## 3 Theoretical Analyses
### Theoretical Model  
The hydraulic structures, including dams, spillway guide walls, hydraulic gates, are complex multi-degrees-of-freedom (MDOF) systems with complicated connections and damping conditions. For ease of calculation, the complex MDOF system should be simplified appropriately. This paper mainly concerns the vibrational response of the guide wall and the dam so that the system can be transformed into a two-degrees-of-freedom system. It is worth mentioning that the guide wall has a lighter weight compared to the weight of the large-scale concrete dam. To investigate the simplified form of the vibration system in this circumstance, the existing research of the hydraulic gate, another type of lightweight hydraulic structure, can provide a reference. To investigate the hydraulic gate vibration, [PERSON] [12] established a two-degrees-of-freedom theoretical model, shown in Figure 6. The model is inspired by the dynamic vibration absorber (DVA), which is commonly employed in the transportation field to control the vibration of vehicles [33]. The two-degrees-of-freedom system consists of two parts: the lightweight accessory structure and the heavyweight primary structure. While the primary structure is under the external excitation \(P_{1}\), the vibration component induced by \(P_{1}\) is transferred from the primary structure to the accessory structure through the simplified stiffness \(K_{2}\) and damping \(C_{2}\), leading to an intensity increase in the accessory structure. This kind of vibration can be defined as associated vibration (AV), and the two-degrees-of-freedom system can be consequently named as the AV system. In practice, the hydraulic gate can be treated as the accessory structure, and the dam can be treated as the primary structure. By analyzing the dynamic responses of the accessory structure in this model, the mechanism of intense vibration that occurred in the surface outlet radial gate of the Jinping arch dam in China has been successfully elucidated. Figure 6: A two-degrees-of-freedom theoretical model for a hydraulic gate: the associated vibration (AV) system. It is noted that the spillway guide wall has a lighter weight than the dam, and can also be considered as an accessory structure connected to the dam. However, the theoretical model for the vibration mechanism of the spillway guide wall is quite different from that for the hydraulic gate vibration mechanism. The reason is that the hydraulic gate is rigidly connected to the dam, while the guide wall is connected to not only the dam but also the foundation. Moreover, the connection between the guide wall and the dam is usually in a flexible form, such as sealing devices, diving joint, and the connection between the guide wall and the foundation is similar to the dam-foundation connection. With regard to these characteristics, the former model can be modified as shown in Figure 7, with the constraint between the guide wall and the foundation taken into account. Note that only horizontal constraints are considered in the theoretical model. The reasons are that in prototype results, the vertical displacements of the dam and the guide wall show less synchronization for structural resonance frequencies, which is analyzed in Section 2. As shown in Figure 8, the vertical displacement of the guide wall is measured as being smaller, which, in practice, is not a major threat to safety [13]. The EOMs for this modified two-degrees-of-freedom system are as follows [34; 35]:  
\[\left\{ \begin{array}{l}M_{1}\ddot{x}_{1}+C_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2} \Big{)}+K_{2}(x_{1}-x_{2})+C_{1}\dot{x}_{1}+K_{1}x_{1}=P_{1}\\ M_{2}\ddot{x}_{2}+C_{2}\Big{(}\dot{x}_{2}-\dot{x}_{1}\Big{)}+K_{2}(x_{2}-x_{1}) +C_{3}\dot{x}_{2}+K_{3}x_{2}=P_{2}\end{array} \right.