## 1 Introduction  
The extraction of modal parameters is essential for the dynamic analysis, damage detection, and model updating of bridges. Recently, significant advancements have been made in the experimental modal analysis of bridges, particularly in utilizing moving vehicles as an excitation source for damage detection [1, 2, 3]. Compared to natural frequencies, mode shapes are more sensitive to local variations along the bridge caused by degradation or damage [2, 4]. To improve the accuracy of bridge mode identification and effectively capture modal variations induced by structural deterioration, advanced signal processing techniques have been increasingly employed. Specifically, empirical mode decomposition (EMD) [5, 6, 7] and variational mode decomposition (VMD) [8] have been employed to extract intrinsic mode functions (IMFs), providing a robust and effective framework for identifying modal changes associated with bridge damage [9, 10]. Compared with the widely used impulse load and steady-state load, a vehicle moving along the bridge is a cost-effective and efficient mean to excite the full bridge span in a single trip. Bridge responses can be measured directly via sensors installed on the bridge or indirectly through vehicle-mounted sensors. While direct measurements on the bridge can provide accurate data, they require extensive and costly on-site instrumentation tailored to each bridge. Alternatively, the moving vehicle serves as both an excitation source and a data collector. However, vehicle-mounted measurements primarily reflect the coupled vehicle-bridge dynamics, encompassing the bridge responses, the surface roughness, and the interaction between the vehicle and the bridge. A variety of novel methods has been continuously developed to distinguish the bridge response from the vehicle-induced signals. [PERSON] et al. [11] proposed a new second-order blind identification method by combining the vehicle transfer function and ensemble empirical mode decomposition (EEMD) to extract the vibrations caused by surface roughness and identify bridge modes. [PERSON] et al. [12] tried to eliminate the interaction between the vehicle axle and the bridge surface by measuring their relative motions. Additional strategies enhance the response of the bridge by applying artificial forces [11; 13] and heavy trucks [14] and employing subtraction between two consecutive vehicle responses to facilitate mode shape identification [15; 16]. The contact point between a vehicle and a bridge has emerged as a critical location for damage detection [17; 18] and mode recognition [13; 19]. Although it is physically associated with the vehicle, the contact point's motion could reflect the bridge's dynamic response at time-varying positions along its length as the vehicle moves. Additionally, serving as the transmission between the vehicle and the bridge, the contact point allows for the more efficient extraction of bridge dynamics by suppressing vehicle-induced interference. Instead of one single contact point, the residual response between consecutive vehicular contact points is adopted to remove road roughness. [PERSON] et al. [20] investigated the dynamic response relationship between the front and rear contact points of a dual-axle scanning vehicle. Furthermore, a recursive formula was developed to mitigate damping distortion. [PERSON] et al. [21] recognized a bridge beam mode based on the contact response of dual-axle test vehicles in wavelet transforms. In addition, in a moving reference-based manner, the local short-time Fourier transform (STFT) amplitudes of these residual accelerations along the frequency ridges [13] benefit mode recognition. However, the previous studies have demonstrated [13; 22] that the majority of the energy in both the vehicle and the bridge responses is concentrated in the lower frequency range around the bridge's fundamental natural frequency under moving loads. Consequently, the dynamical characteristic of this interaction system is not sufficiently excited at the higher frequency range.