Effect of Water-added-mass on Modal Behavior of Shaft-line of Large Hydroelectric Generator Unit

The water-added-mass to a hydro turbine-shaft-line can significantly affect its modal behavior including the vibration modes and their natural frequencies. So it is of great significance to consider the effect of added water mass on the modal behavior of a hydro turbine to assure the safe and efficient operation of the hydro turbine-shaft-line system. In this investigation, the effect of water-added-mass on the shaft-line of one large prototype hydroelectric generator unit has been investigated in detail via the finite element tool. First, a CAD model of the shaft-line was created based on actual dimensions. Then a finite element model of the entire shaft-line was generated with high-quality hexahedral and tetrahedral cells. A first round of finite element simulation of the shaft-line in the air was performed without considering the effect of surrounding water, but the results were not realistic. On this basis, a finite element-based water-structure coupling simulation was carried out to analyze the effect of water-added-mass on the modal behavior of the shaft-line of the large hydroelectric generator unit. The result is more realistic to reality than that in the air. The results of this study show that when the water surrounding the turbine runner is added to the hydro-turbine shaft-line, it increases the overall mass of the shaft-line system, thereby reducing the natural frequencies of the system. The decrease in natural frequencies can cause the shaft-line to approach its critical speed during operation, vibrate at higher amplitudes than usual, and even lead to potential damage to the shaft-line system. According to the results calculated with the finite element water-structure coupling method, the countermeasures to improve the evaluation of the water-added-mass effect on the structural modal behavior of the shaft-line for the hydroelectric generator units have been provided. The conclusion can also apply to other hydraulic turbine and pump units.


Introduction
Energy is crucial to the progress of human civilization and economic and social development and has a considerable bearing on the survival and development of humankind.Over the years, fossil fuels have become the dominant energy sources used in industry, transportation and daily life, meeting the needs of human life and economic development.However, fossil fuels were found to be a major contributor to the energy crisis and environmental problems [1].Clean and renewable energy, such as hydropower and wind power, plays a crucial role in addressing these challenges and is increasingly being developed.
Hydropower, which harnesses the energy of flowing water to provide large amounts of reliable and predictable electricity, has become the base load of many national grids.The efficient and stable operation of large-scale hydroelectric generator units in hydropower plants is vital to the safety of power grids [2].The shaft-line of a large hydroelectric generator unit transmits the rotational motion from the turbine to the generator and consists mainly of the runner of the turbine and turbine-shaft, the generator rotor and its shaft, and other structures.During operation, failure and damage of the hydraulic turbine-shaft-line occur frequently [3][4][5].The researchers normally built analytical models and 1D simplified models to calculate the structural characteristics of shaft-lines of the hydraulic generating units [6][7][8][9].However, since the turbine runner is submerged in water, the water-added-mass effect on the modal behavior of the shaft-line must be taken into consideration in the 3D model.
In this work, the 3D CAD model for the shaft-line of a large hydroelectric generator unit is built first.Then the finite element (FE) model of the shaft-line and the surrounding water of the runner are created.After this, the mode shapes and their natural frequencies of the shaft-line considering the water-added-mass effect are analyzed and discussed.

Methodology
For turbine runners that are submerged in water, the fluid imparts an effective mass to the structure due to its motion, and thus the water-added-mass becomes an important factor in modal analysis.Finite Element Analysis (FEA), a powerful numerical calculation tool, is often adopted to model the water-added-mass effect.
According to the finite element method, the equilibrium equation for the structural dynamic response of the turbine shaft-line can be written as where q is the structural displacement.Ms, Cs, Ks are the mass, damping and stiffness matrices of the shaft-line, and Fs(t) is the external force over time t.Fsw(t) =− is the pressure load p from the water at the water-structure coupling interface.is the water-structure coupling stiffness matrix.
Assuming that the water in the Francis runner of the shaft-line is homogeneous, slightly compressible, and non-viscous, and that the average density is the same, the momentum equation and the continuity equation of the fluid can be simplified to Eq. 2.
where Mw, Cw, Kw are the water equivalent mass, water equivalent damping and water equivalent stiffness matrices, respectively.Fsw(t) =− is the loads generated by structural displacements at the water-structure coupling interface.Mws is the water-structure coupling mass matrix.
Combining the dynamic response equilibrium equations of the structure and the water, the mathematical equations solving the water-structure coupling problem considering the water-addedmass effect are expressed by the following finite element discrete equations [10].

Unit Parameters and Shaft-line Model
The hydroelectric generator unit in the study is a typical middle-head Francis unit with a rated power of 180MW.The main parameters for the hydroelectric generator unit are listed in Table 1.The hydroelectric generator unit is with a vertical shaft, and the shaft-line mainly consists of the runner, turbine-shaft, generator and generator-shaft (Fig. 1).The density of the steel shaft-line is 7700kg/m 3 , the Poisson's ratio is 0.3 , and Young's modulus of the structures around 206GPa, respectively.The shaft-line is supported by the upper generator bearing, turbine bearing in the radial direction, and supported by thrust bearing in the axial direction.The turbine runner is submerged in water.To perform the numerical modal analysis of the shaft-line, the FE model of the shaft-line and the surrounding water of the runner have been generated with a hybrid of high-quality tetrahedral elements and hexahedron elements (Fig. 2).

Results and Discussion
Considering the water-added-mass effect, the modal behavior of the hydroelectric generator shaft-line, including the mode shapes and their natural frequencies, is analyzed by finite element analysis.To quantitatively assess the effect of the water-added-mass, the structural modal behavior of the shaft-line without water is also calculated.The natural frequency of the first bending mode for the shaft-line within and without water is tabled in Table 2.
Table 2.The natural frequencies of the shaft-line with and without water.

Mode shape
Natural frequency (Hz) Frequency reduction ratio (%) 100* (f0 -f1)/ f0 without water (f0) with water (f1) First bending mode 6.9 4.9 29.0 It can be seen that the the surrounding water of the runner increases the effective mass of the shaftline, and decreases the natural frequency for the first bending mode from 6.9Hz to 4.9Hz.The frequency reduction ratio is around 39.1%, so the result without consideration of the water-addedmass effect is too optimistic.
Comparing the natural frequencies for the shaft-line within water to the rated rotational frequency and runaway frequency listed in Table 1, it is clear that the natural frequency for the first bending mode is approximately 2.94 times the rated rotational frequency and 1.4 times the runaway rotational frequency.so the hydroelectric generator unit will not be in resonance under normal operating conditions.

Conclusions
This paper studied the water-added-mass effect on the modal behavior of the shaft-line of one large prototype hydroelectric generator unit with the finite element water-structure coupling method.The approach presented in this paper is more accurate than the analyses without consideration of the influence of the water.This study concludes that the model of the surrounding water must be considered to accurately assess the modal behavior of the hydroelectric generator shaft-line, including the natural frequency and vibration modes.
The most important mode of the shaft-line is the first bending mode with the largest deformations on the generator and the runner.The motion of the runner and generator is in counter-phase.For the first bending mode, the water surrounding the runner dramatically reduces the natural frequencies of the shaft-line, by around 29.0%, but it has almost no influence on the mode shapes.The natural frequency for the first bending mode of the shaft-line is much higher than the rated rotating frequency and the runaway rotating frequency, which indicates that the shaft-line of the studied hydroelectric generator unit will not be in resonance under normal operating conditions.
The 3D shaft-line modal behavior analysis method based on water-structure coupling presented in this work can also be used to evaluate the water-added-mass effect on other hydraulic machinery, e.g.other types of hydraulic turbine units, pump units, pump-turbine units, and marine machinery.In the next step, the dynamic stresses and deformations in the shaft system of hydraulic machinery can be analyzed based on this method.

Figure 1 .
Figure 1.The shaft-line model for the studied hydroelectric generator unit.

Figure 2 .
Figure 2. The finite element model for the hydroelectric generator shaft-line.

Figure 3 .
Figure 3.The model shapes of the hydroelectric generator shaft-line.

Fig. 3
Fig.3 demonstrates the first 2 mode shapes of the shaft-line with the repeated natural frequencies.According to the motion of the shaft-line, these 2 mode shapes can be classified as the first bending mode, where the generator and the runner have larger displacement than other parts of the shaft-line.And the motion of the runner and generator is in counter-phase.The surrounding water of the runner has almost no influence on the first bending mode.

Table 1 .
The parameters of the hydroelectric generator unit.