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This paper uncovers the frequency variation in plate-type bridges under moving vehicles, highlighting the transverse characteristics stemming from the bridge’s spatial configuration. First, the mode shapes are determined for plates with two opposite sides simply supported and the other two free (S-F-S-F bridge) by Levy’s method. Then, analytical solutions are derived for the frequencies of the vehicle-plate system subjected to a moving vehicle. Through numerical analysis, it is demonstrated that the frequencies of the vehicle-plate system exhibit spatiotemporal variations, contingent upon both the transverse spatial path and longitudinal time-varying distance of the plate traveled by the vehicle. The “edging effect” was identified wherein maximum frequency variation will occur when the vehicle traverses along one side edge of the plate. Moreover, the transverse variation ratio of the plate-type bridge frequency increases with the decrease in the bridge’s span/width ratio. These findings underscore the significance of the bridge’s transverse characteristics when using the moving vehicle to measure the frequencies of plate-type bridges, particularly for short-span bridges.

Acoustic Emission (AE) and hysteresis parameters were studied from PZT-5A (soft) and PZT-8 (hard) ceramics during the application of ac fields in the frequency range of 0.2 Hz to 5 Hz. AE was found to occur mainly due to domain switching in these ceramics during the application of ac fields. In PZT-5A, a threshold field was observed for the AE activity to begin in both the direction of the applied field and in general, it was found to decrease with increasing frequency. Apart from this, AE activity was found to decrease with increasing number of applied ac cycles, and is attributed to domain pinning. In the case of PZT-8, AE was found to occur even at zero field at higher frequencies and AE activity was found to increase with increasing number of field cycles. These are explained on the basis of the defect dipole — domain interaction in these ceramics.

Modal analysis of bridge under high-speed trains is essential to the design and health monitoring of bridge, but it is difficult to be implemented since the vehicle–bridge interaction (VBI) effect is involved. In this paper, the time–frequency analysis technique is performed on the non-stationary train-induced bridge responses to estimate the frequency variations. To suppress the interference terms in time–frequency analysis but preserve the time-variant characteristics of responses, the enhanced variational mode decomposition (VMD) is proposed, which is used to decompose the train-induced dynamic response into many of envelope-normalized intrinsic mode functions (IMFs). Then the short-time Fourier transform is applied to observe the time–frequency energy distribution of each IMF. The train-induced bridge signals measured from a large-scale high-speed railway bridge are analyzed in this paper. The IMFs associated with the pseudo-frequencies caused by train or the resonant frequencies of bridge are distinguished. And, frequency variations are captured from the time–frequency energy distributions of envelope-normalized IMFs. The results show the proposed method can extract the frequency variations of low-energy IMFs effectively, which are hard to be observed from the time–frequency energy distribution of train-induced bridge response. The instantaneous frequency characteristics extracted from the train-induced bridge response could be the important support for investigating the VBI effect of train–bridge system.

The spatial-varying frequency of a vehicle-bridge interaction (VBI) system subjected to a moving mass is theoretically derived and numerically investigated through a three-dimensional VBI model, in which the effects of moving mass are introduced through the inertial force and centrifugal force in the equation of motion of the bridge. For a large vehicle-to-bridge mass ratio, it has been known that the frequency of a VBI system could change with respect to the location of a moving vehicle. As such, this study derives the analytical solution based on a moving mass-beam system to account for frequency variation and further builds the numerical model with detailed implementation for practical applications. The numerical results show the following findings: (1) The frequency of a VBI system is a function of velocity and location of a moving vehicle. (2) The reduction of spatial-varying frequency ratio for a particular mode decreases with respect to the mode of higher order. (3) The maximum reduction of spatial-varying frequency ratio of the first mode in a moving mass-beam system occurs in the location where the bridge has the maximum deflection as a result of local mode excitation. (4) For the VBI system with high suspension stiffness and large vehicle-to-bridge mass ratio, the absolute variation of spatial-varying frequency ratio of the first mode can be up to 30–40%.

As the frequency variation in a moving mass-plate system is seldom discussed in the previous work, this study first formulates the semi-analytical solutions of a system consisting of a moving mass on a square plate. Based on the moving mass-plate system, several finite element models with designed plate-type bridges are established to further investigate the effect of moving mass on a plate, the influence of bridge local modes, and the influence of moving loads on a bridge. The numerical results have shown that the frequency variation of a plate-type bridge can reach 30% for the first three modes. The unique phenomenon of wave propagation is exhibited in the acceleration responses of the plate-type bridge, while it is uncommon for a beam-type bridge. For design purpose, the following are remarked: (1) The use of Rayleigh damping on the stiffened bridge can effectively suppress the influence of local modes on the bridge deck. (2) The high stiffness in the moving load can be served as the damper to reduce the dynamic responses of a vehicle. (3) In the design stage of a plate-type bridge subjected to moving loads, the moving force is the most conservative one among the three moving loads.