Matrix (A) shows the dynamic behavior of a high speed free truck running on an aligned track under optimal geometric conditions. 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 for which the real part is small and positive. The increase of this critical speed has been observed by an increase of the connection stiffness between car-body and truck. In fact, the car-body truck connection includes more than the classical connection. It includes hydraulic damping devices whose force-displacement diagram is similar to that of a friction damping device. After considering the elastic deformation of the 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. The initiation of cycles with partially constant magnitudes and frequencies, while the truck is running, enables the use of the first harmonic method. Therefore, this method is applied, and a complex term is added to the a33 element of the first matrix (A).

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