numerics,1997), are implemented and their influence on the outcome of the results of the rollercoaster models is discussed here. The complete characterisation of the track model alsorequires the definition of the cant angle variation along the track. For flat tracks, the cantangle at a given point is measured in the plane perpendicular to the centreline, between a


3/31

472 J. Pombo and J. Ambrsio

line that seats on both rails and the horizontal plane. For tracks with a full spatial

geometry, a new definition of this angle is introduced. It is proposed that the oscillatingplane (Mortenson, 1985) plays the role of the horizontal plane of the flat track whenmeasuring the cant angle (Pombo and Ambrsio, 2001, 2003a).

In many applications, when cubic splines are used to interpolate a set ofcontrol points describing the track centreline, undesired oscillations are noticed(Ambrsio, 1997). For instance, if a spline interpolation is used to parameterise a trackmodel composed by a straight segment followed by a circular segment, in the vicinity ofthe transition, the result is not a perfect straight line but simply a curve that oscillatesabout the original line. Akima splines and shape-preserving splines are alternativeinterpolation schemes that are expected to minimise the undesired oscillations of theinterpolated curve.

The use of analytical segments to parameterise the track centreline (Pombo and

Ambrsio, 2001, 2003a), made of straight, circular and transition curves, does notintroduce unwanted oscillations on the track model. Nevertheless, this description israther complex to model tracks with large slopes or with vertical curves. Some of thecommercial codes that adopt this description impose that the tracks are basicallyhorizontal in order to avoid difficulties (MDI, 1995, 1998, 1999). Therefore, the trackdescription with analytical segments cannot be used to parameterise fully spatialgeometries like the ones used in roller coasters.

Regardless of the method used to parameterise the track geometry, the general spatialcurve kinematic constraint, formulated by Pombo (2004) and Pombo and Ambrsio(2001, 2002b, 2003a), can be applied. This constraint is proposed to be used here as thebasis for the definition of the vehicletrack interaction in the roller coaster multibodysystem. When the wheelsets of the vehicle model move along the track, the kinematicconstraint enforces not only that the railway centreline is followed, but also that thespatial orientation of the wheelsets is prescribed with respect to the moving Frenet frame(Mortenson, 1985) associated to the track. According to this approach, in the dynamicanalysis of roller coaster vehicles, the wheelrail contact forces are obtained assumingpermanent contact. Therefore, it is considered that the wheelsets of the roller coaster carsfollow exactly the track geometry. By introducing the track in this form in the multibodyformulation, the reaction forces are associated to the Lagrange multipliers, whose valuesresult from enforcing the track geometric constraints (Nikravesh, 1988).

The position coordinates of the roller coaster centreline, the Frenet frame vectors andtheir respective derivatives are required for the definition of the general spatial curvekinematic constraint. Their online calculation during vehicle dynamic analysis can beextremely costly. Therefore, these quantities are pre-processed and included in a table asa function of the track length. During roller coaster dynamic analysis, the quantities

involved in the general curve constraint are obtained by linear interpolation of thetabulated values (Pombo and Ambrsio, 2003a).

2 Parameterisation of the track centreline geometry

In many applications, including those associated with roller coasters, the motion of thewheelsets of a vehicle is defined to be constrained to move on a given spatial line. Such isa kinematic constraint that is described in detail in references (Pombo, 2004; Pombo andAmbrsio, 2001, 2002b, 2003a) and will only be summarised here. Therefore, the


4/31


emphasis is to discuss the interpolating schemes adopted to represent the track centreline

and their influence on the outcome of the dynamic analysis of the system.

2.1 Equations of motion for a multibody system

The position and orientation of each body i of a multibody system is described by a

position vector ri and a set of rotational coordinates pi, organised in a vector as

(Nikravesh, 1988)

= { , }T T Ti i .q r p (1)

A multibody system with nb bodies is described by a set of coordinates in the form:

{ }1 2= , , , .T

T T T

nbq q q q (2)

The dependencies among system coordinates, which result from the existence of

mechanical joints interconnecting the several bodies, are defined through the introduction

of kinematic relationships involving the coordinates, which are designated by kinematic

constraints. In order to guide the system during the analysis, driving constraints are also

defined to control the system degrees-of-freedom along the 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 (Nikravesh, 1988)

( , ) =t q 0 (3)

where q is the generalised coordinates vector, defined in equation (2), and tis the time

variable, resulting from the existence of driving constraints. The second time derivative

of equation (3) with respect to time yields the accelerations equations:

( , , , t) = = q q q q 0 q (4)

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

For an unconstrained mechanical system (Nikravesh, 1988), the matricial form of the

equations of motion is given by:

=M q f (5)

where M is the global mass matrix, containing the mass and moments of inertia of all

bodies, and f is the force vector, containing all forces and moments applied on system

bodies, as well as the gyroscopic forces.

The system kinematic constraints (3) are added to the equations of motion (5) using

the Lagrange multipliers technique (Nikravesh, 1988). Defining by , the vector of theunknown Lagrange multipliers, the equation of motion for a constrained mechanical

system is:

= .

T

q

q

M q f

0

(6)


5/31


The Lagrange multipliers, associated with the kinematic constraints, are related to the

reaction forces generated between the bodies interconnected by kinematic joints by(Nikravesh, 1988):

( ) .c T= qf (7)

According to this methodology, the dynamic analysis of multibody systems involves thecalculation of the vectors fand , for each time step. Equation (6) is then used to calculatethe system accelerations .q These accelerations together with the velocities q areintegrated in order to obtain the new velocities q and positions q for the time step. Thisprocess proceeds until the complete description of the system motion is obtained, for theselected time interval.

2.2 Track following kinematic constraint

Consider a point R, located on a rigid body i, that has to follow the specified path, asdepicted in Figure 1. The path is defined by a parametric curve g(L), which is controlledby a global parameterL, which represents the length travelled by the point along thecurve until the current location of pointR. The kinematic constraint is written as:

( )3 = ( ) =pmc, Ri L 0 r g 0 (8)

where = + r r A sR Ri i i i represents the coordinates of point R with respect to the globalcoordinate system (x, y, z), ri is the vector that defines the location of the body-fixedcoordinate system (, , )i, Ai is the transformation matrix from the body-i-fixedcoordinates to the global reference frame and s Ri represents the coordinates of pointR

with respect to the body-fixed reference frame. The vector g(L) = {x(L),y(L),z(L)}

T

represents the Cartesian coordinates of the curve where point R is constrained to moveand L is the curve parametric variable. For notational purposes, (.) means that (.) isexpressed in body-fixed coordinates.

Figure 1 Prescribed motion constraint

It should be noticed that the constraint requires the introduction of the new coordinate Lin the multibody system, which is the length of the curve travelled by point R from thestart of the curve up to its current position. Therefore, the velocity and accelerationvectors also include the time derivatives of this parameter.


6/31


The velocity equation is obtained as the time derivative of equation (8) with respect to

time

( , 3) = ( ) =pmc R Ri i id

LdL

L

=

rg

0 r g 0 I s A 0

(9)

where the Jacobian matrix is

( , 3) =pmc R

i i

d

dL

q

g I s A (10)

in equation (10), I is a 3 3 identity matrix and = s A sR Ri i i represents the coordinates ofpointR with respect to the (, , )i coordinate system, written in global coordinates.

The acceleration equation is obtained as the derivative of equation (9) with respect totime. The resulting equation is

2( , 3) 2

2= +

Lpmc

L

LL

=

rg g

0 I s A A s

(11)

where { }=T

i ix y zr are the translational accelerations of body i, { }

T

i i =

represents the angular acceleration of the body-fixed coordinate system (, , )i,expressed in local coordinates and L is the second time derivative of the curveparametric variable. The contribution of the constraint for the right-hand-sideof the accelerations equation is, given the 3 1 vector, written as:

2# = + .R 2i i 2

dL

dL

g s (12)

Equations (8), (10) and (12) represent the quantities that must be implemented in theconstraint module of the computer code.

The second part of the constraint ensures that the spatial orientation of body i remainsunchanged with respect to the Frenet frame associated with the reference path curve, asrepresented in Figure 2.

Consider a rigid body i where (u, u, u)i represent the unit vectors associated withthe axis of the body-fixed coordinate system (, , )i. Also consider the Frenet frame ofthe general parametric curve g(L), which is defined by the principal unit vectors (t, n, b)L,as depicted in Figure 2. Assume that, at the initial time of analysis, the relative orientation

between the body vectors (u, u, u)i and the curve local frame (t, n, b)L are such that thefollowing equations hold

( , 3)

= = .

T

lfac T

T

a

b

c

n u

0 b u

n u

(13)


7/31


Figure 2 Local frame alignment constraint

The kinematic constraint ensures that this alignment will remain constant throughout theanalysis. The transformation matrix from the body-i-fixed coordinates to the globalcoordinate system is written as:

= [ ] .i i A u u u (14)

With the purpose of having a more compact notation, the following unit vectors aredefined

{ } { } { }2 3= 1 0 0 ; = 0 1 0 ; = 0 0 1 .T T T

1u u u (15)

Equation (13) is now rewritten as

1

( , 3)1

3

= = .

T

i

lfac T i

T

i

a

b

c

n A u

0 b A u

n A u

(16)

The velocity equation for this constraint is obtained as the time derivative ofequation (16), expressed as

1 1

( , 3)1 1

3 3

= =

T

T T

i i

T

lfac T T

i i

T

T T

i i

d

dL

d.

dLL

d

dL

n0 n A u A u

rb

0 0 b A u A u 0

n

0 n A u A u

(17)

The contribution of frames alignment constraint (17) to the Jacobian matrix is thesubmatrix 3 7, given by


8/31


1 1

( , 3)1 1

3 3

=

T

T T i i

T

lfac T T

i i

T

T T

i i

ddL

d

dL

d

dL

q

n0 n A u A u

b 0 b A u A u

n0 n A u A u

(18)

where 0T is a 1 3 null vector. The acceleration equation is the time derivative ofequation (17), and it is written as

1 1

( , 3)1 1

3 3

22

2

2 + +

T

T T

i i

T

lfac T T

i i

T

T T

i i

TT

T

i i i i i i

d

dLd

dLL

d

dL

d d' ' 'L L

dL dL

=

=

n0 n A u A u

rb

0 0 b A u A u

n0 n A u A u

n nA n A A

1

22

12

22

32

2 + +

2 + +

TT

T

i i i i i i

TT

T

i i i i i i

d d' ' 'L L .

dL dL

d d' ' 'L L

dL dL

u

b bA b A A u

n nA n A A u

(19)

The contribution of the local frames alignment acceleration equation, described byequation (19) for the right-hand-side of the accelerations equation, is the 3 1 vector,written as:

22

12

2# 2

12

22

2

2 + +

= 2 + +

2 + +

TT

T

i i i i i i

TT

T

i i i i i i

TT

T

i i i i i i

d d' ' 'L L

dL dL

d d' ' 'L L

dL dL

d d' ' 'L L

dL dL

n nA n A A u

b b A b A A u

n nA n A A

3

.

u

(20)


9/31


The complete set of quantities that is necessary to implement computationally in the

general spatial curve constraint is described by equations (8), (10), (12), (16), (18)and (20), which represent the quantities that must be implemented in constraint module ofthe computer code.

2.3 Overview of the track centreline pre-processor

The purpose of the track centreline pre-processor is to define the track model in the formof a parameterised curve that represents its spatial geometry. In the pre-processor, threealternative parametric track descriptions are implemented (Pombo, 2004; Pombo andAmbrsio, 2001, 2002a, 2002b, 2003a), using cubic, Akima and shape-preservingsplines. The information is organised in a database where all quantities necessary todefine the general spatial curve kinematic constraint, formulated by Pombo (2004) and

Pombo and Ambrsio (2001, 2002b, 2003a), are obtained as a function of the track lengthmeasured from its origin to the actual position of a point representing a vehicle wheelset.The methodologies used by the pre-processor are summarised as

The user defines the track model using a piecewise cubic interpolation scheme toparameterise the geometry of the track centreline. For that, the user has to define aset of control points, representative of the centreline geometry, and the cant angle ateach one of the control points.

Regardless of the interpolation approach selected, the track centreline isparameterised as a function of the track length.

The track cant angle is also parameterised as a function of the track length, and theprincipal unit vectors, which define the Frenet frame associated with the track

centreline after the cant angle rotation, are calculated. A database containing all parameters necessary to define the kinematic constraint

associated with the geometric characteristics of the track centreline is created.The track-length step of the database is defined by the user. During dynamicanalysis, the track centreline database is used by the multibody code that performslinear interpolation on the points contained in the table to find the necessarygeometric information for the curve kinematic constraint.

A schematic representation of the methodology used in the track centreline pre-processoris presented in Figure 3. It must be noted that by ensuring very small steps for thedatabase, it is possible to ensure that two successive time steps do not use the sameinterval in the linear interpolation for a given velocity of the vehicle. The quality of the

interpolation, from the numerical point of view, is the same for any degree of thepolynomial used. Therefore, linear interpolation of the database is used because it leadsto smaller computational costs. The track-length step used for the database must besmaller than the product of the vehicle maximum velocity by the integration time stepused. The database scheme described is general, being suitable for any track interpolationscheme envisaged.


10/31


Figure 3 Schematic representation of the track centreline pre-processor used to build the track

centreline database (described in Figure 5)

2.4 Track centreline parameterised by piecewise cubic interpolation

The polynomial interpolation of complex line geometries is generally realised bypiecewise polynomial schemes. These methods construct curves that consist ofpolynomial pieces of the same degree that have a prescribed overall smoothness.The input data usually includes the interpolation points and parameters values, in whichtangent information may be added as well (Farin, 1990). The advantage of this type ofinterpolating procedures is that they exhibit local geometrical control, i.e., the variation ofthe position of a control point only affects the neighbourhood of that point, maintaining

unchanged the rest of the curve.The type of polynomials more popular in many applications are the piecewise cubiccurves, which have a C1 orC2 continuity. The C1 continuity implies that adjacent cubicpieces share a common point and a tangent line at the junction of the two curves, whereasthe C2 continuity also implies that the two curves possess equal curvature at their joint(Mortenson, 1985). It must be noted that the C2 continuity implies the continuity of theaccelerations between two adjacent spline segments. The non-enforcement of the C2continuity for the splines would lead to significative impulses in the segment transitionsthat do not exist in the system. For curves with C2 continuity, the local control of thecurve is more problematic, which means that if a data point is changed, the interpolating


11/31


curve may also change in the vicinity of other data points not close to this one

(Farin, 1990).

2.4.1 Track parameterisation by cubic splines

The track centreline can be described using cubic spline segments that interpolate a set ofcontrol points given by the user. The parametric cubic curve is defined as (Anand, 1994):

3 23 2 1 0( ) = + + +u u u ug a a a a (21)

where g(u) is a point on the curve, u is the parametric variable and ai are the unknownalgebraic coefficients that must be calculated.

Let a set of points gi, representing the track centreline, be defined by their coordinates(x, y, z). When the cubic-spline segments are used to represent the interpolation curves,

the algebraic coefficients ai in equation (21) are written explicitly in terms of theboundary conditions, i.e., segment end points and tangent vectors to each cubic segment,leading to (Anand, 1994):

3 2

2 2 1 1 (0)

3 3 2 1 (1)( ) = 1

0 0 1 0 (0)

1 0 0 0 (1)

u u u u

g

gg

g

g

(22)

where the spline local parameteru [0, 1] and g(0) and g(1) represent the coordinates ofthe end points of each segment. The spline derivatives g(0) and g(1), at the end points,are calculated in order to ensure C2 continuity between spline segments. When the second

derivatives of the first and last points, of the set to be interpolated, are null, the cubicspline is said to be a natural spline (Anand, 1994; de Boor, 1978). In this case, the firstderivatives in all other control points are obtained by solving:

0 1 0

1 2 0

2 1 3

1 1 2

3( )2 1

3( )1 4 1

3( )1 4 1

3( )1 2n n n

n n n

.

=

g g g

g g g

g g g

g g g

(23)

In order to implement the spatial curve kinematic constraint in the computer code, it isnecessary to find the value of the cubic spline parametric variable u that corresponds to a

prescribed track lengthL. Consider the parametric variable uP

, corresponding to a pointP, located on the kth cubic spline segment and to which is associated a curve length ,PkL measured from the kth segment origin. In this case, the parameteruP is obtained using theequation (Pombo and Ambrsio, 2001, 2003a):

0

= 0

P

Tu

u u P

k k kdu L g g (24)

where gu is the derivative ofg with respect to u. This non-linear equation is solved in thepre-processor using the NewtonRaphson method (Nikravesh, 1988; Pina, 1995).


12/31


2.4.2 Track parameterisation by Akima splines

The Akima spline interpolation scheme, developed by Akima (1970), minimises theundesired wiggles in the interpolant observed in several interpolation methods(Akima, 1970; de Boor, 1978). This scheme interpolates a given set of data points so thatthe resultant curve not only passes through such points, but also appears smooth andnatural. This interpolation scheme is based on a piecewise function composed of a set ofpolynomials Pi, each of the third degree, applicable to successive intervals of the datapoints. The slope of the curve is determined locally at each point. Each polynomial,representing a portion of the curve between a pair of given points, is determined by thepoints coordinates and by the slopes at such points (Akima, 1970). This method has localcontrol, which means that the ith piecePi depends only on the information from, or near,the interval [xi, xi+1], where xi and xi+1 represent the coordinates of the end points of eachsegment (de Boor, 1978).

The pre-processor uses the FORTRAN routine DCSAKM (Visual Numerics, 1997),implemented in the IMSL math library, to compute the Akima splines that interpolate thesets of data points (ui,xi), (ui,yi) and (ui,zi), where xi, yi and zi are the Cartesiancoordinates of the control points, written as:

{ }= ; 1, ,T

i ix y z i = ... Ng (25)

ui = 1, ,N being the parametric variable of the cubic curve correspondingto each control point gi and N the number of control points to interpolate.The parametric curve obtained consists in three piecewise cubic polynomials, defined as(Visual Numerics, 1997):

2 33 4

1 2

12 33 41 2

2 33 41 2

+ ( ) + ( ) + ( )2 6( )

( ) = ( ) = + ( ) + ( ) + ( ) ;= 12 6

( )+ ( ) + ( ) + ( )

2 6

x x

x x i i

i i i i i

y yi iy y i i

i i i i i

z z

z z i i

i i i i i

c c

c c u u u u u ux u

u u uc cu y u c c u u u u u u

iz u

c cc c u u u u u u

+


13/31


In order for the Akima cubic splines to be used in the pre-processor for the

description of the track centreline, their geometric characteristics must be expressed as afunction of the track length L and not as a function of the curve parametric variable u.The relationship betweenL and u is calculated by using equation (24).

2.4.3 Track parameterisation by shape-preserving splines

The shape-preserving cubic splines are functions that preserve the shape of theinterpolation data (Irvine et al., 1986; Micchelli et al., 1985; Visual Numerics, 1997).For example, if it is required to interpolate a function f that is monotone or convex onsome interval, it is advantageous to use a spline interpolation functions that has the sameproperties. To achieve this goal, spline functions with more parameters than the knots atthe data points are used. The additional parameters are selected to ensure the desiredproperties of the spline interpolation function s. In the formulation considered here,the additional parameters are simple knots distinct from the data points (DeVore andLorentz, 1993).

The pre-processor uses the FORTRAN routine DCSCON (Visual Numerics, 1997),implemented in the IMSL math library, to compute the shape-preserving cubic splinesthat interpolate the sets of data points (ui,xi), (ui,yi) and (ui,zi), xi, yi and zi being theCartesian coordinates of the control points, ui= 1, ,N the parametric variable of thecubic curve corresponding to each control point and N the number of control points tointerpolate. The parametric curve obtained consists in three piecewise cubic polynomials,written as (Visual Numerics, 1997):

2 33 41 2

3 42 31 2

2 33 41 2

+ ( ) + ( ) + ( )2 6

( )( ) = ( ) = + ( ) + ( ) + ( )2 6

( )+ ( ) + ( ) + ( )

2 6

x xx x x x xi ii i i i i

y yj jy y y y y

j j j j j

z zz z z z zk kk k k k k

c cc c u u u u u u

x u c cu y u c c u u u u u u

z uc c

c c u u u u u u

g (27)

with

1

1

1

; 1, ... , 1

; 1, ... , 1

; 1, ... , 1

x x

i i x

y y

j j y

z z

k k z

u u u i = N

u u u j = N

u u u k = N

+

+

+

<

< <

(28)

In equation (27), g(u) is a vector that describes the coordinates of a point on the curve, uis the local parametric variable and , andx y zmi mj mk c c c are the local polynomial coefficientsthat must be calculated. The quantities , andx y zi j ku u u are, respectively, theNx,Ny andNzbreak points forx(u), y(u) and z(u), respectively, of the shape-preserving cubic splines,which are assumed to be strictly increasing. On the splines produced by this formulation,it is not possible, a priori, to predict the number of break points of the resultinginterpolant (Visual Numerics, 1997). In most cases, there are break points at otherlocations than those of the data abscissas ui, which means that

x y z

i j ku u u andNxNyNz. The method is non-linear and, although the interpolant is a piecewisecubic, cubic polynomials are not reproduced, but linear polynomials are reproduced


14/31


(Visual Numerics, 1997). A detailed description of the theoretical issues inherent to this

interpolation scheme is out of the scope of this work. The interested reader is referred toIrvine et al. (1986) and Micchelli et al. (1985).

The advantage of the shape-preserving splines parameterisation is that it is consistentwith the concavity of the data, which is rather useful when it is important to preserve theconvex and concave regions implied by the control points. This interpolation scheme alsoensures C2 continuity between spline segments, i.e., it guarantees that the parameterisedcurve has a continuous curvature.

In order for the shape-preserving cubic splines to be used in the pre-processor for thedescription of the track centreline, their geometric characteristics must be expressed as afunction of the track length L and not as a function of the curve parametric variable u.The relationship betweenL and u is evaluated using equation (24).

2.5 Cant angle for the track model

In horizontal curves, the outer rail of a railway track is usually raised, as shown inFigure 4, in order to reduce the effects of the centrifugal acceleration on rail-guidedvehicles (Andersson et al., 1998; British-Rail, 1997; Dukkipati and Amyot, 1988;Garg and Dukkipati, 1984; Wayne, 1998). The difference in heights between the two railshtis called cant; the cant angle is given by:

arcsin th

D =

(29)

whereD is the distance between the nominal wheelrail contact points. The cant variationhas to be taken into account by the pre-processor when creating the track model.

Figure 4 Cant and cant angle

For flat tracks, the cant angle is defined with respect to the horizontal plane. For trackswith a full spatial geometry, it is not clear which is the reference plane relative to whichthe cant angle should be defined. From the analytic properties of parametric curves(Mortenson, 1985), it is known that the osculating plane, at a given pointPon a curve, isthe plane of closest contact to the curve in the neighbourhood ofP, as depicted inFigure 5. Thereby, it is proposed here that the osculating plane, associated with thetrack centreline, plays the role of the horizontal plane in flat tracks, when measuring thecant angle.


15/31


Figure 5 The moving Frenet frame associated with the curve point

For the parametric curve to be used in the general spatial curve kinematic constraint(Pombo and Ambrsio, 2001, 2003a), which enforces the vehicle wheelsets tomove along the track with a prescribed angular orientation, it is necessary to define amoving frame represented by a set of three orthogonal vectors t, n and b (Farin, 1990;Mortenson, 1985; Yamaguchi, 1988). This frame, known as Frenet frame, provides anappropriate curve referential at every point. The unit vectors that characterise the Frenetframe are the unit tangent vector t, the principal unit normal vector n and the binormalvectorb. Such vectors, defined in the intersection of the normal, rectifying and osculatingplanes shown in Figure 5, are written as:

; ;

u

u= = =

g k

t n b tnkg

(30)

with the auxiliary vectorkgiven by:

2

Tuu uuu u

u=

g gk g g

g

(31)

where gu and guu represent, respectively, the first and second derivative of the parametriccurve g(u) with respect to the parametric variable u. For notation purposes, thesuperscript (.)T indicates the transpose of a vector and the tilde placed over a vectorindicates that the components of the vector are used to generate a skew-symmetric matrix

(Nikravesh, 1988), which is used to define the vector product, i.e., b t n=

is expressedas ,=b t n where 3 2 3 1 2 1[0 ; 0 ; 0].t t t t t t = t

The pre-processor, developed to parameterise the track centreline, automaticallyaccounts the contribution of the track cant for the calculation of the curve geometricparameters. The cant angle is measured in the normal plane as the angle between theprincipal unit normal vector n, which lies in the osculating plane as represented inFigure 5, and the unit normal vectornc, which is associated with the new curve movingframe (tc, nc, bc) after the rotation, as shown in Figure 6.


16/31


Figure 6 Cant angle contribution to the track model

The new components of the principal unit vectors, after the cant angle rotation is applied,are expressed as (Pombo and Ambrsio, 2001, 2002a, 2002b, 2003a):

=

= cos( ) sin( )

= sin( ) + cos( ).

c

c

c

t t

n n b

b n b

(32)

In this formulation, the user must set the cant angle that corresponds to each one of thenodal points used to parameterise the track centreline. Regardless of the parameterisationscheme chosen, the values of the cant angle are linearly interpolated between thetrack segments end points (Pina, 1995). With the complete information available,the pre-processor program uses equation (32) to calculate the geometric parametersthat define the geometry of the track centreline as a function of the track length. Theremaining equations required to define thoroughly the special curve kinematic constraintare described in Pombo (2004) and Pombo and Ambrsio (2001, 2002b, 2003a).

2.6 Information included in the track centreline database

The general spatial curve kinematic constraint (Pombo, 2004; Pombo and Ambrsio,2001, 2002b, 2003a) is used in the framework of a dynamic analysis program where thevehicles may have a large number of wheelsets constrained to move along the track.Therefore, the direct use of the formulation online during dynamic analysis is notefficient from the computational point of view. An alternative approach is to use apre-processor to construct a lookup table where all quantities, appearing in the definitionof the track centreline or on the kinematic constraint, are tabulated as a function of thetrack length L. Once the database of the track centreline is built, the track model iscompletely defined. Therefore, the user can assemble it with multibody models ofrail-guided vehicles in order to perform the dynamic analysis of the whole system.During the dynamic analysis, the multibody program only has to interpolate the databasein order to obtain all required geometric characteristics of the track to set up the spatial

curve kinematic constraint.

Table 1 Structure of the track centreline database

L x y z dx/dL dy/dL dz/dL d2x/dL2 d2y/dL2 d2z/dL2 tcx d2bcy/dL

2 d2bcz/dL2

0.00

0.05

0.10


17/31


In Table 1, the structure of the track centreline database obtained with the pre-processor

program is presented. The database consists of a table with 37 columns, where each onecorresponds to a railway geometric parameter. The first column of the databasecorresponds to the track length L with a step size L, the corresponding Cartesiancoordinates being (x, y, z) of the track centreline stored in the three columns that follow.Columns 5 through 10 store the first and second derivatives of the Cartesian coordinateswith respect to L. The next nine columns contain the information about the Cartesiancomponents of the unit tangent vectortc and the first and second derivatives of the vectorcomponents with respect to L. Columns 20 through 28 of the track centreline databasestore the Cartesian components of the principal unit normal vector nc as well as thecomponents of the first and second derivatives with respect to L. The last nine columnscontain the Cartesian components of the binormal vector bc and the first and secondderivatives of the vector components with respect toL.

3 Application cases

The discussion of the methodology proposed for the construction of general track models,with emphasis on the interpolation schemes used, is carried here based on ademonstrative application. It concerns a three-dimensional track model with a geometrythat contains all the ingredients used on roller coasters designs (Wayne, 1998). In thisstudy, the general spatial curve kinematic constraint is used for the definition of thevehicletrack interaction. It enforces the wheelsets of the vehicle model to followthe track with its spatial orientation prescribed with respect to the moving Frenet frameassociated with the track centreline. The interested reader is referred to the work ofPombo and Ambrsio (2001, 2002b, 2003a) for a detailed description on how to describeand set up such a constraint. It is assumed here that the wheels of the roller coaster carsare permanently in rigid contact with the rails and, therefore, no wheelrail contactformulation is required.

3.1 Roller coaster track geometry

The vehicle used in this work moves on a three-dimensional track with the geometryillustrated in Figure 7, whereHi refers to the height of the point i,Rj designates the radiusof curve segment j, k is the angle that each track segment kmakes with the horizontalplane andLyij refers to the distance between two points i andj. In Figure 8, a view of theroller coaster, as used in the application, is presented.

The roller coaster model is assembled considering transition curves with 60 m length

for the connections between the horizontal curve GI and the straight segments FG andIJ. The cant angle for the circular curve GI is 1.014 rad (58.1), and it varies linearlyin each of the transition segments. The track cant angle adopted corresponds to theequilibrium cant, i.e., the cant for zero track plane acceleration.

Three alternative roller coaster models are built using cubic, Akima andshape-preserving splines. In order to compare the different descriptions of the rollercoaster geometry and their impact on the accuracy of the models developed, two sets ofcontrol points, with a length step of 1 m and 5 m, are used to parameterise the centrelinecurve. The track models are pre-processed and their relevant geometric information isincluded in a track centreline database.


18/31


Figure 7 Roller coaster geometry with the track length parameter for each point in parenthesis

(in meter)

Figure 8 View of the roller coaster as used in the simulation

In Figure 9, the z Cartesian coordinate of the roller coaster model built using cubic,Akima and shape-preserving splines is presented. This graphic is drawn as a function ofthe track length L, in the neighbourhood of point B, represented in Figure 7, and for a

control point increment of 1 m. The curve parameterised with cubic splines leads tooscillations in the transition between the straight segment and the circular segment.The other two parameterisation schemes do not exhibit oscillations and preserve theshape of the data. The behaviour shown in Figure 9 is also observed in all other parts ofthe track where transitions between segments with different curvatures exist.

Figure 9 zCartesian coordinate of the roller coaster model


19/31


3.2 Multibody model of the roller coaster vehicle

In general, a railway vehicle consists of a collection of bodies and mechanical elementsmoving along the track. Owing to their high structural stiffness, the carbody and thewheelsets are considered as rigid bodies. These bodies are connected by sets of kinematicjoints, responsible to control their relative motion, and by groups of rigid or flexiblelinks, which are used to model the suspensions of the vehicle. The important informationrequired to assemble the vehicle model includes the mass, the inertia tensor and theCentre of Mass (CM) location for each rigid body. Furthermore, the suspension elementscharacteristics and the location of their attachment points are also required during themodel-assembling procedure.

The roller coaster vehicle consists of a train with three cars that are interconnected bylinking bars, as represented in Figure 10. The multibody model of the vehicle is

assembled using 11 rigid bodies, identified with the numbers 111, corresponding to 3carbodies, 6 wheelsets and 2 linking bars. The origin of the global coordinate system(x,y,z) is located at the height of the track surface, as depicted in Figure 10. The train isassembled in such a manner that the rear wheelset centre is placed on the origin of thex-axis. Note that the model used here does not constitute that of a real vehicle. All data isapproximate and used here for demonstration purposes only.

Figure 10 Multibody model of the roller coaster vehicle

A local reference frame (, , ) is rigidly attached to the CM of each body. The spatialorientations of these reference frames are such that they are aligned with the principalinertia directions of the respective rigid bodies. The initial position of each rigid body isgiven by the location of its CM with respect to the global reference frame (x,y,z).The mass, the inertia properties with respect to the three principal axes and the initialposition of each rigid body are presented in Table 2. The numbers in the first column ofthis table identify the rigid bodies in the vehicle model, presented in Figure 10. The massand the inertia properties of the bodies are estimated based on the geometry of each body.

Four spherical joints are used in the roller coaster vehicle model to connect thelinking bars to the carbodies in order to assemble the train. Each spherical joint imposesthree kinematic constraints between the two connected bodies in order to control theirrelative motion, allowing only three relative degrees of freedom (Nikravesh, 1988).Hence, only relative rotations are allowed between the bodies connected byspherical joints.


20/31


Table 2 Mass, inertia properties and initial positions of each rigid body

Inertia properties (Kg.m2) Initial position (m)

ID Rigid bodiesMass(Kg) I I I x0 y0 z0

1 Rear wheelset of rear car 30 1.675 0.150 1.675 0.00 3.0 80.1

2 Front wheelset of rear car 30 1.675 0.150 1.675 1.00 3.0 80.1

3 Carbody of rear car 290 48.333 120.833 120.833 0.50 3.0 80.5

4 Rear wheelset of middle car 30 1.675 0.150 1.675 2.30 3.0 80.1

5 Front wheelset of middle car 30 1.675 0.150 1.675 3.30 3.0 80.1

6 Carbody of middle car 290 48.333 120.833 120.833 2.80 3.0 80.5

7 Rear linking bar 4 3.8 103 32 10

3 32 103 1.65 3.0 80.1

8 Rear wheelset of front car 30 1.675 0.150 1.675 4.60 3.0 80.1

9 Front wheelset of front car 30 1.675 0.150 1.675 5.60 3.0 80.1

10 Carbody of front car 290 48.333 120.833 120.833 5.10 3.0 80.5

11 Front linking bar 4 3.8 103 32 103 32 103 3.95 3.0 80.1

Six general spatial curve kinematic constraints are used to assemble the multibody model

of the roller coaster vehicle. These kinematic constraints are defined between the vehicle

wheelsets and the track centreline in order to enforce the wheelsets to move along the

roller coaster track; their spatial orientation is prescribed according to the track geometry.

The type of kinematic constraints used to assemble the roller coaster vehicle model,

the bodies connected by them and the local coordinates of the attachment points are

presented in Table 3.

Table 3 Kinematic constraints used in the multibody model of the roller coaster vehicle

Attachment points local coordinates (m)

Bodies connected Body i Body j

ID Kinematic constraints i j i i i j j j

1 Spherical joint 3 7 1.0 0.0 0.4 0.15 0.0 0.0

2 Spherical joint 6 7 1.0 0.0 0.4 0.15 0.0 0.0

3 Spherical joint 6 11 1.0 0.0 0.4 0.15 0.0 0.0

4 Spherical joint 10 11 1.0 0.0 0.4 0.15 0.0 0.0

The suspension of the roller coaster vehicle consists of a group of flexible links that

ensure the transmission of forces between the carbodies and the wheelsets. The model ofthe suspension of each car consists of a three-dimensional spring element, assembled at

each wheelset extremity, which connects the wheelsets to the carbody, as shown in

Figure 11. The model also uses four vertical dampers, assembled between the wheelsets

and the carbody.


21/31


Figure 11 Schematic representation of the suspension model used in each roller coaster car

The characteristics of the spring elements used in the multibody model of a roller coastervehicle are estimated assuming that the vehicle has eigenfrequencies similar to the onesin the bogies of railway vehicles. Hence, considering the railway vehicle ML95(Pombo and Ambrsio, 2000), it is assumed here that the eigenfrequencies in the vertical,lateral and longitudinal directions of each roller coaster car are equal to 9 Hz, 11 Hz and25 Hz, respectively. Under these conditions, the eigenfrequencies of each car can bewritten as (Beer and Johnston, 1990):

; 2= =eq

n n n

K f

m(33)

whereKeq is the equivalent stiffness of the system, m is the mass of the carbody and fn isthe eigenfrequency of the system. As shown in Figure 11, in the suspension model of

each car, there are four springs acting in parallel in each direction. Therefore,the relationship between each spring in the car suspension and the equivalent stiffness ofthe system is:

; = , ,4

eq

ll

KK l x y z.= (34)

Combining equations (33) and (34) leads to the stiffness of the spring elements as:

2 2 ; = , ,l nlK m f l x y z.= (35)

By using equation (35), the spring stiffness of the suspension systems of the roller coastervehicles are obtained as Kx = 1789 kN/m, Ky= 346 kN/m and Kz= 2317 kN/m.The characteristics of the vertical dampers, Cz, used in the suspension model are

estimated assuming that the roller coaster vehicle has a damping factor equal to 0.9(Beer and Johnston, 1990). Under this assumption, the damping coefficient of each rollercoaster car is written as (Beer and Johnston, 1990):

= = 2eq CR eqC C mK (36)

where is the damping factor and CCR is the critical damping coefficient. As shownin Figure 11, in the suspension model of each car, there are four vertical dampers actingin parallel. Therefore, the damping coefficient of each vertical damper is written as:


22/31


1.4 2

eqeq

z

CC mK= = (37)

By using equation (37) and the spring stiffnessKzobtained before the vertical damping isCz= 7380 Ns/m. As noted before, more realistic values for the suspension characteristicscan be set if the data is supplied by a roller coaster manufacturer.

3.3 Dynamic analysis results

With the three-dimensional track model of the roller coaster parameterised and thevehicle model assembled, the dynamic analysis of the system is performed. Comparativeoutput parameters for the track models built using cubic, Akima and shape-preservingsplines are summarised in Table 4, for distances between control points of 1 m and 5 m.

The length parameter stepL adopted for the construction of all track centrelinedatabases is L = 0.05 m. The simulations are performed in a computer with a Pentium

42.5 GHz processor with 256 Mb of RAM. The motion resulting from the dynamicanalyses, observed using two different viewpoints, is sketched in Figure 12. Table 4shows that the integration time step is not sensitive to the track model. This is notsurprising because all computational costs associated with the interpolation of the trackcentreline database only differ in the pre-processor. Different track models could inducemore or less oscillations in the curves, which may lead to different integration time stepsizes when variable time-stepping algorithms are used. In the track model used here, thiseffect has not been an issue. Another aspect that is shown in Table 4 is that the distancetravelled during dynamic analyses is not exactly the same for the three trackparameterisation schemes. This reflects that the different track models have slightlydifferent lengths resulting from the different parametric descriptions adopted. A finalobservation from Table 4 is that the use of shape-preserving splines leads to minimumCPU times.

Table 4 Comparative results of the dynamic analyses performed with different control pointsincrement

TrackParameterisation

Analysistime

(seconds)

Initialvelocity

(m/s)

Averagetime step(seconds)

CPUtime

Maximumconstraintviolations

Distancetravelled

(m)

Control points increment of 1 m

Cubic splines 49.5 2 102 5m 03s 4.9 102 1067.8

Akima splines 49.5 2 102 5m 31s 3.3 101 1067.9

Shape-preservingsplines 49.5 2 102

4m 57s 4.9 102

1067.8

Control points increment of 5 m

Cubic splines 49.5 2 102 3m 44s 4.9 102 1068.5

Akima splines 49.5 2 102 4m 03s 2.5 101 1065.7

Shape-preservingsplines

49.5 2 102 3m 43s 4.9 102 1068.8


23/31


Figure 12 Views of the roller coaster motion resulting from the dynamic analyses in different

points of the track (see Figure 7 for each location of the vehicle on the track)


24/31


When comparing the maximum constraint violations obtained during dynamic analyses,

using the different track model descriptions, it is observed that the track curvesparameterised with Akima splines produce maximum constraint violations almost tentimes larger than the violations obtained with the other interpolation schemes. This can beexplained by the fact that the track centreline described by Akima splines only has C1continuity, not ensuring the continuity of the curvature, which leads to a physicaldifficulty of following a non-smooth geometry. Figure 13 presents thex component of thefirst derivative of the unit tangent vectort in the neighbourhood of point B, referenced inFigure 7, for all track parameterisation schemes and for a control point increment of 1 m.In the curve parameterised with Akima splines, the first derivative exhibits a goodnumber of kinks, which lead to discontinuities in the second derivative. Note also that thefirst derivative oft, for the curve obtained with the cubic splines, has oscillations in thevicinity of point B, not observed for the shape-preserving splines.

Figure 13 First derivative of thex component of the unit tangent vectort

For a better visualisation of the consequences of the position constraint violations,consider the case of a single body, representing a car, travelling along the roller coastertrack parameterised with Akima splines. Figure 14 shows a view of the car at the entranceof the loop, i.e., at point D in Figure 7. During dynamic analysis, a maximum constraintviolation of 2.9 101 is obtained at this point. This violation occurs in the second

equation of the local frames alignment constraint, which is used by Pombo (2004) andPombo and Ambrsio (2001, 2002b, 2003a) to define the general spatial curve kinematicconstraint. This means that the perpendicularity condition between the unit vector u,associated with the -axis of the body-fixed coordinate system (, , ), and the binormalvectorb, of the track centreline Frenet frame, is not enforced. Since the results obtainedusing Akima splines for track parameterisation are not satisfactory, this interpolationmethod is not considered hereafter. Note that the constraint violation, pictured inFigure 14, can be wrongly mistaken for a derailment. However, the methodologyimplemented here cannot represent derailment conditions directly.


25/31


Figure 14 Roller coaster car at the entrance of the loop using Akima splines for track

parameterisation (on point D of Figure 7)

The rear wheelset acceleration in thezdirection of the rear roller coaster car, for the trackmodels parameterised with cubic and shape-preserving splines, with distances betweencontrol points of 1 m, are presented in Figure 15. As shown, there is a good agreementbetween the results obtained with both track models. The large peaks of accelerationobserved are direct results of the sudden change of the vertical curvature between parts ofthe track with different geometric characteristics. These sudden changes reflect the factthat no vertical transition curves are used in this roller coaster track model.

Figure 15 Rear wheelset acceleration inzdirection, of the rear roller coaster car, for distancesbetween control points of 1 m

In order to investigate the influence of the discretisation of the spline control points onthe acceleration response, new simulations are performed with a control point incrementof 5 m. Figure 16 shows the acceleration of the rear wheelset of the rear car inzdirection,obtained for the two track descriptions. Comparing Figures 15 and 16, it is clear that asmaller increment for the control points stepping leads to a slightly smoother track.


26/31


Figure 16 Rear wheelset acceleration inzdirection, of the rear roller coaster car, for distances

between control points of 5 m

The smaller perturbations, observed in the accelerations, are a result of the oscillationsinherent to the interpolation processes and, therefore, do not have any correspondingphysical meaning for the real vehicles. This statement can be confirmed from theaccelerations results in the neighbourhood of the region BD of Figure 7 shown inFigure 17. As the difference between the two curves show, the small perturbations in theaccelerations correspond to the wiggles intrinsic to the interpolation schemes. It is also

observed that these undesired oscillations are especially present in the track modelparameterised with cubic splines.

Figure 17 Rear wheelset acceleration inzdirection, of the rear roller coaster car, for distancesbetween control points of 5 m. A zoom of the neighbourhood of the region BD of thetrack model


27/31


The acceleration perceived by a passenger travelling in the roller coaster vehicle is

illustrated in Figure 18 as the acceleration inzdirection in the carbody of the rear car, forboth track models parameterised with cubic and shape-preserving splines and for adistance between control points of 1 m. The similarity between Figures 15 and 18 meansthat the suspension elements ensure the transmission of forces between the carbody andthe wheelsets, but do not isolate the passengers compartment from the vibrationsinduced by the track. During the simulation, the two small peaks observed in the verticalacceleration, at 35 and 40 seconds, are obtained when the vehicle passes through thehorizontal transition curves where the track cant angle has a linear variation.

Figure 18 Rear car acceleration inzdirection for distances between control points of 1 m

One of the important features of the methodology now proposed is its ability to calculatethe reaction forces that develop between the track and the vehicle wheelsets. By usingkinematic constraints between the wheelsets and the track, the joint reaction forces thatdevelop, i.e., the vertical and lateral forces and the roll moments, are directly associatedwith the Lagrange multipliers given by equation (7) for the track following constraints.Figure 19 represents the vertical reaction forces on the wheelsets of the rear rollercoaster car, and Figure 20 shows the roll reaction moments for the same wheelsets,i.e., the reaction moment that develops between the wheelset and the track, whichprevents the vehicle from rolling out of the track. These results are obtained for a track

model parameterised with shape-preserving splines with distances between control pointsof 1 m.The large peaks of vertical reaction forces observed in Figure 19 result from the

changes of the vertical curvature between parts of the track with different geometriccharacteristics. These large vertical loads on the roller coaster track reflect the fact that novertical transition curves are used in the track model. Note also the similarity in thecurves for the forces, given in Figure 19, and the rear wheelset accelerations shown inFigure 17.


28/31


Figure 19 Vertical reaction forces that develop between the track and the wheelsets of the rear

roller coaster car. Track model parameterised with shape-preserving splines withdistances between control points of 1 m

Figure 20 Roll reaction moments that develop between the track and the wheelsets of the rearroller coaster car. Track model parameterised with shape-preserving splines withdistances between control points of 1 m

The roll reaction moments between the track and the wheelsets of the rear roller coastercar, illustrated in Figure 20, show that these values are only significant when the vehiclepasses through the horizontal transition curves. In these track sections, the wheelsets areenforced to roll in order to have a spatial orientation compatible with the track cant anglethat has a linear variation.


29/31


4 Conclusions

In this work, the development and implementation of a methodology for the accuratedescription of a roller coaster track spatial geometry was presented. For this purpose,several parameterisation schemes were proposed to describe the track geometryand the moving reference frame associated with the track centreline. For the completecharacterisation of the track, the cant angle, defined here with respect to the osculatingplane, is also considered. The formulation used to represent the wheelsettrackinteraction enforces that the wheelsets move along the track with a prescribed orientationrelative to the Frenet frame associated with the centreline.

The results presented showed that the track parameterisation with cubic splinesor with shape-preserving splines can be used to describe the geometry of fullythree-dimensional roller coaster tracks. The major drawback of the cubic splines

formulation was that it lead to undesired oscillations in the track model, which could beperceived as track irregularities during dynamic analysis. The use of shape-preservingsplines leads to the best representation of the track model. A track parameterisation withAkima splines, also presented, lead to high constraint violations in the transitionsbetween track segments with different geometric characteristics. These violations werelarge enough to lead to results that were not consistent with the track regularity required,i.e., C2 continuity. Therefore, the use of Akima splines parameterisation is notrecommended whenever the calculation of the second derivatives associated with thespatial curve is required in a computer formulation.

The results of the dynamic analyses showed that the track model is sensitive to thediscretisation of the spline control points used to parameterise the track geometry.It was shown that a smaller distance between the control points lead to a smoother track.It was also shown that the small perturbations observed in the results, perceived asoscillations in the track model, were inherent to the interpolation processes and,therefore, did not have a physical meaning. These undesired wiggles in the trackmodel should be avoided in order not to confuse them with measured track irregularities.The results also showed the ability of the formulation to calculate the reaction forces thatdevelop between the vehicle wheelsets and the track.

The methodology presented in this work for the parameterisation of the trackcentreline can also be used for the description of the railway since it provides the movingframe associated with the curve and the osculating plane relative to which the cant angleis defined. Using this approach, the geometry of the rails can be described with respect tothe moving frame, which provides one of the basic ingredients for the use of themultibody code in the context of railway dynamics applications. The results obtainedwith the applications to the roller coaster models allow for recommending the use of

shape-preserving splines for the track centreline representation.

Acknowledgements

The support of Fundao para a Cincia e Tecnologia (FCT) through the grantwith the reference BPD/19066/2004 made this work possible, and it is gratefullyacknowledged.


30/31


References

Akima, H. (1970) A new method of interpolation and smooth curve fitting based on localprocedures,Association for Computing Machinery, Vol. 17, No. 4, pp.589602.

Ambrsio, J. (1997) Trainset Kinematic: A Planar Analysis Program for the Study of the GangwayInsertion Points, Technical Report IDMEC/CPM/97/005, Institute of Mechanical Engineering,Instituto Superior Tcnico, Lisbon, Portugal.

Anand, V.B. (1994) Computer Graphics and Geometric Modelling for Engineering, J. Wiley,New York.

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

Beer, F.P. and Johnston, E.R. (1990) Vector Mechanics for Engineers: Dynamics, McGraw Hill,London, United Kingdom.

British-Rail, R. (1997) Introduction to Railway Vehicle Dynamics, AEA Technology plc.,London, UK.

de Boor, C. (1978)A Practical Guide to Splines, Springer-Verlag, New York.

DeVore, R.A. and Lorentz, G.G. (1993) Constructive Approximation, Springer-Verlag, Berlin,Germany.

Dukkipati, R.V. and Amyot, J.R. (1988) Computer-Aided Simulation in Railway Dynamics,M. Dekker Inc., New York.

Farin, G.E. (1990) Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide ,Academic Press, Boston, Massachussetts.

Garg, V.K. and Dukkipati, R.V. (1984) Dynamics of Railway Vehicle Systems, Academic Press,New York.

Irvine, L.D., Marin, S.P. and Smith, P.W. (1986) Constrained interpolation and smoothing,Constructive Approximation, Vol. 2, pp.129151.

MDI, M.D. (1995)ADAMS/Rail 9.1: Technical Manual, Ann Arbor, Michigan.

MDI, M.D. (1998)ADAMS/View 9.0: Training Documentation, Ann Arbor, Michigan.MDI, M.D. (1999)ADAMS/Rail 9.1.1 ADtranz Milestone I: Release Notes, Ann Arbor, Michigan.

Micchelli, C.A., Smith, P.W., Swetits, J. and Ward, J.D. (1985) Constrained Lp approximation,Constructive Approximation, Vol. 1, pp.93102.

Mortenson, M.E. (1985) Geometric Modelling, Wiley, New York.

Nikravesh, P.E. (1988) Computer-Aided Analysis of Mechanical Systems, Prentice-Hall,Englewood Cliffs, New Jersey.

Pina, H.L.G. (1995)Mtodos Numricos (Numerical Methods), McGraw-Hill, Lisbon, Portugal.

Pombo, J. (2004) A Multibody Methodology for Railway Dynamics Applications, PhDDissertation, Instituto Superior Tcnico, Lisbon, Portugal.

Pombo, J. and Ambrsio, J. (2000) Avaliao do Desempenho dos Modelos dos VeculosFerrovirios nas Vias da REFER e do Metropolitano de Lisboa, Technical Report No. 20 ofPEDIP Project No. 25/00379 on Dinmica de Veculos Ferrovirios, Lisbon, Portugal.

Pombo, J. and Ambrsio, J. (2001) General rail track joint for the dynamic analysis of rail guidedvehicles, in Ibrahimbegovic, A. and Schiehlen, W. (Eds.): Proceedings of the EuromechColloquium No. 427 on Computational Techniques and Applications in Nonlinear Dynamicsof Structures and Multibody Systems(in CD), Cachan, France, 2427 September, pp.123.

Pombo, J. and Ambrsio, J. (2002a) Development of a roller coaster model, in Goicolea, J.,Soares, C.M., Pastor, M. and Bugeda, G. (Eds.): Proceedings of the Mtodos Numricos enIngeniera V (in CD), Madrid, Spain, 36 June, pp.127.

Pombo, J. and Ambrsio, J. (2002b) Development of a roller coaster model, in Schiehlen, W. andValasek, M. (Eds.): Proceedings of the NATO-ASI on Virtual Nonlinear Multibody Systems,Prague, Czech Republic, 23 June3 July, Vol. 2, pp.195203.


31/31


Pombo, J. and Ambrsio, J. (2003a) General spatial curve joint for rail guided vehicles: kinematics

and dynamics,Multibody Systems Dynamics, Vol. 9, pp.237264.Pombo, J. and Ambrsio, J. (2003b) A general track model for rail guided vehicles dynamics,

in Barbosa, J.I. (Ed.): Proceedings of the VII Congresso de Mecnica Aplicada eComputacional, vora, Portugal, 1416 April, Vol. 2, pp.4756.

Pombo, J. and Ambrsio, J. (2003c) A wheel-rail contact model for rail guided vehiclesdynamics, in Ambrsio, J. (Ed.): Proceedings of the ECCOMAS Thematic Conferenceon Advances in Computational Multibody Dynamics (in CD), Lisbon, Portugal, 14 July,pp.126.

Visual Numerics, I. (1997) IMSL Fortran 90 Math Library 4.0: Fortran Subroutines forMathematical Applications, Huston, Texas.

Wayne, T. (1998) Roller Coaster Physics: An Educational Guide to Roller Coaster Design andAnalysis for Teachers and Students, Charlottesville, Virginia.

Yamaguchi, F. (1988) Curves and Surfaces in Computer Aided Geometric Design,

Springer-Verlag, Berlin, Germany.

2007 Paper IJVD JPombo JAmbrosio

Documents

Transcript of 2007 Paper IJVD JPombo JAmbrosio