# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 1 Introduction  
The relentless drive towards the modernization of railway systems, epitomized by high-speed rail networks for inter-city transportation and urban rail transit systems encompassing subways, light rails, and trams, has been flourishing [1; 2]. Nonetheless, the pressing issue of environmental noise pollution stemming from the advancement of railroad transport remains an imperative concern. The impact of railway noise is profound, directly influencing the planning and construction of new transportation network routes, and necessitating the minimization of high-intensity noise emissions in residential areas [3]. Extensive information indicates that exposure to noisy environments can lead to physiological and psychological discomfort, as well as health issues. Furthermore, it disrupts normal work routines and sleep patterns, thereby diminishing work efficiency and giving rise to negative emotions [4; 5]. Wheel-rail noise constitutes the primary acoustic disturbance generated during the ordinary operation of trains and stands as a pivotal component of railroad noise [6]. This category of noise, in line with its genesis, is classified by [PERSON] into three distinctcategories: rolling noise, impact noise, and curve squeal noise. Among these, curve squeal noise commands particular attention due to its high-frequency tonal characteristics, making it the predominant contributor to overall wheel-rail noise levels. Regrettably, as of present, no comprehensive solution has emerged to mitigate curve squeal noise. Research findings indicate that sound pressure levels can reach 100-110 dB at a distance of 7.5 m from the track center, and in proximity to the wheels, levels may soar to 130 dB, frequently exceeding rolling noise levels by 10-30 dB [7]. This noise phenomenon is notably pronounced in small-radius curve lines, necessitated by terrain constraints, particularly prominent in urban rail transit systems. Consequently, curve squeal noise has evolved into a pivotal source of wheel-rail noise. At its core, curve squeal noise represents a self-excited vibration triggered by friction. To unveil the mechanics of this self-excited frictional vibration and devise strategies for source suppression, numerous scholars have developed a spectrum of friction models to investigate and elucidate the phenomenon of self-excited frictional vibration. These models encompass over a dozen approaches, including the stick-slip friction model [8], friction-relative sliding velocity negative slope friction model, self-locking-sliding friction model [9], modal coupling friction model, friction time-delay friction model [10], chaotic behavior model with friction oscillator [11], and others. Initially, squeal noise's mechanism was ascribed to the negative damping theory. [PERSON] et al. asserted that the negative slope of the friction curve between the wheel and rail introduced a detrimental negative damping effect within the system, thereby instigating squeal noise generation [12]. [PERSON] et al. devised a two-degree-of-freedom mathematical model, considering the linear friction's negative slope characteristic, to scrutinize the influence of damping on system stability [13]. This model was subsequently deployed to explore the impact of coupling stiffness on system stability [14]. [PERSON] et al. constructed a mathematical model delineating the wheel squeal noise generation mechanism grounded in the negative damping theory [15]. Furthermore, the existence of negative damping was confirmed through friction creep curve assessments obtained from experimental measurements [16]. This mathematical model, in conjunction with the friction creep curve, was applied to elucidate why the sound pressure level of curve squeal increases with variations in the stroke angle and rolling speed [17]. However, experimental findings by [PERSON] et al. contradicted the prevailing notion by demonstrating that squeal noise could still manifest within regions characterized by a positive frictional slope [18]. Similarly, [PERSON] et al.