## 1 Introduction  
Curve squeal is a highly disturbing type of railway noise that occurs when the vehicle is in motion at tight curves under certain operation and environmental conditions. It is characterized by a high-pitched intense and tonal sound, with sound energy being concentrated at pure frequencies, typically beyond 500 Hz. The relative lateral velocity between the wheel and the rail, induced by a high angle of attack due to an imperfect curving behavior, is widely accepted as the main cause of curve squeal and it is commonly explained by two potentially coexisting mechanisms: instabilities due to falling friction and geometrical instabilities at constant friction [1]. Falling friction was first introduced by [PERSON] [2] in the 1970s and has been used since as the basis for other more detailed curve squeal models [3, 4, 5, 6]. It suggests that a decreasing friction coefficient at the wheel-rail interface with respect to lateral sliding velocity acts as an equivalent negative damping that feeds energy into the system, leading eventually to self-sustained vibrations and hence curve squeal. As explained by [PERSON] [6], three phases may be distinguished in the time domain vibrations: first, the wheel slides with respect to the rail and, due to falling friction, the equivalent damping is negative, so that the lateral velocity starts to grow at a rate governed by the modal growth rates until the saturation velocity is reached. In the second phase, the wheel contact point begins to creep, as the nonlinearities of the friction force slow down the amplitude growth (the equivalent damping is positive in some phases of the oscillation); the unstable modes are in competition until one or some of them become dominant. Finally, the lateral velocity stabilizes and the solution converges to a stationary regime, or limit cycle. On the other hand, the geometrical instabilities mechanism, introduced later, suggests that, unlike on falling friction, instabilities can still occur at a constant friction coefficient, thanks to the coupling between the tangential and the normal dynamics [7; 8; 9; 10; 11; 12; 13], where the rail may play an important role. As [PERSON] [7] explained, the normal force fluctuations influence the lateral force, but not vice versa, leading to an asymmetric stiffness matrix, which can induce mode-coupling instabilities. Moreover, [PERSON] et al. [10], as well as [PERSON] et al. [12] showed that single wheel modes can couple with the rail and produce squeal, in a way that is somewhat different than the mode coupling phenomenon, due to the large apparent damping provided by the rail at the contact point. In squeal models, the occurrence of the phenomenon is often determined by a stability analysis. One commonly used method which is well adapted to a modal description of the structures is the analysis of the complex eigenvalues of the system [6]. Other methods exist, however, in which the structures are described in the frequency domain via their mobilities and the Nyquist's criterion is used to determine the unstable frequencies of the system [5]. Stability analysis involves the linearization of the contact forces and, therefore, it can only be regarded as a first step into the study of curve squeal, as it cannot determine the vibration levels and excitation frequency of a system subjected to self-sustained vibrations; determination of the nonlinear self-sustained vibrations is then required. Two families of methods can be used: those based on the integration (typically numerical) of the dynamic equations of the system in the time domain [6; 10; 14; 15] and those involving a direct limit cycle search in the time or in the frequency domain, such as the shooting method [16] and the Constrained Harmonic Balance Method [17].