3. Dynamics of the Wheelset
3.8 Alternative Methods of Guidance
The early mining railways employed "peg in the slot" as a guidance system and whilst this was superseded by the coned wheelset early in the 19th Century there is a continuous record of invention of alternative guidance systems. In the past many alternative systems were considered because the running gear of railway vehicles often performed badly; rapid wear of wheels and rails, hunting and derailments oc-curred as a result of the lack of understanding of how a wheelset should be incorpo-rated in a vehicle. As discussed above, the smallest radius for flange free curving of
a wheelset with conventional profiles well adapted to conventional rail profiles is about 150 m, and the longitudinal movement of a coned wheelset in taking up a ra-dial position on sharply curved track can represent a design problem. As a target curve radius for some innovative urban transit systems can be as low as 10 m there is scope for alternative methods of guidance of railway vehicles.
For configurations in which the wheels are fixed on a common axle, two condi-tions, positive centering action for small amplitudes within the flangeway clearance and positive retention or static stability for large displacements, can be made the basis for selection of wheel rail geometry. The possibilities are shown in Figure 3.13. Configurations (c) and (e) have positive conicity and rate of change of contact slope with lateral displacement and hence satisfy the basic requirements. It is inter-esting that one conforms to the conventional wheel rail combination whilst the other resembles a plateway, a system much used in the later 18th century.
For some applications it is possible to separate out the functions of support and guidance completely, and provide separate sets of wheels, exerting only normal forces. There are four basic possibilities as shown in Figure 3.14 and some of these have been used both with steel wheels on steel rails and with pneumatic tyres on beams. However, the multiplicity of wheels and the ponderous nature of the switch-ing arrangements make this approach relatively unattractive.
Independently rotating wheels have been frequently proposed as they eliminate the classical hunting problem. Some of the possibilities have been surveyed by Frederich [27]. The essential difference between a conventional wheelset and inde-pendent wheels lies in the ability of the two wheels to rotate at different speeds and therefore, strictly, there is an additional degree of freedom. However, the effect in
(a) (b) (c)
(d) (e) (f)
ε < 0, λ > 0 ε > 0, λ < 0 ε > 0, λ > 0
ε < 0, λ < 0 ε > 0, λ > 0 ε > 0, λ < 0
Figure 3.13 Basic possibilities for wheel-rail geometries. ε (equivalent contact slope) and λ (equivalent conicity) are the linearised geometric parameters defined in equations (2.3.36) and (2.3.37).
terms of the linear wheelset model is to render f11 = 0. The kinematic oscillation of a conventional wheelset is therefore eliminated (the feedback loop of Figure 2.19 is broken) but a measure of guidance is then provided by the lateral component of the gravitational stiffness (reduced by the lateral force due to spin creep) which becomes the flange force when the flangeway clearance is taken up, but this leads to slow self centering action. Extensive experimental experience has shown that indeed the kine-matic oscillation is absent but that one or other of the wheels runs in continuous flange contact [28]. The detailed analysis of a system embodying independently rotating wheels requires that the calculation of creepages take into account the differing wheel rotational speeds, Ωr and Ωl, and this particularly affects the longitudinal creepage. If there is only one point of contact between each wheel and the rail then, neglecting the rotational acceleration of the wheels, each wheel will be rolling with zero longitudinal force and zero longitudinal creepage in accordance with the simple model. However, if a wheel contacts the rail at two distinct points, as is common with freely rotating wheels, the analysis is more complex. Good agreement between calculation and ex-periment is demonstrated in [29, 30, 31].
An attempt to increase the effect of the lateral resultant gravitational force but re-duce the amount of spin is to incline substantially from the horizontal the axis of rota-tion of the wheels, Figure 3.15(a), as put forward by Wiesinger [32]. A generic wheel-set model including the effect of modest amounts of camber has been studied theoreti-cally and experimentally by Jaschinski and Netter [33].
One approach in applying freely rotating wheels is to provide a pivot ahead of the wheels. The wheelset mounted on a leading or trailing arm, a ‘pony’ axle, was used on many steam locomotives and the lack of fore and aft symmetry has a significant effect on the dynamics of the system. In concept this is closely linked to the peg in the slot type of system to be discussed later, in that stability depends on the lack of fore-and-aft symmetry. A similar effect is achieved by a linkage, or the equivalent, to the preceding vehicle in a train. An example of this is the original version of the Figure 3.14 Possible configurations of support and guidance wheels depending on normal
forces only.
A C S
(a) (b)
(c) (d)
Figure 3.15 Some further alternative methods of guidance using independently rotating wheels (a) cambered wheels (b) Talgo train with vehicles guided by vehicle in front (c) Koyanagi’s system with guidance from a central rail ahead of the wheels (d) guidance from a trace using a sensor S, controller C and steering actuator A.
d
ψ 2fψ 2fλdψ/r0
2fψ
2fλdψ/r0 ψ
Figure 3.16 Conventional wheelset mounted on a leading or trailing arm.
Talgo train [34], Figure 3.15(b). The same effect is achieved by guiding the pivot point along a central rail as in the concept, Figure 3.15(c), analysed by Koyanagi [35]. More complex but related arrangements used in light rail applications have been considered in [36, 37]. A logical extension of these concepts is the form of guidance, using independently rotating wheels, shown in Figure 3.15(d). Not only does the dynamics of this system have some interesting features but a modern im-plementation using active controls shows considerable promise, [38]. Guidance is achieved by measuring the lateral displacement from a reference trace on the track, and sending a steering command to an actuator through a controller.
One aspect of the dynamical behaviour of these unsymmetric fore-and-aft con-figurations can be illustrated by considering the conventional wheelset mounted on a leading or trailing arm, Figure 3.16. The elastic stiffness matrix becomes
E k dk
dk d k
y y
y y
=⎡
⎣⎢ ⎤
⎦⎥
2 (1)
In the absence of suspension damping, assumed for simplicity, the equations of mo-tion for the wheelset are otherwise unaltered. A stability diagram for this system is shown in Figure 3.17 in which the stability boundaries are plotted as a function of arm length d and speed. Mounting the spring either forward or aft is strongly stabi-lising as far as the dynamic stability of the system is concerned for the dominant effect is that of increasing the yaw stiffness. As the source of instability is the pres-ence of assymmetric coupling terms in the equations of motion, between lateral translation and yaw, it might be thought that an obvious way of stabilising the sys-tem is to negate one or other of these terms by adding lateral stiffness offset either in front of or behind the wheelset. This is similar to the technique known as ‘mass bal-ancing’ much used in Aeroelasticity. Unfortunately the creep stiffness coupling terms are large in relation to the elastic stiffness couplings and impracticably large arm lengths
-3 -2 0 1 2 3
50 100
d (m) V
(m/s)
0
D
O
Figure 3.17 Stability diagram for conventional wheelset mounted on trailing or leading arm. Parameters from Table 2.1 except ky = 2 MN/m and kψ =0.
or stiffnesses would be necessary. However, if the spring is moved too far forward static instability or divergence occurs. The condition for static stability is that in the character-istic equation p0 > 0 [9] or
( ) /
− = /
−
⎛
⎝⎜ ⎞
⎠⎟
d f
k
l r
y l r 2
1
0 0
λ
λ (2) The system can only be statically unstable if either λl/r0 > 1 and d > 0 or λl/r0 < 1 and d < 0. Thus, a leading arm with low conicity is prone to divergence. This form of instability has been experienced on locomotives in the past, and is a potential problem for articulated vehicles as discussed in Chapter 8.
As mentioned above, the case of freely rotating wheels mounted on a leading or trailing arm may be covered by putting f11 = 0 in the linear equations of motion. In this case the criterion for static stability becomes simply d > 0 so that the wheels must be behind the pivot point for static stability. In fact, more complex arrange-ments make it possible to provide stability in both directions and various ways of exploiting a lack of fore-and-aft symmetry are discussed in Chapter 9.
A simple model of a four wheeled vehicle with independently rotating wheels and guided by peg-in-slot and linkage reveals the basic aspects of the dynamics of such vehicles. These considerations equally apply to the system of Figure 3.15(d). Refer-ring to Figure 3.18, it can be seen that guidance is obtained by steeRefer-ring the wheels (the rear wheels in the opposite sense to those of the front) of the simple vehicle shown by a linkage connected to a reference line (such as a peg in a slot) distance L
ahead of the vehicle centre of mass. Assuming a general form of linkage, then for
2a
y
tL δ δ
y ψ
ε
Figure 3.18 Simple model of vehicle with independently rotating wheels guided by peg-in-slot and linkage.
small displacements, the steering law can be written as
δ = −G y( +Lψ−yt) (3) where G is the gain dependent on the form and dimensions of the linkage. The linear equations of motion are easily formulated and take the form, writing f for f22,
my + 4f( y /V - ψ ) = 0 (4)
Izψ+4fa2ψ /V−4faδ =0 (5) Substituting from (3)
(ms2 + 4fs/V)y - 4fψ = 0 (6) 4fGay + ( I
zs2+ 4fa2s/V + 4fGaL)ψ = 4fGyt (7) The similarity of these equations to the wheelset equations of motion should be noted. The analysis of Section 3 can now be repeated. For the present system, the coefficients of the characteristic polynomial are
p4= mI p3= 4f(ma2+ I)/V p2= 4fGaLm + 8f2a2IV2 (8) p1 = 16f2GaL/V p0 = 16f2Ga
A typical locus of the eigenvalues as speed is varied is shown in Figure 3.19 for various values of the gain G. At low speeds, as in the case of the wheelset, the char-acteristic equation can be approximately factorised in the form
(s2 + 2µ
1s + ω
1 2) (s + µ
2) (s + µ
3) = 0 (9) where
µ2= - 4f/mV (10)
µ3= - 4fa2/IV (11)
µ1 2 2
2 4
= GV − +
a{L (ma I V) / fa} (12)
ω1= V G a/ (13)
For small values of G, the quadratic factor represents a steering oscillation labelled A in Figure 3.19. This has wavelength a G/ independent of speed, in which there is zero creep. For example, if G = 1 rad/m, L = 2 m, a = 1.5 m and V = 10 rn/s then the wavelength of the steering oscillation is 7.69m corresponding to a frequency of 1.3 Hz and the decay rate is 0.81 of critical. The decay rate of this oscillation de-pends on L being positive; in this simple vehicle, as is intuitively obvious, the peg must be placed forward of the centre of the vehicle. In reverse motion, V negative, such a vehicle would be unstable. The two roots given by (10) and (11), labelled B in Figure 3.19 correspond to subsidences in lateral translation and yaw respectively, exactly analogous to the wheelset. At higher speeds these two real roots coalesce to form a well damped oscillation.
For G > 4a/L2and low speeds the steering oscillation is replaced by two sub-sidences. Equation (12) shows that as speed is increased, the inertia forces have the effect of reducing the decay rate of the steering oscillation. The damping of the steering oscillation vanishes at an approximate speed VB when one root of the char-acteristic polynomial is s = iωA. From equation (12)
VA2
= 4fa2L/(ma2+ I) (14) but this approximate result is only valid for small G. A more exact speed VB is easily obtained by substituting s = iω in the characteristic polynomial and equating real and imaginary parts, to give
VB2 =4fa L2 / (ma2+I){1−GL m a2 2 3/ (ma2+I) }2 (15) It can be seen that increasing the gain is stabilising and the vehicle will be stable for all values of V if
G > (ma2+ I)2/m2L2a3 (16)
-1000 -50 0 50
50 100
µ ω
0 100 200
0 50 100
V (m/s)
ω
AA
B B
Fig 3.19 Variation of eigenvalues with speed for peg-in-slot system: a = 1.25 m; f = 10 MN.;
G = 1; L = 2 m; m = 1250 kg.; Iz = 700 kgm2.
Also the vehicle is stable for all values of G if
V < 2fa2L(ma2+ I) (17)
The similarity of the behaviour of the system to that of a railway wheelset is carried over to its curving performance. In steady motion on a uniform curve the equations of motion (4) and (5) reduce to
−4fψ = −mV2 /R0 (18)
4fa2 /R0 = −4faδ (19) so that the vehicle yaws through an angle
ψ = mV2 /4fR0 (20) to react the centrifugal force, and the required steer angle is
δ = a R/ 0 (21) In order to generate these steer angles the vehicle must move laterally through a distance y given by substitution into the steering law, equation (3) from (20) and (21) given by
y=L2 R0−a gR0−LmV fR
2
2 2 0
/ / / (23) It can be seen that at low speeds if G > a/2L2 the vehicle is displaced towards the centre of the curve and if G < a/2L2 it is displaced outwards. As speed increases fur-ther outward movement takes place in order to allow the vehicle to yaw whilst gen-erating the correct steer angle.
Another approach to the improvement of the wheelset as a guidance element is intermediate between a conventional wheelset and one with independent wheels.
This provides a torque connection such as a damper or clutch between the two wheels of a wheelset. Such a scheme was proposed by Benington [39] and a similar scheme has been analysed by Choromanski and Kisilowski [40]. A 'creep controlled' wheelset has been developed [41] which uses a controlled magnetic coupling in the centre of the axle of the wheelset. Various control laws have been used for the con-trol of the coupling, including feedback of creep measurements. Essentially the wheels have a good torque connection at low frequencies so that curving ability is maintained but at high frequencies, typical of wheelset kinematic frequencies, the wheels are more or less uncoupled so that instability does not arise. The stability of wheelsets with independently rotating wheels using a control engineering approach has been considered by Goodall and Li [42].
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4
Guidance of the Two-Axle Vehicle
4.1 Introduction
As discussed in Chapter 1, guidance is the ability of a vehicle to follow the
As discussed in Chapter 1, guidance is the ability of a vehicle to follow the