Bends, Jogs, And Wiggles for Railroad Tracks and Vehicle Guide Ways

Transcription

1 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways Louis T. Klauer Jr., PhD, PE. Work Soft 833 Galer Dr. Newtown Square, PA Preprint, June 4, 00 Copyright 00 by Louis T Klauer Jr. International Patent Pening ABSTRACT In traitional esign for railroa track other than special work the geometrical path of a track is compose as a sequence of straight lines, circular arcs, an spirals. The present paper efines some new geometrical shapes that can be incorporate into alignments for railroa tracks an other vehicle guie ways. Where the new shapes are appropriate, alignments that incorporate them will give ynamic performance superior to corresponing alignments whose only curve elements are spirals an circular arcs. The simplest of the new shapes are referre to as Ben, Jogs, an Wiggles. While these shapes have been a part of every-ay language for a long time, they o not appear to have been previously efine for or use in geometrical esign of tracks an guie ways. The new shapes are efine within the framework of a recently evelope metho for esign of improve railroa track spirals. This paper reviews that metho, notes situations in which the new shapes can provie improve geometry, presents mathematical formulae by which the shapes can be efine, an shows by examples how the shapes can be applie in some typical railroa track situations. It is shown that the new shapes can be efine in terms of Gegenbauer polynomials an that the known properties of those polynomials contribute both to unerstaning an application of the shapes. Among ways that the new shapes can be use to improve railroa track there are three that are particularly encourage. First, the Jog shape can be use to efine turnout an crossover geometry that is ynamically better than the geometry in current use. Secon, when existing curves are being re-aligne an their spirals are foun to be too istorte for immeiate restoration of ieal geometry, the spirals can be moifie by amixture of the new shapes an tamping with limite track throws can then achieve smoothe alignments whose ynamic characteristics are relatively optimal. Thir, when existing curves are being upgrae for higher spee an transition lengths nee to be increase without relocation of the curves, transitions base on combinations of spirals an Ben will be ynamically better than corresponing compoun transitions base on separate arcs an spirals. The new shapes are applicable not only to railroa tracks but also to other vehicle guie ways incluing, for example, maglev guie ways (magway, roller coaster tracks, an bobsle runs. OUTLINE 1 Introuction Review of Improve Spiral Design Metho

2 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. (Preprint, June 4, 00) 3 Constructing the roll acceleration for a Ben by overlapping two spirals 4 Constructing the roll acceleration for a Ben by inserting a factor 5 Constructing the roll acceleration for a Jog by overlapping three spirals 6 Constructing the roll acceleration for a Jog by inserting a factor 7 Jogs as shapes for turnouts an crossovers 8 Constructing the roll acceleration for a Wiggle by inserting a factor 9 Simplification an generalization by means of Gegenbauer polynomials 10 Using spiral, Ben, an Jog combinations to upgrae curves with inaequate offset 11 Classification of ajacent arc relationships 1 Moifie Gegenbauer series for alignments that avoi obstacles 13 Moifie Gegenbauer series for maintenance of track geometry 14 Conclusions 15 References. 1. Introuction Recent publications (references 1,, & 3) have presente an emonstrate an improve way of thinking about an calculating the geometry of transition spirals for railroa curves. Those publications applie the improve metho to the most common situation where a spiral makes a simple connection between two ajacent segments of track with iffering constant values of curvature. The present paper explores the application of the improve metho to more complex transition shapes referre to as Ben, Jogs, an Wiggles. Ben will be of value for track layout at locations where the track nee to change irection by a small amount. The backgroun is as follows. When an attempt is mae to use the normal spiral curve spiral sequence to accomplish a small turn, the spirals themselves accomplish the turn so that the sequence changes to spiral spiral. If the spirals employe are traitional linear spirals, then the two spiral sequence has worse than usual ynamic characteristics where the two spirals meet. That problem can be ameliorate by use of improve spirals. However, it will be more logical to employ a shape esigne for this situation. The Ben is esigne to be such a shape. It is efine using a conceptual framework first evelope for improving the esign of spirals an will give goo ynamic performance in small turn situations. Jogs are efine below for situations where the track nee to curve quickly in one irection an then quickly in the other irection an where the curvature of the track nee to keep changing throughout both curves. Jogs can be use for the esign of turnouts an crossovers. A Jog can also be use in continuous track where a clearance obstruction on one sie is followe fairly closely by an obstruction on the other sie. A Wiggle ben to the right, then to the left, an then to the right, or vice-versa. A Wiggle can be use where an obstacle on one sie requires a local eviation by what woul otherwise be a straight path. Shapes like Wiggles but with more than three ben can also be efine. This paper gives formulae for Ben, Jogs, an Wiggles an illustrates track shapes which can be obtaine from them. The author expects to be able in a future paper coauthore with others to present results of simulations like those of reference 3 that will show how preicte vehicle ynamic responses

3 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 3 (Preprint, June 4, 00) to Ben, Jogs, an shapes efine by Gegenbauer polynomial series compare with preicte vehicle responses to corresponing traitional geometries. The new shapes are applicable to any guie ways on which vehicle spee are high enough relative to guie way curvatures so that centripetal accelerations are important. Other examples of guie ways in that category are guie ways for maglev vehicles, roller coasters tracks, an bob-sle runs.. Review of Improve Spiral Design Metho The present paper is base on an improve metho for esigning track geometry shapes in which the curvature varies with istance. This Section gives an overview of that metho as applie to the esign of the spiral, which is the simplest such shape. The main novelty of the improve metho is that esign is begun not by consieration of competing shapes but rather by consiering competing forms for the roll of the track as a function of istance. The rationale for the improve esign metho is the proposition that the primary job of a shape element in which curvature changes with istance is to cause a vehicle that traverses it to have its roll angle change from one steay value to another with the least fluctuation of lateral force applie to the rails an with the least fluctuation of lateral an roll accelerations applie to vehicles. This premise focuses attention on the vehicle s rotation about its roll axis as it traverses the spiral an on the character of the roll an linear accelerations to which the vehicle is subjecte in that process. For reasons explaine in references 1,, 3, 4, an 5 it is normally avantageous to have the longituinal axis for roll of the track raise above the plane of the track to the height of a typical vehicle center of gravity or higher. Within the improve metho, after a roll motion has been chosen, there is a nee to be able to compute the shape that correspon to that roll motion. The computation begins with the generally accepte premise that the curvature of the path of the roll axis shoul be such that at esign spee the centripetal acceleration ue to the spee an curvature at any given point balances the component of gravity ue to the roll (i.e., bank or superelevation angle) of the track at that point. Looking at the components of centripetal acceleration an gravity in the rolle plane of the track, this premise is expresse by the ifferential equation g b _ axis( = tan( r _ angle( ) (1) v In equation (1) b_axis( enotes the bearing angle of the path of the roll axis. Its erivative with respect to istance along the path of the roll axis is by efinition the curvature of that path. g is the acceleration of gravity. v is the vehicle spee for which the gravitational an centripetal acceleration components are to be in balance, an r_angle( is the roll angle of the track as a function of istance along the path of the roll axis. Once a roll motion is selecte so that track roll is specifie as a function of istance, integrating equation (1) once yiel the bearing angle of the roll axis as a function of istance, an integrating the sine an cosine of the roll axis path bearing angle with respect to istance yiel Cartesian coorinates of points on the path of the roll axis as functions of istance along it. With the roll angle of the track an the path of the roll axis both known as functions of istance along the path of the roll axis, the path of the track itself can be inferre by simple trigonometry as illustrate in Figure 1.

4 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 4 (Preprint, June 4, 00) FIGURE 1. Illustration of elevation of roll axis height above plane of track. A track shape obtaine as just outline base on a roll motion selecte as an initial guess will be unlikely to connect properly to the ajacent track segments that the shape is intene to connect. The metho becomes practical when the roll motion is efine by formulae that have ajustable parameters an when there is a computational proceure by which the parameter values can be ajuste so that the resulting shape oes connect properly with the segments of track that are ajacent to it. References an 3 explain the computational proceure for obtaining improve spirals an give examples of plausible roll motions an of the spiral shapes that they generate. The computational proceure is also emonstrate in Section 10 below. The following example of a roll motion for a spiral an of the spiral shape that the roll motion generates is taken from reference 3. The shapes of the roll functions are illustrate in Figure.

5 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 5 (Preprint, June 4, 00) FIGURE Orer {,1} roll function; the roll acceleration has a simple zero at the mi point an a n orer zero at each en. The roll angle an each of its erivatives is given by a single function that applies throughout the length of the spiral. The roll function shown is referre as orer {,1}. The {,1} esignation is use to inicate that the roll acceleration has a n orer zero at each en an a 1 st orer zero at the mipoint. Denote istance along the path of the roll axis of the spiral by s, let the spiral exten from a istance s = - a to a istance s = a, an let roll_change enote the change in the track roll angle over the length of the spiral. Then the formula for the roll acceleration (meaning the n erivative of track roll angle in raians with respect to istance along the path of the roll axi is r _ angle( 105 roll _ change ( a + s ( a = () 7 16 a Figure 3 illustrates the spiral that is obtaine from the above roll function for connecting tangent track to a curve with curvature of 1.0 eg per 100 ft chor an that is offset from the tangent by ft. The track roll axis is at a height.44 m (8 ft) above the (unrolle) plane of the track. In this figure the improve spiral obtaine from the above roll function is compare with a traitional linear spiral connecting the same tangent an curve. It may be seen that the improve spiral is a little over twice as long as the traitional spiral an that it is smoother in character than the linear spiral in the vicinity of the en points of the latter. (The ynamic isturbance cause by the linear spiral woul be reuce if the length of the linear spiral were increase, but such an increase woul cause an increase in the offset between the spiral an the curve an woul require relocation of the curve.)

6 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 6 (Preprint, June 4, 00) FIGURE 3 Plots of curvature, alignment, an superelevation for an orer {,1} spiral with.44 m (8 ft) roll axis height an for a corresponing traitional linear spiral. Both spirals connect tangent track to a 1.0 eg curve elevate for balance at 90 mph. (The two alignments shown in the central part of the plot are too close together to be istinguishe at this scale. The isplacement from one to the other is inicate by the track throw.) The lengths of the traitional an improve spirals illustrate are 15.4 m (500 ft) an m (108.8 ft) respectively. This Section conclues with two observations about the behavior of the track curvature. The first observation pertains to the behavior of the track curvature at the en of a track shape. As explaine in reference, rail vehicle motion simulations have shown that a iscontinuity in the first erivative of track curvature can excite episoes of hunting on the part of some trucks. We woul therefore like to see how to insure that the first erivative of track curvature will be continuous (which is to say, zero) at each en of a track shape. In light of Figure 1, at corresponing points on the track an on the path of the roll axis, the compass bearing of the track is relate to the compass bearing of the path of the roll axis by the formula b _ track ( s ) = b _ axis ( s ) arctan h sin ( r _ angle( ) (3) where h represents the roll axis height. The formula for the curvature of the track shape as a function of istance along the path of the roll axis can be obtaine by ifferentiating equation (3) with respect to istance along the track. Denoting istance along the track by z an for the moment abbreviating

7 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 7 (Preprint, June 4, 00) r_angle( as r(, that curvature is h sin ( r ( ) b _ track ( b _ track ( = b _ axis ( z z z 1 + h sin ( r ( ) Looking further at Figure 1 one can obtain the relation = (4) z = 1+ h sin ( r ( ) b _ axis ( + h cos( r ( ) r ( (5) Our purpose here is to establish a conition uner which the first erivative of the track curvature will z be zero at the en of the shape. It is apparent that = will always be close to unity so that z we can ignore that factor an look just at how to insure that b _ track(, the n erivative of the track bearing with respect to istance along the path of the roll axis, will be zero at the en of a shape. Taking avantage of equation (1) we can write h sin ( r ( ) g b _ track ( = tan( ( ) ) r s (6) v 1 + h sin ( r ( ) 1 Differentiating once more we obtain g b _ track( = v r( h sin( r( ) + cos ( r( ) 1 + h sin( r( ) Coupling equation (7) with the simple formula 3 h sin( r( ) h sin( r( ) (7) sin( r( ) = sin( r( ) r( r( + cos( r( ) r( (8) 3 two things can be observe. First, if the roll axis is not raise above the plane of the track so that h = 0, then the first erivative of the track curvature with respect to istance will be zero at the en of a shape if the roll velocity, r(, is zero there. Secon if the roll axis is raise above the plane of the track so that h > 0, then to insure that the first erivative of the track curvature with respect to istance will be zero at the en of a shape we will nee to restrict consierations to roll functions for

8 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 8 (Preprint, June 4, 00) which the angular velocity, angular acceleration, an angular jerk, namely r(, r(, an 3 r(, are all zero there. 3 In the roll acceleration of equation () the n orer zero at each en causes the angular jerk to be zero at each en, an this feature makes that roll acceleration suitable for use with the roll axis raise above the plane of the track. In Section 7 below we will look later at a situation in which it oes not appear practical to raise the roll axis above the plane of the track. In that situation we will look at a roll acceleration function that has only a 1 st orer zero at each en. The secon observation regaring the behavoir of the track curvature is for some transition shapes there are regions in which it will be greater than the curvature of the path of the roll axis. When that is the case it may appear that the balance between centripetal an gravitational force components unerlying equation (1) is not being realize. We therefore note that the call in equation (1) for balance base on the curvature of the path of the roll axis rather than on the curvature of the track is eliberate in relation to what is locate at the height to which the roll axis is raise when that is a vehicle center of gravity or the shouler of an typical seate passenger. 3. Constructing the roll acceleration for a Ben by overlapping two spirals. As note above, if between two segments of tangent track there is nee for a small turn, then in traitional practice the turn is accomplishe by placing two spirals back to back. A turn forme in this way has suboptimal ynamic characteristics, especially if the spirals in question are traitional linear spirals. If attention is focuse on the roll motion of a vehicle through the turn, then it makes sense to look for a single roll acceleration function that covers the whole turn. An alignment that provies a transition between two non parallel tangents an that is obtaine from a continuously varying roll acceleration function that is symmetric about its mi point is referre to herein as a Ben. We will look at two ifferent ways of forming roll acceleration functions for Ben. The first way takes the roll acceleration functions of two spirals like those just illustrate with one raising the curvature an the other lowering it, an positions them so that they partially overlap. As the constituent roll accelerations are type {,1} we will enote the combination as -{,1}. The two roll acceleration functions being combine will have opposite signs an will apply in ifferent ranges of istance along the track. We combine them with the help of the auxiliary function BOX(a,s,b) efine as 1 if a <= s <= b an 0 otherwise. A single roll acceleration function multiplie by BOX(a,s,b) will contribute only in the range of s values in which it applies. We move one roll acceleration function backwar a istance q from the center of the ben an move the other forwar from the center by the same amount. This will give a Ben with a length of (q + a). We write the formula for the sum of the roll accelerations as r/ = - BOX(-a - q, s, a - q) j (s + q) (s + q - a) (s + q + a) + BOX(-a + q, s, a + q) j (s - q) (s - q - a) (s - q + a) (9) The overall factor of j can be replace by an expression proportional to the maximum magnitue of the roll angle, which occurs in the mile of the Ben.

9 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 9 (Preprint, June 4, 00) Figures 4, 5, an 6 illustrate the shapes of the roll motions efine by the forgoing equation when a is set to 1.0, the maximum roll is set to 0.1 raians, an q is set successively to 0.1*a, 0.7*a, an 0.9*a Figure 4. Roll functions for -{,1} Ben with q = 0.1*a. (In this an following figures that show roll acceleration, velocity, an angle together the length of the shape is mae artificially small so that the three curves have comparable heights.)

10 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 10 (Preprint, June 4, 00) Figure 5. Roll functions for -{,1} Ben with q = 0.7*a Figure 6. Roll functions for -{,1} Ben with q = 0.9*a Note with respect to equation (1) above that in railroa practice the roll angle will not normally be more than about 0.1 raians (6 inches elevation relative to a gage of about 60 inche an that as a

11 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 11 (Preprint, June 4, 00) result the tangent of the roll angle will be nearly the same as the roll angle itself. Equation (1) thus inicates that the curvature at each point along the Ben will be approximately proportional to the roll angle at that point an hence that the total change of track bearing angle accomplishe by a Ben will be approximately proportional to the integral of roll angle over the length of the Ben. Comparing the above three figures it can be observe that the area uner the roll angle curve, an thus also the total turn angle of the Ben, ecreases as q ecreases. Note also that q cannot be lowere to 0, since with q = 0 the constituent roll accelerations woul cancel an there woul be no curvature at all. Illustrations of track shapes erive from roll functions are provie for some of the roll functions that are efine below. However, the basic iea that applies in all cases can be seen via comparison of the plots of roll angle in Figure an superelevation in Figure 3 with the plots of curvature an alignment in Figure 3. The roll function family escribe above coul be use in practice. There are some small unaesthetic inflections in the roll acceleration function in Figure 5 near s = 0, but they woul have little effect on the ynamic performance of a corresponing Ben. However, we will present another approach that appears more attractive. 4. Constructing the roll acceleration for a Ben by inserting a factor. In orer to fin another way to construct a roll acceleration function that will generate a Ben, we compare the simple spiral roll acceleration of Figure with the Ben roll acceleration of Figure 4. In Figure the roll acceleration crosses the s axis at s = 0 as a result of the presence of the factor s in equation (). Looking at Figure 4 we observe that the roll acceleration function for a Ben crosses the s-axis once to the left of s = 0 an again to the right of s = 0. We can cause an s-axis crossing at s = - f by inserting into equation () a factor of (s + f ), we can cause another s-axis crossing at s = f by inserting a factor of (s f ), an we can remove the crossing at s = 0 by ropping the factor of s. The resulting roll acceleration formula can be written as r _ angle( = j ( s + a) ( s a) ( s + f ) ( s f ) (10) where j is a multiplier to be etermine. This roll acceleration exten from s = - a to s = a an is eviently symmetric about its mi point. It therefore qualifies as the roll acceleration for a Ben. We label this roll acceleration function in accorance with the orers of its zeros as {,} where the first inicates that there is a secon orer zero at each en an the secon inicates that there are two first orer zeros in the interior. Applying the constraint that the roll velocity must return to zero at the en of the Ben at s = a we fin that it is necessary to have f = a / 7, an allowing a reefinition of j the formula for the roll acceleration becomes r _ angle( = j ( a s ) ( a 7s ) (11) The shapes of this roll acceleration function an of the corresponing roll velocity an roll angle functions are illustrate in Figure 7.

12 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 1 (Preprint, June 4, 00) Figure 7. Roll functions for {,} Ben. It may be observe that the roll functions of Figure 7 are similar in character to their counterparts in Figure 4. Figures 8 an 9 show an example of a Ben alignment that was obtaine by integrating equation (11) an that connects two ajacent sections of tangent track. In this example the angle of turn between the two tangents is 0.1 raians (= 5.73 egree, the maximum superelevation of the track is 0.1 raians (= about 6 inches superelevation), the balancing spee of the Ben is set as 90 mph, an the height of the roll axis above the track is set at 8 feet.. FIGURE 8. Alignment of a Ben connecting two tangents whose bearings iffer by 0.1 raians (= 5.73 eg.) with track bank angle of 0.1 raians (about 6 inches superelevation) at the mi point, with length such that the balance spee is 90 mph, an with the track roll axis 8 feet above the track. The y axis is the symmetry line of the two tangents an the x axis is the base line that passes through the point of intersection of extensions of the tangents.

13 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 13 (Preprint, June 4, 00) FIGURE 9. Track roll angle versus istance along the base line for the Ben of Figure 8. For a Ben with parameters like these it is practical to introuce three approximations that simplify the mathematics. First, the tangent function in equation (1) is replace by its argument (the bank angle in raian. Secon an thir, when integrating the sine an cosine of the track bearing angle, b_axis(, to obtain respectively the y an x coorinates of points on the path of the roll axis, the sine of the bearing angle is replace by the bearing angle itself an the cosine of the bearing angle is replace by unity. The effect of the secon an thir simplifications taken together is that b_axis( ceases to be the bearing angle an becomes in stea the tangent of the bearing angle. Therefore, when these simplifications are being applie b_axis( will be rename as bt_axis( as a reminer. Alignments obtaine base on these simplifications will iffer from corresponing alignments obtaine when the integrations are carrie out numerically on the conceptually correct integran. However, as long as roll angles an bearing angle changes o not excee about 0.1 raians the ifferences of shape will be small an will not have averse effects on the motions of vehicles traversing the Ben. The forgoing simplifications were use to obtain the alignment illustrate in Figure 8. The algebra for this simplifie application is as follows. The track roll angle as a function of istance obtaine by integrating equation (11) twice can be written as where k is a constant of convenience. ( a s ) 4 r _ angle( = k (1) As a result of the thir of the three approximations the integral for the x coorinate of a point on the path of the roll axis becomes trivial, an assuming the axes shown in Figure 8 we have the result x = s. This means that the parameter s no longer measures istance along the path of the roll axis an instea measures istance along the x axis. We therefore change the parameter in equation (1) from s to x. Continuing in the coorinate system illustrate in Figure 8, noting that the tangent of the bearing angle along the path of the roll axis will to be antisymmetric in x, we write g k x = g x bt _ axis( x) = t r _ angle( t) (13) v ( 315 a - 40 a x +378 a x -180 a x +35 x ) 315 v Pursuant to the secon of the simplifications, the offset of the path of the roll axis from the base line (i.e., along the y axis in Figure 8) is given by the integral

14 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 14 (Preprint, June 4, 00) -g k = ( x) t bt _ axis( t) = x a y _ axis (14) ( 193 a 315a s + 10a s - 16a s + 45 a s 7 s ) 630 v As alreay note, with the path of the roll axis efine by equation (14) the compass bearing along it is given by b _ axis( x) = arctan y _ axis( x) = arctan( bt _ axis( x) ) (15) x Returning to the formula for the isplacement of the Ben from the base line, it may be observe that expression (14) is zero at each en of the Ben. To obtain the y coorinates of points on the path of the roll axis relative to the base line through the intersection of the two tangents as shown in Figure 8 it is necessary to a the y imension from the base line to the points where the Ben meets the tangents, namely turn tan a (16) where turn enotes the compass bearing of the secon tangent relative to the first tangent. The track alignment is obtaine from the path of the roll axis by subtracting the overhang illustrate in Figure 1, namely ( r _ angle( x) ) o _ hang( x) = h sin (17) where h represents the height of the roll axis above the track. Thus with the axes shown in Figure 8, the formula for the y coorinate of a point on the track becomes turn y _ track( x) = y _ axis( x) + a tan h sin ( r _ angle( x) ) (18) The primary constraint is that turn angle of the Ben be equal to turn. In light of equation (15) the equation that expresses that constraint is turn bt _ axis( a) = tan (19) an solving that equation for the multiplier k we obtain k turn 315 tan v 18 a g = 9 (0) There are two seconary constraints that place lower limits on the value of the half length a. One is that the roll angle of the track not excee a maximum value enote max_roll. That constraint is expresse by the equation r _ angle(0) = max_ roll. Solving that equation for a provies a lower

15 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 15 (Preprint, June 4, 00) limit of turn 315 v tan a _ roll _ lim = (1) 18 g max_ roll The other seconary constraint is that the roll velocity along the track not excee a value corresponing to the maximum allowe value of the twist of the track. That constraint is expresse by the equation a r _ veloc = max_ r _ veloc () 7 where r_veloc(x) is the erivative of r_angle(x) with respect to x. Solving that equation for a the corresponing lower limit on the value of a is foun to be 1/ 4 turn 9 (308700) v tan a _ twist _ lim = (3) 98 g max_ r _ veloc The formulae for the minimum value of the half length, a, that follow from the seconary constraints on maximum roll angle an maximum roll velocity show simple epenence on the turn angle, turn, the balancing spee, v, an on the maximum roll angle or the maximum roll velocity. It is generally esirable to choose a value for a that is greater than both of the lower limits if the circumstances of the right of way so allow. To obtain the istance along the track as a function of the x coorinate along the base line it is necessary to carry out a numerical integration. In light of equations (3) an (15) the formula is = x 1 s _ track( x) z (4) 0 cos arctan( bt _ axis( z) ) arctan h sin( r _ angle( z) ) z Stepping back to compare the two forms of Ben at which we have looke, it can be note that the roll acceleration formula in equation (11) is simpler than equation (9) that governs Figures 4, 5, an 6 partly because it oes not have a parameter like the parameter q of equation (9) that provies an aitional egree of freeom for constructing Ben shapes. Comparing the roll acceleration function of Figure 7 with that of Figure 6 we can observe that the turn angle of a Ben erive from the roll acceleration function of Figure 7 coul be increase if we coul fin a convenient way to put a smooth ip in the magnitue of the roll acceleration near s = 0. We can o that by aing to equation (11) a factor of (1 + q s ) where q is an ajustable parameter. Again applying the constraint that the roll velocity shoul be zero at the en of the Ben an solving for the constant multiplier in terms of the maximum value of the roll angle we obtain the formula - 10 max_roll (a + (a - (a 4 q + 3 a (1 - q s ) - 1 s ) (q s + 1) r/ = (5) a 8 (a 4 q + a q + 45)

16 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 16 (Preprint, June 4, 00) Setting the parameter q to zero causes equation (5) to become equivalent to equation (11). Increasing the parameter q from zero prouces a family of Ben with increasing total turn angles. Figure 10 illustrates the roll motions obtaine with a =.0, with the maximum roll angle set to 0.1 raians, an with q set to 5.0. Figure 10. Roll functions for hybriize {,} Ben with q = 5.0. While formula (5) coul be use as the basis for a family of Ben shapes, it will be preferable in practice to use the more general approach that will be set forth in Section 9 below. 5. Constructing the roll acceleration for a Jog by overlapping three spirals The term Jog as use herein refers to an alignment shape that begins tangent to one straight line, that moves smoothly away from that line towar a secon straight line that is parallel to but not collinear with the first one, that en tangent to the secon line, an that is antisymmetric about its mi point. A crossover between ajacent parallel straight tracks provies an example of what a Jog looks like. Recall that the roll acceleration for the -{,1} Ben was forme by aing the roll acceleration of a spiral to the roll acceleration of another spiral with partial overlap. Analogously, by aing the roll acceleration of a {,1} spiral to that of a -{,1} Ben with the same partial overlap we obtain the roll acceleration of a 3-{,1} Jog. The corresponing formula is r/ = - BOX(-a - q, s, a - q) j (s + q) (s + q - a) (s + q + a) + BOX(-a, s, a ) j (s ) (s - a) (s + a) - BOX(-a + q, s, a + q) j (s - q) (s - q - a) (s - q + a) (6)

17 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 17 (Preprint, June 4, 00) Because of the complexity of this formula it was not possible with the computing resources available uring preparation of the present article to obtain a general formula for the values of s at which the magnitue of the roll associate with equation (6) has its maximum values. However, that woul not prevent use of the formula in esign work. A sense of what the roll functions look like can be gleane from Figures 11 an 1 in which the functions are evaluate for a =, j = 0.1, an q = 1.4 (65% overlap) an. (45% overlap) respectively. Figure 11. Roll functions for 3-{,1} Jog with 65% overlap.

18 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 18 (Preprint, June 4, 00) Figure 1. Roll functions for 3-{,1} Jog with 45% overlap. For overlap values below 45% or above 65% the roll acceleration curves become less smooth an hence presumably less esirable for esign of Jogs for railroa track. Even with the parameters of Figures 11 an 1 that are usable, one can observe unaesthetic inflections in the acceleration curves. The combination of that lack of smoothness an the mathematical complexity of the algebra of equations like equation (6) make Jogs of the above type unappealing in comparison to those that are presente in the following Sections. 6. Constructing the roll acceleration for a Jog by inserting a factor. We now look at a secon way to form the roll acceleration function for a Jog. Just as each of the Ben formulae has one more root (or s-axis crossing) than the corresponing Spiral formula, so, each of the three curves (roll acceleration, roll velocity, an roll angle) efining the roll motion of a Jog nee to have one more root than the corresponing curve for a Ben. Starting from equation (10) which was the initial equation for a Ben an which is symmetric about the mi point, we can both a another root an make the resulting function antisymmetric by inserting a factor of s so that the ae root is at the mi point of the function. The result is r _ angle( = j ( s + a) ( s + f ) s ( s f ) ( s a) (7) where f is between 0 an a. Integrating that equation twice to obtain the formula for the change in roll angle over the length of the Jog an requiring that the change in roll angle be zero at the en of the Jog, we fin that we must set f = a. If we then solve for the constant multiplier in terms of 3 the maximum roll angle in the Jog, max_roll, the formula for the multiplicative Jog becomes

19 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 19 (Preprint, June 4, 00) max_roll s ( a s ) ( a 3 s ) r _ angle ( = (8) 9 51a Figure 13 illustrates the shapes of the roll functions for the multiplicative Jog. Figure 13. Symmetric (i.e., antisymmetric) Jog roll functions via 5 factor roll acceleration. The overall turn angle of the antisymmetric Jog can be seen to be zero because the curve for the roll angle is antisymmetric about the mi point of the Jog. 7. Jogs as Shapes for Turnouts an Crossovers This Section configures a jog intene to serve as the alignment for a crossover between two parallel tangent (i.e., straight) tracks. As the track bearing angles relative to the two tangents an the track roll angles are small, it is reasonable to use the simplifie treatment set forth in Section 4 above. The formula for the roll angle of the symmetric Jog as a function of length along the track is obtaine by integrating equation (8) twice an can be written in the form ( a s ) 4 r _ angle( = k s (9) where k is a constant of convenience. The corresponing form of equation (8) for the roll acceleration is

20 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 0 (Preprint, June 4, 00) ( a s ) ( 3 s a ) r _ accel( = 4 k s (30) The minus sign is inserte so that for k positive, at the left en of the Jog where s < 0, the initial bank an turn will be positive which is interprete as being to the left. Applying the first small angle approximation, the integral for the tangent of the bearing angle along the path of the roll axis versus istance, x, along the tangents becomes 5 g x g k ( ) ( ) ( a - x ) x = t r _ angle t = bt _ axis (31) a v 10 v Applying the secon small angle approximation, the integral for the coorinate of a point on the roll axis along a y axis normal to the two tangents an with zero value mi way between them is ( x) = x t bt _ axis( t) y _ axis (3) 0 = g k x ( 693 a a 8 x a x 6930 v a 4 x a x 8 63 x 10 ) In this situation the primary constraint is that the lateral isplacement over the length of the Jog, enote jog_ist, shoul equal the specifie istance between the centerlines of the two parallel tangent tracks. With the roll angle zero at each en of the Jog there is no overhang at either en, an this constraint takes the form Solving that constraint for k we fin jog _ ist y _ axis( a) = (33) k 3465 jog _ ist v = (34) g a The seconary constraints are that the roll angle an twist of the track shoul nowhere excee the respective limits chosen for those two properties. (The roll velocity etermines the track twist.) The maximum value of roll angle occurs at s = -a/3 an s = a/3 so that this constraint is expresse as r _ angle a / 3 = max_ (35) ( ) roll an the value of shape half length such that the maximum magnitue of the roll angle is max_roll is foun to be given by

21 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 1 (Preprint, June 4, 00) jog _ ist v a _ roll _ lim = (36) 81 g max_ roll The magnitue of the roll velocity has its maximum value at s = 0 an the value of the shape half length such that the magnitue of the roll velocity there equals max_r_veloc is foun to be given by 1/ 3 1/ 3 / jog _ ist v a _ twist _ lim = (37) 1/ 3 1/ 3 8 g max_ r _ veloc Both of the above expressions for the half length are evaluate, an the larger value is use so that both of the seconary constraints are satisfie. Both formulae for the Jog s half length show that the length of the Jog will epen in a simple way on the balancing spee, v, the Jog s lateral isplacement, jog_ist, an either the maximum bank angle, max_roll, or the maximum roll velocity, max_r_veloc. Since we are looking here at crossovers we nee to take account of the fact that physical crossovers begin an en with physical track switches. Physical track an guie way switch esign is an immense fiel in which many concepts for balancing cost an performance have been evelope. What is note here is that costs of constructing an maintaining a switch are increase when there is an increase in the length of the assembly that must move when the setting of the switch is change. That length increases when there is an increase in the track length over which geometry prevents a guie rail or rails from simultaneously being in the working location for the both routes. This means that it is esirable to arrange to have the initial lateral separation of the iverging path from the through path evelop as rapily as vehicle ynamics will allow. We have observe that raising the height of the vehicle roll axis above the track is always ynamically beneficial. However, raising the vehicle roll axis also causes an increase in the istance from the start of a Jog to the point at which the Jog reaches a given lateral isplacement towar the final tangent. Therefore, in contrast to the application of the Ben in Section 4 above, for this application of a Jog as a crossover the roll axis is not raise an the path of the track is given by equation (3) itself. We thereby make some sacrifice of ynamic performance in orer to reuce the cost of the crossover. Figure 14 illustrates such a crossover for the conitions that the two ajacent sections of tangent track have a centerline separation of 0 feet, that the maximum superelevation in the crossover is 0.05 raians (about 3 inches superelevation), an that the balance spee of the crossover is 90 mph. The crossover exten for 781 feet in each irection from the center of symmetry. Figure 16 shows the track roll angle profile corresponing to Figure 14 as given by equation (30). (Formulae for Figures 15 an 17 are presente following Figure 17.)

22 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. (Preprint, June 4, 00) Figure 14. A Jog constructe from the roll acceleration of equation (30) which has a n orer zero at each en. This Jog is configure as a crossover between tangent tracks with centerline spacing of 0 ft, with maximum roll angle of 0.05 raians (corresponing to maximum superelevation of about 3 inche, an with imensions chosen so that the balancing spee is 90 mph. The total length of the crossover measure along the tangents is 1,56 ft. (Note that in North American practice crossovers are not usually esigne for high spee operation an o not usually inclue superelevation.) FIGURE 15. A Jog constructe from the roll acceleration of equation (30) with a 1 st orer zero at each en. Other esign parameters are the same as those for Figure 14. Because the roll angle buil more quickly at each en than is the case when the roll acceleration has n orer zeros at the en, this crossover is shorter than the one in Figure 14. The total length of this crossover measure along the tangents is 1,44 ft. FIGURE 16. Track roll angle for Jog of Figure 14.

23 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 3 (Preprint, June 4, 00) FIGURE 17. Track roll angle for Jog of Figure 15. We look next at the rationale of Figures 15 an 17. As note at the en of Section above, when the roll axis is not raise above the plane of the track it is reasonable to look at roll acceleration functions that have 1 st rather than n orer zeros at the en. Starting with the counterpart of equation (7) but with 1 st orer rather than n orer zeros at s = - a an s = a an repeating the sequence of steps that lea from equation (7) to equation (8), one fin that the counterpart of equation (8) with a 1 st orer zero at each en is ( a + ( a ( 3a 7 s ) max_ roll r _ angle( = (38) 7 36 a The formula for the roll angle of the symmetric Jog obtaine by integrating equation (38) twice can be written in the form where k is another constant of convenience. ( a s ) 3 r _ angle( = k s (39) Applying the simplifie treatment as in the previous case the integral for the tangent of the bearing angle versus istance, x, along the tangents becomes g x g k ( ) ( ) ( a - x ) x = t r _ angle t = bt _ axis (40) a v 8v Applying the secon small angle approximation, the integral for the istance along a y axis normal to the two tangents is 4 = ( x) = x t bt _ axis( t) y _ axis (41) ( 315 a 40 a x a x 180 a x 35 x ) g k x + 50v The primary an seconary constraints an the manner in which they are use to etermine k an a are the same as in the previous case. The solution for k is 315 jog _ ist v k = (4) 9 3 a g

24 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 4 (Preprint, June 4, 00) The lower limit for a such that the max_roll constraint is satisfie is ( 77175) 1/ 4 9 jog _ ist v a _ roll _ lim = (43) 98 g max_ roll an the lower limit for a such that the max_r_veloc constraint is satisfie is a _ twist _ lim 1/ 3 1/ 3 / jog _ ist v = (44) 1/ 3 1/ 3 4 g max_ r _ veloc The constraints on the Jog s half length show the same epenence as before on the balancing spee, the Jog s lateral isplacement, an on the maximum bank angle or maximum roll velocity but with ifferent constant factors. Figure 15 illustrates a crossover base on the above formulae for the conitions that the two ajacent sections of tangent track have a centerline separation of 0 feet, that the maximum superelevation in the crossover is 0.05 raians (about 3 inches superelevation), an that the balance spee of the crossover is 90 mph. The crossover exten for 71 feet in each irection from the center of symmetry. Figure 17 shows the track roll angle profile corresponing to Figure 15 as given by equation (39). In contemporary North American railroa practice turn-outs from tangent tracks o not incorporate superelevation an therefore o not have efine balancing spee. Construction of a switch that incorporate superelevation as prescribe by the formulae of this Section woul require progressive lowering of the rail seats of the low rail of the iverging route an woul require a novel machining of the point for the through route. The points woul also be longer than the points of conventional railroa switches. 8. Constructing the roll acceleration for a Wiggle by inserting a factor The term Wiggle as use herein refers to an alignment shape that begins tangent to some straight line, that makes a smooth lateral excursion away from an then back towar that straight line, that en again tangent to that straight line, an that is symmetric about its mi point. (We will see in Section 9 below that asymmetry can be accommoate by aition of higher orer shapes.) As note in the introuction, a Wiggle can be an effective way for an otherwise straight alignment to circumvent a local obstruction that intrues from one sie. We construct a roll acceleration formula that can exhibit the features of a Wiggle by aing another linear factor to the Jog roll acceleration formula in equation (7). The initial formula is r _ angle( = j ( s + a) ( s + p) ( s + f )( s q)( s i)( s a) (45) The parameters p an i are eliminate by applying the constraints that the roll velocity an roll angle both return to zero at s = a. The asymmetry of the resulting roll acceleration polynomial is controlle by (f q). It is therefore convenient to replace f an q by the new variables b = (f +

25 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 5 (Preprint, June 4, 00) q)/ an c = (f q)/. We want these shapes to be symmetric so that c is set to zero an rops out. This makes the roll acceleration a polynomial in s that epen on j, a, an b. The following figures illustrate the shape of the roll angle function for two values of the parameter b. Figure 18. Symmetric roll functions via 6 factor roll acceleration with b = 1.0.

26 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 6 (Preprint, June 4, 00) Figure 19. Symmetric roll functions via 6 factor roll acceleration with b = 1.1. Looking in Figure 18 at the areas between the roll angle curve an the istance axis one can see that the amounts of curvature to the right an to the left are approximately equal so that the total angle of turn over the length of the shape will approximate the esire value of zero. By way of contrast, the roll angle curve of Figure 19 is biase to one sie so that the resulting shape will look much like a Ben an not much like a Wiggle. The roll angle function corresponing to equation (45) is a 10 th orer polynomial. To obtain a close form algebraic expression for the constraint that the compass bearing of the Wiggle be the same at the en as at the beginning woul require putting that 10 th orer polynomial roll angle function into equation (1) an then obtaining the compass bearing angle as the integral of equation (1) in close form. As that is consiere impossible we will provie an illustration using the simplifie metho escribe in Section 4 above. The formula for the tangent of the bearing angle on the path of the roll axis is then available in close form. Imposing the requirement that the tangent of the bearing angle be the same at the en as at the beginning fixes the value of b, an in the context of the simplifie treatment the equation for the roll acceleration of a Wiggle (with j reefine) becomes 4 4 r _ angle( = j ( a s ) ( a 18a s + 33s ) (46) What comes next is an example of application of a Wiggle to avoi a single obstacle in an otherwise straight section of track. The illustration makes use of the mathematical simplifications that were explaine in Section 4 above. The roll angle function corresponing to equation (46) can be written as 4 r _ angle( = k ( a s ) (11 s a ) (47) where k is a constant of convenience. Integrating the simplifie version of equation (1) yiel 5 g k x ( a x ) bt _ axis ( x) = (48) v an evaluating the simplifie form of the integral for y_axis yiel 6 g k ( a x ) y _ axis ( x) = (49) 1 v The y coorinate along the path of the track is ( arctan( bt _ axis ( x) )) o _ hang ( x) y _ track ( x) = y _ axis ( x) cos (51) where o_hang(x) is the overhang as escribe previously. The primary constraint is that the lateral excursion from the general tangent have a specifie value that we enote by swing_ist an takes the form

27 Ben, Jogs, An Wiggles for Railroa Tracks an Vehicle Guie Ways, Louis T Klauer Jr. 7 (Preprint, June 4, 00) y _ track (0) = swing _ ist (5) It is evient from equation (48) that bt_axis(0) = 0. Therefore when equation (51) is use in equation (5) the cosine factor is unity an can be roppe. To get the constraint into a form that can be solve algebraically we can introuce another approximation that is in keeping with the simplifie treatment. Namely, in the equation for the overhang we replace the sine of the roll angle by the roll angle itself an write We can then solve for k an fin o _ hang ( x) = h r _ angle( x) (50) k 1 swing _ ist v = (53) 10 a ( a g + 1 h v ) There are two seconary constraints. One is that the maximum roll angle that occurs at s = 0 not excee a specifie value we enote as max_roll. The lower limit for a obtaine by solving that constraint is 3 v ( h max_ roll + swing _ ist) a _ roll _ lim = (54) g max_ roll The other seconary constrain is that the maximum roll velocity (which correspon to maximum allowe track twist) which occurs at a not excee a value we enote as max_r_veloc. The lower limit for a obtaine by solving that constraint is h theta a _ twist _ lim = 4 1 v sin (55) g 3 where ( h g) swing _ ist theta = arcsin (56) h max_ r _ veloc v The values use for the example illustrate in Figures 0 an 1 are: swing istance = 0.0 feet; balancing spee = 90 mph; roll axis height = 8.0 feet; maximum roll velocity corresponing to a maximum change of cross level in 6 feet = 1. inches; an the acceleration of gravity = 3.17 feet/secon_square. With the selecte parameters the roll angle oes not get as large as the typical limit of 0.1 raians. While the alignment given by this simplifie construction is not ientical to the alignment that woul be obtaine if all of the trigonometric functions of the metho were fully evaluate, its utility an ynamic characteristics will be just as goo as those of a corresponing