## 5 Conclusions  
In this paper, a new vibration mechanism is presented to analyze the flow-induced vibration (FIV) of the spillway guide wall in a hydropower station, with the new concepts of associated vibration (AV) and associated-forced coupled vibration (AFCV) defined. The prototype tests on the left guide wall of the Three Gorges Hydropower Station (TGHS) indicate that there are several similar dominant frequencies of the horizontal displacements between 1# dam block and guide wall blocks. To investigate the mechanism of the vibration synchronization, a two-degrees-of-freedom theoretical model of the AFCV system is proposed, considering the lightweight spillway guide wall as an accessory structure installed in both the large-scale primary structure (the dam) and the foundation. Afterwards, the practicability of this theoretical model is verified by mutual analyses. Using the cross-wavelet transform (XWT), the correlation between the vibrational signals measured on the dam block and guide wall blocks, respectively, is illustrated, which confirms the existence of AFCV. With the vibrational characteristics calculated by numerical simulation, the vibration mechanism is clarified by the theoretical model.  
In brief, the guide wall vibration is not only a simple independent forced vibration (FV) but also a vibration influenced by the dam, of which the effect can be named as AFCV. This kind of vibration derives from the effect of the shear force through joints and thermal expansion. The contribution level of the AV on the guide wall depends on the relative position of the guide wall, the dam, and the position of the dissipation zone. When the guide wall is positioned closer from the dam and further to the dissipation zone, the guide wall vibration appears to possess synchronized characteristics with the dam, proving the existence of AV; the synchronized vibrational frequency band is narrow but concentrated, which indicates AV is a dominant vibration component in this case. On the contrary, when the guide wall is positioned further from the dam and closer to the dissipation zone, the guide wall vibration appears to be with a broad frequency band and possesses more independent FV characteristics, although the vibration intensity tends to increase because of the stronger fluctuation in the flow.  
It is noted that all of the above changes in the vibrational characteristics can be elucidated by varying the parameters in the theoretical model, such as \(\lambda_{12}\), \(\xi_{2}\), \(\gamma\), which indicates the validity of the theoretical model. Furthermore, by changing parameters using practical engineering measures, the model can provide a certain beneficial reference regarding vibration reduction.  
Moreover, it should be pointed out that due to the complexity of the engineering structures and FSI effect, the mechanism of vibration propagation and the engineering measures to reduce propagation still require further investigation.  
Conceptualization, J.L. and C.L.; data curation, B.M.; formal analysis, Y.Z.; funding acquisition, J.L. and B.M.; methodology, Y.Z. and C.L.; software, Y.Z.; validation, J.L.; writing--original draft preparation, Y.Z.; writing--review and editing, J.L. and C.L.  
This research was funded by National Key R&D Program of China (No. 2016 YFC0401905), Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51621092), National Natural Science Foundation of China (No. 51579173, No. 51779167, No. 51809194), China Postdoctoral Science Foundation (2019M652550), and Funds for Postdoctoral Scientific Research in Henan Province (in 2019).  
The authors declare no conflict of interest.