# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 1 Introduction
observed the persistence of wheel noise under experimental conditions that eliminated negative slope [19]. Consequently, it is evident that the conventional negative damping theory struggles to furnish a cogent explanation for these instances of squeal noise generation occurring without negative damping conditions. Recently, modal coupling theory has been employed to elucidate curve squeal noise within the framework of non-negative frictional slopes, a context where system instability remains achievable even in the absence of negative slope. [PERSON] et al. expound the modal coupling mechanism using a simplified single-mass two-degree-of-freedom mathematical model. In this framework, when the friction coefficient surpasses a critical threshold, two modes with comparable frequencies but differing phases converge into a stable mode and an unstable mode, both bearing the same frequency [20]. This model is subsequently leveraged to investigate the influence of damping on modal coupling instability [21]. Modal coupling theory offers a rationale for the occurrence of squeal noise under conditions of non-negative damping, a phenomenon that defies the explanatory power of negative damping theory. Modal coupling theory-based analysis techniques encompass the discrete lumped mass model and the finite element complex modal analysis method. Presently, complex modal analysis based on finite elements stands as the predominant approach employed in both industry and academia for the prediction and analysis of frictional squeal noise. This method relies on finite element computations to extract complex eigenvalues and modal shapes of the system. The real part of the eigenvalues or the damping ratio is then utilized to assess system stability and predict the propensity for squeal noise occurrence [22; 23]. This methodology has evolved since its inception in 1989 when LILES employed finite elements to scrutinize the complex eigenvalues of braking systems. Subsequently, numerous researchers have adapted it to investigate curve squeal noise [24]. These studies have explored the influence of factors such as friction coefficient and contact position by establishing a two single-wheel contact model and individually analyzing their complex eigenvalues [25]. Building upon this foundation, some researchers have further developed a finite element model of a wheelset with double rails, elucidating the pivotal role of the friction coefficient in modal coupling mechanisms. Additionally, [PERSON] introduced a time-domain curve noise model that accounts for the coupling effect of vertical wheel-rail forces and lateral dynamics, revealing that constant friction can also induce curve squeal [26]. The modal coupling mechanism was experimentally verified by [PERSON] et al. using a double-disk test rig [27]. In parallel, [PERSON] et al. employed a waveguide finite element (WFE) model to investigate the mass and damping behavior of infinite tracks as a potential third mechanism contributing to whistling noise occurrence [28]. Furthermore, [PERSON] et al. utilized a one-degree-of-freedom model to identify the equivalent damper behavior of the infinite track as the source of instability within a wheel-rail system [29]. While prior research has made substantial strides in examining wheel-rail curve squeal noise from the perspective of frictional characteristics, few scholars have ventured into the realm of wheel-rail stiffness matching. The present study homes in on the vibrational attributes of the wheel-rail system in small-radius curved segments. Given the limitations of existing research methodologies, we establish a computational model of the wheel-rail system within the small half-diameter curved section of urban rail transit. Our investigation systematically delves into the mechanics of wheel-rail friction-induced self-excited vibration, the modes of instability within the wheel-rail system, and the factors influencing wheel-rail curve squeal noise.