The air spring is the main part of the secondary suspension of passenger railway vehicles. The aim of this paper is to review existing modelling techniques for air springs in order to check if challenges set in the past decade for available models have been met. The advantages and disadvantages of different air spring models (phenomenological/mechanical, thermodynamic, analytic, FEM) are summarised and discussed from the point of view of: model accuracy, multiphysics interaction, influence of structural and material non-linearities, obtention of parameters, frequency range and the balance between accuracy and computational effort. The first conclusion is that current research is mainly focused on the vertical behaviour with less attention paid to the lateral performance. Moreover, it is concluded that further research is needed to include non-linearities of the bellow and to consider fluid-structural interaction; this would allow improving the model of vertical behaviour and evaluating better the lateral performance of the pneumatic system. FEM models might be an interesting tool that allows performing a more complete analysis of air springs (combining different physics, including material non-linearities, considering the real shape of the bellow and reinforcing fibres, etc) favouring the comfort analysis and including the lateral dynamics of the air spring.

An engineering design formula for the torsion stiffness of a filled rubber bushing in the frequency domain, including the amplitude dependence, is presented. It is developed by applying a novel separable elastic, viscoelastic, and friction material model to an equivalent strain of the strain state inside the bushing, thus leading to an equivalent shear modulus that is inserted into an analytical formula for the torsion stiffness. The rubber model is the result of extending the force-displacement relation established in a sound rubber component model to the stress-strain level. Unlike other simplified methods, this procedure takes into account the variation in the properties inside the bushing owing to non-homogeneous strain states. Moreover, as this formula depends on the bushing geometry in addition to the material properties, it is a fast engineering tool to design the most suitable rubber bushing to fulfil user requirements. Furthermore, it is shown - by dividing the considered bushing into several slices, consequently each equivalent shear modulus is closer to the true value - that the approach of working with only one equivalent shear modulus for the whole bushing is accurate enough.

The finite length nature of rail-pad supports is characterised by a Timoshenko beam element formulation over an elastic foundation, giving rise to the distributed support element. The new element is integrated into a vertical track model, which is solved in frequency and time domain. The developed formulation is obtained by solving the governing equations of a Timoshenko beam for this particular case. The interaction between sleeper and rail via the elastic connection is considered in an analytical, compact and efficient way. The modelling technique results in realistic amplitudes of the ‘pinned–pinned’ vibration mode and, additionally, it leads to a smooth evolution of the contact force temporal response and to reduced amplitudes of the rail vertical oscillation, as compared to the results from concentrated support models. Simulations are performed for both parametric and sinusoidal roughness excitation. The model of support proposed here is compared with a previous finite length model developed by other authors, coming to the conclusion that the proposed model gives accurate results at a reduced computational cost.