Rotational equations usable for railway wheelsets

Abstract

The Euler equation is a correct way for writing rotational moments of solids. But it is simple only if written in rotating frames. Applying it to railway wheelsets is difficult because it necessitates using the Euler angles, or Euler parameters, combined to rotation matrices or, numerically more stable, quaternions. Euler angles can be avoided in railway specific codes, by writing dynamical equations in track frames. However, academic literature [Landau LD, Lifshitz EM. Mechanics (Institute of Physical Problems, USSR Academy of Sciences, Moscow), Vol. 1, Course of theoretical physics. 21st English ed. Oxford (UK): Elsevier; 1960; Shabana AA, Zaazaa KE, Sugiyama H. Railroad vehicle dynamics. CRC Press; 2008.] does not provide simple solutions as to how properly writing equations of gyroscopic moments in no rotating frames. This paper describes how it is possible, owing to an approximation validated for railway applications, to avoid Euler angles and rotation matrices, while correctly taking into account gyroscopic effects. Using a most severe example, emphasising gyroscopic effects, it is demonstrated that a fast specific code using the approximation provides results equivalent to those of an multi body system generalised code with no approximation.

Keywords:

• Euler equation
• gyroscopic moments
• Coriolis moments
• solid dynamics
• railway codes

Acknowledgements

This study was made possible with the help of the US Federal Railroad Administration, and the Volpe Center.

Notes

1. The wording ‘Inertial Moments’ in this paper refers to the moments products of quadratic velocities by inertia moments, i.e. gyroscopic and Coriolis moments.

2. From the French mathematician G.-G. Coriolis (1792–1843) who gave his name to the ‘Coriolis acceleration’. ‘Coriolis forces’ are applied to solids of which the velocity on the earth's surface has a north–south component.

3. These velocities are measured with respect to the global reference frame.

4. ‘Model 2’ and ‘Model 3’ will be introduced in Section 7, they differ only by wheelsets inertia moments.

5. For codes delivering wheelsets angles with respect to the track, the track situation should not be omitted for dynamics purposes.

6. Passing from Δ V to moments necessitates using a length ‘h’, close to 0.75 m: .h can be better estimated as follows: , where y is the modulus of wheelset lateral displacement, here y=0.0062 m.

7. If setting I too small, I=[1, 1, 1] for example, numerical stability can be difficult

8. But, due to springs models, M_E can depend on codes (see note a).

9. Equations (5), (10), and (11). If the vehicle is not quite stabilized, a relative error <10% is acceptable. At first ϵ can be ignored.

10. These two figures do not depend on “n”, they do not depend on wheelsets inertia moments

11. This benchmark case and the generalized MBS code, Sams-Rail®, were developed by the University of Illinois at Chicago with the support of the Federal Railroad Administration.