## 3 High-Frequency TMD-Based Vibration Control Methodology for PSPPs  
A tuned mass damper is typically composed of a properly tuned spring, an auxiliary mass, and damper elements, providing frequency-dependent hysteresis that increases damping in the primary structure. The mechanism of suppressing structural vibrations by attaching a TMD is to transfer the vibration energy of the structure to the TMD and dissipate the energy at the TMD [29]. Due to the high-frequency vibration characteristics of the PSPP powerhouse structure, the natural vibration frequency is higher; therefore, it is necessary to design a higher-frequency TMD device to reduce the structural vibration. The existing TMD device is mainly used for low-frequency control and the coil spring is used as the stiffness unit. However, coil springs that can be produced currently do not have enough stiffness to control the high-frequency vibration. If the number of springs is multiplied, this not only increases the need for structural space but also limits the normal production and lifespan. In addition, many tuning devices of these springs are difficult to coordinate under high-frequency vibrations, and low reliability makes it difficult to realize the desired damping effect. Therefore, a suitable TMD device should be designed for high-frequency vibration control of powerhouse structures according to the pressure pulsation characteristics of PSPPs. Based on pressure water pulsation characteristics, high-frequency vibration of the powerhouse structure is mainly transmitted to the surrounding concrete via the solute, and subsequently through supporting members such as beams and columns to the floor [30; 31; 32]. Due to the large space that is occupied by the large generator rotor and the equipment for the structure of a hydropower station, the floor is not suitable for installing TMD devices. Combined with the propagation path of the pressure water pulsation, reducing the vibration of the floor by controlling the structural column is considered such that the vibration energy is dissipated in the transmission path. Therefore, this paper proposes a new high-frequency TMD vibration damping device that is suitable for PSPP structures. It is based on the bending stiffness of the beam and the satisfactory deformation performance of the spring steel. By absorbing the vibration energy of the dissipative structural column, the objective of vibration reduction is realized. A schematic diagram of the device is shown in Figure 1. The device consists of a spring plate, a counterweight plate, and a connecting plate. The spring plate is fixed at both ends to the connecting plate so that it can be fixed to the structural column through the connecting plate. The axial stiffness of the steel spring plate is much larger than its bending stiffness, hence the first axial modal frequency of the steel spring plate is much larger than the first bending modal frequency of the system. No torsional modes occur in the target frequency range. The weight plate is supported by a steel spring plate and vibrates along its lateral direction to form a tunable mass damper system. To simplify the analysis, the steel plate is assumed to be made of the same material, which has cantilever length \(L\), width \(b\), height \(h\), cross-sectional area \(A\), Young's modulus \(E\), density \(\rho\), and quality \(m_{t}\). The natural frequency of the TMD is calculated via the Rayleigh method [33]. The natural frequency \(\omega_{t}\) of the TMD can be calculated as follows:  
\[\omega_{t}=\sqrt{\frac{3 EI}{m_{t}L^{3}+\frac{33}{140}\rho AL^{4}}} \tag{8}\]  
For the PSPP powerhouse structure, high-frequency pulsation water pressure will cause high-frequency resonance of the structure, and the structural resonance response will be readily observable.