1. Introduction
At present, along with the continuous increase of the unit capacity and operating head of the Hydraulic-Turbine Generator Unit (HGU), the safety and stability of the HGU have become increasingly highlighted, and the vibration is one of the important factors affecting the stable operation of the Unit. Excessive vibration not only causes fatigue damage to materials, shortens the service life of the equipment, but also causes the HGU to fail to operate normally. In severe cases, it may cause damage accidents to HGU and eventually lead to causing huge economic losses [
1,
2]. In recent years, various extents of vibration problems have occurred in hydropower plants (HPP), such as Xiaolangdi, Yantan, Wuqiangxi, and Wanjiazhai HPP in China; Guri HPP in Venezuela, and Grand Coulee HPP in the United States. Due to the vibration, some cause cracks in the runner blades, tearing of the draft tube; some cause fatigue of parts or welds, which leads to form or expand cracks until they break; and some even cause resonance in the powerhouse and adjacent hydraulic structures, which eventually endangers the safe and stable operation of the HPP [
3,
4,
5]. Therefore, the predictive analysis and control of the vibration phenomenon of the HGU is a significant factor that must be considered during the design, construction, and operation of the HPP [
6].
In the actual operation of the HGU, there should be more or less vibration due to the high-speed rotation of the shafting, the research on which has always been highly valued by the academic and engineering circles. Due to the particularity of the working medium of the HGU, the causes of the vibration are much more complicated than other rotational machinery. In addition to the vibration caused by mechanical unbalance, the influence of hydrodynamic pressure of the fluid and the unbalance of the electromagnetic force should also be considered. Therefore, in the actual operation of the Unit, the combination of mechanical unbalance factors, hydraulic unbalance factors and electromagnetic unbalance factors jointly cause the vibration of the shaft system [
7]. In recent years, the research on the vibration of the HGU has made important progress in the depth and breadth of research compared with the traditional research methods. By drawing on mathematics, physics and other basic disciplines and combining them with computational technology, many new research fields have been explored, and thus research and experimental methods have become more modernized, and with it, more and more in-depth and significant results and findings have been disclosed on the mechanism of the HGU’s vibration, which contributes to pushing ahead with the theory and practice in engineering [
8].
In terms of mechanical factors, due to mechanical unbalance caused by manufacturing, installation, maintenance and other factors, the shafting and supporting structure of the HGU deviate from the symmetry, which results in vibration and unstable operation of the Unit. Numerous studies have shown that the main reasons for mechanical unbalance are: the main shaft is not straight; the mass of the rotating parts is unbalanced, such as the mass unbalance of the turbine runner and the generator rotor; the rotational parts are not concentric with the fixed parts, resulting in friction or collision; clearance of guide bearing pad is excessive or lack of lubrication; thrust bearing adjustment is not appropriate or the surface of thrust bearing pads is not level, etc. These factors cause mechanical vibration [
9,
10,
11,
12,
13]. It has also been found in practice that due to the improper manufacturing or installation quality of the generator, or the bending and deformation of the generator shaft system, the outer circle of the generator rotor is eccentric relative to the inner circle of the stator. This eccentricity also leads to a non-uniform air gap that results in an unbalanced magnetic pull. Unbalanced magnetic pulling force and centrifugal force act on the shafting, which causes lateral vibration and torsional vibration of the shafting. The vibration caused by mechanical defects or faults has common characteristics in that the vibration frequency is mostly equal to the rotational frequency or a multiple of the rotational frequency, and the unbalanced force is generally horizontal and radial. In addition, the natural vibration characteristics of the shafting system determined by the structural design of the shafting system, such as the natural vibration frequency, the main vibration mode and the critical speed, have a fundamental effect on the vibration characteristics of the shafting system. The stiffness of the shafting structure and the stiffness coefficient of the guide bearing also affect the natural frequency, critical speed and main vibration mode of the shafting. Because as long as various excitation sources interacted with the natural vibration characteristics of the shafting structure, the actual vibration response characteristics are just shown [
14,
15,
16,
17].
In terms of hydraulic factors, all the energy driving the HGU comes from the kinetic energy of the fluid, which interacts with the rotational parts of the HGU. Due to the irregular flow channel shape of the turbine and the uncertainty of the rotation phenomenon, the internal flow field is so complex that it is very difficult to grasp the hydraulic vibration. Therefore, hydraulic vibration is one of the main vibration sources of the shaft system. More and more researches illustrate that the main factors that cause hydraulic vibration are as follows: the vibration caused by the Karman vortex; the vibration caused by the low-frequency vortex in the draft tube; the vibration caused by the unstable flow at the outlet of the stay vane; the vibration caused by unreasonable matching between the number of runner blades and the number of stay vanes of the hydraulic turbine; as well as the vibration caused by the cavitation effect, etc. Hydraulic factors have a significant impact on the shafting vibration, and its mechanism is also very complex due to the strongly coupled interaction with the turbulent flow, the research of which carried out by scholars in the past reveals the aforementioned results [
18,
19,
20]. In addition, by summarizing the previous research results, it also can be found that hydraulic vibration varies with the working conditions of the HGU. If the working water head of the turbine is constant during operation, the amplitude of hydraulic vibration varies with the change of flow and load, while the load is constant, the hydraulic vibration under the condition of different heads is also different. However, if the water head and flow rate are constant, the hydraulic vibration is different due to the different cavitation coefficients of the turbine [
21,
22].
In terms of electrical factors, during the operation of the HGU, because the lines of magnetic force pass through the air gap between the rotor and the stator, a pulling force is generated between the stator and the rotor, i.e., magnetic pulling force. When the magnetic field is uniform under ideal conditions, the magnetic pulling force distributed on each point in the radial direction of the rotor is also uniform, and their resultant force should be zero so that the shafting could rotate in a balanced state. However, if there is an eccentricity between the rotor and the stator, the magnetic pulling force is also unbalanced, and the unbalanced magnetic pull could affect the dynamic characteristics of the shafting system which makes the vibration frequency change with the shafting rotation. Therefore, the vibration caused by electromagnetic factors mainly means that the vibration exciting force comes from the electromagnetic force of the electrical parts of the generator, which is characterized in that the vibration increases with the increasing of the excitation current [
23,
24,
25]. The electromagnetic vibration of the HGU can be divided into two categories: one is rotational frequency vibration and the other is extreme frequency vibration (its frequency is usually up to 100 Hz). The frequency of rotational frequency vibration is usually equal to the Unit’s rotational frequency or its integral multiple, which is one of the main vibration sources of HGU with a large diameter. The main causes of the rotational frequency vibration are as follows: the air gap between the rotor and the stator is not uniform; the outer circle of the rotor or the inner cavity of the stator is not round; the rotor geometric centre is inconsistent with the rotational centre; there are a dynamic or static imbalance of the rotor as well as the short circuit between coil turns, etc. However, the extreme frequency vibration is usually formed by the interaction of the rotor magnetic field and the stator magnetic field, and the large or abnormal extreme frequency vibration usually occurs in the form of resonance. The extreme frequency vibration of the stator is a natural vibration that usually occurs in the yoke part of the stator core. The main causes of the extreme frequency vibration are as follows: the sub-harmonic magnetic potential of the stator fractional slots, which induces a vibration frequency of 100 Hz, and its amplitude increases with the increase of the load current; the magnetic potential generated by the circulating current in the parallel-circuit branch of the stator, which can induce a series of asymmetric sub-harmonic potentials, and cause extreme frequency vibration with a frequency of 100 Hz; the reverse magnetic potential induced by the negative sequence current, which causes the stator core to vibrate as a standing wave; as well as either the stator is not round, or the split stator is not tightly seamed. Up to now, much research and experiments regarding the unbalanced magnetic pull force carried out by many scholars have illustrated and testified the beneficial results hereinbefore, which have played a positive role in analyzing and understanding the electrical vibration mechanism [
26,
27,
28].
In the actual operation of the HGU, these three factors are coupled with each other, showing complex characteristics of vibration response. In fact, the vibration of the HGU is mainly caused by the high-speed rotation of the shafting, so it is necessary to take the whole shafting as a specific object of study and analysis. Although many scholars tried to consider the above three factors in the predictive research of shafting vibration characteristics as much as possible, accurately simulating the vibration by considering all the above factors in one numerical calculation model still cannot be fully realized at present. Therefore, it is feasible and customary to select the representative influencing factors according to the actual situation of HGU and consider them in the specific model of the corresponding study [
29,
30]. In addition, the aforementioned factors caused by the unbalance of manufacturing, installation, maintenance and so on could be regarded as external vibration exciting sources, and their research results are focused too much on how to control the influence of external excitation sources with neglecting the exploration on the internal and inherent mechanism of shafting vibration. However, moderate vibration of the Unit is allowed in practical engineering, and only when the external excitation source is close to the natural vibration characteristics of the shafting structure, the resonance phenomenon that is most harmful to the Unit would appear. Long-time resonance will inevitably endanger the safe and stable operation of the Unit, which is not allowed in engineering. Therefore, in order to predict and evaluate whether the shafting could generate resonance or not, in addition to controlling the influence of the external excitation source, it is also necessary to fundamentally analyze and study the inherent vibration characteristics of the shafting structure, i.e., the study of the natural vibration characteristics, which are mainly determined by the structural design of the shafting.
The study of natural vibration characteristics includes the contents of traditional rotor dynamics research such as numerical calculation of natural frequency, critical speed, and mode shape, as well as the influence analysis of various factors on natural vibration characteristics [
1,
2,
31,
32,
33]. Among them, the finite element modal analysis is not only a modern method to study the dynamic characteristics of the natural vibration of the structure, but also the application of the system identification method in the field of engineering. The modal analysis uses the modal coordinates corresponding to the main vibration modes of the undamped system to replace the physical coordinates, and decouple the differential equations with coupled coordinates into differential equations with independent coordinates, so as to calculate the modal parameters. Modes are the natural vibration characteristics of mechanical structures, and each mode has a specific natural frequency, damping ratio, and mode shape. These modal parameters could be obtained by numerical calculation or experimental analysis, and thus such a process of calculation or experimental analysis is called modal analysis. Vibration modes are inherent and integral characteristics of elastic structures. If the main modal characteristics of the structure in a certain susceptible frequency range are figured out by the modal analysis method, it is possible to predict the actual vibration response of the structure under the action of various external or internal vibration exciting sources within this frequency range, so as to make the structure design avoid resonance or vibrate at a specific frequency. Therefore, the modal analysis would be an important method to guide the dynamic design of structures and fault diagnosis of equipment in the future [
34].
In conclusion, in order to explore and reveal the natural vibration characteristics of the shafting structure of the HGU, and combine with the needs of the research on the operation stability faced by the author in the actual hydropower project, this paper will conduct the finite element modal analysis research on the shafting structure of an HGU for a newly built hydropower plant in Turkey. By calculating and simulating the natural vibration characteristics of the shafting structure, it is able to predict the possibility of resonance occurring in the shafting under the action of relevant vibration sources, thereby evaluating the stability and safety of the Unit’s structural design, as well as guide the optimal design of the shafting.
4. Result and Analysis
4.1. Material Defining and Meshing
The physical and mechanical properties of the materials of each component of the shafting have an important influence on the modal characteristics of the entire shafting. Before the calculation, the material parameters of each component of the shafting should be defined according to the physical and mechanical parameters listed in
Table 3.
In this paper, the method of automatic meshing is used to discretize the entity model. Before meshing, it is necessary to define the type, size, and density of the elements. Then the system will automatically select an optimized global control parameter to control the size and position of the tetrahedral element, and meanwhile, the mesh is automatically encrypted where the gradient of the domain variable is large or the stress is concentrated such as runner, lower guide bearing and thrust bearing, etc. After meshing, the finite element calculation model of the shafting is generated, which contains a finite number of elements and nodes as shown in
Figure 3, and its relevant meshing data of the model are shown in
Table 4.
4.2. Boundary Conditions and Calculation Settings
The shaft system of the HGU is mainly constrained by its axial and radial degrees of freedom through three guide bearings and one thrust bearing. Among them, the water guide bearing and the upper and lower guide bearings of the generator constrain the radial swing and rotation of the shaft system respectively, that is, the radial displacement degree of freedom and the rotation degree of freedom around the corresponding coordinate axis, while the thrust bearing bears the weight of the entire shaft system and constrains the degree of freedom of axial displacement. Therefore, bearing constraints are given at the three guide bearings to constrain the radial displacement degrees of freedom and the rotational degrees of freedom around the shaft, while the axial displacement is constrained at the thrust bearings, and the rotational degrees of freedom around the shaft axis are released.
During the calculation, the prestress is not considered, but the influence of damping is considered, and then the first ten modes of the shafting are extracted and the natural frequency is calculated. To obtain the critical rotational speed of the shafting, five points of rotational velocity are selected to draw the Campbell diagram. Considering the constraints on the shafting, the rotational velocity at each point around the X, Y, and Z directions is shown in
Table 5.
4.3. The First Ten-Order Mode Shapes
Through calculation and solution, the first ten-order modal analysis cloud diagram and mode shape of the shaft system are finally obtained as shown in
Figure 4. The dashed shadow in each mode shape diagram represents the initial state of the rotor, and the relative displacement between the solid body and the dashed shadow represents the amplitude of the rotor in this mode.
It could be seen from the modal shape diagram of the shaft system that the first and second order modes represent the radial translation of the shaft system, the third and fourth order modes represent the oscillation of the shaft system, and the fifth and sixth orders are the bending deformation of the shaft system, the seventh order is the torsional deformation of the rotor, the eighth and ninth orders are the more complex continuous bending deformation of the shaft system, and the tenth order is the torsional deformation of the runner, and meanwhile, the amplitude of the bending and torsional deformation gradually increases with the increase of the order. That is to say, shafting has the changing laws of translation-oscillating-bending-torsion-more complex continuous bending and torsion. The higher the order, the more complex the bending and torsional deformation of the shafting. That is because the higher the order and the higher the natural frequency, the greater the exciting force required to cause resonance, and the greater the deformation amplitude caused by those forces acting on each structural element. In addition, because the mass and stiffness, loads and constraints, as well as boundary conditions of each component are usually different, which leads to different forced conditions and responses on each structure, so that the local displacement of the structural element is significantly different, thus it shows complex mode shapes on the macro.
In addition, the first four orders of the vibration mode of the shaft system are mainly regarded as rigid body modes with very low natural frequencies, while the subsequent modes are regarded as elastic modes. This is mainly because the rigid body in space has six degrees of freedom, respectively for three translational and three rotational degrees of freedom. An elastic continuum actually has an infinite number of degrees of freedom, but the finite element analysis makes the continuous infinite number of degrees of freedom subject discretized into a finite number of degrees of freedom subject. Thus, the degrees of freedom of the structure are also finite [
36]. In this paper, the translational and rotational degrees of freedom in the Y direction are constrained by the setting of boundary conditions, so the first four-order rigid body modes are mainly represented as translation and oscillation along the X and Z directions, which conforms to the fundamental theory. However, for a structural system with a free boundary, not only the elastic mode but also the rigid body mode are required to fully describe its dynamic characteristics [
37]. Nevertheless, in the testing process of practical engineering, the rigid body modes are always ignored and not considered as a part of the elastic mode, so it is difficult to obtain the rigid body mode of the structure. In this paper, the rigid body mode of the shaft system can be directly restored through numerical calculation and finite element modal analysis, which conforms to the inherent dynamic characteristics of the rotational structure.
4.4. The First Ten Order Natural Frequency and Critical Speed
According to the given velocity point and the calculated natural frequencies of each order, a Campbell diagram is drawn, in which the abscissa is the rotational velocity, and the ordinate is the frequency. The velocity-frequency curve is obtained as shown in
Figure 5. The intersection of the velocity-frequency curve of each order and a straight line with a slope of 1 is the critical speed point of the mode, i.e., the velocity corresponding to the intersection is the critical speed.
Table 6 gives the first ten order natural frequencies and critical speeds of the shafting.
The critical speed of the rotating shafting refers to the speed when the value is equal to the natural frequency of the shafting, that is, the speed when resonance occurs. The shaft system of the HGU is subjected to various vibration exciting sources during operation. If the frequency of a certain vibration exciting source is equal to the natural vibration frequency of the shaft system, the shaft system would be likely to resonate. In order to avoid the harm caused by resonance, it is so necessary to calculate and analyze the critical speed of the shafting so that the working speed of the HGU could be kept away from the critical speed, so as to avoid resonance.
It can be found from
Table 6 that the natural frequencies of the first four-order rigid body modes of the shaft system are very low at any rotational velocity, while from the fifth-order elastic mode, the critical speed of the shaft system increases gradually with the increase of modal order. Usually, the elastic mode of the structural system is the common cause of all vibration and noise problems. Therefore, in the structural design of the shaft system, the rated working speed of the shaft system is normally required to be lower than the first-order critical speed of the elastic mode, or the intermediate value between the first-order critical speed and the second-order critical speed. In addition, in order to ensure that the structure does not resonate within the range of working speed, the structural design of the shaft system should meet the requirement that the working speed should deviate from the critical speed with a safety margin of at least 15%~25% [
38].
The rated speed of the shafting studied in this paper is 600 r/min according to
Table 1, i.e., 62.80 rad/s, while the first-order critical speed of the elastic mode (corresponding to the fifth-order mode in
Table 6, the same as below) is 104.45 rad/s. Therefore, the safety margin of the rated working speed deviating from the critical speed is 39.9%, which fully meets the natural vibration characteristics requirements of the structural design, so the shaft system of the HGU could operate stably at the rated speed, and resonance could not occur due to the rotational frequency.
However, the runaway speed of the Unit is 1064 r/min, i.e., 111.37 rad/s, which is between the first-order critical speed of 104.45 rad/s and the second-order critical speed of 130.21 rad/s (corresponding to the sixth-order mode in
Table 6). When the working condition of runaway occurs, the Unit needs to cross the first-order critical speed, and the safety margin of the runaway speed deviating from the first-order critical speed is too small, only 6.6%, thus there would be the possibility of the resonance due to rotational frequency. Therefore, this kind of situation should be prevented during the structural design of the shafting.
The critical speed is mainly related to some factors such as the elasticity and mass distribution of the shaft system. For a discretized rotating system with a finite number of lumped masses, the number of critical speeds is equal to the number of lumped masses; while for an elastic rotating system with continuous mass distribution, there are infinitely many critical speeds, and their magnitude is positively related to the elasticity of the shafting [
39]. Therefore, in view of the problem mentioned hereinbefore that resonance is likely to be caused by crossing the first-order critical speed of the elastic mode when the Unit is a runaway, materials with larger elastic modulus are proposed to be selected during the optimal design of the shafting structure so as to increase the stiffness of the shafting, which makes the natural frequency of the first-order elastic mode of the shafting moved up and meanwhile the first-order critical speed is also increased accordingly. In this way, the runaway speed would be always lower than the first-order critical speed and maintains a certain safety margin. In addition, it can also be considered to improve the mass distribution of the shafting structure during structural design to make the overall mass distribution more balanced.