Note: Descriptions are shown in the official language in which they were submitted.
~
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
4
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.
5
CA 02155914 2000-04-07
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.
6
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
7
2
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.
8
CA 02155914 2000-04-07
1
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;
9
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
47-65
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
11
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
12
~~~~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
13
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
14
CA 02155914 2000-04-07
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
16
~~.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
17
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
18
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.
18
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.
21
~~.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
22
~~~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.
23
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,
24
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
CA 02155914 2000-04-07
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
26
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,
27
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.
28
