CA2155914C - Train tire profile - Google Patents
Train tire profile Download PDFInfo
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- CA2155914C CA2155914C CA002155914A CA2155914A CA2155914C CA 2155914 C CA2155914 C CA 2155914C CA 002155914 A CA002155914 A CA 002155914A CA 2155914 A CA2155914 A CA 2155914A CA 2155914 C CA2155914 C CA 2155914C
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Classifications
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B60—VEHICLES IN GENERAL
- B60B—VEHICLE WHEELS; CASTORS; AXLES FOR WHEELS OR CASTORS; INCREASING WHEEL ADHESION
- B60B3/00—Disc wheels, i.e. wheels with load-supporting disc body
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B60—VEHICLES IN GENERAL
- B60B—VEHICLE WHEELS; CASTORS; AXLES FOR WHEELS OR CASTORS; INCREASING WHEEL ADHESION
- B60B17/00—Wheels characterised by rail-engaging elements
Abstract
A section of a train wheel tread is formed into a continuous hyperbolic curve to respond asymptotically to changes in the path of a rail, thus,dampening lateral oscillation of the train wheels back and forth across rail. The hyperbolic curve at a minimum extends from a taping line to a wheel lift point. However, for further dampening effects the hyperbolic curve is extended from a fillet section across the taping line toward the field side of the wheel. Two tires having the same hyperbolic profile work in conjunction as a feedback system to dampen lateral oscillation while at the same time saving energy, reducing tire and rail wear.
Description
~
2~~59~4 TRAIN TIRE PROFILE
BACKGROUND 0P' THE INVENTION
This invention relates generally to trains and train wheels and more particularly to a novel tire profile that improves train stability and energy efFiciency.
Train tire profile includes several sections. A flange section protrudes downward from the aide of the train wheel and extends over the lateral side of a train track. A fillet extends upward along a field side of the flange providing transition to a straight conical wheel tread section. The wheel tread sectien serves as the major load bearing surface that supports the train wheels on a txain track. The art uses tread profile of two opposiag tires each on one of two rails to steer.
Two opposing tires are a wheelset, The flange provides steering when 16 rail curve exceeds capability of treads to steer without flange contact.
Two main factors must be considered when designing tire profiles for use with railed devices. The first is the dynamic stability of the vehicle at various speeds throughout its operating speed range. When in transit, a train experiences lateral oscillations known as "hunting".
Wheel hunting results in the wheels oscillating laterally back and forth between the wheel flanges. The maximum speed or critical speed of the train is determined by the onset of unstable, undesirable wheelset hunting. For eyample, if the train saes too fast, the force of the lateral oscillations wffl overcome the flange barrier and cause the train to 26 derail. Hunting is caused by the dynamics between the wheel tread profile and the rail. Increasing the slope of the wheel tread too fast toward flange increases forces causing hunting and, therefore, lowers the critical speed of the vehicle. Decreasing slope of wheel txead toward flange decreases steering farces, also lowering the critical ~
2~~a~~.4 hunting speed. This ie the measure of mismatch, These limits define critical mismatch threshold.
A second factor involved with train stability is the ability of the vehicle to negotiate track curves. This curving ability is determined d primarily by the ability of the opposing tires of wheelsets to follow the track curves. Optimally, the wheelsets should perform a purely rolling motion in the track curves without any contact between the wheel flanges and the rails. This requires steering forces to be generated by the sloped wheel tread independently of the wheel flange permitting the wheelset to yaw or rotate about a vertical axis which may be through it~ center. Oscillation of steering forces happen around vertical axis through its center of gravity(mass). This oscillation is a metric space. The oscillation of wheelset results in hooting. The steering forces move the train wheelsets into a more xadial 16 position(axle 82 FlCr.1'l) with respect to the track curves, thus, increasing train stability around curves.
A wheelset includes two opposite wheels that may be joined together by an sale. With a conical (straight taper) wheel tread the conicity remains virtually constant with lateral deflection. of a wheelset 2b relative to the track. That is, straight taper wheel treads have a constant elope. In other words, the conicity of each wheel remains the same irrespective of whether the wheelset rune centrally on the track or is deflected closer to one rail. Increasing the conicity of the wheel tread improves the steering ability of the wheelsets because of the 25 irxcreased steering force. However, increased conicity also increases the oaeillation of the wheelset. Oscillation of wheelset results in hunting. Therefore, with regard to the conicity of wheel treads, there is a conflict between the requirement for hunting stability and increased vehicle speed and for good curving ability of the wheelsets.
U.S. Patent No. 4,294,482 to Scheffel et al., discusses a profiled wheel tread that is made up of a combination of discrete circle and line segments. Ia tha art the term "profiled" is used in relation to wheels having a curved tread section and distinguishes such wheels from conventional wheels having a straight conical (linear sloped) tread section. The profiled tread in Scheffel et. al., utilizes multiple discrete curve sections each having a separate radius that are combined to fornx a non-continuous curve. The term conicity is imprecise. 'fhe curve radii of the profile increase from taping line to the flange. mhis is thought to reduce conicity of the profile. It reduces conicity of profile when compared to profile of constant radius between taping line and fillet. Thus, the tire tread conicity has essentially a "droop"characteristic with increasing lateral deflection toward the flange compared to tread of constant radius. A train profile with 16 multiple curve discontinuities and a "drooping" characteristic will initiate vibration between the wheelset and the train. Because the wheel tread also has relatively high conicity of tread slope change at the taping line, the tire is also more likely to hunt in relation to a straight taper. The minimum radius is at the taping line with larger radii toward fillet.
U.S. Patent No. 5,295,824 to Ziethen et al. discusses means for varying the train rail profile to extend wheel wear and track durability. Ziethen et al., however, does not suggest means for reducing vibration and hunting in train treads. The definition of concern ig that radius of tire tread is larger than radius of contacted rail. Also definition of concern is that there would not be two point contact between tire profile and rail. In FIG. 1 of Ziethen the lower continuous line(10) looks like it goes horizontal at coordinate origin Eline 57 col 8j thus indicating a discrete radius of tread in accordance SO with that invention to the extent shown in their figure. This meets 2~~~9~4 their requirement that tread radius is larger than contacted rail radius. Therefore, in a manner similar to Scheil~el, FxG. 1 in Ziethen suggests using discrete change of tread section. Therefore, the Ziethen tread profile begins with a linear wheel tread and then !~ abruptly changes to a radius in tread toward the fillet. The abrupt change in curvature in the tread section induces vibration and increases wheel resistance when the wheelset is laterally displaced on standard rail, This is not as absorptive barrier. Thus, the profile in Zisthen is not energy efficient and further does not provide additional steering forces different than prior art.
U.S. Patent No. 1,298,628 to Coda describes a wheel profile including two separate sections that intersect at the taping line. The profile is a wheel tread having a coniCal(linear) outer portion tangent to a portion of continually increasing conicity to the wheel flange. U.S.
lg Patent No. 1,788,705 to Emerson shows a substantially linear tread profile that is profiled to match the curvature of a corresponding inner munded corner of railhead. U.S. Patent No. 99,850 to Vial, discusses a wheel having both a concave throat section and a separate convex tread seetion'toward the fillet. Coda, Emerson and Vial, similar to the . other patents previously listed, have either straight conical tread profiles or discontinuous tread that include a combination of different sections each having a different curvature formula. Tread profiles with elliptical or circular profiles include a harmonic frequency that causes periodic oscillation under certain track conditions. Thus, the 2!i problems of vibration, limited steering forces and hunting still exist.
Another problem with train tires is excessive wear both at the tread and flange. Far example, linear tread profiles typically exhibit excessive wear next to the flange. Contrary to normal expectations, an -increase in the contact area between the wheel and the rail does not 80 necessarily decrease tread wear. Research has now shown that the rate of wear of the tread, in fact, increases when the shape of the tread approaches the shape of the rail head. This is because the wheel-tread /rail-head contact area increases to such an extent that there is a disadvantageous increase of the creep forces or slip of the wheel on the rail. The increased creep forces significantly increase the wear rats offsetting reductien in wear rate due to increased contact area. Thus, it is predicted that tread profiles that match the railhead, such as Ziethen, have a tendency to wear faster according to Schefl'el.
Many standard train tire treads have linear profiles but, due to wear, rapidly become nonlinear. This small amount of nonlinear wear causes hunting as described by John F. Leary in paper entitled America Adopts Worn Wheel Profiles in Railway Gazette International, 3uly 1990. Additional information is provided by John F. Leary, Stephen N. Handle arid Britto Rsjkumar in a paper entitled:
1b "Development of Freight Car Wheel Profiles - A Case Study", in Wear, 144 (I991) pager 363-862, Association of American Railroads, Pueblo, Colorado. The, nonlinear circle segments used for defining the tire profiles in both Scheffel and Ziethen also suggest prof"~ling the tire tread into nonlinear shapes. Thus, present nonlinear profiled tire treads have increased steering forces that induce hunting.
Accordingly, a need remains for a train tire profile that is both resistant to hunting and vibration, provides steering forces during turns and is energy efficient.
It is, therefore, an object of the invention to increase the stability of trains while traveling along rails.
Another object of the invention is to the increase energy efficiency of trains while at the same time increasing the maximum critical speed 90 that the trains can safely travel oa a rail.
A further object of the invention is to increase the standardization of profiled tire tread designs that have both superior curving ability and that are also resistant to hunting.
The present invention takes into account feedback theory to S develop a tread profile that is resistant to hunting, provides added steering forces through curved rail and is energy efficient. Rail technology does not presently use feedback theory to effectively model and control the interaction between train tires and rails.
Correspondingly, present wheel profiles do not maximize train performance. Feedback theory and how it is used in control systems is discussed by Sandeep in article entitled "Deterministic Controllers for a Class of Mismatched Systems", Journal of Dynamic Systems, Measurement and Control; March 1994 Vol. 116 pp 17. The important concepts are measure of mismatch and critical mismatch threshold.
Elements of control system theory are discussed in many different texts under different labels. Ferenc Szidarovsky and A. Terry Bahill discuses elements of Lyapunov stability theory in book Linear Systems Theory and is herein incorporated by reference. The feedback system must be a continuous mathematical function. There must be a unique global minimum at average position within feedback system.
And finally for any state trajectory the function of that perturbed trajectory must be decreasing with time. These design criteria are taken into account when designing the tread profile of the present invention to control energy stored in the train tire. A tire profile is correspondingly designed essentially as a feedback system operating between two opposite tires in a train wheelset. The result is a tread profile that increases train wheel performance by reducing hunting and vibration while at the same time increasing steering capacity and energy efficiency.
The wheel profile effectively incorporates an absorptive barrier LCxelbauml that redistributes wear away from the flanges. The absorptive barrier provides an asymmetric response iu relation to different lateral displacements of the wheelaet that effectively maintains the wheelset at a centered position. The tread profile exhibits slight resistive forces when the wheel moves laterally across the rail in locations relatively close to a center wheel position.
However, wheel pair exert significantly increased resistive forces when the wheels are moved across the rail to location significantly far from the centered wheel position. Thus, when the wheelset is off center, a net force is exerted on the train truck wheelset in an asymptotical manner in the direction of the centered position.
Specifically, the invention incorporates a hyperbolidasymptotic tread profile on the tread section of each wheel in a wheelset. The rb profile decreases asymptotically in amplitude as it extends further from the flange. The asymptotic profile effectively dampens lateral vibrations in the wheelset in a manner similar to an asymptotically decreasing response in an electronic feedback system. The profile of the wheel exhibits a "continuous" asymptotic curve over substantially the entire tread section of the wheel to reduce vibration and ensure a proper wheel/rail feedback response. The asymptotic profiles from each wheel in the wheelset, when combined, farm a catenary that maintains the wheelset in a centered position. A load supported by the wheelset will exhibit a local minimal when the wheelset is in a 26 centered position, Thus, vertical gravitational forces tend to maintain the centered position. The hyperbolic profile provides a phase lag in the wheelset when navigating rail curves, Thus, the tire treads of opposite tires are an absorptive barrier that efFectively absorb and dampen lateral oscillations. Lateral oscillations increase tire wear and use SO energy to wear tires thus reducing energy efficiency. The hyperbolic profile is incorporated into either just the tread section of the wheel or extended in a continuous curve from the tread section up into the fillet section. Thus, extreme off center displacements of the wheelset still reside within the asymptotical feedback system of the wheel profile.
The asymptotic tire profile is derived using a matrii algebraic relationship between various selected points on the wheel to generate a reliable and consistent profiled shape. The selection of certain reference points on the tread ensures smooth transitions between different wheel sections, in turn redudng vibration and ensuring proper wheel response in relation to varying rail radii.
The foregoing and other objects, features and advantages of the invention will become more readily apparent from the following detailed description of a preferred embodiment of the invention which proceeds with reference to the accompanying drawings.
Rg , D ~Q ,R1PTION OF err'. T_~RAWIN :g FIG. 1 is a schematic diagram representing a cross section of s prior art linear tread profile for a wheelset.
FIG. 2 shows a schematic diagram of a tread profile with two linear slopes.
FIG. 3 shows a schematic diagram of a wheel prefile according to the invention.
FIG. 4 is a schematic top view of the standard wheelget in FIG. 1 shown on rail curve.
FIG. 8 is an expanded cross section of the asymptotic tread profile shown in FIG. 3.
FIG. 6 is an expanded cross section of the profile spawn in FIG. 6 with additional reference points.
FIG. 7 is part of a prior art a combination of a linear and circular profile taken as an expanded view in the same relative positions of 80 points 44, 46 and 47 is FIG. S.
FIG. 8 is a schematic showing a prior art worn linear tread profile.
FIG. 9 shows the lift created by different tread profiles of opposing treads on a load.
FIG. 10 is a schematic showing a centered wheelset with linear tread profiles.
FIG. 11 is the wheelset of FIG. 10 shown in off center position.
FIG. 12 is a schematic showing the circular displacement of a load due to the linear/circular profile shown n FIG. 7.
FIG. 13 is a schematic showing the force vectors exerted on a hyperbolic profile by a rail.
FIG. 14 is a graph showing a relationship between the stability of different tire profiles according to curve periodicity.
FIG. 15 is a graph showing the steering forces of several tire profiles.
FIG. 16 shows boundary limits of critical mismatch threshold where tread of hyperbolic form is an absorption boundary.
FIG. 17 is a schematic showing the phase angles for a wheelset traveling through a curved track section.
DETAILED DESCRIPTION
THEORY
The logical and mathematical basis for the present invention is based on classical mathematical and physical relationships that are known to those skilled in the art, and therefore, are not described in detail. The following is a list of publications discussing these physical and mathematical relationships:
1. Rorres, C. and Anton, H., Applications of Linear Algebra, (1984), pages 4-6;
2. Friedberg, S. and Insel, A, Introduetio to Linear A1_gebra wi . y ~~~ 1' a .ions, (18$6), pages 31, 32, 4$, 49, 226~254;
8. Shenk, A.,. ~'!alCUlu6 nd "alvtic C~~ometrv, (197?), page 418 subject LORAN;
b 4. Stone, D. and Leery J., Rai wav Aae, Feb. (1992), "Reinventing The Wheel' ;
5. McLachlan, P. Eng., 1?ail Profiles Can make A Difference, a paper presented to: American Public Transit Assoaation (1992) Rapid Transit Conference, Los Angeles, Cal, June 16, 1892;
6. Hay, William W., $ailro~d En;ineering, Vol 1, pages 7. Thomas and Finney, ale plus and nahi~tie Cx_~a~netrv. 6th Ed, (1978), pages 494-498, '7th Ed, pages 559-60; subject is effect of curvature an periodicity per Huygen and his clock;
$. Elsgalts, L., Tlifferential E~mations and Calcmlus of Mir Publishers, Moscow, pages 244-247, subject is stability under constantly operating perturbations;
9. Mayr, Otto, The Ori.;ins of F .dba .k .ontrel, {1970), history;
10. Kellogg, Dimon Oliver, FeLndations of Potential, (1953);
11. Leery, John F., Elkins, John A,, Comvuter Solutions to WheeL~tai1 Interaetiq~n Problems, presented at APTA June 13-17, (1992), flange pressure and other subjects;
12. Murphy, Gordon John, Rasie Automatic Control Theorv , (1957), pages 1~16, so, 31;
18. Pitman, R.J.G., ALtnmatic Coil .rol 9vstams ~xnlain , (1968), good basic mechanical explanation of feedback theory particularly chapter "Inertia and Friction in Control Systems' ;
14. Ditto, William L. and Pecora, Louis M., ~s~iSnt3Sc 80 ~, August (1993), Vol 289, Number 2, "Mastering Chaas';
IO
15. Txaub, Joseph >i'., Wozniakowski, Henryk, Scientific AmeriCari, January (1994), "Breaking Intractability' ;
18. Schroeder, M. R., Fro .alashaos, power T_.aws: Minutes From an Infinite Universe. (1991);
6 17. Dorf, Richard C.,1!4odern_Cnn .rot 4vs .ms. (1974) pages 129-131, concept of stability;
18. Magid, Andy R., ~Rplied Matrix Models a second coLrse in linear Algebra with ~!omp ~ . r Ap, li a ions; (1985), a method of handling over determined systems;
19. Law, E. H, ~'ransactione of the ASME J~wr_n_al of >i n n ri for Industry, November, (1974) pages 1168-74, "Nonlinear Wheelset Dynamic Response to Random Lateral Rail Irregularities' ;
20. Hedrick, J. Karl, Arslan, A.V., ~,n ~r al o ~ .,..,ami ,'~ s gyms g;a g ~ men . and ContraL Sept, (1979}, 1~ Vol 101, pages 230-7"N'onlinear Analysis of Rail Vehicle Forced Lateral Response and Stability' ;
21. Friee, R.H., Cooperrider, N.K., Law, E.H., Tr nsa . .'ens of the ASME loyal of D ami ya ms, easL?re~, and Control, Sept, (1881), Vol 105, pages 201-210; " Experimental Investigation of Freight Car Lateral Dynamics' ;
22. Fries, R.H., Cooperrider, N.K., Law, E.~T., Transactions of ~h_e A4ME Journal of Dy amic ~, s pms.Measurement'nds',,9~n .rfll~
December, (1978), Vol 100, pages 238-2b1.
FIG. 1 is a schematic diagram representing a cross section of a 2~ prier art linear tread prol3le for a wheelset 18. The tread profile for both wheels 18 and 14 comprise a linear conical tread 17 with a dowawardly directly flange 19. The wheels 18 and 14 are joined together by an axle 13 that is corresponding connected to train truck (sot shown) to support a load. Wheelset 18 is shown in a centered position which is defined as having the flange of wheels 18 and 14 substantially equal distances 26 from rails 22 and 20, respectively. In the centered position, wheels 18 and 14 make contact with rails 22 and 20, respectively at points 24 which are defined as taping lines. A
distance 12 exists between taping lines of wheels 18 and 14. Taping line may be contact area of centered travel. Taping line has become defined measuring location rather than center of travel. The hereina$er-described field side 53 can also be used as a means for aligning the template.
FIG. 4 is a schematic showing a top view of the standard linear IO wheelset 16 in FIG. I. Referring to both FIGS. 1 and 4, the distance between the taping lines of wheels 14 and 18 in a typical embodiment is approximately 1500 millimeters. The flanges 19 for a single wheelset 16 will contact either rail 20 or 22 when the radius R2 of inside rail 20 is approximately 1074 feet and the radius Rl of outside rail 22 is approximately 1079 feet This is with 28 inch tires.
When the flanges 19 make contact with the rail, resistance between the flange 19 and the rails 20 and 22 reduce train energy efficiency. More specifically, the flange 19 has approximately a seventy degree inside slope that acts as a wedge when traveling through rails 20 and 22. The wedging action creates wear near the flange 19 that both reduces tire life. The wedging action increases wheel resistance. Wedging action exits in rail curves.
The limit of lateral movement of the wheelset 16 is set by the gauge tolerance of the wheels. The gauge tolerance is defined as the distance between the rail and a given location on a flange fillet area (see FIG. 5). For example, in FIG. 1, the distance 26 between rail 20 and flange 19 defines how far the wheelset 16 can move laterally before flange 19 strikes rail 20. The rails 20 and 22 are not straight and rail curves are not uniform. In addition to the retarding force exerted on the flange 19 when making contact with the rails 20, 22, the force of the rails against the treads I4 and 18 cause the train car on curved or tangent track to zigzag back and fourth fi-om one rail to the other. The ~~~~J~~~
zigzag is as much due to change of rail path as to response to rail path.
To reduce tire wear and tirelrail resistance, flange pressure must he minimized when there im contact with the rail. The flange contact pxessure, however, changes with wear on the tire tread. Thus, in addition to designing the txead profile to simply minimize rail contact, the life cycle of the tread profile must also be considered.
The linear tread pmfile for the wheelset shown in FIG. 1 does not significantly reduce hunting as will be further explained below, thus, each tread profile has reduced energy efficiently and increased tire wear. Further, the linear profile does not provide adequate steering forces to assist the train around the curved track as described above.
Flange contact results in wear and use of energy to overcome friction.
FTG. 2 shows a schematic diagram of a wheelset 28 including wheels 30 each having a tread profile with a first slope 32 and a second slope 34. The increased conicity of slope 34 in relation to slope 32 increases resistive forces that prevent rails 20 and 22 from striking against flanges 38. Thus, the flanges 3fi axe less likely to make contact with the rails during turns. The problem is that the change between slope S2 and 34 is not continuous. Discoatinuous tread profiles do not establish continuous feedback systems that will effectively dampen oscillation as described by hyapunov. Thus, wheelset 28 will oscillate when sufficient lateral force moves the rail onto slope 34 and further against flange 38. Further, the tread profile shown in li IG. 2 induces increased train vibration when traversing between the two 26 noncontinuous slopes 32 and 34.
FIG. 3 is a schematic diagram showing a wheelset 38 according to the invention. The wheelset 38 includes wheels 39 each comprising mirrored images of an asymptotic curve. A catenary is defined as curve theoretically formed by a perfectly flexfble, uniformly dense and thick, inextendable cable suspended from two points. For example, the form of a telephone wire when suspended between two telephone poles.
The eatenary form of each wheel has the hyperbolic relationship y~cosh x. The relationship in the catenary is formed from the addition of two curves y=e"(x!2) and y=e~(-x/2).
The catenary can also be defined here as a curve formed from the sum of the mirror images of two continuous asymptotic hyperbolic curves. In the present context, a crontiauous curve refers to a curve having derivatives that are smooth and continuous. A hyperbola has no harmonics and ie aperiodic, in turn, dampening vibration in train tires. The tread profile of wheels 39 are farmed of half hyperbolas laid on their sides. The hyperbolic profile is used on opposing tire treads in each wheeleet providing an overall wheelset feedback system that dampens tire oscillations. The gradual asymptotic change created by the hyperbolic slope ie significant since it effectively produces a x5 recentering correction force in the tread. The recentering forces of the wheeiset 38 will be explained in more detail below in FICI. 13.
The invention keeps natural and forced vibration frequencies far apart to minimize oscillation. Far e~cample, the natural frequency[Huygenl is defined as the frequency between opposing fixes in a wheelset. An example of tire tread with natural frequency is where mirror images of opposing treads add up to catenary cycloid. A
forced frequenay is the frequency imparted by the guiding rails. The forced frequency depends on train speed and rail path variation. Speed and rail path variations (forced frequencies) continuously change sad are,therefore, unknown. However, a range of forced frequencies can be determined.
The natural frequency of the two tires is determined according to tread profile and is, therefore, known. Thus, the natural and forced frequencies are kept far apart by careftal choice of the asymptotic tread 80 profile. Far example, the natural frequency of the tread profile is selected to have a long aperiodic frequency much longer than the period of the forced frequencies. Thus, the opposing natural and forced frequencies act to dampen as opposed to increase oscillation.
This is presented in Vector Mechanics for Engineers by Ferdinand P.
Beer and E. Russell Johnston, Jr. 1988.
PROFILE DESIGN
FIG. 5 is an expanded cross section of the asymptotic tread profile 39 previously shown in FIG. 3. The profile 39 is formed at the left end into a flange 50 having a vertical left side 54, a semicircular top 55 and a linear sloping inside face 57. The inside face 57 in a preferred embodiment has an approximately 70 degrees slope that extends to a fillet(transition) section. The transition section extends from point 40 to the beginning of a tread section at point 43. The tread section extends from point 43 to an extreme wear point 47. A field side 53 of profile 39 includes a bevel 52 that extends up to point 51. The different points on the tread profile are defined as follows:
40 defines the beginning of linear inside face 57 of flange 50;
41 transition gauge point located within the transition section;
42 gauge point used as reference point for measuring(flange thickness) distance between the flange and the rail when the tire is in a centered position;
43 first curvature point in the tread section;
44 lift point defined as geometric point used to determine consistent curve to form tread. Lift point gives curve of two opposite tires characteristic of lifting load as absorptive barrier when off center and recentering asymptotically;
45 second curvature point;
48 taping line defined as center position of wheel on rail when wheelset in a center positibn;
47 extreme wear point is point located as far from flange as normal wear is expected;
48 first reference point within the transition section is point derived from tread formula continues to this point;
48 second reference point within the transition section is point derived #~om tread formula continued toward fillet.
Points 40, 41, 42, 48, and 49 are located in the fillet section defined as the transition section between tha flange 50 and the tread section. Points 43, 44, 45, 4B and 47 are located within the tread section where the tire tread normally makes contact with the rail.
However, points 48, 44 and 45 do not contact rails before tread has worn--this is a defining difference from Ziethen. Portions of the wheel to the right of point 47 define a tread transfer section used for supporting the train load during frog and switch crossings. Points 48, 44 and 45 define a minimum accepted tread radius. Points 48, 44 and 45 are a measure of the mismatch of the rail and tire system. That these paints not contact rail is thought to be necessary element of tread design. The total width of the profile firom the left most side 64 of flange 50 to field side 53 is typically 126 millimeters For narrow tire.
The minimum tread radius in a preferred embodiment is not less than the railhead and nezt to the fillet.
In a first embodiment of the invention, the tread section from 26 point 48 to vicinity of taping line 46 is machined into the shape of a hyperbola. Ia a second embodiment of the invention, the hyperbolic prof?~le is extended acmss both the transition section and the tread section from point 40 or 42 to vicinity of taping line 48. In another embodiment, the hyperbola extends over the entire tread section from 80 point 43 to point 47. Thus, in various embodiments of invention ~~.559~.4 various sections of profile 39 contain hyperbolic curvature to increase ~avheel efficiency. FIG. 5 shows the hyperbolic curve extending completely across the transition section and the tread section from point 40 to point 51.
r, ~r r~rrr ~mrON OT,~ pOTN't'S ON HYPE'.RBOLTG 'p''a'.OFILE
FIG. 8 is an expanded cross section of the profile 89 shown in FIGS showing additional reference points that may be used for deriving the hyperbolic curvature of the tire tread according to the invention. Pofnts 48A and 48B are selected arbitrarily close to point 46 no that a slope can he defined at point 46 by using points 48A and 468.
Points 42A and 42B are selected arbitrarily close to point 42 to define slope at gauge point 42.
Any curve can be used to provide the dampening features of the invention as long as the curve is asymptotic to the rail head, One of the easiest curves to work with is the hyperbola. A hyperbola is represented in equation 1 as follows:
Ax~2 + Bxy + Cy~2 * Dx + Ey + F = 0 (Equation 1);
where x is equal to the horizontal distance from the left most vertical line 54 of flange 60 (FIG. 5) and y is equal to the vertical distance from reference point 51 (FIG. 5). A hyperbola is defined as having a discriminate having the following value;
B~2-4AC a 0.
By selecting Hve separate points an the tread and loading each point into a hyperbolic equation as stated in equation 1, the coefficients A to F are calculated. The coefB.cients A, H, C, C, E and F are co-factor matrixes. From this information, the correct values for points 48, 44, 46, 46 and 4't containing the tread section are selected to form a hyperbolic profile. Another embodiment of the invention forms a $0 hyperbola from slope values at taping line and the gauge point 42 and finally using lift point 44. The choice of points on the tire tread determine the boundary values. The change of elope in the tread are one boundary value kind. The term boundary value generally means slope at a point boundary.
Boundary values are two things. The first is the slope at a particular point. The second ie that the whole asymptotic formula of one profile is a boundary value. The catenary of two treads form final and distinct boundary value, These values give the profiles the characteristic of mismatch between tread and rail, and the additional characteristic of an absorptive boundary. Additional derived characterlatics are asymptotic figure and asymptotic reaction. In a first embodiment of the invention, the formula of "flat" hyperbola does not extend to points lying on the fillet (i.e., to the right of point 44).
The flat hyperbola has minimum radius of branch at point 44. The 18 slope between lift point 44 and taping line 46 is a relatively flat hyperbola,therefore, the profile is very asymptotic in the sense of dynamics. This means that the slope does not change very rapidly between taping line and the lift point. That is, there is least amount and consistent amount o~ damping. The absorptive barrier of two treads resist movement off center of rail with slight resistance to movement. The two treads recenter with slight force when given no opposition. The profile is finished by using points within the transition section to provide a smooth transition from the tread section to the linear sloping section 57 of the flange fi0. The fillet section may have 26 a variety of different selected curvatures. P'or example, the fillet section can comprise twc separate curves comprising points 42A, 42, 428, 48, and 48 and paints 40, 41, 42A,42, 42B. The curvature of the fillet section is generally chosen as one or two conic curves(may be ellipses). The choice of the five points determine the curve.
SO Alternatively, the fillet section may be contained within the continuous hyperbolic shape of the tread section. This could be done by using only points 40, 42, 44, 4B, and 47. Alternately more points could be chosen such that A, B, C, D, E, and F represent variables rather than constants. This is a more complex equation meeting the intent of the 6 invention.
For example, the set of pouts 4d, 42, 44, 4S and 47 must form a tent value called discriminate of a hyperbola. This teat places a limit of choice of lift point, point 44. This limit is placed on all profiles of this invention. Lift point 44 is less than a critical mismatch threshold.
lU Limits of choice of lift point are defined by FIG. 16. Value for point 44 outside this range do not have the characteristics of this invention.
Points 43, 44 and 45 define a minimum acceptable tread radius.
The importance of minimum radius is that said radius needs to be to oae aide of tangent travel. The location next to flange and fillet makes 16 a consistent and smooth function of tread. Too small of minimum radius may create oscillation. Size of radius also effects useful life of profile. The hyperbolic profile can be limited from taping line 4$ to Lift point 44 and still provide similar dampeninE effects. The lift point value should be less than enough to drive two opposing tires side to 20 side. Lift point 44 defines the critical mismatch thxeshold point between the tread section and rail. Once the lift point is high enough, there is enough stored energy to drive opposite tires side to side in oscillation. This fs the critical mismatch of too much lift.
Since the asymptotic profile is also beneficial when used only in 26 limited sections of the tire profile. It is within the context of the invention to use asymptotic curve segments within sections or over the entire tire profile 39 between the gauge point 42 and the field end 68.
More mathematically complex profiles can be designed that exhibit similar dampening effects as provided by the hyperbolic curve. As SO such,FIG. 6 is only one simple representation of present invent5.on.
s FIG. 7 is a partial prior art combination of a circular and linear profile taken as an expanded view is the same relative positions of points 44, 46 and 47 in FIG. 5. The profile segment from point 46 to 47 is a straight line. Line below point 48 is horizontal reference line.
Point 56 is a radius point far circle segment that extends from point 44 to point 46. The radius is 350 millimeters. The radii from, point 56 to point 44 and from point 56 to paint 46 are not to aeale. The figure formed by points 56, 4B and 47 is a right triangle. The line from paint 46 to 47 is tangent to the arc from point 44 to point 46. The linearlcircular profile shown in FIG. 7 is similar to the profile previously shown in Schef~el, for illustrative purposes.
FIG. 8 comprises five points A to E taken from a worn profile.
The points are oriented in an expanded view at the, same relative positions of points 44, 46 and 47 in FIG. 5. FIGS. ? and $ are shown to illustrate differences in relation to the hyperbolic profile shown in FIG.
5. For example, FIG. 9 shows lift on a train load created by different tire profiles. The lift is measured according to the offset distance of a wheelset from a center position. A vertical center line 58 indicates a zero lateral displacement of a wheel from a predetermined center reference point. The center point is selected at 62.5 millimeters from the far left side of the flange (e.g., side 54 in FIG. 5). Eaeh vertical line to the right of center line 58 is one millimeter of lateral displacement of the wheel away from reference point 58. The vertical scale is expanded fifteen time greater than the horizontal scale. Line 80 represents the vertical displacement of a load carried on two tires having the circular/linear profile shown in FIG. 7. Line 62 represents the vertical displacement of a load carried on the worn tire profile shown in FIG. 8.
Line 64 shows the vertical displacement for a load carried on a sat of tires having the hyperbolic profile shown in FIG. 5. Line 66 is a 80 horizontal line used as a reference.
It can be seen that the profile in FIG. 7, represented by line 80, has significant lift as the wheelset moves further from center line 58.
Thus, the wheelset will experience substantial oscillation and vibration. The worn profile shown by line 62 has significantly less vertical lift than the linear/circular profile represented by line 60 but still exhibits more lift than the hyperbolic proflJ.e represented by line 84. A lower amount of vertical lift results in less train vibration and increased train stability. Far eaample,FIG.10 is a front view of the wheelset 18 previously shown is FIG. 1. The wheelset 18 is shown in a centered position between rails 22 and 20. The tread of each wheel 18 And 14 has a completely linear profile. Point 81 is a reference location at the center point of axle 19.
In the centered position shown in FIG. 10, the reference location 81 has zero rise (x = 0) and zero lateral deflection {y=0). Deferring to FIG.11, the wheelset 18 is oft'set to the right, for example, when proceeding around a curved section of rail. Because, the tread profile for both wheels 18 and 14 are linear, the vertical rise at the taping line 24 in wheel 14 is substantially equal to the vertical dmp at the taping line 24 in wheel 18. Thus, reference point 81 experiences a net horizontal shift(x to xl), but remains at substantially the same vertical position (yry2). A load on axle 1H of FIG. 1 will, therefore, experience zero net vertical change for relative lateral movementsloscillations df the wheelset relative to the rail.
Even with an essentially zero vertical change, the linear tread still exhibits lateral oscillation. A linear tread profile does not effectively reduce hunting. Steering is achieved as result of feedback.
Thus, if the wheelaet starts moving laterally back and forth, the load will accordingly move back and forth out of phase with the wheelset.
The load then begins to oscillate creating an unstable system.
~~.5~~~4 Further,the linear profile is FIG. 10, as mentioned above, does not provide the necessary steering forces to turn is curved track.
A nonlinear profile, such as the prn~'~le shown in FIG. 9, provides added steering forces for steering the train around curves. However, a lateral offset in a wheelset having the tire profile shown in FTG. 7, will cause the taping line 24 on one tire to rise a different amount than the corresponding vertical drop in the opposite tire. Thus, the reference point B1 will experience a net vertical rise that raises the load supported on axle 18. The effect is a "push" between the wheelset and the load that creates an oscillation. Since the train also moves laterally back and forth on the rail, a rotational push is effectively transferred to the load. This is the wheelset yaw or rotation about a vertical axis of the wheelset.
FIG. 12 shows a typical rotational oscillation at the reference point B1 for a nonlinear wheel profile. The azle reference point 61, in a centered position, is positioned at the intersection of the x and y axis.
As the wheelset moves laterally from the center position, the reference point B1 moves both latQrally to the left and rises upward. This resulting circular rotation at the wheeiset causes an associated circular rotation in the load supported on axle 61. However, the rotation in the wheelset is out of phase with the resulting rotation in the train load causing oscillation.
The hyperbolic profile of the invention, makes gradual changes in both the vertical and horizontal displacement of the wheels. Thus, the 26 wheeleet can wander significantly from a centered position without significantly changing the overall displacement of the load. In other words,an asymptotic tread profile allows large variations in wheel displacements with minimal effective displacement transfer of the associated load. Because, the hyperbolic profile ef~ctively dampens 3D oscillation, even minor changes in the overall wheelset displacement do ~~~J~~~~
not result in load oscillation. The asymptotic profile also returns to the initial state asymptotically.
The feedback response of the hyperbolic tread profile is also asymmetrical. As a result, increasingly greater lateral and vertical forces are exerted against the wheel when moving laterally toward the flange. However, a proportionally slower decrease in Force is exexted as the rail moves laterally toward the field side o~ the opposite wheel.
For example, FIG. 13 shows a leFt rail 90 and a right rail $8 that support wheels 92 and 94, xespectivel,~Y. The wheelset comprising wheels 92 and 94 are shown laterally offset to the left. Each wheel has a hyperbolic profile according to the present invention.
The leflt wheel 92 is positioned closer to the flarxge and, therefore,experiences lateral and vertical vector resistance forces from rail 90 resulting in an overall vector force 96. Alternatively, the flange 135 of wheel 94 is positioned further from the rail than the Range of wheel 92. The vertical and horizontal forces exerted on wheel 94 by rail 88 are combined to exert an overall force 98.
since the feedback response of the hyperbolic tread profile is asymmetrical, the increase in the overall force vector 96 on wheel 92 is not equal to the decline in the force vector $$ from wheel 94.
8pecifically,the overall lateral and vertical force (lift) exerted by the rail 80 against wheel 92 increases as the wheelset moves laterally towaxd the wheel flange. Thus, the tread prnSle provides more correction as the wheelset moves further from a centered position. The combination of lateral and vertical forces against the wheel decrease as the wheelset moves beak into a centered position with the flanges away from the rail. Thus, a net force 100 is directed toward wheel 92 forcing the wheelset to the right back into a centered position.
Since the asymptotic profile is asymmetric, there is a given 80 amount of lift.upon lateral displacement, as shown by line 84 in FIG. 9.
i 2~.~~~~~
However,the vextical lift is small to reduce circular vibrational effects.
The two opposite tires 90 and 94 are also designed to have local minima when located in a centered position between two rails. Local minima in the present contest is defined as the location where the load S exhibit the lowest potential eaexgy of gravity. Thus, when wheels 92 and 94 are in a centered position, the corresponding load is at its lowest point in relation to the ground.
Traditional profiles, such as BhoWn in F1G. 1, have a linear profile that does not effectively reduce hooting. As a result, the wheel flanges come in frequent contact with the rail lacreasingllocalizing tread wear near the flange. This results in traditional tread profiles exhibiting more wear next to the flange than on Held side of the wheel.
Gauge variation of rails spread wear out on field side of tread. The flange localizes wear to itself; The tread profile according to the 16 invention, however, sgreads wheel to rail contact back toward centered contact thus reducing tread wear next to the flange. The asymptotic profile tends to take the shortest distance relative to the rail. Thus there is Less tread wear due to less distance traveled by treads on rail.
FIG. 14 is a graph showing a relationship between the stability of different tire profiles according to curve periodicity. Completely aperiodic profiles such as a worn liner profile ox a linear profile are incompletely effective at reducing lateral vibration thus, resulting in significant flange contact and lateral oscillation as described above.
Highly periodic profiles such as a circle or ellipse will exhibit circular 26 vibration at the tread resulting in circular oscillation. The hyperbolic profile of the invention, however, exhibits both aperiodic and periodic characteristics. The aperiodic characteristic is not induced vibration.
The periodic characteristic is as increased curving ability. That is iacreased curving ability with higher induced hunting speed. Thus, 2~~~91~
the hyperbolic profile has increased stability over present profile shapes.
FIG. 15 is a graph showing the steering forces of several tire profiles. Vertical center line 58 is the same as center line 58 in FIG. 9 and each vertical line to the right of line 88 represents a displacement of 1 millimeter from a center wheelset position, The steering force is a product of the vertical (lift) vector and the change in slope in the tread at a given wheelaet offset from the centered position, Line BS
represents the steering force for the linearlcircular profile shawa in FIG. 9. Line 70 represents the steering forces for the hyperbolic profile according to the invention. Line 72 represents the steering forces for a profile having a lfnes.r slope of 1:80. The linear elope of line ?2 has a constant value of 0.0333. This is one divided by thirty. That number has no units. Line ?4 represents the steering Forces for the worn 18 profile shown in FIG.7.
From FIG. 15 it can be seen that the steering forces of the hyperbolic curve in Line 70 are substantially constant with lateral displacement. Thus, there is no sudden change of steering force with induced oscillation. Relative movement of rail is damped by profile either partially ignoring change or restoring centered position.
Relative movement fmm centered position is resisted with almost constant but slightly increasing force.
Fltl. 16 defines limits of choice of point 44 to form a continuous absorptive barrier of a hyperbolic tire pro$le. The asymptotic profile is the present invention effectively creates an absorptive barrier having a continuous surface curvature. Noncontinuous curvatures create new barrier limits that can induce oscillation. Absorptive barriers are described as absorbing barriers for the random walk in a problem in Linear Algebra Basics Practice and Theory by Bernard R, Gelbaum 1989 Elsevier Science Publishing Company on page 546.
The limit of choice for point 42 is only herein termed absorption boundary. In other words, choice of point outside boundary of figure in FIG.16 results in other than asymptotic response.
The absorption boundary extends from gauge point 42 to the taping line 46 (FIG.S). Lift point 44 according to the invention is located somewhere within the absorption boundary 56. A profile extending above line 54 is essentially an ellipse. Thus, a profile having a lift point above curve 54 will have a negative discriminate and, in turn, induces the periodic circular oscillation inherent in elliptical profiles. Negative discriminate indicates a hollow profile. Line/curve 54 defines a parabola with a discriminate of zero. Curve 54 is upper bound for choice of point 44. A profile taken along line 58 is of course linear and, therefore, exhibits the oscillation effects discussed above.
When the design of the tire profile uses point 44 within absorption barrier 56, an asymptotic profile is formed. Wear is distributed over the entire profile. The result is that the wheel is more durable and performs better over a longer period. Point 44 chosen closer to line 58 is more aperiodic. Point chosen closer to curve 56 is more periodic.
In FIG. 17 the direction of rolling is toward axle 84. Axle 82 is a tangent response. Axle 84 leads a tangent response. Axle 86 lags a tangent response. Leading a tangent response results in oscillation.
Since feedback mechanisms tend to oscillate, attempting to maintain exactly a tangent response results in oscillation. Consequently, the best axle position is one that slightly lags a tangent response. It is also important to note that the combined profiles of each wheelset provide a phase angle that lags a tangent response. Phase angle is important in the relationship between the storage of kinetic energy as potential energy of gravity. Thin ie a source of energy that could result in oscillation, Due to the profile of the tread, the amount of potential energy stored by the train wheel exhibits a lag in phase angle in relation to the tangent response. The phase angle lags so that the 8 Least amount of energy is stored by the wheel. There is less resultant oscillation.
To explain further, FIG. 17 shows the relationship between the opposing asymptotically profiled tires 83. For example, a train track 80 is rounded in a semicircular fashiozx. A wheelset when in position i0 82 is traveling in a tangent orientation in relation to track 80 is said to have a zero phase angle. When the left tire 83 is dropped back from the tangent position 82 to position 88, the wheelset has a lagging or negative phase angle. When left tire of the wheelset Leads the right wheel as shown in position 84, the wheelset has a positive or leading 15 phase angle.
A hyperbolic profile exhibits a negative phase angle similar to wheelset 86 and, therefore, allows the load to move laterally with corresponding lateral movements of the wheels. Thus, the lagg'mg phase response of the asymptotic profiled wheelset dampens the lateral 20 oscillation of the load. A positive phase angle, on the other hand,cauaes oscillation as the wheelaet seeks an equilibrium state such as shown by wheelset 82.
The lagging wheelaet response happens to have the lowest potential energy of the three response modes. Specifically, the 2~ potential energy stored in raised wheelset must not be Large enough to overcome the frictional forces between the rail and tire, By making the stored kinetic energy have an asymptotic response, absorptive damping is exhibited in the wheel in addition to the normal frictional damping provided by a straight conical tread profile, 2~.5~9~.~
The profile according to the invention can not correct for defects in machining accuracy, incorrectly configured rail, incorrectly configured vehicle or vehicles that operate too rapidly for existing conditions. Investor assumes no liability for such cases and assumes no liability for unlicensed attempts to use product.
Having described and illustrated the principles of the invention in a preferred embodiment thereof, it should be apparent that the invention can be modified in arrangement and detail without departing from such principles. Further, use of word asymptotic profile is inclusive of term hyperbolic form. The application is not just to trains, but all railed devices including manufacturing processes. I claim all modifications and variation coming within the spirit and scope of the following claims.
BACKGROUND 0P' THE INVENTION
This invention relates generally to trains and train wheels and more particularly to a novel tire profile that improves train stability and energy efFiciency.
Train tire profile includes several sections. A flange section protrudes downward from the aide of the train wheel and extends over the lateral side of a train track. A fillet extends upward along a field side of the flange providing transition to a straight conical wheel tread section. The wheel tread sectien serves as the major load bearing surface that supports the train wheels on a txain track. The art uses tread profile of two opposiag tires each on one of two rails to steer.
Two opposing tires are a wheelset, The flange provides steering when 16 rail curve exceeds capability of treads to steer without flange contact.
Two main factors must be considered when designing tire profiles for use with railed devices. The first is the dynamic stability of the vehicle at various speeds throughout its operating speed range. When in transit, a train experiences lateral oscillations known as "hunting".
Wheel hunting results in the wheels oscillating laterally back and forth between the wheel flanges. The maximum speed or critical speed of the train is determined by the onset of unstable, undesirable wheelset hunting. For eyample, if the train saes too fast, the force of the lateral oscillations wffl overcome the flange barrier and cause the train to 26 derail. Hunting is caused by the dynamics between the wheel tread profile and the rail. Increasing the slope of the wheel tread too fast toward flange increases forces causing hunting and, therefore, lowers the critical speed of the vehicle. Decreasing slope of wheel txead toward flange decreases steering farces, also lowering the critical ~
2~~a~~.4 hunting speed. This ie the measure of mismatch, These limits define critical mismatch threshold.
A second factor involved with train stability is the ability of the vehicle to negotiate track curves. This curving ability is determined d primarily by the ability of the opposing tires of wheelsets to follow the track curves. Optimally, the wheelsets should perform a purely rolling motion in the track curves without any contact between the wheel flanges and the rails. This requires steering forces to be generated by the sloped wheel tread independently of the wheel flange permitting the wheelset to yaw or rotate about a vertical axis which may be through it~ center. Oscillation of steering forces happen around vertical axis through its center of gravity(mass). This oscillation is a metric space. The oscillation of wheelset results in hooting. The steering forces move the train wheelsets into a more xadial 16 position(axle 82 FlCr.1'l) with respect to the track curves, thus, increasing train stability around curves.
A wheelset includes two opposite wheels that may be joined together by an sale. With a conical (straight taper) wheel tread the conicity remains virtually constant with lateral deflection. of a wheelset 2b relative to the track. That is, straight taper wheel treads have a constant elope. In other words, the conicity of each wheel remains the same irrespective of whether the wheelset rune centrally on the track or is deflected closer to one rail. Increasing the conicity of the wheel tread improves the steering ability of the wheelsets because of the 25 irxcreased steering force. However, increased conicity also increases the oaeillation of the wheelset. Oscillation of wheelset results in hunting. Therefore, with regard to the conicity of wheel treads, there is a conflict between the requirement for hunting stability and increased vehicle speed and for good curving ability of the wheelsets.
U.S. Patent No. 4,294,482 to Scheffel et al., discusses a profiled wheel tread that is made up of a combination of discrete circle and line segments. Ia tha art the term "profiled" is used in relation to wheels having a curved tread section and distinguishes such wheels from conventional wheels having a straight conical (linear sloped) tread section. The profiled tread in Scheffel et. al., utilizes multiple discrete curve sections each having a separate radius that are combined to fornx a non-continuous curve. The term conicity is imprecise. 'fhe curve radii of the profile increase from taping line to the flange. mhis is thought to reduce conicity of the profile. It reduces conicity of profile when compared to profile of constant radius between taping line and fillet. Thus, the tire tread conicity has essentially a "droop"characteristic with increasing lateral deflection toward the flange compared to tread of constant radius. A train profile with 16 multiple curve discontinuities and a "drooping" characteristic will initiate vibration between the wheelset and the train. Because the wheel tread also has relatively high conicity of tread slope change at the taping line, the tire is also more likely to hunt in relation to a straight taper. The minimum radius is at the taping line with larger radii toward fillet.
U.S. Patent No. 5,295,824 to Ziethen et al. discusses means for varying the train rail profile to extend wheel wear and track durability. Ziethen et al., however, does not suggest means for reducing vibration and hunting in train treads. The definition of concern ig that radius of tire tread is larger than radius of contacted rail. Also definition of concern is that there would not be two point contact between tire profile and rail. In FIG. 1 of Ziethen the lower continuous line(10) looks like it goes horizontal at coordinate origin Eline 57 col 8j thus indicating a discrete radius of tread in accordance SO with that invention to the extent shown in their figure. This meets 2~~~9~4 their requirement that tread radius is larger than contacted rail radius. Therefore, in a manner similar to Scheil~el, FxG. 1 in Ziethen suggests using discrete change of tread section. Therefore, the Ziethen tread profile begins with a linear wheel tread and then !~ abruptly changes to a radius in tread toward the fillet. The abrupt change in curvature in the tread section induces vibration and increases wheel resistance when the wheelset is laterally displaced on standard rail, This is not as absorptive barrier. Thus, the profile in Zisthen is not energy efficient and further does not provide additional steering forces different than prior art.
U.S. Patent No. 1,298,628 to Coda describes a wheel profile including two separate sections that intersect at the taping line. The profile is a wheel tread having a coniCal(linear) outer portion tangent to a portion of continually increasing conicity to the wheel flange. U.S.
lg Patent No. 1,788,705 to Emerson shows a substantially linear tread profile that is profiled to match the curvature of a corresponding inner munded corner of railhead. U.S. Patent No. 99,850 to Vial, discusses a wheel having both a concave throat section and a separate convex tread seetion'toward the fillet. Coda, Emerson and Vial, similar to the . other patents previously listed, have either straight conical tread profiles or discontinuous tread that include a combination of different sections each having a different curvature formula. Tread profiles with elliptical or circular profiles include a harmonic frequency that causes periodic oscillation under certain track conditions. Thus, the 2!i problems of vibration, limited steering forces and hunting still exist.
Another problem with train tires is excessive wear both at the tread and flange. Far example, linear tread profiles typically exhibit excessive wear next to the flange. Contrary to normal expectations, an -increase in the contact area between the wheel and the rail does not 80 necessarily decrease tread wear. Research has now shown that the rate of wear of the tread, in fact, increases when the shape of the tread approaches the shape of the rail head. This is because the wheel-tread /rail-head contact area increases to such an extent that there is a disadvantageous increase of the creep forces or slip of the wheel on the rail. The increased creep forces significantly increase the wear rats offsetting reductien in wear rate due to increased contact area. Thus, it is predicted that tread profiles that match the railhead, such as Ziethen, have a tendency to wear faster according to Schefl'el.
Many standard train tire treads have linear profiles but, due to wear, rapidly become nonlinear. This small amount of nonlinear wear causes hunting as described by John F. Leary in paper entitled America Adopts Worn Wheel Profiles in Railway Gazette International, 3uly 1990. Additional information is provided by John F. Leary, Stephen N. Handle arid Britto Rsjkumar in a paper entitled:
1b "Development of Freight Car Wheel Profiles - A Case Study", in Wear, 144 (I991) pager 363-862, Association of American Railroads, Pueblo, Colorado. The, nonlinear circle segments used for defining the tire profiles in both Scheffel and Ziethen also suggest prof"~ling the tire tread into nonlinear shapes. Thus, present nonlinear profiled tire treads have increased steering forces that induce hunting.
Accordingly, a need remains for a train tire profile that is both resistant to hunting and vibration, provides steering forces during turns and is energy efficient.
It is, therefore, an object of the invention to increase the stability of trains while traveling along rails.
Another object of the invention is to the increase energy efficiency of trains while at the same time increasing the maximum critical speed 90 that the trains can safely travel oa a rail.
A further object of the invention is to increase the standardization of profiled tire tread designs that have both superior curving ability and that are also resistant to hunting.
The present invention takes into account feedback theory to S develop a tread profile that is resistant to hunting, provides added steering forces through curved rail and is energy efficient. Rail technology does not presently use feedback theory to effectively model and control the interaction between train tires and rails.
Correspondingly, present wheel profiles do not maximize train performance. Feedback theory and how it is used in control systems is discussed by Sandeep in article entitled "Deterministic Controllers for a Class of Mismatched Systems", Journal of Dynamic Systems, Measurement and Control; March 1994 Vol. 116 pp 17. The important concepts are measure of mismatch and critical mismatch threshold.
Elements of control system theory are discussed in many different texts under different labels. Ferenc Szidarovsky and A. Terry Bahill discuses elements of Lyapunov stability theory in book Linear Systems Theory and is herein incorporated by reference. The feedback system must be a continuous mathematical function. There must be a unique global minimum at average position within feedback system.
And finally for any state trajectory the function of that perturbed trajectory must be decreasing with time. These design criteria are taken into account when designing the tread profile of the present invention to control energy stored in the train tire. A tire profile is correspondingly designed essentially as a feedback system operating between two opposite tires in a train wheelset. The result is a tread profile that increases train wheel performance by reducing hunting and vibration while at the same time increasing steering capacity and energy efficiency.
The wheel profile effectively incorporates an absorptive barrier LCxelbauml that redistributes wear away from the flanges. The absorptive barrier provides an asymmetric response iu relation to different lateral displacements of the wheelaet that effectively maintains the wheelset at a centered position. The tread profile exhibits slight resistive forces when the wheel moves laterally across the rail in locations relatively close to a center wheel position.
However, wheel pair exert significantly increased resistive forces when the wheels are moved across the rail to location significantly far from the centered wheel position. Thus, when the wheelset is off center, a net force is exerted on the train truck wheelset in an asymptotical manner in the direction of the centered position.
Specifically, the invention incorporates a hyperbolidasymptotic tread profile on the tread section of each wheel in a wheelset. The rb profile decreases asymptotically in amplitude as it extends further from the flange. The asymptotic profile effectively dampens lateral vibrations in the wheelset in a manner similar to an asymptotically decreasing response in an electronic feedback system. The profile of the wheel exhibits a "continuous" asymptotic curve over substantially the entire tread section of the wheel to reduce vibration and ensure a proper wheel/rail feedback response. The asymptotic profiles from each wheel in the wheelset, when combined, farm a catenary that maintains the wheelset in a centered position. A load supported by the wheelset will exhibit a local minimal when the wheelset is in a 26 centered position, Thus, vertical gravitational forces tend to maintain the centered position. The hyperbolic profile provides a phase lag in the wheelset when navigating rail curves, Thus, the tire treads of opposite tires are an absorptive barrier that efFectively absorb and dampen lateral oscillations. Lateral oscillations increase tire wear and use SO energy to wear tires thus reducing energy efficiency. The hyperbolic profile is incorporated into either just the tread section of the wheel or extended in a continuous curve from the tread section up into the fillet section. Thus, extreme off center displacements of the wheelset still reside within the asymptotical feedback system of the wheel profile.
The asymptotic tire profile is derived using a matrii algebraic relationship between various selected points on the wheel to generate a reliable and consistent profiled shape. The selection of certain reference points on the tread ensures smooth transitions between different wheel sections, in turn redudng vibration and ensuring proper wheel response in relation to varying rail radii.
The foregoing and other objects, features and advantages of the invention will become more readily apparent from the following detailed description of a preferred embodiment of the invention which proceeds with reference to the accompanying drawings.
Rg , D ~Q ,R1PTION OF err'. T_~RAWIN :g FIG. 1 is a schematic diagram representing a cross section of s prior art linear tread profile for a wheelset.
FIG. 2 shows a schematic diagram of a tread profile with two linear slopes.
FIG. 3 shows a schematic diagram of a wheel prefile according to the invention.
FIG. 4 is a schematic top view of the standard wheelget in FIG. 1 shown on rail curve.
FIG. 8 is an expanded cross section of the asymptotic tread profile shown in FIG. 3.
FIG. 6 is an expanded cross section of the profile spawn in FIG. 6 with additional reference points.
FIG. 7 is part of a prior art a combination of a linear and circular profile taken as an expanded view in the same relative positions of 80 points 44, 46 and 47 is FIG. S.
FIG. 8 is a schematic showing a prior art worn linear tread profile.
FIG. 9 shows the lift created by different tread profiles of opposing treads on a load.
FIG. 10 is a schematic showing a centered wheelset with linear tread profiles.
FIG. 11 is the wheelset of FIG. 10 shown in off center position.
FIG. 12 is a schematic showing the circular displacement of a load due to the linear/circular profile shown n FIG. 7.
FIG. 13 is a schematic showing the force vectors exerted on a hyperbolic profile by a rail.
FIG. 14 is a graph showing a relationship between the stability of different tire profiles according to curve periodicity.
FIG. 15 is a graph showing the steering forces of several tire profiles.
FIG. 16 shows boundary limits of critical mismatch threshold where tread of hyperbolic form is an absorption boundary.
FIG. 17 is a schematic showing the phase angles for a wheelset traveling through a curved track section.
DETAILED DESCRIPTION
THEORY
The logical and mathematical basis for the present invention is based on classical mathematical and physical relationships that are known to those skilled in the art, and therefore, are not described in detail. The following is a list of publications discussing these physical and mathematical relationships:
1. Rorres, C. and Anton, H., Applications of Linear Algebra, (1984), pages 4-6;
2. Friedberg, S. and Insel, A, Introduetio to Linear A1_gebra wi . y ~~~ 1' a .ions, (18$6), pages 31, 32, 4$, 49, 226~254;
8. Shenk, A.,. ~'!alCUlu6 nd "alvtic C~~ometrv, (197?), page 418 subject LORAN;
b 4. Stone, D. and Leery J., Rai wav Aae, Feb. (1992), "Reinventing The Wheel' ;
5. McLachlan, P. Eng., 1?ail Profiles Can make A Difference, a paper presented to: American Public Transit Assoaation (1992) Rapid Transit Conference, Los Angeles, Cal, June 16, 1892;
6. Hay, William W., $ailro~d En;ineering, Vol 1, pages 7. Thomas and Finney, ale plus and nahi~tie Cx_~a~netrv. 6th Ed, (1978), pages 494-498, '7th Ed, pages 559-60; subject is effect of curvature an periodicity per Huygen and his clock;
$. Elsgalts, L., Tlifferential E~mations and Calcmlus of Mir Publishers, Moscow, pages 244-247, subject is stability under constantly operating perturbations;
9. Mayr, Otto, The Ori.;ins of F .dba .k .ontrel, {1970), history;
10. Kellogg, Dimon Oliver, FeLndations of Potential, (1953);
11. Leery, John F., Elkins, John A,, Comvuter Solutions to WheeL~tai1 Interaetiq~n Problems, presented at APTA June 13-17, (1992), flange pressure and other subjects;
12. Murphy, Gordon John, Rasie Automatic Control Theorv , (1957), pages 1~16, so, 31;
18. Pitman, R.J.G., ALtnmatic Coil .rol 9vstams ~xnlain , (1968), good basic mechanical explanation of feedback theory particularly chapter "Inertia and Friction in Control Systems' ;
14. Ditto, William L. and Pecora, Louis M., ~s~iSnt3Sc 80 ~, August (1993), Vol 289, Number 2, "Mastering Chaas';
IO
15. Txaub, Joseph >i'., Wozniakowski, Henryk, Scientific AmeriCari, January (1994), "Breaking Intractability' ;
18. Schroeder, M. R., Fro .alashaos, power T_.aws: Minutes From an Infinite Universe. (1991);
6 17. Dorf, Richard C.,1!4odern_Cnn .rot 4vs .ms. (1974) pages 129-131, concept of stability;
18. Magid, Andy R., ~Rplied Matrix Models a second coLrse in linear Algebra with ~!omp ~ . r Ap, li a ions; (1985), a method of handling over determined systems;
19. Law, E. H, ~'ransactione of the ASME J~wr_n_al of >i n n ri for Industry, November, (1974) pages 1168-74, "Nonlinear Wheelset Dynamic Response to Random Lateral Rail Irregularities' ;
20. Hedrick, J. Karl, Arslan, A.V., ~,n ~r al o ~ .,..,ami ,'~ s gyms g;a g ~ men . and ContraL Sept, (1979}, 1~ Vol 101, pages 230-7"N'onlinear Analysis of Rail Vehicle Forced Lateral Response and Stability' ;
21. Friee, R.H., Cooperrider, N.K., Law, E.H., Tr nsa . .'ens of the ASME loyal of D ami ya ms, easL?re~, and Control, Sept, (1881), Vol 105, pages 201-210; " Experimental Investigation of Freight Car Lateral Dynamics' ;
22. Fries, R.H., Cooperrider, N.K., Law, E.~T., Transactions of ~h_e A4ME Journal of Dy amic ~, s pms.Measurement'nds',,9~n .rfll~
December, (1978), Vol 100, pages 238-2b1.
FIG. 1 is a schematic diagram representing a cross section of a 2~ prier art linear tread prol3le for a wheelset 18. The tread profile for both wheels 18 and 14 comprise a linear conical tread 17 with a dowawardly directly flange 19. The wheels 18 and 14 are joined together by an axle 13 that is corresponding connected to train truck (sot shown) to support a load. Wheelset 18 is shown in a centered position which is defined as having the flange of wheels 18 and 14 substantially equal distances 26 from rails 22 and 20, respectively. In the centered position, wheels 18 and 14 make contact with rails 22 and 20, respectively at points 24 which are defined as taping lines. A
distance 12 exists between taping lines of wheels 18 and 14. Taping line may be contact area of centered travel. Taping line has become defined measuring location rather than center of travel. The hereina$er-described field side 53 can also be used as a means for aligning the template.
FIG. 4 is a schematic showing a top view of the standard linear IO wheelset 16 in FIG. I. Referring to both FIGS. 1 and 4, the distance between the taping lines of wheels 14 and 18 in a typical embodiment is approximately 1500 millimeters. The flanges 19 for a single wheelset 16 will contact either rail 20 or 22 when the radius R2 of inside rail 20 is approximately 1074 feet and the radius Rl of outside rail 22 is approximately 1079 feet This is with 28 inch tires.
When the flanges 19 make contact with the rail, resistance between the flange 19 and the rails 20 and 22 reduce train energy efficiency. More specifically, the flange 19 has approximately a seventy degree inside slope that acts as a wedge when traveling through rails 20 and 22. The wedging action creates wear near the flange 19 that both reduces tire life. The wedging action increases wheel resistance. Wedging action exits in rail curves.
The limit of lateral movement of the wheelset 16 is set by the gauge tolerance of the wheels. The gauge tolerance is defined as the distance between the rail and a given location on a flange fillet area (see FIG. 5). For example, in FIG. 1, the distance 26 between rail 20 and flange 19 defines how far the wheelset 16 can move laterally before flange 19 strikes rail 20. The rails 20 and 22 are not straight and rail curves are not uniform. In addition to the retarding force exerted on the flange 19 when making contact with the rails 20, 22, the force of the rails against the treads I4 and 18 cause the train car on curved or tangent track to zigzag back and fourth fi-om one rail to the other. The ~~~~J~~~
zigzag is as much due to change of rail path as to response to rail path.
To reduce tire wear and tirelrail resistance, flange pressure must he minimized when there im contact with the rail. The flange contact pxessure, however, changes with wear on the tire tread. Thus, in addition to designing the txead profile to simply minimize rail contact, the life cycle of the tread profile must also be considered.
The linear tread pmfile for the wheelset shown in FIG. 1 does not significantly reduce hunting as will be further explained below, thus, each tread profile has reduced energy efficiently and increased tire wear. Further, the linear profile does not provide adequate steering forces to assist the train around the curved track as described above.
Flange contact results in wear and use of energy to overcome friction.
FTG. 2 shows a schematic diagram of a wheelset 28 including wheels 30 each having a tread profile with a first slope 32 and a second slope 34. The increased conicity of slope 34 in relation to slope 32 increases resistive forces that prevent rails 20 and 22 from striking against flanges 38. Thus, the flanges 3fi axe less likely to make contact with the rails during turns. The problem is that the change between slope S2 and 34 is not continuous. Discoatinuous tread profiles do not establish continuous feedback systems that will effectively dampen oscillation as described by hyapunov. Thus, wheelset 28 will oscillate when sufficient lateral force moves the rail onto slope 34 and further against flange 38. Further, the tread profile shown in li IG. 2 induces increased train vibration when traversing between the two 26 noncontinuous slopes 32 and 34.
FIG. 3 is a schematic diagram showing a wheelset 38 according to the invention. The wheelset 38 includes wheels 39 each comprising mirrored images of an asymptotic curve. A catenary is defined as curve theoretically formed by a perfectly flexfble, uniformly dense and thick, inextendable cable suspended from two points. For example, the form of a telephone wire when suspended between two telephone poles.
The eatenary form of each wheel has the hyperbolic relationship y~cosh x. The relationship in the catenary is formed from the addition of two curves y=e"(x!2) and y=e~(-x/2).
The catenary can also be defined here as a curve formed from the sum of the mirror images of two continuous asymptotic hyperbolic curves. In the present context, a crontiauous curve refers to a curve having derivatives that are smooth and continuous. A hyperbola has no harmonics and ie aperiodic, in turn, dampening vibration in train tires. The tread profile of wheels 39 are farmed of half hyperbolas laid on their sides. The hyperbolic profile is used on opposing tire treads in each wheeleet providing an overall wheelset feedback system that dampens tire oscillations. The gradual asymptotic change created by the hyperbolic slope ie significant since it effectively produces a x5 recentering correction force in the tread. The recentering forces of the wheeiset 38 will be explained in more detail below in FICI. 13.
The invention keeps natural and forced vibration frequencies far apart to minimize oscillation. Far e~cample, the natural frequency[Huygenl is defined as the frequency between opposing fixes in a wheelset. An example of tire tread with natural frequency is where mirror images of opposing treads add up to catenary cycloid. A
forced frequenay is the frequency imparted by the guiding rails. The forced frequency depends on train speed and rail path variation. Speed and rail path variations (forced frequencies) continuously change sad are,therefore, unknown. However, a range of forced frequencies can be determined.
The natural frequency of the two tires is determined according to tread profile and is, therefore, known. Thus, the natural and forced frequencies are kept far apart by careftal choice of the asymptotic tread 80 profile. Far example, the natural frequency of the tread profile is selected to have a long aperiodic frequency much longer than the period of the forced frequencies. Thus, the opposing natural and forced frequencies act to dampen as opposed to increase oscillation.
This is presented in Vector Mechanics for Engineers by Ferdinand P.
Beer and E. Russell Johnston, Jr. 1988.
PROFILE DESIGN
FIG. 5 is an expanded cross section of the asymptotic tread profile 39 previously shown in FIG. 3. The profile 39 is formed at the left end into a flange 50 having a vertical left side 54, a semicircular top 55 and a linear sloping inside face 57. The inside face 57 in a preferred embodiment has an approximately 70 degrees slope that extends to a fillet(transition) section. The transition section extends from point 40 to the beginning of a tread section at point 43. The tread section extends from point 43 to an extreme wear point 47. A field side 53 of profile 39 includes a bevel 52 that extends up to point 51. The different points on the tread profile are defined as follows:
40 defines the beginning of linear inside face 57 of flange 50;
41 transition gauge point located within the transition section;
42 gauge point used as reference point for measuring(flange thickness) distance between the flange and the rail when the tire is in a centered position;
43 first curvature point in the tread section;
44 lift point defined as geometric point used to determine consistent curve to form tread. Lift point gives curve of two opposite tires characteristic of lifting load as absorptive barrier when off center and recentering asymptotically;
45 second curvature point;
48 taping line defined as center position of wheel on rail when wheelset in a center positibn;
47 extreme wear point is point located as far from flange as normal wear is expected;
48 first reference point within the transition section is point derived from tread formula continues to this point;
48 second reference point within the transition section is point derived #~om tread formula continued toward fillet.
Points 40, 41, 42, 48, and 49 are located in the fillet section defined as the transition section between tha flange 50 and the tread section. Points 43, 44, 45, 4B and 47 are located within the tread section where the tire tread normally makes contact with the rail.
However, points 48, 44 and 45 do not contact rails before tread has worn--this is a defining difference from Ziethen. Portions of the wheel to the right of point 47 define a tread transfer section used for supporting the train load during frog and switch crossings. Points 48, 44 and 45 define a minimum accepted tread radius. Points 48, 44 and 45 are a measure of the mismatch of the rail and tire system. That these paints not contact rail is thought to be necessary element of tread design. The total width of the profile firom the left most side 64 of flange 50 to field side 53 is typically 126 millimeters For narrow tire.
The minimum tread radius in a preferred embodiment is not less than the railhead and nezt to the fillet.
In a first embodiment of the invention, the tread section from 26 point 48 to vicinity of taping line 46 is machined into the shape of a hyperbola. Ia a second embodiment of the invention, the hyperbolic prof?~le is extended acmss both the transition section and the tread section from point 40 or 42 to vicinity of taping line 48. In another embodiment, the hyperbola extends over the entire tread section from 80 point 43 to point 47. Thus, in various embodiments of invention ~~.559~.4 various sections of profile 39 contain hyperbolic curvature to increase ~avheel efficiency. FIG. 5 shows the hyperbolic curve extending completely across the transition section and the tread section from point 40 to point 51.
r, ~r r~rrr ~mrON OT,~ pOTN't'S ON HYPE'.RBOLTG 'p''a'.OFILE
FIG. 8 is an expanded cross section of the profile 89 shown in FIGS showing additional reference points that may be used for deriving the hyperbolic curvature of the tire tread according to the invention. Pofnts 48A and 48B are selected arbitrarily close to point 46 no that a slope can he defined at point 46 by using points 48A and 468.
Points 42A and 42B are selected arbitrarily close to point 42 to define slope at gauge point 42.
Any curve can be used to provide the dampening features of the invention as long as the curve is asymptotic to the rail head, One of the easiest curves to work with is the hyperbola. A hyperbola is represented in equation 1 as follows:
Ax~2 + Bxy + Cy~2 * Dx + Ey + F = 0 (Equation 1);
where x is equal to the horizontal distance from the left most vertical line 54 of flange 60 (FIG. 5) and y is equal to the vertical distance from reference point 51 (FIG. 5). A hyperbola is defined as having a discriminate having the following value;
B~2-4AC a 0.
By selecting Hve separate points an the tread and loading each point into a hyperbolic equation as stated in equation 1, the coefficients A to F are calculated. The coefB.cients A, H, C, C, E and F are co-factor matrixes. From this information, the correct values for points 48, 44, 46, 46 and 4't containing the tread section are selected to form a hyperbolic profile. Another embodiment of the invention forms a $0 hyperbola from slope values at taping line and the gauge point 42 and finally using lift point 44. The choice of points on the tire tread determine the boundary values. The change of elope in the tread are one boundary value kind. The term boundary value generally means slope at a point boundary.
Boundary values are two things. The first is the slope at a particular point. The second ie that the whole asymptotic formula of one profile is a boundary value. The catenary of two treads form final and distinct boundary value, These values give the profiles the characteristic of mismatch between tread and rail, and the additional characteristic of an absorptive boundary. Additional derived characterlatics are asymptotic figure and asymptotic reaction. In a first embodiment of the invention, the formula of "flat" hyperbola does not extend to points lying on the fillet (i.e., to the right of point 44).
The flat hyperbola has minimum radius of branch at point 44. The 18 slope between lift point 44 and taping line 46 is a relatively flat hyperbola,therefore, the profile is very asymptotic in the sense of dynamics. This means that the slope does not change very rapidly between taping line and the lift point. That is, there is least amount and consistent amount o~ damping. The absorptive barrier of two treads resist movement off center of rail with slight resistance to movement. The two treads recenter with slight force when given no opposition. The profile is finished by using points within the transition section to provide a smooth transition from the tread section to the linear sloping section 57 of the flange fi0. The fillet section may have 26 a variety of different selected curvatures. P'or example, the fillet section can comprise twc separate curves comprising points 42A, 42, 428, 48, and 48 and paints 40, 41, 42A,42, 42B. The curvature of the fillet section is generally chosen as one or two conic curves(may be ellipses). The choice of the five points determine the curve.
SO Alternatively, the fillet section may be contained within the continuous hyperbolic shape of the tread section. This could be done by using only points 40, 42, 44, 4B, and 47. Alternately more points could be chosen such that A, B, C, D, E, and F represent variables rather than constants. This is a more complex equation meeting the intent of the 6 invention.
For example, the set of pouts 4d, 42, 44, 4S and 47 must form a tent value called discriminate of a hyperbola. This teat places a limit of choice of lift point, point 44. This limit is placed on all profiles of this invention. Lift point 44 is less than a critical mismatch threshold.
lU Limits of choice of lift point are defined by FIG. 16. Value for point 44 outside this range do not have the characteristics of this invention.
Points 43, 44 and 45 define a minimum acceptable tread radius.
The importance of minimum radius is that said radius needs to be to oae aide of tangent travel. The location next to flange and fillet makes 16 a consistent and smooth function of tread. Too small of minimum radius may create oscillation. Size of radius also effects useful life of profile. The hyperbolic profile can be limited from taping line 4$ to Lift point 44 and still provide similar dampeninE effects. The lift point value should be less than enough to drive two opposing tires side to 20 side. Lift point 44 defines the critical mismatch thxeshold point between the tread section and rail. Once the lift point is high enough, there is enough stored energy to drive opposite tires side to side in oscillation. This fs the critical mismatch of too much lift.
Since the asymptotic profile is also beneficial when used only in 26 limited sections of the tire profile. It is within the context of the invention to use asymptotic curve segments within sections or over the entire tire profile 39 between the gauge point 42 and the field end 68.
More mathematically complex profiles can be designed that exhibit similar dampening effects as provided by the hyperbolic curve. As SO such,FIG. 6 is only one simple representation of present invent5.on.
s FIG. 7 is a partial prior art combination of a circular and linear profile taken as an expanded view is the same relative positions of points 44, 46 and 47 in FIG. 5. The profile segment from point 46 to 47 is a straight line. Line below point 48 is horizontal reference line.
Point 56 is a radius point far circle segment that extends from point 44 to point 46. The radius is 350 millimeters. The radii from, point 56 to point 44 and from point 56 to paint 46 are not to aeale. The figure formed by points 56, 4B and 47 is a right triangle. The line from paint 46 to 47 is tangent to the arc from point 44 to point 46. The linearlcircular profile shown in FIG. 7 is similar to the profile previously shown in Schef~el, for illustrative purposes.
FIG. 8 comprises five points A to E taken from a worn profile.
The points are oriented in an expanded view at the, same relative positions of points 44, 46 and 47 in FIG. 5. FIGS. ? and $ are shown to illustrate differences in relation to the hyperbolic profile shown in FIG.
5. For example, FIG. 9 shows lift on a train load created by different tire profiles. The lift is measured according to the offset distance of a wheelset from a center position. A vertical center line 58 indicates a zero lateral displacement of a wheel from a predetermined center reference point. The center point is selected at 62.5 millimeters from the far left side of the flange (e.g., side 54 in FIG. 5). Eaeh vertical line to the right of center line 58 is one millimeter of lateral displacement of the wheel away from reference point 58. The vertical scale is expanded fifteen time greater than the horizontal scale. Line 80 represents the vertical displacement of a load carried on two tires having the circular/linear profile shown in FIG. 7. Line 62 represents the vertical displacement of a load carried on the worn tire profile shown in FIG. 8.
Line 64 shows the vertical displacement for a load carried on a sat of tires having the hyperbolic profile shown in FIG. 5. Line 66 is a 80 horizontal line used as a reference.
It can be seen that the profile in FIG. 7, represented by line 80, has significant lift as the wheelset moves further from center line 58.
Thus, the wheelset will experience substantial oscillation and vibration. The worn profile shown by line 62 has significantly less vertical lift than the linear/circular profile represented by line 60 but still exhibits more lift than the hyperbolic proflJ.e represented by line 84. A lower amount of vertical lift results in less train vibration and increased train stability. Far eaample,FIG.10 is a front view of the wheelset 18 previously shown is FIG. 1. The wheelset 18 is shown in a centered position between rails 22 and 20. The tread of each wheel 18 And 14 has a completely linear profile. Point 81 is a reference location at the center point of axle 19.
In the centered position shown in FIG. 10, the reference location 81 has zero rise (x = 0) and zero lateral deflection {y=0). Deferring to FIG.11, the wheelset 18 is oft'set to the right, for example, when proceeding around a curved section of rail. Because, the tread profile for both wheels 18 and 14 are linear, the vertical rise at the taping line 24 in wheel 14 is substantially equal to the vertical dmp at the taping line 24 in wheel 18. Thus, reference point 81 experiences a net horizontal shift(x to xl), but remains at substantially the same vertical position (yry2). A load on axle 1H of FIG. 1 will, therefore, experience zero net vertical change for relative lateral movementsloscillations df the wheelset relative to the rail.
Even with an essentially zero vertical change, the linear tread still exhibits lateral oscillation. A linear tread profile does not effectively reduce hunting. Steering is achieved as result of feedback.
Thus, if the wheelaet starts moving laterally back and forth, the load will accordingly move back and forth out of phase with the wheelset.
The load then begins to oscillate creating an unstable system.
~~.5~~~4 Further,the linear profile is FIG. 10, as mentioned above, does not provide the necessary steering forces to turn is curved track.
A nonlinear profile, such as the prn~'~le shown in FIG. 9, provides added steering forces for steering the train around curves. However, a lateral offset in a wheelset having the tire profile shown in FTG. 7, will cause the taping line 24 on one tire to rise a different amount than the corresponding vertical drop in the opposite tire. Thus, the reference point B1 will experience a net vertical rise that raises the load supported on axle 18. The effect is a "push" between the wheelset and the load that creates an oscillation. Since the train also moves laterally back and forth on the rail, a rotational push is effectively transferred to the load. This is the wheelset yaw or rotation about a vertical axis of the wheelset.
FIG. 12 shows a typical rotational oscillation at the reference point B1 for a nonlinear wheel profile. The azle reference point 61, in a centered position, is positioned at the intersection of the x and y axis.
As the wheelset moves laterally from the center position, the reference point B1 moves both latQrally to the left and rises upward. This resulting circular rotation at the wheeiset causes an associated circular rotation in the load supported on axle 61. However, the rotation in the wheelset is out of phase with the resulting rotation in the train load causing oscillation.
The hyperbolic profile of the invention, makes gradual changes in both the vertical and horizontal displacement of the wheels. Thus, the 26 wheeleet can wander significantly from a centered position without significantly changing the overall displacement of the load. In other words,an asymptotic tread profile allows large variations in wheel displacements with minimal effective displacement transfer of the associated load. Because, the hyperbolic profile ef~ctively dampens 3D oscillation, even minor changes in the overall wheelset displacement do ~~~J~~~~
not result in load oscillation. The asymptotic profile also returns to the initial state asymptotically.
The feedback response of the hyperbolic tread profile is also asymmetrical. As a result, increasingly greater lateral and vertical forces are exerted against the wheel when moving laterally toward the flange. However, a proportionally slower decrease in Force is exexted as the rail moves laterally toward the field side o~ the opposite wheel.
For example, FIG. 13 shows a leFt rail 90 and a right rail $8 that support wheels 92 and 94, xespectivel,~Y. The wheelset comprising wheels 92 and 94 are shown laterally offset to the left. Each wheel has a hyperbolic profile according to the present invention.
The leflt wheel 92 is positioned closer to the flarxge and, therefore,experiences lateral and vertical vector resistance forces from rail 90 resulting in an overall vector force 96. Alternatively, the flange 135 of wheel 94 is positioned further from the rail than the Range of wheel 92. The vertical and horizontal forces exerted on wheel 94 by rail 88 are combined to exert an overall force 98.
since the feedback response of the hyperbolic tread profile is asymmetrical, the increase in the overall force vector 96 on wheel 92 is not equal to the decline in the force vector $$ from wheel 94.
8pecifically,the overall lateral and vertical force (lift) exerted by the rail 80 against wheel 92 increases as the wheelset moves laterally towaxd the wheel flange. Thus, the tread prnSle provides more correction as the wheelset moves further from a centered position. The combination of lateral and vertical forces against the wheel decrease as the wheelset moves beak into a centered position with the flanges away from the rail. Thus, a net force 100 is directed toward wheel 92 forcing the wheelset to the right back into a centered position.
Since the asymptotic profile is asymmetric, there is a given 80 amount of lift.upon lateral displacement, as shown by line 84 in FIG. 9.
i 2~.~~~~~
However,the vextical lift is small to reduce circular vibrational effects.
The two opposite tires 90 and 94 are also designed to have local minima when located in a centered position between two rails. Local minima in the present contest is defined as the location where the load S exhibit the lowest potential eaexgy of gravity. Thus, when wheels 92 and 94 are in a centered position, the corresponding load is at its lowest point in relation to the ground.
Traditional profiles, such as BhoWn in F1G. 1, have a linear profile that does not effectively reduce hooting. As a result, the wheel flanges come in frequent contact with the rail lacreasingllocalizing tread wear near the flange. This results in traditional tread profiles exhibiting more wear next to the flange than on Held side of the wheel.
Gauge variation of rails spread wear out on field side of tread. The flange localizes wear to itself; The tread profile according to the 16 invention, however, sgreads wheel to rail contact back toward centered contact thus reducing tread wear next to the flange. The asymptotic profile tends to take the shortest distance relative to the rail. Thus there is Less tread wear due to less distance traveled by treads on rail.
FIG. 14 is a graph showing a relationship between the stability of different tire profiles according to curve periodicity. Completely aperiodic profiles such as a worn liner profile ox a linear profile are incompletely effective at reducing lateral vibration thus, resulting in significant flange contact and lateral oscillation as described above.
Highly periodic profiles such as a circle or ellipse will exhibit circular 26 vibration at the tread resulting in circular oscillation. The hyperbolic profile of the invention, however, exhibits both aperiodic and periodic characteristics. The aperiodic characteristic is not induced vibration.
The periodic characteristic is as increased curving ability. That is iacreased curving ability with higher induced hunting speed. Thus, 2~~~91~
the hyperbolic profile has increased stability over present profile shapes.
FIG. 15 is a graph showing the steering forces of several tire profiles. Vertical center line 58 is the same as center line 58 in FIG. 9 and each vertical line to the right of line 88 represents a displacement of 1 millimeter from a center wheelset position, The steering force is a product of the vertical (lift) vector and the change in slope in the tread at a given wheelaet offset from the centered position, Line BS
represents the steering force for the linearlcircular profile shawa in FIG. 9. Line 70 represents the steering forces for the hyperbolic profile according to the invention. Line 72 represents the steering forces for a profile having a lfnes.r slope of 1:80. The linear elope of line ?2 has a constant value of 0.0333. This is one divided by thirty. That number has no units. Line ?4 represents the steering Forces for the worn 18 profile shown in FIG.7.
From FIG. 15 it can be seen that the steering forces of the hyperbolic curve in Line 70 are substantially constant with lateral displacement. Thus, there is no sudden change of steering force with induced oscillation. Relative movement of rail is damped by profile either partially ignoring change or restoring centered position.
Relative movement fmm centered position is resisted with almost constant but slightly increasing force.
Fltl. 16 defines limits of choice of point 44 to form a continuous absorptive barrier of a hyperbolic tire pro$le. The asymptotic profile is the present invention effectively creates an absorptive barrier having a continuous surface curvature. Noncontinuous curvatures create new barrier limits that can induce oscillation. Absorptive barriers are described as absorbing barriers for the random walk in a problem in Linear Algebra Basics Practice and Theory by Bernard R, Gelbaum 1989 Elsevier Science Publishing Company on page 546.
The limit of choice for point 42 is only herein termed absorption boundary. In other words, choice of point outside boundary of figure in FIG.16 results in other than asymptotic response.
The absorption boundary extends from gauge point 42 to the taping line 46 (FIG.S). Lift point 44 according to the invention is located somewhere within the absorption boundary 56. A profile extending above line 54 is essentially an ellipse. Thus, a profile having a lift point above curve 54 will have a negative discriminate and, in turn, induces the periodic circular oscillation inherent in elliptical profiles. Negative discriminate indicates a hollow profile. Line/curve 54 defines a parabola with a discriminate of zero. Curve 54 is upper bound for choice of point 44. A profile taken along line 58 is of course linear and, therefore, exhibits the oscillation effects discussed above.
When the design of the tire profile uses point 44 within absorption barrier 56, an asymptotic profile is formed. Wear is distributed over the entire profile. The result is that the wheel is more durable and performs better over a longer period. Point 44 chosen closer to line 58 is more aperiodic. Point chosen closer to curve 56 is more periodic.
In FIG. 17 the direction of rolling is toward axle 84. Axle 82 is a tangent response. Axle 84 leads a tangent response. Axle 86 lags a tangent response. Leading a tangent response results in oscillation.
Since feedback mechanisms tend to oscillate, attempting to maintain exactly a tangent response results in oscillation. Consequently, the best axle position is one that slightly lags a tangent response. It is also important to note that the combined profiles of each wheelset provide a phase angle that lags a tangent response. Phase angle is important in the relationship between the storage of kinetic energy as potential energy of gravity. Thin ie a source of energy that could result in oscillation, Due to the profile of the tread, the amount of potential energy stored by the train wheel exhibits a lag in phase angle in relation to the tangent response. The phase angle lags so that the 8 Least amount of energy is stored by the wheel. There is less resultant oscillation.
To explain further, FIG. 17 shows the relationship between the opposing asymptotically profiled tires 83. For example, a train track 80 is rounded in a semicircular fashiozx. A wheelset when in position i0 82 is traveling in a tangent orientation in relation to track 80 is said to have a zero phase angle. When the left tire 83 is dropped back from the tangent position 82 to position 88, the wheelset has a lagging or negative phase angle. When left tire of the wheelset Leads the right wheel as shown in position 84, the wheelset has a positive or leading 15 phase angle.
A hyperbolic profile exhibits a negative phase angle similar to wheelset 86 and, therefore, allows the load to move laterally with corresponding lateral movements of the wheels. Thus, the lagg'mg phase response of the asymptotic profiled wheelset dampens the lateral 20 oscillation of the load. A positive phase angle, on the other hand,cauaes oscillation as the wheelaet seeks an equilibrium state such as shown by wheelset 82.
The lagging wheelaet response happens to have the lowest potential energy of the three response modes. Specifically, the 2~ potential energy stored in raised wheelset must not be Large enough to overcome the frictional forces between the rail and tire, By making the stored kinetic energy have an asymptotic response, absorptive damping is exhibited in the wheel in addition to the normal frictional damping provided by a straight conical tread profile, 2~.5~9~.~
The profile according to the invention can not correct for defects in machining accuracy, incorrectly configured rail, incorrectly configured vehicle or vehicles that operate too rapidly for existing conditions. Investor assumes no liability for such cases and assumes no liability for unlicensed attempts to use product.
Having described and illustrated the principles of the invention in a preferred embodiment thereof, it should be apparent that the invention can be modified in arrangement and detail without departing from such principles. Further, use of word asymptotic profile is inclusive of term hyperbolic form. The application is not just to trains, but all railed devices including manufacturing processes. I claim all modifications and variation coming within the spirit and scope of the following claims.
Claims (19)
1. A profile for a train wheel, comprising:
a flange section having a protuberance for extending over an edge of a rail;
a fillet section extending from field side surface of the flange; and, a tread section extending from the fillet section toward a field side of the train wheel, the tread section including a lift point adjacent to the fillet section, a taping line and a wear point and having a continuous asymptotic profile between the lift (design) point and the taping line including an asymptotically increasing radius that increases continuously from the lift point toward the taping line thereby causing the train wheel to respond asymptotically to changes in direction.
a flange section having a protuberance for extending over an edge of a rail;
a fillet section extending from field side surface of the flange; and, a tread section extending from the fillet section toward a field side of the train wheel, the tread section including a lift point adjacent to the fillet section, a taping line and a wear point and having a continuous asymptotic profile between the lift (design) point and the taping line including an asymptotically increasing radius that increases continuously from the lift point toward the taping line thereby causing the train wheel to respond asymptotically to changes in direction.
2. A profile according to claim 1 wherein the continuous asymptotic profile extends at least partially between the taping line and the extreme wear point.
3. A profile according to claim 1 wherein the continuous asymptotic profile extends at least partially across the fillet section.
4. A profile according to claim 1 wherein the asymptotic profile comprises a continuous asymptotic curve that extends from the fillet section to the wear point.
5. A profile according to claim 4 wherein at least a portion of the fillet section includes a profile having an elliptical curvature.
6. A profile according to claim 1 wherein the lift point comprises minimum radius on the tread section.
7. A profile according to claim 4 including a tread transfer section, the asymptotic profile extending continuously through the wear point in the tread section to a field side of the wheel.
8. A profile for a train wheel according to claim 1 wherein the curvature of the asymptotic profile is defined by a given expression:
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0 ;
where x is equal to a horizontal distance for every point on the tread measured from a given reference point and y is equal to a vertical distance for every point from said reference point; and wherein a discriminate value for the given expression for every point has the following value:
B~2-4AC > 0.
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0 ;
where x is equal to a horizontal distance for every point on the tread measured from a given reference point and y is equal to a vertical distance for every point from said reference point; and wherein a discriminate value for the given expression for every point has the following value:
B~2-4AC > 0.
9. A train wheelset for a train rail, comprising:
a first wheel and a second wheel, each wheel having a flange section, a fillet section and a tread section, the first and second wheel each having a field side oriented in opposite directions on the train rail;
and, the tread section for the first and second wheel each having a continuous asymptotic curvature with a radius that increases continuously toward the field side for creating a feedback response between the first and second wheels that responds asymptotically to lateral movement of the first and second wheel on the train rail.
a first wheel and a second wheel, each wheel having a flange section, a fillet section and a tread section, the first and second wheel each having a field side oriented in opposite directions on the train rail;
and, the tread section for the first and second wheel each having a continuous asymptotic curvature with a radius that increases continuously toward the field side for creating a feedback response between the first and second wheels that responds asymptotically to lateral movement of the first and second wheel on the train rail.
10. A train wheelset according to claim 9 wherein the curvature of the asymptotic profile is defined by a given expression:
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0 ;
where x is equal to a horizontal distance for every point on the tread section measured from a given reference point and y is equal to a vertical distance for every point from said reference point; and, wherein a discriminate value for the given expression for every point has the following value:
B~2-4AC > 0.
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0 ;
where x is equal to a horizontal distance for every point on the tread section measured from a given reference point and y is equal to a vertical distance for every point from said reference point; and, wherein a discriminate value for the given expression for every point has the following value:
B~2-4AC > 0.
11. A train wheelset according to claim 10 wherein the tread profile comprises a machined asymptotic outline.
12. A train wheelset according to claim 9 wherein the first and second wheel tread have hyperbolic profiles that form opposite mirrored sections of catenary, the tire profiles operating in conjunction to provide asymmetric resistance to lateral displacement of the wheelset in relation to the train rail.
13. A train wheelset according to claim 12 wherein the first and second wheels have a natural frequency substantially longer than a forced frequency exerted by the rail against the wheels.
14. A train wheelset according to claim 9 wherein a local minima for a load supported by the first and second wheels exists when the first and second wheels align in a centered position between the rails.
15. A train wheelset according to claim 9 wherein the first and second wheels exert a phase lag in relation to a load supported by said wheels when traversing about a curved rail.
16. A method for forming a profile on a train wheel having a flange section, a fillet section extending from the flange, a tread section extending from the fillet section and a transfer section, the tread section having a lift (design) point adjacent to the fillet section, a center taping line and a wear point adjacent to the transfer section, the method for forming the train wheel profile comprising the following steps:
selecting a reference point in relation to the train wheel;
identifying five points on the tread section;
selecting each of the five points that satisfy a given equation as follows:
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0;
where x is equal to a horizontal distance for each of the selected points from the reference point and y is equal to a vertical distance for each of the selected points form the reference point;
selecting the five points such that a discriminate value for the given equation has the following value:
B~2-4AC > 0; and, forming the tread section to intersect each of the five points on the tread section thereby forming a tread profile having a continuous hyperbolic curvature.
selecting a reference point in relation to the train wheel;
identifying five points on the tread section;
selecting each of the five points that satisfy a given equation as follows:
Ax~2 + Bxy + Cy~2 + Dx + Ey + F = 0;
where x is equal to a horizontal distance for each of the selected points from the reference point and y is equal to a vertical distance for each of the selected points form the reference point;
selecting the five points such that a discriminate value for the given equation has the following value:
B~2-4AC > 0; and, forming the tread section to intersect each of the five points on the tread section thereby forming a tread profile having a continuous hyperbolic curvature.
17. A method according to claim 15 including selecting the five points such that the radius of the hyperbolic curvature increases toward the taping line and a given minimum radius of the hyperbolic curvature is located adjacent to the fillet section.
18. A method according to claim 17 including selecting the five points such that a given minimum radius of the hyperbolic curvature is located on the fillet.
19. A method according to claim 16 wherein every location on the tread section and portions of locations on the fillet section and transfer section satisfy the given expression and discriminate value.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA002155914A CA2155914C (en) | 1995-08-11 | 1995-08-11 | Train tire profile |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA002155914A CA2155914C (en) | 1995-08-11 | 1995-08-11 | Train tire profile |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CA2155914A1 CA2155914A1 (en) | 1997-02-12 |
| CA2155914C true CA2155914C (en) | 2000-10-31 |
Family
ID=4156411
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA002155914A Expired - Fee Related CA2155914C (en) | 1995-08-11 | 1995-08-11 | Train tire profile |
Country Status (1)
| Country | Link |
|---|---|
| CA (1) | CA2155914C (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| DE102011121785A1 (en) * | 2011-12-21 | 2013-06-27 | Voith Patent Gmbh | Wheel disc for a rail vehicle |
-
1995
- 1995-08-11 CA CA002155914A patent/CA2155914C/en not_active Expired - Fee Related
Also Published As
| Publication number | Publication date |
|---|---|
| CA2155914A1 (en) | 1997-02-12 |
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| EEER | Examination request | ||
| MKLA | Lapsed |