## 1 Introduction
[18] found that the pressure difference between the inlet and outlet of the runner and the pressure difference between the pressure side and the suction side of the blade was large, which led to the unbalanced force and torque on the turbine main shaft. Although the pressure pulsation and exciting force of the internal flow field of the hydraulic turbine have been studied deeply, the influence of the blade wear degree on the flow field is rarely taken into account. Therefore, this paper will analyze the flow field of runner blades under different wear degrees, explore the relationship between the fluid exciting force and runner blade wear degrees, and provide a basis for the subsequent analysis of the shafting vibration characteristics of hydropower units in this paper. The shafting vibration of hydropower units belongs to the category of rotor dynamics and is a research field involving nonlinear vibration caused by complex exciting forces in the operation of the main shaft system. [PERSON] et al. [19] found that the random fluid exciting force has a great impact on the water guide bearing and runner system, but it is mainly manifested in vibration form and stability, rather than vibration amplitude. [PERSON] et al. [20] analyzed the bend-torsional coupled vibration response of the shafting of hydropower units, but only considered the linear steady fluid exciting force on the runner, without taking into account the transient flow characteristics of the turbine. Based on the Lagrange equation, [PERSON] et al. [21] studied the influence of parameters such as clearance and the friction coefficient at different rotational speeds and found that some parameter values would produce special friction-related features. Based on the Lagrange equation, [PERSON] et al. [22] established a mathematical model of coupling the adjustment system and the spindle system and analyzed the influence of the gyroscopic effect and axial deviation on the stability of the system. [PERSON] et al. [23] found that when the nonlinear sealing force model is adopted and there is no other excitation influence, the axis locus of the runner does not exceed the seal clearance even if the runner is unstable, and the friction probability between the runner system and the seal system is small. [PERSON] et al. [24] derived the energy equation of the runner and blade of a hydraulic turbine and found that with the continuous increase in the mass of the runner blade, the bifurcation phenomenon would occur in the runner system. [PERSON] et al. [25] considered the combined effects of hydraulic, mechanical,and electromagnetic forces and found that the hydraulic instability caused by changes in the blade outlet flow angle, outlet diameter, and guide blade opening determines the overall trend of the dynamic characteristics of shafting. However, the influence of the fluid exciting force change caused by the blade wear degree on spindle vibration characteristics was not considered in the present study. Therefore, the relationship between the turbine blade wear degree and the vibration response of the shaft system was studied in this paper. A turbine model with blades exhibiting various degrees of wear was developed to analyze the effects of blade wear on the fluid exciting force. The exciting force was subsequently integrated into a dynamic model of the hydropower units shaft system to analyze the vibration response under different wear conditions. A mathematical relationship between the blade wear and the vibration response of the units was established. This relationship was intended to utilize measurable vibration characteristics to assess the wear level of turbine runner blades, providing a foundation for future fault prediction in hydropower units.