The dynamic behaviour of a high speed free truck of railway carriage running on an aligned track with optimal geometric conditions can be tabulated in ful1 matrix A(7, 7) or A(17, 17).

The maximum running speed will be lower than the critical speed inherent to the system which is determined by the first eigenvalue of the matrix A(7, 7) and A(17, 17) for which the real part is small and positive. In fact, the car-body truck connection includes hydraulic damping devices whose force-displacement diagram is similar to that of a friction damping device. After considering the elastic deformation of a connecting system ensured by the damping devices between truck and car-body, and the connecting springs between truck and car-body mounted in parallel, we observe a drawback torque relative to the truck. This characteristic can be compared to an hysteresis cycle. In taking account the angular stiffness of the hydraulic damping devices, the critical speeds are looked for. The results obtained are comparable with those achieved from matrix A(7, 7) and A(17, 17).