## 4 Experiment Validation
### Experimental Result Analysis  
In this study, the inclination data collected from the five cross-sections were utilized to compute the surrogate inter-section transfer matrix for bridge acceleration. Further, the acceleration data obtained at +2.4 m and 0 m were employed to estimate the acceleration at \(-\)2.4 m. It is important to note that, aside from the geometric coordinates of these cross-sections (namely, \(\pm\)2.4 m, 0 m, and \(\pm\)4.4 m, along with the support locations), knowledge of other bridge parameters--such as stiffness, density, moment of inertia, and the distance between sensors and the beam's central axis--is not required for the proposed algorithm. Furthermore, information regarding the mass and speed of the electric vehicle is also unnecessary. The method does not rely on finite element models or any prior knowledge as a prerequisite. A representative set of measured inclination time histories is depicted in Figure 15, revealing that the inclination responses are predominantly influenced by low-frequency components. In this study, we further assumed that the first two vibration modes are the primary contributors to the structural responses. Consequently, the FastICA algorithm was employed to isolate two signal sources and the corresponding mixing matrix derived from the data of the five inclinometers. The separated signal sources are visualized in Figure 16. Given the proximity of the frequencies of these  
Figure 14: Layout of sensors on the middle span (triangle: inclinometer; cylinder: accelerometer). Figure 13: Electric vehicle (**a**), geometric parameter (unit: cm) (**b**), and mass blocks (**c**). two sources (by counting the distance between peaks), it is inferred that both sources are composed of the first two principal frequency components, aligning with Equation (18a,b). The accuracy of the signal separation was also scrutinized. Figure 17 presents a comparison between the measured responses at +2.4 m and the reconstructed responses. The reconstructed responses are obtained by multiplying the mixing matrix with the separated signal sources. The comparison demonstrates a close match between the two sets of responses, affirming that the FastICA algorithm effectively isolates accurate signal sources and their corresponding mixing matrices. Another test of the accuracy of the signal separation was also applied focusing on the mixing matrix. Utilizing the mixing matrix, a surrogate transfer matrix for inclination from the +2.4 m and 0 m cross-sections to the \(-\)2.4 m cross-section can be established. In accordance with Equation (12a), elements of the mixing matrix are directly utilized to construct this surrogate matrix. Subsequently, the measured inclination time histories at the +2.4 m and 0 m cross-sections are compiled into an input matrix, which is then multiplied by the surrogate inclination transfer matrix. The result of this operation effectively estimates the inclination time history at the \(-\)2.4 m cross-section. Figure 17b, labeled as \"TM (transfer matrix) estimation,\" displays the comparison between the estimated and actual inclination measurements at \(-\)2.4 m. The close correspondence between the estimated and measured results substantiates Equation (20b), confirming that the surrogate inter-section inclination transfer matrix is equivalent to the actual one, which validates the second key principle. Additionally, the favorable comparison outcome also suggests that the measured inclination data contains a minimal level of noise. Figure 16: Two separate signal sources with five inclinometers with FastICA. Figure 15: Typical time history of bridge inclination. Having established the accuracy of the FastICA algorithm, the surrogate inclination mode shapes are presented in Figure 18a. Subsequently, the surrogate displacement mode shape function was derived through discrete integration and displayed in Figure 18b. The first and surrogate mode shapes, for both inclination and displacement, appear to be analogous.