Receptance of a semi-infinite periodic railway track and an equivalent multi-rigid body system for use in truncated track models

In many problems involving railway track dynamics, the infinitely long railway track has to be truncated, generating a finite track model. To account for high-frequency vibration, high train speed and the presence of multiple carriages, the track model may need to be very long and demand too great computational effort. Alternative modelling strategies are required so that the length of the track model is limited while it is still sufficiently able to exhibit dynamical characteristics of the infinite track. This may be achieved by dividing the track into three parts, two semi-infinite parts (two semi-infinite tracks) on either side of a central finite track model, treating the semi-infinite parts to be periodic in the unbounded direction and then replacing them with an equivalent multi-rigid body system. The multi-rigid body system may be determined based on the requirement that it provides the same receptance to the track model as the semi-infinite track. This paper is concerned with the calculation of the receptance and the determination of the equivalent system for vertical vibration of the track. The receptance is calculated based on the two semi-infinite tracks joined together as an infinitely long periodic structure subject to specific harmonic loads, in combination with the relationship between the internal forces and displacements of a Timoshenko beam. To estimate the parameters of the equivalent system, the structural layout of the system must be created first. The structural layout is created based on the characteristic frequencies and 'modes' of the semi-infinite track. The parameters of the equivalent system are estimated by letting the modal frequencies of the equivalent system be the same as the characteristic frequencies of the semi-infinite track and by minimising the relative difference in receptance between the multi-rigid body system and the semi-infinite track at a set of pre-defined frequencies, including the characteristic frequencies of the semi-infinite track. The above procedure is demonstrated for a typical high-speed slab track and a typical ballasted track.