General Spatial Curve Joint for Rail Guided Vehicles: Kinematics and Dynamics

Transcription

1 Multibody System Dynamics 9: , Kluwer Academic Publishers. Printed in the Netherlands. 237 General Spatial Curve Joint for Rail Guided Vehicles: Kinematics and Dynamics JOÃO POMBO and JORGE A. C. AMBRÓSIO IDMEC/IST, Av. Rovisco Pais, P Lisboa, Portugal; (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 kinematic constraint is proposed, which enforces that a point of a body follows a reference path while the body maintains a prescribed orientation relative to a Frenet frame associated to the spatial track curve. Depending on the specific application, the tracks of rail-guided vehicle are described by analytical line segments or by parametric curves. For railway and light track vehicles, the nominal geometry of the track is generally done by putting together straight and circular track segments, interconnected by transition track segments that ensure the continuity of the first and second derivatives of the railway in the transition points. For other applications, the definition of the track is done using parametric curves that interpolate a given number of control points. In both cases, the complete characterization of the tracks also requires the definition of the cant angle variation, which is done with respect to the osculating plane associated to the curve. The track models for multibody analysis must be in the form of parameterized curves, where the nominal geometry is obtained as a function of a parameter associated to the curve length. The descriptions adopted here ensure, not only that the type of continuity of the original track definition is maintained, but also that no unwanted deviations from the nominal track geometry are observed, which can be perceived in the dynamic analysis as track 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 to undesired oscillations in the model. The parameterization of analytical segments of straight, circular and cubic polynomial track segments does not introduce such oscillations on the track geometry but it is rather complex for the description of railways with large slopes or with vertical curves. Splines with tension minimize the undesired oscillations of the interpolated curve that describes the railway track nominal geometry, but the curve segment parameters are not proportional to the length of the track. It is proposed here that the nominal geometry of the track is described by a discreet number of points, which are organized in a tabular manner as function of a parameter that is the length of the track measured from its origin to a given point. For each entry, the table also includes the vectors defining the Frenet frames and the derivatives required by the track constraint. The multibody code interpolates such table to obtain all required geometric characteristics of the track. With applications to a roller coaster, the suitability of this description is discussed in terms of the choice of original parametric curves used to construct the table, the size of the length parameter step adopted for the table and the efficiency of the computer implementation of the formulation. Key words: railway dynamics, Frenet frame, spatial curve geometry, prescribed motion constraint.

2 238 J.POMBO AND J.A.C. AMBRÓSIO Figure 1. Two body arrangement to model the track foundation flexibility. The base body has a prescribed motion while the track element has a motion relative to it described by the in-plane degrees-of-freedom, i.e. two translations and one rotation. 1. Introduction The dynamic analysis of railway [1 3], roller coaster [4] or any other type of rail guided vehicles requires an accurate description of the track geometry. The track is composed of two rails, which can be viewed as two parallel line defined in a plane that sits in a spatial curve, defined hereafter as the reference path. The basic ingredient to define the track is therefore the geometry of the reference path, which must include the vertical gradients, lateral curves and cant. Any track irregularities can be perceived as deviations from the reference path parallel lines, representing the rails. Typically, these are modeled by adding to the track perfect geometry small perturbations 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 to a track reference, as depicted by Figure 1. There, the track reference element has to move along the reference path and have an orientation compatible with it. The objective of this work is to present a description of the spatial geometric features of the tracks and its computational implementation in a form suitable to the multibody methodology adopted for the modeling of railway systems. The introduction of the track irregularities and the flexibility of the track foundations are not considered in this work. Depending on the specific application, the reference path of the track geometry can be described by a number of types of parametric curves [5]. For railway and light track vehicles the description of the nominal geometry of the track is generally done by putting together straight and circular track segments, interconnected by transition segments that ensure the continuity of the first and second derivatives of the railway in the transition points [1, 6]. Moreover, these transition elements are 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 segments of the same type with different radius or orientations. For other applications,

3 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 239 parametric curves that interpolate a given number of control points are commonly used to define the track geometry. In any case, the complete characterization of the tracks also requires the definition of the cant angle variation along the reference path. For flat tracks, the cant angle in a given point of the reference path is measured in the plane perpendicular to the reference path, between a 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 osculating plane of the reference track plays the role of the horizontal plane of the flat track in measuring the cant angle. In this work two types of track geometric descriptions are discussed in the framework of the multibody models for railway dynamics analysis, i.e., analytical segments [7] and cubic splines [10, 11]. The application of cubic splines to interpolate a set of control points describing the track geometry leads to undesired oscillations in the track model [12]. For instance, if a spline interpolation is used to 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 a curve that oscillates in turn of the original lines. Another drawback, in the direct application of this approach, is that the local parameter used in each spline segment is not linearly related with the length of the segment, e.g., a given point of the reference path for which the local curve parameter is half of the parameter interval is not necessarily located half way along the curve. Other methodologies using splines with tension or Akima splines [11], are alternative techniques for the parameterization of the reference path geometric description in railway applications. Although these alternative forms of interpolation have the potential to minimize the 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 on the track geometry. However, this description is rather complex to model railways with large slopes or with vertical curves. Some of the commercial codes that adopt this description impose that the tracks are basically horizontal in order to avoid difficulties [7 9]. Regardless of the form in which the reference path geometry is described a suitable kinematic constraint must be defined in order to enforce not only that a particular point of given body of the multibody systems follows the reference path but also that the body orientation does not change with respect to a Frenet-frame associated to the curve. The methodology proposed here for the general spatial curve joint can use any descriptive form for the curve. The position, the Frenet-frame vectors 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 origin to the actual point position. Therefore, during the dynamic analysis the quantities involved in the general spatial curve joint are obtained by linear interpolation of the tabulated values. The length parameter step is small enough to ensure that for any reasonable speed of the rail guided vehicle not more than once a time step the

4 240 J.POMBO AND J.A.C. AMBRÓSIO Figure 2. Cant and cant angle. quantities are obtained by interpolation with the same point limits. The constraint is implemented 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 guided systems are examined here. Special emphasis is put in the description of the methods used to derive the analytic properties of parametric curves required to establish the general spatial curve kinematic constraint CANT AND CANT ANGLE When traveling in horizontal curves, rail guided vehicles are influenced by centrifugal forces, which act away from the center of the curve and tend to overturn the vehicles. The sum of a vehicle weight and its centrifugal force produces a resultant force directed toward the outer rail. In order to counteract this force, the outer rail in a curve is raised, which is called cant or superelevation, h t, and is defined according to Figure 2 [3]. The base for the definition of cant is the distance 2b 0, between the nominal wheel-rail contact points. Angle ϕ t, in Figure 2, called cant angle, is given by: ϕ t = arcsin(h t /2b 0 ). (1) A curve is designed as being balanced at the equilibrium speed when the traveling vehicles produce a resultant force through the centerline of the track. Under this condition, the vertical rail forces are equal, so that maximum utilization of traction effort 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 reference plane. For a spatial curve it is not clear what the reference plane is, relative to which

5 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 241 Figure 3. Transition curves and superelevation ramps. the cant angle should be defined. In this work it is assumed that the cant angle is defined with respect to the osculating plane, which is presented later together with the descriptive geometry of the reference path curve TRANSITION CURVES AND SUPERELEVATION RAMPS When trains are operated at normal speeds, a circular curve with cant cannot be followed directly by a tangent track, i.e., a straight track segment. A transition between these two types of elements, designated by transition curve, is required in order to minimize the change of lateral acceleration of the vehicles. Usually, the radius of a transition curve is changed continuously, decreasing from an infinite radius at the tangent end to a radius equal to that of the circular curve at the other end. This change of radii provides a smooth transition from tangent to curve segments, and vice versa, also allowing for the superelevation to change gradually over its length. Therefore, the cant is also changed continuously leading to the socalled superelevation ramp. In general, the transition curve and the superelevation ramp have the same start and end points, i.e., the curvature and the superelevation in transition curves correspond to each other, as illustrated in Figure 3 [1, 3] ANALYTIC PROPERTIES OF PARAMETRIC CURVES The definition of the general spatial curve kinematic constraint requires a rather elaborate geometric description of the properties of the parametric curves used. The analytic properties of the curves are classified as intrinsic or extrinsic [5]. The intrinsic properties are local properties that vary from point to point, thus, they are only computed at specific points. These properties include the principal vectors, designated by tangent, normal and binormal vectors, the principal planes, desig-

6 242 J.POMBO AND J.A.C. AMBRÓSIO 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 characteristics of a geometric element. For a given curve these include the arc length and whether or not it is a plane curve or a straight line. In this section the focus is only on some of the local and global properties, which are important for the definition of the kinematic constraint. The interested reader can obtain additional information in [5] Parametric Curves A parametric curve consists on a point-bounded collection of points that have Cartesian coordinates given by continuous, one-parameter, single-valued mathematical 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 two definitive end points corresponding to the interval limits of the parametric variable u. The coordinates of any point on a parametric curve are treated as the components of the vector g(u) given by x(u) g g(u) = y(u). (3) z(u) In order for the curve, represented in Figure 4, to be used in the kinematic constraint it is required that the moving frame represented by vectors t, n and b is defined. The unit vectors that characterize the frame, known as Frenet frame, are defined in the intersection of the different planes represented in Figure 4. For a straight line it is assumed that the osculating plane is either horizontal, if the line is in the XY plane, or that its orientation is such that its intersection with the XY plane is perpendicular to the straight line Unit Tangent Vector On a parametric curve, the tangent vector at point g is denoted by g u and it is found by differentiating g(u) with respect to the parametric variable. Thus, g u (u) = dg(u) du. (4)

7 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 243 Figure 4. The moving frame. Note that when u appears as a superscript, it indicates differentiation with respect to u. It should be also noticed that the relationship between the parametric derivatives and 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 curve at point g(u), which is given by: t = gu g u. (6) Principal Unit Normal Vector The principal normal vector at point g(u) is normal to the curve and consequently it must lie in the plane normal to the unit tangent vector [5]. However, among the many possible normal vectors, the unit principal normal vector points towards the spatial center of curvature of the curve. Given the parametric expression for a curve, the principal normal vector is found by [5]: k = g uu guut g u g u 2 gu, (7) where g uu is the second derivative of g(u) with respect to the parameter u. Finally, the principal unit normal vector, is obtained as: n = k/ k. (8)

8 244 J.POMBO AND J.A.C. AMBRÓSIO Binormal Vector In order to define the Frenet frame associated to the reference path another vector normal 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 point on a curve have been developed. They are intrinsic properties since they vary from point to point. In Figure 4 it is clear that these elements form a local, threedimensional orthogonal coordinate system consisting of three axis vectors. This coordinate system, known as Frenet frame, is also designated by moving trihedron of the curve in some literature [5] Arc Length The arc length is an extrinsic property of the curve since it is a global characteristic that doesn t vary from point to point. The length of a parametric curve is given by [5] L = u 2 u 1 g ut g u du, (10) where u 2 >u 1 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 the current position of the point represented by the parameter. Within the framework of the application of the parametric description of the spatial curve in the definition of the general spatial path kinematic constraint a replacement of parameter u by a length representative parameter is necessary. This is discussed together with the computer 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 multibody systems. The multibody methodology, based on Cartesian coordinates, is briefly described in order to introduce the formulation of the general spatial curve constraint. Finally, the new constraint is formulated and its implementation aspects are discussed.

9 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS A multibody system is defined as a collection of rigid and/or flexible bodies constrained by kinematic joints, which control their relative motion, and eventually acted upon by a sets of internal and/or external forces. The position and orientation of each body i in the space can be described by a position vector r i and a set of rotational coordinates p i, which are organized in a vector as [13] q i ={r T, p T } T i. (11) According with this definition, a multibody system with nb bodies is described by a set of coordinates in the form: q ={q T 1, qt 2,...,qT nb }T. (12) 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 [13] (q,t)= 0, (13) where q is the generalized coordinates vector, defined in (12), and t is the time variable, resulting from the existence of driving constraints. The second time derivative of Equation (13) with respect to time yields the accelerations equations: (q, q, q,t)= 0 q q = γ, (14) where q is the Jacobian matrix of the constraints, q is the acceleration vector and γ is the vector that contains all contributions that depend on the velocities and on time. For an unconstrained mechanical system [13], the matricial form of the equations of motion are given by M q = f, (15) 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 (13) can be added to the equations of motion (15) using the Lagrange multipliers technique [13]. Defining by the vector of the unknown Lagrange multipliers, the equations of motion for a constrained mechanical system can be written as a system of differential and algebraic equations as [ M T q q 0 ] { } { q f = λ γ }. (16)

10 246 J.POMBO AND J.A.C. AMBRÓSIO The Lagrange multipliers are associated to the kinematic constraints and are physically related with the reaction forces generated between the bodies interconnected by kinematic joints. These reaction forces, due to the kinematic joints, are given by [13] f (c) = T q λ. (17) According to this methodology, the dynamic analysis of multibody systems involves the calculation of the vectors f and γ, for each time step. Equation (16) is then used to calculate the system accelerations q. These accelerations together with the velocities q are integrated in order to obtain the new velocities q and positions q for the time step. This process proceeds until the complete description of the system motion is obtained, for the selected time interval. The usual procedures to handle the integration of sets of differential-algebraic equations must still be applied in this case in order to eliminate the constraint drift or to maintain it under control [13] GENERAL SPATIAL CURVE KINEMATIC CONSTRAINTS The general spatial curve kinematic constraint equations are derived now and the resulting formulation is implemented in the computer program DAP-3D [13]. This constraint is the basis of the definition of the tracks for the rail guided vehicles by enforcing that a body moves along the railway. When such body travels along the track, not only the railway path has to be followed, but also its spatial orientation has to be prescribed, according to railway characteristics. The formulation used to implement these kinematic constraints that define the Frenet frame is described next 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 on a rigid body i, that has to follow the specified path, as depicted in Figure 5. The path is 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 current location of point R. This parameter L should not be confused with parameter u used in Equations (2) through (10). The kinematic constraint is written as (pmc,3) = 0 r R i g(l) = 0, (18) where r R i = r i + A i s R i represents the coordinates of point R with respect to the global coordinate system (x,y,z), r i is the vector that defines the location of the body-fixed coordinate system (ξ,η,ζ) i, A i is the transformation matrix from the body i fixed coordinates to the global reference frame and s R i represents the coordinates of point R with respect to the body-fixed reference frame. The vector

11 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 247 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. For notational purposes (.) means that (.) is expressed in body-fixed coordinates. The constraint equations are assigned with a superscript of two indices where the first denotes the type of constraint and the second defines the number of independent equations that it involves. It should be noticed that the constraint requires the introduction of the new coordinate L in the multibody system, which is the length of the curve traveled by point R from the start of the curve up to its current position. Therefore, the velocity and acceleration vectors also include the time derivatives of this parameter. The velocity equation is obtained as the time derivative of Equation (18) with respect to time [ (pmc,3) = 0 ṙ R i ġ(l) = 0 I s R i A i dg ] ṙ ω = 0, (19) L where the Jacobian matrix is (pmc,3) q = [ I s R i A i dg ] (20) in Equation (20), I is a 3 3 identity matrix and s R i = A i s R i represents the coordinates of point R with respect to the (ξ,η,ζ) i coordinate system, written in global coordinates. The acceleration equation is obtained by the derivative of Equation (19) with respect to time. The resulting equation is (pmc,3) = 0 [ I s P i A i dg ] r ω = ω i ω i A i s P i + d2 g L L 2, (21) 2

12 248 J.POMBO AND J.A.C. AMBRÓSIO Figure 6. Local frame alignment constraint. where r i = {ẍ ÿ z} T i are the translational accelerations of body i, ω i = { ω ξ ω η ω ζ } T i represents the angular acceleration of the body-fixed coordinate system (ξ,η,ζ) i, expressed in local coordinates, and L is the second time derivative of the curve parametric variable. The contribution of the constraint for the right-hand-side of the accelerations Equation is given the 3 1 vector, written as γ # = ω i ṡ R i + d2 g L 2. (22) 2 Therefore, Equations (18, 20) and (22) represent the quantities that must be implemented in constraint module of the computer code 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 reference path curve, as represented in Figure 6. Consider a rigid body i where (u ξ, u η, u ζ ) i represent the unit vectors associated to the axis of the body-fixed coordinate system (ξ,η,ζ) i. Consider also that the Frenet frame of the general parametric curve g(l), is defined by the principal unit vectors (t, n, b) L, as depicted in Figure 6. Assume that, at the initial time of analysis, the relative orientation between the body vectors (u ξ, u η, u ζ ) i and the curve local frame (t, n, b) L are such that the following equations hold (lfac,3) = 0 n T b T n T u ξ u ξ = u ζ a b c. (23) The kinematic constraint ensures that this alignment will remain constant throughout the analysis. The transformation matrix from the body i fixed coordinates to the global coordinate system is written as A i =[u ξ u η u ζ ] i. (24)

13 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 249 With the purpose of having a more compact notation, let the following unit vectors be defined u 1 ={100} T ; u 2 ={010} T ; u 3 ={001} T. (25) Equation (23) is now re-written as (lfac,3) = 0 n T A i u 1 b T A i u 1 n T = A i u 3 a b. (26) c The velocity equation for this constraint is obtained as the time derivative of Equation (26), expressed as (lfac,3) = 0 ( ) dn T 0 T n T A i ũ 1 A i u 1 ( ) db T ṙ 0 T b T A i ũ 1 A 1 u 1 ω = 0. (27) ( ) dn T L 0 T n T A i ũ 3 A i u 3 The contribution of frames alignment constraint (27) to the Jacobian matrix is the submatrix 3 7givenby ( ) dn T 0 T n T A i ũ 1 A i u 1 ( ) (lfac,3) db T q = 0 T b T A i ũ 1 A i u 1, (28) ( ) dn T 0 T n T A i ũ 3 A i u 3 where 0 T is a 1 3 null vector. The acceleration equation is the time derivative of Equation (27), and it is written as (lfac,3) = 0 ( ) dn T 0 T n T A i ũ 1 A i u 1 ( ) db T 0 T b T A i ũ 1 A i u 1 ( ) dn T 0 T n T A i ũ 3 A i u 3 r ω L

14 250 J.POMBO AND J.A.C. AMBRÓSIO [ ( dn 2 L [ ( db = 2 L [ ( dn 2 L ) T ( d A i ω i + nt A i ω i ω i + L 2 2 n 2 ) T ( d A i ω i + bt A i ω i ω i + L 2 2 b 2 ) T A i ω i + nt A i ω i ω i + L 2 ( d 2 n 2 ) T A i ] ) T A i ] ) T A i ] u 1 u 1 u 3, (29) The contribution of the local frames alignment acceleration equation, described by Equation (29) for the right-hand-side of the accelerations equation [13] is the 3 1 vector, written as [ ( ) dn T ( d 2 L A i ω i + nt A i ω i ω i + L 2 ) T ] 2 n A i u 1 [ ( db γ = 2 L [ ( dn 2 L ) T ( d A i ω i + bt A i ω i ω i + L 2 2 b 2 ) T A i ω i + nt A i ω i ω i + L 2 ( d 2 n 2 2 ) T ] A i ) T A i ] u 1 u 3. (30) The complete set of quantities that is necessary to implement computationally in 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 of the computer code. 4. Pre-Processor for Railway Geometric Description For multibody analysis, the track models must be defined in the form of parameterized curves. Here, two different parametric track descriptions, using analytical functions and cubic splines, are presented. A pre-processor program uses these parametric descriptions in order to define the nominal geometry of a railway using a discrete number of points as function of the curve length parameter. This information is organized in a database where all quantities, necessary to define the spatial curve constraint, are obtained as a function of the track length, measured from its origin, i.e., from the point where the analysis starts CURVE FOR THE REFERENCE PATH BY ANALYTICAL FUNCTIONS The pre-processor program, developed to parameterize the reference path geometry, supports the three types of analytical segments identified before, i.e., tangent, transition and circular segments. These segments are defined analytically and are characterized by their length, horizontal/vertical curvature and cant angle. The formulation of the analytical segments is similar to the track description adopted by

15 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 251 the commercial codes, such as ADAMS/Rail [7 9]. The reference path tangential analytical segments are defined by: x(l) = L, y(l) = 0, x(l) = 0. (31) The transition curves are expressed as x(l) = 2 ( ) Kb sin K h 2 L, y(l) = 2 ( )] Kh [1 cos K h 2 L, z(l) = 2 ( )] Kv [1 cos K v 2 L. (32) While the circular curve segments are depicted by x(l) = sin(k h L)/K h, y(l) = [1 cos(k h L)]/K h, z(l) = [1 cos(k v L)]/K v, (33) where L is the distance traveled while K h and K v represent, respectively, the horizontal and vertical track curvatures given by K h = R 1 h ; K v = R 1 v (34) R h and R v 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 Equations (31) through (33), L is a local parameter and the position of a point moving on the curve, obtained by these equations, must be transformed to global coordinates by using an appropriate coordinate transformation. Moreover, it should be noticed that Equations (31, 32) and (33) are based on the simplification that the horizontal travel distance is approximately equal to the total travel distance. This assumption is valid as long as the track grade is relatively small [7]. According to the presented formulation, the reference path is obtained by assembling a number of analytical segments. The actual form how these track segments are ordered and their characteristics are left for the user to define. However, in order to ensure the smooth transition between railway segments, the introduction of transition curve segments between a straight and circular curve segments, or between segments of the same type, is required if comfort is a concern. This issue,

16 252 J.POMBO AND J.A.C. AMBRÓSIO Figure 7. Points interpolated by cubic spline segments. that is application dependent, is left for the user to decide. A detailed description on the track geometries and their reasoning in terms of comfort, vehicle wear and normalization is out of the scope of this work and it can be found in [6] CURVE FOR THE REFERENCE PATH BY CUBIC SPLINES The reference path curve can also be described using cubic spline curves that interpolate a set of control points given by the user. The advantage of these curves is that 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 that can be associated to, but it is not, the length traveled over the curve. A parametric cubic curve is defined as [10] g(u) = a 3 u 3 + a 2 u 2 + a 1 u + a 0, (35) where g(u) is a point on the curve, u is the parametric variable and a i are the unknown algebraic coefficients that must be calculated. Equation (35) can be separated into the three components of g(u), such that x(u) = a 3x u 3 + a 2x u 2 + a 1x u + a 0x, y(u) = a 3y u 3 + a 2y u 2 + a 1y u + a 0y, z(u) = a 3z u 3 + a 2z u 2 + a 1z u + a 0z. (36) Let a set of points g i, representing the reference path, be defined by their coordinates (x,y,z) i as represented in Figure 7. When the cubic spline segments are used to represent the interpolation curves, the algebraic coefficients a i in Equation (35), are written explicitly in terms of the boundary conditions, i.e., segment end points and tangent vectors [10]. In this sense, each spline segment is g(u) =[u 3 u 2 u 1] 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 coordinates of the end points of each segment. The spline derivatives, g (0) and g(1),

17 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 253 at the end points are calculated in order to ensure C 2 continuity between spline segments. Assuming that the second derivatives of the first and last points, of the set to be interpolated, are null, the cubic spline is referred to as natural [10]. In this case, the first derivatives in all other control points are obtained as g 0 g 1 g 2. g n 2 g n 1 = 3(g 1 g 0 ) 3(g 2 g 0 ) 3(g 3 g 1 ). 3(g n 1 g n 3 ) 3(g n 1 g n 2. (38) Once the g i values are obtained by Equation (38), they are used (Equation (37)) to obtain the points coordinates anywhere in the cubic splines segments. In order for the cubic splines to be used in the kinematic constraint their geometric characteristics must be expressed as function of the reference path length L, measured from its origin, and not as a function of the parameter u. The relation between L and u is given by k 1 L(u) = L 0 n + Lactual k (u), (39) n=1 where k is the number of the spline segment where the point is actually located, u is the spline parametric variable and L 0 i corresponds to the length of the ith spline segment that, referring to Equation (10), is given by L 0 = 1 0 g ut g u du. (40) Note that, according to the cubic splines formulation, the local parametric variable u [0, 1]. The parameter L actual (u) represents the length of the actual spline segment from its origin to the actual location of the point and it is defined as L actual (u) = u 0 g ut g u du. (41) In order to implement the kinematic constraint in the computer code it is necessary to find the value of the cubic spline parametric variable u that corresponds to a prescribed segment length L. It is clear from Equations (39) and (41), that the relation between these two parameters is not linear. Consider the parametric variable, u R, corresponding to a point R, located on the kth cubic spline segment,

18 254 J.POMBO AND J.A.C. AMBRÓSIO Figure 8. Cant angle contribution to the track model. and to which it is associated a curve length L R k, measured from the kth segment origin. In this case, the parameter u is obtained from the parameter L using ur 0 g ut g u du L R k = 0. (42) This non-linear equation is solved, in a pre-processor, using the Newton Raphson method [11, 13] INTRODUCTION OF A PRESCRIBED CANT ANGLE IN THE KINEMATIC CONSTRAINT As referred before, in horizontal curves the outer rail is usually raised in order to reduce the effects of the centrifugal acceleration on vehicles. In this sense, the railway superelevation has to be taken into account when creating a track model. The pre-processor program, developed to construct railway databases, automatically accounts the contribution of the track cant for the calculation of the curve geometry. The cant angle is defined here as the angle between vector n R and the osculating 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 designated by ϕ R. Assume that the reference path moving frame is defined by its principal 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 the t axis by an angle ϕ R, as shown in Figure 8. Therefore, it is necessary to calculate the new components of the principal unit vectors (t R, n R, b R ) L of the curve moving frame after the rotation. Such vectors obtained as t R = A L t R ; n R = A L n R ; b R = A L b R, (43) where A L represents the transformation matrix from the parametric curve local frame to the global reference frame (x,y,z)given by A L =[t n b] L. (44)

19 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 255 The relationship between the principal unit vectors before and after a rotation ϕ R about the t axis is t R = R ϕt ; n R = R ϕn ; b R = R ϕb, (45) where R ϕ is the rotation transformation matrix for rotations around t axis given by R ϕ = 0 cosϕ R sin ϕ R (46) 0 sinϕ R cos ϕ R 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 ={001} T. (47) Notice that, in keeping with the right-hand convention, ϕ R is positive in a counterclockwise sense when viewed from a point on the positive t axis and toward the origin [5]. Substituting (45) in (43), and after rearranging, the new components of the principal unit vectors, after the cant angle rotation, are expressed as: t R = t, n R = n cos(ϕ R ) + b sin(ϕ R ), b R = n sin(ϕ R ) + b cos(ϕ R ). (48) According with this formulation, the user must set the cant angle for the beginning and for the end points of every track segment. The values of the cant angle 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 parameters that define the reference path geometry and store them in the railway database TRACK INFORMATION INCLUDED IN RAILWAY DATABASE The direct use, in the general spatial curve constraint, of the equations of the reference path, as obtained by any of the parametric descriptions previously presented, 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 dynamic analysis program, where the rail guided vehicles may have a large number of bodies constrained to move in general spatial curves, the solution of the nonlinear Equations (42) and the sets of Equations (33, 36, 48) and so forth at every time step would be an heavy burden on the code. An alternative implementation of these equations is the construction of a table where all quantities appearing in the definition of the kinematic constraint are tabulated as function of the global length parameter.

20 256 J.POMBO AND J.A.C. AMBRÓSIO Figure 9. Structure of reference path geometric information table. After selecting any of the parametric descriptions of the spatial curve presented before the length parameter step, L adopted for the database construction has to be chosen. Then, the pre-processor program automatically constructs a table with all parameters necessary to define the geometric characteristics of the reference path, taking into account the track cant variation. These geometric parameters are organized in columns as function of the length parameter L of the track, measured from its origin up to the actual point in the track. The multibody program interpolates linearly the table in order to obtain all required geometric characteristics of the track. If the size of the length parameter step L is set to be similar to the product of the vehicle velocity by the average integration time step used during dynamic analysis, then only a few number of interpolations, if any, will be performed in between two successive lines of the table. In Figure 9 it is presented the structure of the railway database obtained with the pre-processor program, where the adopted step size for the track length is L = 0.1 m. As shown, a railway database consists of a table with 37 columns where each one corresponds to a railway geometric parameter. The first column of the database corresponds to the track length L with a step size L being the corresponding 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 coordinates 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 of the unit tangent vector t, which is defined in Equation (6). Columns 14 through 19 store the first and second derivatives, of the unit tangent vector components, with respect to L, required by Equations (28). The next three columns of the railway database contain the Cartesian components of the principal unit normal vector n, which is defined in Equation (8). Columns 23 through 28 store the components of the first and second derivatives, of the vector n, with respect to L. The next three columns contain the Cartesian components of the binormal vector b, which is defined 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 models of the railway, roller coaster or other types of rail-guided vehicles, in order to perform the dynamic analysis of the whole system. For this purpose, the multibody

21 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 257 Figure 10. Horizontal track geometry. program has to interpolate the railway database in order to obtain all required geometric characteristics of the track as described before. 5. Application Examples The discussion of the methodology proposed for the general spatial curve constraint is carried here based on some application examples. The first includes a horizontal track model, with geometric characteristics similar to the ones used in train and tram railway networks. The second one concern the application to a three-dimensional track model with a geometry analogous to the one used on roller coasters designs [4] HORIZONTAL TRACK MODEL APPLICATION In this application example, the horizontal track geometry has the characteristics presented in Figure 10. Since the track has no vertical curves or grade, the limitations of using analytical segments are overcome. Therefore, the track model is built using either analytical functions or cubic splines. In order to compare the different descriptions of the track geometry and their impact in the accuracy of the models developed, three reference paths are modeled using analytical segments and spline curves with different control points increments. As the emphasis of this work is the track model and not the vehicle model itself only a single body, representing the complete vehicle, is considered for the dynamic analysis performed here. The track models 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 of 10 m each. The cant angle for the circular curve is 0.2 rad and null for the tangent track. This angle varies linearly n the transition segment. The track cant angle adopted corresponds to the equilibrium cant, i.e., the cant for zero track plane acceleration at a given curve radius and speed [3, 4]. For the track model

22 258 J.POMBO AND J.A.C. AMBRÓSIO 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 10 2 sec 139 seg m Cubic splines 34.5 sec 10 m/s 10 2 sec 139 seg m ( ) Maximum Figure 11. Acceleration of the vehicle center-of-mass in the y direction for tracks described by cubic splines and by analytical segments. that uses spline segments the distance between the control points is 1 m. In either case the rigid body has a mass of 176 Kg and inertias of I ξξ = Kgm 2, I ηη = 2.2Kgm 2 and I ζζ = 144.5Kgm 2. A initial velocity of 10 m s 1 is assigned for the simulations. Several parameters characterizing the simulations with the two track models are summarized in Table I. A first observation is that the integration time-step is not sensitive to the description adopted for the track model. This is not surprising because all computational costs associated to the interpolation of the railway table are exactly the same regardless of the parametric description adopted. At the most, different track models could induce more or less oscillations in the curves with consequences in the integration time-steps size, when variable time stepping algorithms are used. In all track models used this effect has not been an issue. Another aspect that shows in Table I is that the distance traveled for the 34.5 second of simulation is not exactly the same for both track models. This reflects that the different track models have slightly different lengths resulting from the different parametric descriptions adopted. 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 can be seen, there is a good agreement between the results obtained with the two forms of parameterization. The discontinuities observed at 10 s and at 26 s suggest that

23 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 259 Figure 12. Acceleration of the vehicle center of mass in the y direction for track models described 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 with the track model described by cubic splines are a direct result of the oscillations inherent to the interpolation process of the control points. The influence of the distance between the control points of the splines on the acceleration response is observed in Figure 12 where the vehicle center of mass accelerations in y direction are presented for two track models described by cubic splines, but with distances between control points respectively of 1 m and 10 m. The track with larger distances between control points exhibits a smoother response. However, the acceleration response deviates more clearly from the 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 dynamic response in terms of acceleration that can be perceived in a dynamic analysis of a complete railway vehicle as perturbations of the railway. Therefore, caution must be used in the parametric interpolation curve description selected for the track model.

24 260 J.POMBO AND J.A.C. AMBRÓSIO 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 10 2 sec 11 m 50 s ,010.5 m Distance = 5 m 49.6 seg. 2 m/s 10 2 sec 11 m 52 s ,009.8 m ( ) Maximum 5.2. ROLLER COASTER MODEL APPLICATION The second application example is a three-dimensional track model of a roller coaster with the geometry illustrated in Figure 13. Since the track has vertical curves and grade, its model cannot be parameterised with analytical segments and, therefore, the track model is build using only cubic splines. In this roller coaster example, two track models with distances between control points respectively of 1 m and 5 m are build. In Figure 14, two views of the roller coaster are presented. A single body vehicle model is assembled with the two track models and the dynamic simulations of the systems are performed. The motion resulting from the simulations, observed from two different viewpoints, is sketched in Figure 15. Table II contains comparative parameters that characterize the dynamic simulations performed with the two roller coaster models. These parameters are similar to the ones observed in Table I, which means that the time stepping control of the integration algorithms are not sensitive to the track complexity.

25 GENERAL SPATIAL CURVE JOINT FOR RAIL GUIDED VEHICLES 261 Figure 15. Views roller coaster motion resulting from the simulations.

26 262 J.POMBO AND J.A.C. AMBRÓSIO 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 correlation between the results obtained with both models. The large peaks of acceleration observed are direct results of the sudden change of the vertical curvature between parts of the track with different geometric characteristics. These sudden changes reflect the fact that no transition curves are used in this roller coaster design. The smaller perturbations observed in the results result from the oscillations inherent to the spline interpolation process and, therefore, do not have a physical meaning. 6. Conclusions A kinematic constraint representing a general spatial curve joint has been developed here and its computational implementation has been presented. The strategy adopted for the computer implementation of the joint starts by having the spatial curve expressed in a parametric form. A moving reference frame is defined in the curve such a way that the axes are defined in the intersections of the normal, osculating and rectifying planes. The introduction of the cant angle and of its variation along the curve has also been implemented. The reference plane used in the spatial curve to define the cant angle is the osculating plane, which is the horizontal plane in case of a flat curve. After recognizing that the parameters used in the curve definition are not necessarily related to the arc length of the curve a transformation is proposed. Recognizing the fact that the transformation equations are nonlinear and 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 the curve is suggested. The result of the use of this pre-processor is a table where all quantities involved in the constraint are tabulated as function of the arc length traveled by the constrained point of a system body in the curve. This methodology has the advantage of making the time required for the dynamic simulation of the rail-guided vehicle completely independent of the track