Note: Descriptions are shown in the official language in which they were submitted.
g~
-- 2 --
This invention relates to railroad vehicles
having trucks which are equipped with steerable axles.
Various systems have been proposed in which a
railroad truck comprises a pair of wheelsets with each
5 of the wheelsets being attached to the truck in a
manner that allows the wheelsets to move to a radial
configuration when the truck is travelling on curved
railroad tracks. One such example is Canadian Patent
No. 1,083,886 issued August 19, 1980, to Urban Transportation
10 Development Corporation Limited. Other examples include
U.S. Patent No. 2,071,207 to Heinrich Knecht (1935) and
U.S. Patent No. 356,347 to Brown and Midelton issued 1887.
In the foregoing patents and indeed in most
systems providing a two axle truck having steerable axles
15 the axles are mounted to the truck so as to be pivotal
about a substantially vertical pivotal axis. When the
truck enters a curve, the axles pivot about their
respective pivotal axes so as to be radially aligned.
Radial alignment is desirable for many reasons including
20 reduction of wear of components and noise which would
otherwise be caused by slippage of the wheels along the
rail. It is convenient to refer to a truck having
steerable axles as a steerable truck.
Although several mechanisms have been provided
25 to permit radial alignment of axles very little thought
has been given to prevenking divergent behaviour of these
~t.
.: .~
~2~91~
-- 3
ax]es when the truck is travelling along the track. sy
reason of minor variations in the rails or other problems
there may be slight disturbances which would tend to
turn the truck from its desired position. Analysis
5 mu~t be carxied out to determine whether the truck
will return to its desired position, once deviated
from it, or whether it will continue to deviate further
from the desired position until there is contact
between the flange of the usual railroad wheel and the
10 rail,
~ In most mechanisms including that illustrated
in the aforesaid Canadian patent, the truck is attached
to the railroad vehicle body for relative pivotal
movement about a substantially vertical pivotal axis.
i5 In the ordinary case, where it is desired that the axles
share the load equally, and the pivotal connection
between the truck and the car body also carries the load
of the car to the truck, the pivotal connection is
located centrally between the axles. In some situations
20 where the designers did not intend the wheels to share
equal loads (such as when only one axle is driven) the
pivotal axis is offset from the point midway between the
axles.
We have found that steerable trucks are not
25 necessarily stable. That is, when such steerable trucks
are disturbed from the desired position they may not
-- 4 -
return to the desired position but rather continue to
deviate until there is contact between the rail and
flange of the wheel. We have found that it is possible
to stabilize such a steerable truck by offsetting the
5 pivotal axis between the truck and the railroad vehicle
body by a certain critical amount. Alternatively,
stability may be achieved in such a truck by providing
resilie~t spring-like forces to restrain pivotal movement
of the truck with respect to the car body alone or in
10 combination with offset of the pivotal axis. We have
discovered that there is an important mathematical
relationship between the conicity of the wheels of the
standard railroad vehicle forming part of the typical
wheelset, the amount of off~et oE ~he pivotal axis from the
15 midway point between the two axles of such a truck, and
the resil~ence of the pivotal connection between the
truck and the car body of the vehicle.
According to this invention a railroad vehicle
is provided for movement on rails. The vehicle has at
20 least one body portion and at least one truck. ~he truck
has an outboard and an inboard axle relative to said body
portion. The axles are mounted to said truck for pivotal
movement about respective vertical axes such that said axles
may be moved to a radial position when said vehicle is
25 travelling on curved track.
Each of said axles is adapted to support wheels of a wheel-
set, each wheelset having a conicity ~ .
The truck has a pivotal connection between the truck and
said body portion.
~L2~9~
The truck has steeriny means whereby the axles are moved
to the radial position when said vehicle is travelling on
curved track, the alignment being obtained in response
to the angle through which the truck pivots relative to
said body portion in traversing the curved track. The
ratio of said angle and the resultant angle through which
each axle pivots relative to the truck defines a steering
ratio for the steering means, which ratio is common for
the two axles of said truck.
The railroad vehicle comprises stabilizing means
to inhibit deviation of the truck from the desired
position relative to the rails. The stabilizing means
has
a) a resilient element in said pivotal connection
between said truck and said body portion, said element
havin~ a torsional stiffness KT
and
b) said pivotal connection between said truck and
said body portion is offset a distance X from a point
midway between said axles
wherein KT, ~ , and IX I may each be zero or larger
and KT, ~ , and X are selected such that said truck
acting under the usual forces generated by pivotally
a~fixed wheelsets having said conicity ~ develops
restoring forces under deviation from said desired
position tending to restore the truck to said desired
position.
~2~900
6 --
The invention will now be discussed in greater
detail in association with the follow:Lng drawings which
illustrate schematically a railroad vehicle having
trucks according to a preferred embodiment of the
invention:
Figure 1 is a schematic view of a railroad
vehicle having two trucks according to the invention;
Figure 2 illustrates schematically in plan,
the forces and moments acting on a steerable truck in
which may be provided an elastomeric restraint or tor-
sional spring between the truck and vehicle body;
. .
Figure 3 is an enlarged view of a portion of
Figure 2 showing a steering lever and connections in
close up;
Figure 4 is a view similar to Figure 3
illustrating the functioning of the steering lever in
aligning an axle on traversing a curve;
Figure 5 is a view similar to that of Figure 4
illustrating the additional angular displacement of
an axle resulting from spring element connections between
the steering means and the axle;
Figure 6 illustrates a means of arranging for
an elastomeric restraint between the car body and the
truck frame;
Figure 7 illustrates a means of arranging
for an air bag suspension having torsional stiffness
between the car body and the truck frame;
39~6)
Figure 8 is a sectional view taken in the
direction of ].ines 8-8 of Figure 7;
Fiyure 9 illustrates schematically in plan a
portion of an articulated vehicle having a plurality of
trucks according to the invention; and
Figure 10 is a sectional view taken in the
direction o~ lines X-X of Figure 9.
Figure 1 illustrates a railroad vehicle 2 for
travelling on railroad rails 4. Vehicle 2 comprises two
trucks 5 supporting a body portion or car body 10.
Each truck 5 comprises a pair of wheelsets having
wheels 12. Each wheel 12 has a flange and conical
running surface as ~ypicallv used in railroad
wheelsets. The conicity of the wheelset, being an
average of the conicity of the two wheels 12, is denoted
by ~. ~ = 0 is used to include the case of zero
conicity when each wheel 12 outside of the flange
thereof is essentially cylindrically shaped.
In order to appreciate the forces which develop
on a steered truck, reference is made to Figures 2 to 5.
Figure 2 illustrates the forces and moments on a steered
truck 5. Truck 5, depicted here as a leading truck, has
outboard and inboard axles 20, 22 respectively. Axles 20,
22 are mounted to truck 5 for pivotal movement about
respective vertical axes 24, 26 such that axles 20, 22
can be moved to a radial position when vehicle 2 is travel-
ling on curved track. The truck 5 is provided with a
~ZlB900
pivotal connection 28 shown in Figures 1, 6 and 7 between
truck 5 and car body 10, for pivotal movement a~out a
vertical axis 30.
The vertical axis 30 is not necessarily
located at a point midway between axles 20, 22. In
Figure 2, axis 30 is offset a positive distance x in
the direction toward outboard axle 20 from the geometric
truck centre. However it is assumed throughout that
axis 30 remains at the centre of rails 4. As illustrated
in Figure 6 and in Figures 7 and 8, the pi~otal
connection 28 is preferably provided with an elastomeric
restraint or torsional spring (used here to include a
suspension element with torsional stiffness, such as an
air bag suspension). The elastomeric restraint, shown as
32 in Figure 6 or alternatively the air bag suspension
shown as 33 in Figures 7 and 8, produces a torsional
stiffness, denoted by XT, of pivotal connection 28. The
torsional stiffness KT is greater than or equal to zero,
KT=0 being used to include the case of no elastomeric
restraint or torsional spring provided in pivotal connection
28.
Truck 5 contains outboard and inboard steering
means or mechanisms of a known kind, 34, 36 respectively,
for moving respective axles 20, 22 to the radial position
when the vehicle is travelling on curved track. This
alignment of axles 20, 22 in the radial position is obtained
in response to the angle through which truck 5 pivots
about axis 30, relative to car body 10, in traversing the
,-;
8~
g
curve. The angle through which axle 20 or 22 ~ivots
relative to truck 5 in attaining this alignment, is called
the axle angle. The angle through which truck 5 pivots
relative to car body 10 is called the body angle. The ratio
of the body angle to the axle angle, denoted by k, is a
characteristic of the steering means, known as the steering
ratio.
As shown in Figure 2 outboard steering means
34 co.mprise left and right steering levers 38, each
. 10 having outer end portion 40 connected by connector 4~ - ;
to car body 10 at a lateral distance f from vertical
axis 30. Inner end portions 42 of levers 38 are
connected by connectors 44 to axle 20 at a lateral
distance dlon each side of the centre of axle 20.
Between respective end por~ions 40, 42, steering levers
38 are pivotally connected to truck 5 for movement
about axes 46, at a lateral distance e on each side of
central axis 47-47 of truck 5.
Inboard steering means 36 comprise left and
right steering levers 48, each having outer end portions
50 connected by connector 51 to car body 10 at a lateral
distance h from vertical axis 30. Inner end portions 52
of levers 48 are pivotally connected to truck 5 for
movement about axes 54, at a lateral distance g on each
side of central axis 47-47 of truck 5. Between respective
end portions 50~ 52, steerin~ levers 48 are connected
by connectors 56 to axle 22 at a lateral distance d2
on each side of the centre of axle 22.
Connectors 44 and 56, respective:Ly connecting
steering means 34, 36 to respectlve axles 20, 22 are
each provided with a spring element, the spring elements
having respective stiffnesses denoted by Kl and K2. It is
well known to those skilled in the art that the stiffnesses
of all of the components of the steering means for each
axle may be represented by the single stiffness Kl or K2
1~ as the case may be.
The choice of ~istances f, h from axis 30
to the connection at car body 10 of connectors 41, 51
respectively, determines the value k of the steering ratio
for each axle. In order to appreciate how the value
of f is selected for qiven values of k, ~ and e, refer-
ence is now made to Figures 3 and 4. Figure 3 depicts
one steering lever 38 ~on the left side of Figure 2)
connected to axle 20 and car body 10 and pivotally
connected to truck 5 about axis 46.
The desired operation of steering means
34 in aligning outboard axle 20 to a radial position
is first described. As a result of railroad vehicle 2
travelling on curved track, car body 10 swivels with
respect to truck 5 about vertical axis 30. If railroad
vehicle 2 is travelling around a clockwise curve, car
--10
body 10 will swivel counterclockwise, relative to (lead-
ing) truck 5, about vertical axis 30. Connectors 41 will
then exert a force on levers 38 tending to cause them to
pivot counterclockwise about axes 46, as illustrated for
the left lever 38 in Figure 4. Levers 38 will then apply
cooperating forces through connectors 44 to axle 20.
The result is to force axle 20 to pivot clockwise
about vertical axis 24 into a radial position with
respect to the clockwise curve. Steering means 36
similarly functions to align axle 22 in the radial
., , . , . . . . . ~ .
position.
Suppose that car body 10 rotates about axis
30 an angle a counterclockwise relative to truck 5.
Since connector 41 is attached to car body 10 at a
lateral distance f from axis 30, (left) connector 41
will be displaced a distance approximately f.~ inboard.
Therefore, outer end portion 40 of (left) lever 38
will also be displaced inboard approximately a distance
f.~. Since the steering ratio of steering means 34 is
k, axle 20 will swivel an angle ~/k clockwise relative
to truck 5. Connector 44 attaches to axle 20 at a
lateral distance dlfrom the centre of axle 20. There-
fore (left) connector 44 will be displaced outboard a
89~
- 12 -
distance approximately dI4/k. Hence inner end portion
42 will also be displaced outboard a dlstance approxi-
mately dl.~/k. Therefore, from Figure 4, to a good
approximation,
~ ~ ~ e -
or
" 10'' ' " ' ~ (lr~)~+ll
Therefore,
~-1, = d~ (d~ e)~ 2
(1~ e) ~
~-e ~
!~)
( d~r e)~ t 1,
However, in normal operation there will be
disturbances acting on the truck. In what follows, refer-
ence is again made to Figure 3. Suppose that a torque or
moment Tl exists acting on axle 20 about vertical
axis 24~ (Moments in the clockwise direction are
39
- 13 -
posltive.) This will result in a force F applied to
each lever 38 at a lateral distance dlfrom central
axis 47-47 of truck 5. Levers 38 will thus attempt to
pivot about axes 46, resulting in forces S being
applied to levers 38 by car body 10 through connectors
41. The forces S are applied at a lateral distance f
from central axis 47-47 of truck 5.
Since axis 46 is located at a lateral distance
e from central axis 47-47 of truck 5, taking moments
, . . . . . . . . . .... .. . . . . . . . .
about axis 46 yields
S(~- e) = F
or
S - F (~
The forces S and F on lever 38 are opposed by a
force R at the fulcrum at axis 46. -Therefore,
R = F t s = F ( I ~
~2~8~
Both left and right steering levers 38
each in turn apply a force of amount F through a
connector 44 on axle 20, in opposition to the moment Tl,
so that
2 F1~ - ~
Similarly, the moment on truck 5 due to levers 38 is
.R.~
.. . . . . . . . .. ...
o ~ ~ ) e
Fl
~-e
= l, ~L, T
~ _ c
Therefore by equations (1) and (2)
e~ _ ~ T~
( e - l
= ~ Tl ~3)
I'4-'
~2~
Ml is the moment on leading truck 5 due to a moment T
on leading axle 20.
Similarly, the moment on leading truck 5 dùe
to a torque or rnoment T2 on trailing axle 22 is
M2 ~ (1- ~ ) T~ (4)
The results given in equations (3) and (4) have been derived
for a mechanism involving horizontal levers. It can be
shown that the same relationship exists in any mechanism
lo with a steering ratio of k between the body angle and
the axle angle regardless of the actual geometry chosen
for the mechanism.
Reference is now made to Figure 2 in order to
derive an expression for the net moment on truck 5.
Suppose again that there exists a moment Tl on axle
20 and a moment T2 on axle 22. Consequently axle 20
will pivot an angle ql, and axle 22 an angle q3,
~elative to car body 10. This will result in creep forces
Fl acting on truck 5 along axle 20, and F2 acting on
truck 5 along axle 22, in the direction shown in Figure 2.
Similarly creep torques will act on axles 20 and 22.
Such creep forces have been studied in papers by F.W.
Carter, "On the Action o~ a Locomotive Driving Wheel"
(Proc. Royal Soc. London, Series A, Vol. 112, 1926,
~2~
- 16 -
151-157) and J.~. Kalker, "On the Rolling Contact of
Two Elastic sodies in the Presence of Dry Friction"
~Doctoral Dissertation, Technische Hogeschool, Delft,
Netherlands, 1967). Further study has been done by
D. E. Newland, in the paper "Steering a Flexible
Railway Truck on Curved Track", ASME Paper No. 69-RR-5,
read at Joint Railroad Conference, Montreal, Quebec,
1969). He shows that
~ - 2~ ( ~ ~ 7
.. . .
Fl ~ 3~ ~ )
T, -- - 2 ~, c ( ~ + ~ ) ~ ~)
T2 = 2 $S C ( ~
Here, fL is the lateral creep coefficient,
fT is the longitudinal creep coefficient,
q2 is the angular displacement of truck 5 relative
to car body 10,
a is the distance between vertical axes 24, 30,
b is the distance between vertical axes 26, 30,
c is one-half the rail gauge,
oo
~ is the wheelset conicity,
rl is the wheel radius for the outboard axle,
r2 is the wheel radius for the inboard axle, and
V is the velocity of the railroad vehicle 2.
In the steady state, equations (5) - (8)
become:
t = 2 ~, ~, (9)
F~ 3
T = _~ ~ c )~a ~. .. . .
1~
~ = 2 $~ C ~ ~2 (12)
r~
Since steering means 34, 36 each have steering
ratio k,
~ = &~(lt ~ 3)
~ = ~ (I+ ~) t A~3 (!~)
where ~ ql and ~ q3 are additional axle angular movements
due to the stiffnesses Kl and K2 of the spring elements in con-
25 nectors 44, 56 respectively or their equivalence in any othersteering means. The relationship between A ql and Tl,
I ~
I ~
8~
and between a q3 and T2, is now derived.
Figure 2 illustrates steering means which is
symmetric about the longitudinal central axis 47-47 of
the truc~. If the steering means is not symmetric an
equivalence relationship can be derived to equate it
to the symmetric case. Let the stiffness of the steering
means be KL and the distance of its point of application
from the axle centre be dL for the left hand side of the
steering means and let the corresponding quantities be
KR and dR for the right hand side. The torque required
to produce a displacement ~ will be given by T=KLdL2e +
KRdR e; for the symmetric case the corresponding torque
is given by T=2Kldl e from which it can be seen that
Kl dl KLd~ + KRdR
Clearly this can be extended to include more than one
attachment per side. Because this direct equivalence
relationship exists it is valid to treat all steering
means as if they were symmetric, using equivalent values
for K and d as derived above. All further analyses
recorded herein include only the symmetric case on the
understanding that the unsymmetric case can be substituted
as above.
From Figure 5 it is seen that i a torque T
acts on axle 20 causing it to rotate an additional
angular amount ~ql (illustrated in dotted outline) this
~L2~39~
will result in a change of length dl..~ ql in each con-
nector 44. The resulting force then exerted by each
connector 44 is Kldl. a ql. Since this force is exerted
on each side of axle 20 (in opposite directions) at a
distance dl from vertical axis 24, the resulting moment
is 2Kldl ~ ql, which must equal the momen~ Tl on axle 20.
Therefore,
2 K, 1~2 ~
or, .
2 K,l,
Similarly, it may be shown that
~ ~3 = ~
so that equations (13) and (14) become:
~J g 2 K, 1,
~3 ~ & (I~ ~
~ q q~
8~
- 20 -
sut the torques Tl and T2 are proportional to
the yaw displacement q2 of the truck frame (which may
occur as a result of a disturbance of the truck) as
can be seen from equations (11) and (12). Substitution
yields
g,
g3 ~ C ~ l ~
K~ da . r~ .
Referring again to Figure 2 and equations (3)
and (4), it is seen that the net moment acting on
leading truck 5 is
M, = ~, Q - F~. + ( It;t~ 9)
Substituting into (19) from equations (9) to (12), (17)
and (18), yields
g~ t ~T ~ L ~ ~ J
- 2 fT ~ ~y
- 21 -
Dividing the net moment ML on leading truck 5
by the angular displacement q2 of truck 5 wlth respect
to car body l0 gives the effective torsional stiffness
QL acting on leadiny truck 5 in a direction to increase
angular displacement q2 ~i.e. non-restoring)
= 2 fL { ~(~~~) + ~ ~ ~K~ K 1I r ~
0 ~ 2$T C~(~r - Qr ) - 2~ ( a~ +~ ~ (2.0)
. . .. . . . .
As there are no other forces acting on leading
truck 5 (other than those which may exist between the
car body and truck), it will be stable if the right
side of equation (20) is less than or equal to zero.
Otherwise, once a disturbance has induced a yaw dis-
placement q2 of the truck frame, leading truck 5 may
move so as to increase the displacement from the desired
position until there is contact between rail 4 and the
flange of a wheel 12. However, when the right side of
equation (~0) is positive, stability can still be
ensured by providing pivotal connection 28 between truck
5 and car body l0 with an elastomeric restraint 32 or
a torsional spring such as air bag suspension 33 having
a torsional stiffness greater than or equal to the right
side of equation (20). It is pointed out that when the
~2~391~
right side of equation (20) is not positive, or alter-
nately when pivotal connection 28 is provided with a
suitable elastomeric restraint or torsional spring,
truck 5 will be stable provided that it is the leading
5 truck on railroad vehicle 2.
I~ equation (20), make the following substitu-
tions:
a = ~2 - Y
1 0 .~
.. ..
Here A is the wheel hase of truck 5, so that x measures
the amount of offset between pivotal vertical axis 30
and the point midway between axles 20, 22, the positive
direction being toward outboard axle 20. Thus, equation
(20) becomes
Q = 2~L{t~ 2~) ~ [(A-x) +(A~) K,d,r,
+ ~(( L X)(~ y)(l_~.)L
( 2
Truck 5 will be stable i.f the right side of equation (21)
is less than or equal to zero, in the ahsence of any
25 other forces acting on leading truck 5. Alternatively,
- if the right side of equation (21) is positive, truck
` ~2~89~3~
5 will be stable if an elastomeric restraint or
torsional spring is provided to give a torsional stiff-
ness KT of pivotal connection 28 which is greater than
or equal to the right side of equati.on (21). That is,
truck 5 will be stable as a leading truck provided
that
I~T >~ 2~ f(,~ C) Sd~ , ) +(~ t l~r~ f ((. X~
Examining now t~e sit.uation ~or a trailing truek,
the effective torsional stiffness of the forces acting
can be derived in a similar manner to the above and is:-
4 = 2~ x)-~r~(A_~)~ x)~K~ K~r~ x~lt~)L~ xXI-~)r)~}-
Here x is again positive in the direction towards the
outboard axle (in this case, the trailing axle) from the
geometric truck centre. For stability of the trailing
truck, the right side of equation (23) must be less than
or equal to zero when there are no other forces acting
on the truck. However, when an elastomeric restraint
or torsional spring is provided so that the pivotal
connection between the trailing truck and car body 10
has torsional stiffness KT, the trailing truck will be
9~'
- 24 -
stable if
~A ~ c~ l(A-x)~ttA+x) L~l~ r~ ((A-2X~ )L ~(g~ rL ]3
(~4)
Therefore the truck will be stable as both a
leading and as a trailing truck provided that the
truck satisfies both expressions (22) and (24).
Preferably therefore both leading and trailing trucks
of railroad vehicle 2 will satisfy both expressions
(22) and (24). In this case, the trucks of railroad
vehicle 2 will be stable for either direction of motion .. ..
of railroad vehicle 2. However, it will be appreciated
that the parameters in expressions ~22) and (24) for
one truck need not necessarily have the same values as
the corresponding parameters in these expressions for
the other truck. Preferably, for convenience, railroad
vehicle 2 will be provided with identical trucks, so
that both trucks will have the same values for the
parameters in expressions (22) and (24~.
The above general expressions can be simplified
for the more typical case of trucks in which the steering
means stiffnesses and geometry and the wheel radii are
the same for inboard and outboard axles. If we write:-
1 2
1 2
rO- rl - r2
~L2~
- 25 -
we get, fo.r the leading truck
K~ ~ 2 ~L ~ d~ [ ( " ) t ( ~ )J }
and for the trailing truck
>~ ~ ~L ~ ( ~ Kl~r ~ L
In what follows, expressions are derived to
ensure stability of a truck when used both as a leading
and as a trailing truck. Accordingly, truck 5 may be a
leading or a trailing truck, and it is assumed below
.. that.truck 5 always satisfies both expressions.(22)
and t24).
Consider first the case where the wheels 12 of
truck 5 have no conicity, that is ~ = 0. The truck will
be stable whether used as a trailing or as a leading
truck provided that both expressions ~22) and (24) hold
with ~ set equal to zero. (Note that this is derived
from the equations for the general case.)
Therefore,
K~ L (
and
Kt ~ X )
or combining the latter two expressions
KT ~Y 2 ~L I ~
~L2~8~
- 26 -
When each wheel 12 has no conicity, a -truck
can still be stable as both a leading and as a trailing
truck even without offsetting the pivotal vertical
axis 30 (connecting the truck and the car body) from the
geometric truck centre. Setting A = 0 and x = 0 in
both expressions (22) and t24) yields
2 f, ~ (2~,
o K~ 7~ ;L ~
If expression ~26) is satisfied, truck 5 will be stable
both as aleading and as a trailing truck when the wheelset
eonicity is zero and the pivotal vertical axis 30
eonnecting truck 5 and car body 10 is not offset from
the geometric truck centre.
Consider now the ease when the pivotal eonneetion
28 between truck 5 and car body 10 is not provided with
an elastomerie restraint or torsional spring, that is
when KT = In this case, a truck will be stable both
as a leading and as a trailing truck if the right sides
of expressions (21) and ~23) are not positive.
Suppose in addition that the wheelset eonicity
~ = 0. Then a truck will be stable both as a leading
and as a trailing truek, if in expressions (21) and (23),
~ ~ ~ 6
,
~L21~
2--
and
x ) ~ o
respectively. Therefore,
~ _ A (27)
Suppose again that KT - , so that no elasto-
meric restraint or torsional spring is provided in
pivotal connection 28 between truck 5 and car body 10.
Assume in addition that the vertical axis 30 of pivotal
connection.28 is.not offset ~rom the geometric truck
centre, so that x = O. For the right sides of expressions
(21) and (23) to be less than or equal to zero, then
A _ ~ FT e~ ~ A2 ( l ~ K~ ) ~ K~ J~ r
and
~K~ K~l,r, ~ ~r ~rL (29)
Hence (from 28)
~ r;
t ~-~lr
~3O)
~8g~
- 2~ -
and ~from 2~)
q r;
when K~J~2 ~$ c ¦K~d,~, ((h~ (h-l)) a~ ( K d ~ ] (31)
$, ~'; k r~ , d~,~ r,,
Clearly (29) is satisfied if
K, 1' r, ( ~r, ~ LI~'r,
Thus, when there is no elastomeric restraint or torsional
. spring provided in pivotal connection 28, and when the
vertical axis 30 of pivotal connection 28 is not offset
from the geometric truck centre, the truck will be stable
both as a leading and as a trailing truck provided that
expression (30) holds and in addition either expression
(33) holds or expressions (31) and (32) are true.
Finally, consider the general case where the
torsional stiffness KT is equal to zero in expressions
(22) and (24). In this case, it follows immediately that
$T C )~ X) ~ ) K~l~ r ] > (~ _~x) - TT ~ X)(~ x) (~
and
$ C1~ ~(A )~ > _(A_?t)~$TC~ ¦~A~-x)(~ L't )Arl 1
For the case where all wheels on the truck have a common
radius rO these simplify to
~ [(A-x) ~ ) K~ 2X)(l- $r c~ )
390
- 29 -
and
K l l ~t ~ - X) t (2 + ~) It~ f ~ 2X)(I _ f~ C )~ )
which can be combined to give
f~ ~ > ~ ) ( $T C ~ ( 3 Ll)
' ' r(A~ t(~ K~
and for the fully symmetric case in which Kl = K2 = K
and dl = d2 = d this further simplifies to:-
.. K I ~A-x)~ (A ~
Therefore when there .is no elastomeric restraint or tor-
sional sprinq provided in pivotal connection 28, and
all wheels on the truck have a common radius, the truck
will be stable both as a leading and as a trailing truck
provided that expression (34) holds.
Figure 6 and Figures 7 and 8 illustrate
alternate arrangements of an elastomeric restraint or
a torsional spring such as an air bag suspension
between car body 10 and ~ruck 5, in order to provide
20 the torsional stiffness KT. In Figures 6 and 7,
pivotal connection 28 comprises a vertical axle 60
arranged in a conventional manner to pivot relative to
truck 5 about vertical axis 30. Axle 60 is tapered
outwardly to a greater diameter at 62 for rigid attach-
25 ment to car body 10.
In Figure 6, between truck 5 and the outwardly
8~
tapered portion 62, an elastomeric restraint 32 isprovided to resist a shearing motion when car body 10
pivots relative to truck 5. Assume as above that the
elastomeric restraint 32 has torsional stiffness KT.
5 If truck 5 rotates an ~ngle q2 relative to car body 10,
the resistance to shear by elastomeric restraint 32
will result in a restoring moment equal to KT.q2.
In Figure 7, the air bag suspension 33 is
provided on each side of pivotal connection 28 between,
10 and attaching to, each of car body 10 and truck 5. Air
bag suspensions 33 are such as to resist shearing when
truck 5 pivots relative to car body 10. If for example
truck 5 pivots clockwise relative to car body 10, the
ends of right air bag suspension 33 will be displaced
15 in the direction of the horizontal arrows 64, 66 in the
sectional view of Figure 8, for a horizontal displace-
ment ~ y between the ends of air bag suspension 33.
Suppose that the torsional stiffness due to air bag
suspensions 33 is KT. Then if q2 is the angular dis-
20 placement of truck 5 relative to car body 10, therestoring moment due to air bag suspensions 33 will be
KT.q2.
Accordingly, the result of the arrangement in
Figure 6, or alternatively in Figures 7 and 8, is to
25 provide pivotal connection 28 with a torsional stiffness
KT. It will be appreciated that by suitable choice o:E
the elastomeric restraint or torsional springs, the
- 3~ -
8g~
- 31 -
torsional stiffness KT can be selected appropriately
as set out above so that the truck is stable.
It is pointed out that the invention includes
railroad vehicles of the type known as articulated
vehicles, as shown in plan view in Figure 9. Articu-
lated vehicle 70 comprises a plurality of body portions
72 joined together by articulated joints 74, illustrated
most clearly in Figure 10. Articulated vehicle 70 is
supported at the outboard portions 76 thereof by steer-
able trucks 5 of the type described above. Similarlyeach articulated ~oint 74 is supported by a steerable
truck 5 of the type described above.
Each of the two outboard trucks 5 has a pivotal
connection between the truck and the respective outboard
portion 76. Each of the remaining trucks 5 has a
pivotal connection 28 between the truck and the respec-
tive articulated joint 74. Each of the trucks 5 contain
steering means whereby the axles of trucks 5 are moved
to the radial position when articulated vehicle 70 is
travelling on curved track. The alignment of the axles Of
each truck 5 is obtained in response to the angle to wh`ich
that truck pivots relative to a respective body portion 72 in
traversing the curved track. The ratio of this angle
and the resultant àngle to which each axle of the respec-
tive truck 5 pivots relative to that truck defines asteering ratio k for each steering means. This ratio is
~2~ 0
- 32 -
common for the two axles of the respective truck 5. The
leading out~oard truck 5 will b~ stable provided that ex-
pression (22) is satisfied, and the trailing outboard truck 5
will be stable provided that expre5sion (24) is satisfied
as was set out above. Each remaining t:ruck 5 is stable
provided that it satisfies expression (22) or (24)
respectively depending on whether it is designed as a
steerable leading truck of the respective trailing body
portion or as a steerable traillng truck of the respective
leading body portion. The remainder of the mathematical
relationships set out above which provide for stability - -
when satisfied also apply to the articulated vehicle.
It should be noted that in Figure l, the
left truck is shown provided with an elastomeric
restraint between truck 5 and car body lO, while the
right truck is shown provided with alternate air bag
suspensions between truck 5 and car body lO. However,
it will be appreciated that both trucks may be provided
with identical elastomeric restraints or torsional
springs, or otherwise arranged to have torsional stiff-
ness between the truck 5 and car body lO.
Finally, it will be understood that theparticular arrangements shown here for providing a
torsional stiffness KT in the pivotal connection
between the truck and car body are not central to the
invention. Various other arrangements may be used to
provide a torsional stiffness KT ~ an appropriate
value as set out above so that the truck is stable.
