Simon Iwnicki (Editor)_ Maksym Spiryagin (Editor)_ Colin Cole (Editor)_ Tim McSweeney (Editor) - Handbook of Railway Vehicle Dynamics, Second Edition-CRC Press (2019)

A History of Railway Vehicle Dynamics
17
Thus, while the theoretical foundations had been established, the need for vehicle dynamics was
not, and practicing railway engineers were largely sceptical of theory, particularly when the experimental
basis was very limited. As a result, the next 20 years saw only a few significant contributions
to the science of railway vehicle dynamics.
Rocard [56] in 1935 employed the same form of equations of motion as Carter. In addition to
covering much of the ground as Carter, he considered the case of a massless bogie that is connected
by a lateral spring to the car body and showed that the system could be stabilised. Rocard
also considers the case of the unsymmetric bogie in which the wheelsets have different conicities.
He found that the distribution of conicity could be arranged to give stability in one direction of
motion but not in both. Rocard [57] states that French National Railways made a successful experiment
in 1936.
There were also theoretical contributions by Langer [58] in 1935 and by Cain [59] in 1940 that
involved rather severe assumptions, but, in general, papers concerned with bogie design published
during this period were purely descriptive, reflecting the negligible role played by analysis in this
branch of engineering practice.
However, in 1939, Davies carried out significant model and full-scale experiments with both single
axles and two-axle bogies and demonstrated various forms of the hunting instability. This was
the beginning of roller rig testing for vehicle dynamics [60]. Notably, he presented equations of
motion, including, for the first time, longitudinal and lateral suspension flexibility, later established
to be one of the essential features of a realistic model of railway vehicle dynamics. Though Davies
felt that the laborious arithmetic involved in solutions of the equations was not justified in view of
the assumptions, a particularly valuable contribution was the detailed experimentation and discussion
of the importance of worn wheel and rail profiles to vehicle dynamics.
2.7 WHEEL-RAIL GEOMETRY
Carter assumed that the wheel treads were purely conical. In practice, it had been known from
the earliest days of the railways that treads wear and assume a hollow form, and rails also wear.
It was also known that there was a connection between ride quality and the amount of wheel wear.
For example, in 1855, Clark discussed several examples of wheel profiles and how they varied
between the various wheels of a locomotive, depending on the history of the distribution of applied
forces [27, pp. 181–183]. Conicity varied not only during the running of the locomotive between
maintenance but also between examples of the same nominal design, and it was, of course, a function
of the particular piece of track on which the locomotive was running. Clark also mentions the
high conicity induced by the flat top to the rails caused by the rolling process.
It follows that an important further step in developing a realistic mathematical model was concerned
with the treatment of actual wheel and rail profiles. Whilst new wheel profiles were purely
coned on the tread, usually to an angle of 1:20, in 1934, Heumann [61] emphasised the important
influence of worn wheel and rail profiles on wheelset behaviour in curves. Heumann analysed the
effect of the mutual wheel and rail geometry on the variation of the rolling radius as the wheelset is
displaced laterally and derived the formula for the effective conicity λ 0 of a wheel-rail combination
for small displacements from the central running position, defined as the rate of change of the rolling
radius with lateral displacement of the wheelset
λ0 δ0Rw / Rw −Rr
( 1− r0δ0
/ l)
(2.3)
= ( )
where R w and R r are the wheel and rail radius of curvature, 2l is the distance between the contact
points, δ 0 is the slope of the tread at the contact point and r 0 is the rolling radius of the wheelset
in the central position. Heumann’s expression shows clearly that the effective conicity of a worn
wheelset can be much greater than that of the corresponding purely coned wheelset. Moreover,
Heumann suggested for the first time that profiles approximating to the fully worn should be used