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
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FIGURE 2.10 Limit cycle or bifurcation diagram. δ = nominal flangeway clearance; A = lateral wheelset
amplitude of oscillation; V 0 = non-linear critical speed; q 0 = breakaway yaw angle in yaw spring in series with
dry friction. (From Cooperrider, N.K. et al., The application of quasilinearisation techniques to the prediction
of nonlinear railway vehicle response, In Pacejka, H.B. (Ed.), The Dynamics of Vehicles on Roads and on
Railway Tracks, Proceedings IUTAM Symposium held at Delft University of Technology, 18–22 August 1975,
Swets & Zeitlinger, Amsterdam, the Netherlands, pp. 314–325, 1976; With kind permission from CRC Press:
Handbook of Railway Vehicle Dynamics, CRC Press, Boca Raton, FL, 2006, Iwnicki, S. [ed.].)
Apart from Van Bommel’s work referred to previously, various approaches to the analysis of
non-linear hunting motions have been developed. Cooperrider, Hedrick, Law and Malstrom [104]
introduced the more formal method of ‘quasi-linearisation’, in which the non-linear functions are
replaced by the linear functions so chosen to minimise the mean-square error between the nonlinear
and the quasi-linear response. They also introduced the limit-cycle or bifurcation diagram,
an example of which is shown in Figure 2.10.
This procedure was extended by Gasch, Moelle and Knothe [105,106], who approximated the
limit cycle by a Fourier series and used a Galerkin method to solve the equations. This made it possible
to establish much detail about the limit cycle. It is known that apparently simple dynamical
systems with strong non-linearities can respond to a disturbance in very complex ways. In fact, for
certain ranges of parameters, no periodic solution may exist. Moreover, systems with large nonlinearities
may respond to a disturbance in an apparently random way. In this case, the response is
deterministic but very sensitive to the initial conditions. Such chaotic motions have been studied
for railway vehicles by True et al. [107,108]. Many studies specific to the UIC double-link suspension,
widely used in European two-axle freight vehicles, have been carried out, for example [109].
Piotrowsky [110] pointed out the effect of dither generated by rolling contact in smoothing dry friction
damping.
2.12 LATER RESEARCH ON CURVING
In 1966, Boocock [111] and Newland [112] independently considered the curving of a vehicle by using
the same equations of motion used in stability analyses but with terms on the right-hand side representing
the input due to curvature and cant deficiency. As the wheelsets are constrained by the longitudinal
and lateral springs connecting them to the rest of the vehicle, the wheelsets are not able to take
up the radial attitude of perfect steering envisaged by Redtenbacher. Instead, a wheelset will balance
a yaw couple applied to it by the suspension by moving further in a radial direction so as to generate
equal and opposite longitudinal creep forces, and it will balance a lateral force by yawing further.​