Vehicle-bridge united systemWhat distinguishes the vehicle-bridge united system is that, both the auto-PSD matrix and thecoherence function matrix in the vehicle-bridge united system are time-related. Suppose thatn simulation points are evenly spaced along the bridge at an interval of Δy for a bridge with aspan of ( - ) . As is shown in Figure 3, each segment between two adjacent simulationpoints is further divided to ̃ sections with the first and last ones being ( ̃) ⁄ inlength, and the others ̃ ⁄ for the purpose of avoiding singularity of the XPSD. For eachtime increment dt, the vehicle travels within the re-divided section at the velocity of V.Therefore, the distance of two simulation points along the bridge yields to ̃ .In this analysis, the total travelling period is divided into three sub-periods, i.e., Period 1 (P1),Period 2 (P2) and Period 3 (P3). During P2, the vehicle is operating on the bridge. While inP1 and P3, the vehicle has not arrived at the bridge yet, or have already left the bridge.