Cost-Effective Design Modification of a Sleeve Bearing with Large Bearing Clearance

Abstract

In 2015, a 45 MW vertical hydropower machine exhibited excessive vibration after refurbishment. Measurements revealed a substantial bearing clearance at the lower generator guide bearing. Consequently, the bearing was unable to generate sufficient opposing force to drive the rotor toward the bearing center, resulting in more pronounced overall system vibration. Addressing this challenge required a cost-effective and feasible solution for mitigating the vibration problem. To this end, a design modification was implemented wherein the lower generator guide bearing (originally a sleeve bearing) was modified to a four-lobe bearing by offsetting the two halves of the bearing twice in two axes. Numerical simulations and experimentations were conducted, and the dynamics of the machine before and after the design modification were investigated. Both the simulation and experimental results showed that the machine with the four-lobe bearing improved the system stability and reduced the vibration amplitudes. The numerical simulation result demonstrated that, due to the design modification, the first and second critical speeds were effectively eliminated for a speed range of up to three times the nominal speed. Furthermore, for nominal operation with unbalanced magnetic pull, the four-lobe bearing provided a stability advantage in terms of the modal parameters relative to the original sleeve bearing.

1. Introduction

In rotordynamics, bearings are an intermediary component between the stationary and rotating parts; they are crucial, as the dynamic characteristics of the rotor are profoundly influenced by their properties. It has been reported that classical circular bearings are prone to instability phenomena known as oil whirl, where the rotor operates close to half the operating speed. Due to this, non-circular journal bearings gained significant attention for their ability to ensure stable operation, making them indispensable in a wide range of applications, such as oil and steam turbines, pumps, and high-speed rotating machines. Over the last few decades, numerous research articles have been published regarding the dynamic properties of non-circular journal bearings of different configurations, including elliptical, three-lobe, four-lobe, and offset bearings [1,2,3,4]. Allaire [5] compared the transient unbalance responses of four different types of multilobe bearings, i.e., elliptical, offset, three-lobe, and four-lobe journal bearings. Kumar et al. [6], and Malik et al. and Sinhasan et al. [7,8], provided the design data for two-lobe and three-lobe bearings, respectively, while Someya [9] presented the static and dynamic characteristics of many different bearings, including the four-lobe bearing. Malik [10] studied and compared the static and dynamic performance of the four-lobe journal bearing with symmetric and tilted lobes. The theoretical results show that the bearing with tilted lobes improved overall performance. A similar study was conducted on tilted three-lobe journal bearings [11], which exhibited superior dynamic performance compared with the symmetric bearing. Chauhan [12] presented an extensive literature review of non-circular bearings and discussed the results from theoretical and experimental investigations. In addition, the potential research gaps were identified and suggested for valuable insights and further investigations. Pai and Majumdar [13] studied the stability of a submerged four-lobe bearing and compared the results with those of plain cylindrical bearings under similar conditions. The authors reported that the trajectory of the journal center was damped out in the case of the four-lobe bearing. On the contrary, the motion exhibited significant excursion before reaching stability for the plain cylindrical journal bearing.

Vertically oriented rotors operate under light or no radial static load conditions because the weight acts axially. This typical characteristic of vertical machines makes them susceptible to oil-whirl instability. White et al. [14] studied the rotordynamic characteristics of a vertical twelve-stage multiphase pump, considering both small and large bearing clearances. The results indicated that the large bearing clearance caused more pronounced vibration due to sub-synchronous instability and critical speeds within the operation range. Smith and Woodward [15] studied the field vibration measurements of several large vertical pumps experiencing significant wear in the impellers, wear rings, and seals. This wear was attributed to the operating speed closely aligning with the natural frequency of the pump–rotor system. To address the vibration problem, the authors proposed some potential solutions. Corbo et al. [16] investigated the rotodynamic instability in a vertical pump employed in the nuclear waste processing industry. The bearings, which were originally plain cylindrical bearings, were replaced by tilting pad journal bearings to resolve the vibration problem due to whirling. Leader et al. [17] examined the stability of a vertical sulfur pump undergoing sub-synchronous vibrations caused by “sulfur whirl” at nearly 0.5 times the operation speed. The authors reported that replacing the existing plain bushings with the three-lobe bearing improved the system’s dynamics, reducing the sub-synchronous vibration and increasing the onset speed of instability. Experimental tests were later conducted by Khatri and Childs [18] on bearings with a geometry similar to that described in [17]. However, the authors concluded that the three-lobe bearings demonstrated no stability advantages over the plain cylindrical bearing regarding the whirl frequency ratio (WFR).

The aim of the current study is to examine the dynamics of the vertical hydropower machine exhibiting extensive vibrations, which were attributed to a large bearing clearance at the lower generator guide bearing (LGB). Due to this, the LGB exhibited reduced stiffness and was unable to generate sufficient bearing forces, prompting the machine to rely on the support from the other two bearings. To resolve this problem, the LGB, originally a sleeve bearing, was modified to a four-lobe bearing by offsetting the two halves of the bearing twice in two axes (Figure 1). Hence, the diametric bearing clearance (C_b) was reduced from 1.6 mm to 0.45 mm, which increased the bearing stiffness and enabled the LGB to carry a sufficient load. The design modification is cost-effective, as it allows for a redesign process with reduced financial costs. Replacing the bearing with a new design requires the disassembly of the original and installation of the new bearing, which would have incurred significant costs and an extended maintenance duration. Roughly speaking, the maintenance might take up to three months and interrupt the electricity production, which could potentially lead to a loss of approximately USD 5 million. In contrast, the design modification implemented in this study was executed on-site without the need to dismantle the original bearing, substantially reducing the downtime losses by a factor of ten.

Numerical simulations and experimentations were conducted to investigate if this modification improved the dynamics of the hydropower machine. Numerical simulations were performed for two cases, i.e., operation at nominal speed and over-speeding condition, and the dynamics of the two setups were compared. In the first case, the normal operation of the hydropower unit running at its nominal speed was investigated. The effect of the unbalanced magnetic pull (UMP) produced by the generator and exciter was considered in the model. For the second case, however, the dynamics of the system were examined when the machine encountered a sudden power loss and disconnected from the grid, leading to zero electromagnetic torque and an increase in rotor speed to runaway speed. In this case, the UMP was not considered. Experiments, on the other hand, were conducted on-site for operation at nominal rotor speed, and the vibrations were measured at three bearing locations before and after the design modification.

2. Experimentation

A 45 MW Kaplan hydropower machine was oriented vertically and supported by three bearings, i.e., an upper generator guide bearing (UGB), a lower generator guide bearing (LGB), and a turbine guide bearing (TB). The UGB and original LGB were sleeve bearings, whereas the modified LGB and TB were a four-lobe bearing and a 12-shoe tilting-pad journal bearing (TPJB), respectively. A schematic diagram of the hydropower machine and some technical details of the system are presented in Figure 2 and

Table 1. Specifications and parameters of the hydropower unit.

Description (Unit)	Exciter	Generator	Runner
Mass (kg)	3500	1.51 × 105	53,600
Polar moment of inertia (kg·m2)	3500	1.38 × 106	56,400
Diametral moment of inertia (kg·m2)	1750	6.88 × 105	44,300
Unbalanced magnetic pull (MN/m)	26.62	170
Young’s modulus (N/m2)	2 × 1011

, respectively.

On-site vibration measurement was performed before and after the design modification employed on the LGB. Two sets of measurements were carried out on the machine with the original LGB (MOB) and modified LGB (MMB). In the first case, the original LGB, which was a sleeve bearing with C_b = 1.6 mm, was installed, whereas in the second case, the original LGB was modified to a four-lobe bearing with C_b = 0.45 mm.

Inductive sensors were installed at the three bearing locations to measure the shaft and bearing housing lateral displacements in the x and y axes. During the measurement, the rotor ran at a nominal speed (${\mathsf{\Omega }}_0$ = 150 rpm), and measurements were carried out for different load cases. The data were sampled at 600 Hz, and each measurement was carried out for 30 s.

3. Numerical Simulation

3.1. Rotordynamic Model

The rotor was discretized into 53 Timoshenko beam elements, and the shear, rotary inertia, and gyroscopic effects were considered. The exciter, generator, and runner were added to the model as mass blocks with polar and diametral moments of inertia. As shown in Figure 2, the rotor was radially supported by three bearings, which were further attached to the ground via brackets. They were modeled as springs with isotropic stiffness and no damping. For operation at nominal rotor speed, the UMP of the exciter and generators was considered and modeled as a constant negative stiffness. Equation (1) is the governing equation of the hydropower unit:

$$\mathbf{M}\ddot{\mathbf{q}}+\left(\mathbf{C}+\mathsf{\Omega }\mathbf{G}\right)\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}={\mathbf{f}}_{\mathbf{u}}-{\mathbf{f}}_{\mathbf{b}}$$

(1)

where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, K is the stiffness matrix, $\mathsf{\Omega }$ is the rotor speed, ${\mathbf{f}}_{\mathbf{u}}$ is the unbalance force vector, and ${\mathbf{f}}_{\mathbf{b}}$ is the bearing restoring force vector of the UGB and LGB. Note that the bearing forces of the TB are represented by damping and stiffness coefficients and are included in the C and K matrices, respectively. The magnitude of ${\mathbf{f}}_{\mathbf{u}}$ at the j-node is the product of an unbalance magnitude ($m_i·e_j$) and the square of the rotor speed (Equation (2)). For unbalance response simulations, mass unbalances were introduced at two locations, i.e., at the generator ($m_{GNR}·e_3$ = 400 kg·m) and turbine ($m_T·e_7$ = 200 kg·m), which were more than five times higher than the ISO 1941-1 balance quality grade of G6.3 [19]. This high unbalance is not unusual in hydropower units, and it was found that this level of unbalance could represent the vibration measurement in the actual machine:

$$f_u=m_i·e_j·{\mathsf{\Omega }}^2$$

(2)

where $m_i$ is the mass of the generator or turbine, $e_j$ is the amplitude of the trajectory of the journal center at the j-node, and $\mathsf{\Omega }$ is the rotor speed.

3.2. Model Reduction

Model reduction was employed to improve the computational efficiency of the simulation without significantly affecting the dynamics of the system, particularly the lower frequency modes. Thus, the number of nodes of the full shaft model was reduced from 54 to 7. The shaft nodes corresponding to the three bearings as well as the nodes at the generator and runner were retained, and the DOFs were reordered as master (denoted by an ‘m’ subscript) and slave nodes (denoted by an ‘s’ subscript) in Equation (3):

$$\left[\begin{array}{cc}{\mathbf{M}}_{mm}& {\mathbf{M}}_{ms}\\ {\mathbf{M}}_{sm}& {\mathbf{M}}_{ss}\end{array}\right]\left\{\begin{array}{c}{\ddot{\mathbf{q}}}_m\\ {\ddot{\mathbf{q}}}_s\end{array}\right\}+\left[\begin{array}{cc}{\mathbf{K}}_{mm}& {\mathbf{K}}_{ms}\\ {\mathbf{K}}_{sm}& {\mathbf{K}}_{ss}\end{array}\right]\left\{\begin{array}{c}{\mathbf{q}}_m\\ {\mathbf{q}}_s\end{array}\right\}=\left\{\begin{array}{c}{\mathbf{f}}_m\\ \mathbf{0}\end{array}\right\}$$

(3)

The DOFs of the system were reduced and expressed by the master nodes (${\mathbf{q}}_m$), as shown in Equation (4), by disregarding the inertial terms and considering the lower equation from the stiffness term:

$$\left\{\begin{array}{c}{\mathbf{q}}_m\\ {\mathbf{q}}_s\end{array}\right\}={\mathbf{T}}_s{\mathbf{q}}_m$$

(4)

where

$${\mathbf{T}}_s=\left[\begin{array}{c}\mathbf{I}\\ -{\mathbf{K}}_{ss}^{-\mathbf{1}}{\mathbf{K}}_{sm}\end{array}\right]$$

Similarly, the full mass and stiffness matrices were reduced by applying, ${\mathbf{M}}_{\mathrm{R}}={\mathbf{T}}_s^{\mathrm{T}}\mathbf{M}{\mathbf{T}}_s$ and ${\mathbf{K}}_{\mathrm{R}}={\mathbf{T}}_s^{\mathrm{T}}\mathbf{K}{\mathbf{T}}_s$ multiplications. The static reduction is adequate at zero excitation frequency. However, for large frequencies, the inertial term becomes more significant, and an approximation cannot be valid. In the improved reduction system (IRS) method [20], the static reduction method has been improved by taking the inertia term as pseudo-static forces, seen in Equation (5):

$${\mathbf{T}}_{\mathbf{I}\mathbf{R}\mathbf{S}}={\mathbf{T}}_s+\mathbf{S}\mathbf{M}{\mathbf{T}}_s{\mathbf{M}}_{\mathrm{R}}^{-\mathbf{1}}{\mathbf{K}}_{\mathbf{R}}$$

(5)

where

$$\mathbf{S}=\left[\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathbf{K}}_{ss}^{-\mathbf{1}}\end{array}\right]$$

Like in the static reduction method, the mass, gyroscopic, damping, and stiffness matrices are reduced using ${\mathbf{M}}_{\mathrm{I}\mathrm{R}\mathrm{S}}={\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}^{\mathrm{T}}\mathbf{M}{\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}$, ${\mathbf{G}}_{\mathrm{I}\mathrm{R}\mathrm{S}}={\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}^{\mathrm{T}}\mathbf{G}{\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}$, ${\mathbf{C}}_{\mathrm{I}\mathrm{R}\mathrm{S}}={\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}^{\mathrm{T}}\mathbf{C}{\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}$, and ${\mathbf{K}}_{\mathrm{I}\mathrm{R}\mathrm{S}}={\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}^{\mathrm{T}}\mathbf{K}{\mathbf{T}}_{\mathrm{I}\mathrm{R}\mathrm{S}}$, respectively. To further improve the accuracy, an iteration scheme is commonly employed.

3.3. Bearing Models

As shown in

Table 2. Technical specifications of the bearings.

	UGB	LGB		TB
Geometry
Type	Sleeve bearing	Sleeve bearing	Four-lobe bearing	TPJB
Journal diameter (mm)	1050	2450	2450	1100
Number of segments/pads	-	-	4	12
Dia. bearing clearance (mm)	0.6	1.6	0.45	0.35
Offset ratio (-)	0.5	0.5	1	0.6
Preload ratio (-)	0	0	0.718	0.65
Axial length (mm)	345	185	185	180
Arc length (degree)	-	-	85	18.5
Material
Surface pad material	Babbitt
Density (kg/m3)	7280
Radial thickness (mm)	5	6	6	3
Base pad material	Steel
Density (kg/m3)	7780
Radial thickness (mm)	65	100	100	52
Lubricant Properties
Type	Turboway 68
Inlet oil temperature (°C)	60
Oil supply pressure (MPa)	0.1
Density (kg/m3)	880
Viscosity at 40 °C (mPa·s)	61
Viscosity at 100 °C (mPa·s)	7.7
Bracket	UF	LF		TF
Mass (kg)	10 × 103	10 × 103		10 × 103
Stiffness (N/m)	3 × 108	1.87 × 109		2.2 × 109

, the UGB and the original LGB are sleeve bearings, whereas the modified LGB is a four-lobe bearing. The bearing forces of the UGB and LGBs (with grooves) were calculated by solving Reynolds’ equation, as discussed in Section 3.3.1. On the other hand, the TB is a TPJB, and its bearing coefficients were precalculated and represented by exponential and sine/cosine functions. The procedure for how to define the bearing model is discussed in Section 3.3.2.

3.3.1. The Fluid Film Force of the UGB and LGB

For the rotordynamic simulation of the hydropower machine model defined in Equation (1), the bearing restoring forces (${\mathbf{f}}_{\mathbf{b}}$) of the UGB and LGB had to be calculated at each time step by solving the fluid film lubrication model. These bearing forces were modeled based on Reynolds’ equation, which is a partial differential equation that describes the fluid film pressure in the gap between the journal and bearing surfaces (Equation (6)):

$$\frac{1}{R^2}\frac{\partial }{\partial \psi }\left(h^3\frac{\partial p}{\partial \psi }\right)+\frac{\partial }{\partial z}\left(h^3\frac{\partial p}{\partial z}\right)=6\mu \mathsf{\Omega }\frac{\partial h}{\partial \psi }+12\mu \frac{\partial h}{\partial t}$$

(6)

where p is the fluid film pressure, $\mu$ is the lubricant’s viscosity, h is the fluid film thickness, $\mathsf{\Omega }$ is the rotor speed, and $\psi$ and z are the circumferential and axial axes, respectively. Figure 3 shows the schematics of the four-lobe bearing. The center of the lower left pad is displayed, with the segment radius equal to the sum of the journal’s radius and half of the diametric pad clearance ($R+0.5C_p$). In the case of the sleeve bearing, the center of the pads coincides with the center of the bearing, and the diametric pad clearance ($C_p$) and bearing clearance ($C_b$) are equal. The location of the journal center is represented as a function of the eccentricity angle ($\varphi$) and the trajectory of the journal center ($e_{j,x}$ and $e_{j,y}$) at the j-node. The fluid film thickness, $h\left(\psi \right)$, derived from geometrical relationships, is given in Equation (7) [21,22]. Unlike the sleeve bearing, the equation for fluid film thickness of the four-lobe bearing includes an extra term ($r_{p}cos\left(\psi -{\psi }_c\right)$) and requires treating every lobe separately.

$$h\left(\psi \right)=0.5C_p+e_{j,x}cos\left(\psi \right)+e_{j,y}sin\left(\psi \right)-r_{p}cos\left(\psi -{\psi }_c\right)$$

(7)

where $e_{j,x}$ is the trajectory of the journal center at the j-node in the X-axis, $e_{j,y}$ is the trajectory of the journal center at the j-node in the Y-axis, $r_p=0.5\left(C_p-C_b\right)$ is the pad preload, $\psi$ is the angular displacement measured from the negative X-axis, and ${\psi }_c$ is the angular displacement measured from the negative X-axis to the minimum film thickness (for this case, ${\psi }_c=$0.5$\pi$). The bearing forces ($f_x$, $f_y$) are calculated by integrating the pressure distribution over the fluid film domain using Equation (8):

$$\left[\begin{array}{c}{f}_x\\ f_y\end{array}\right]={\int }_0^{2\pi }{\int }_0^{L}p\left[\begin{array}{c}\cos \psi \\ \sin \psi \end{array}\right]\left(Rd\psi \right)dz$$

(8)

To solve the Reynolds equation and calculate the pressure distribution under the bearing, a finite element method was employed and solved using MATLAB R2020a. In this method, the domain was subdivided into smaller elements, and the solution was estimated using local trail functions. Further description of the method is available in [23]. For our case, the pressure at the boundaries was assumed to be zero. Furthermore, a four-node bilinear quadrilateral element was chosen, and the bearing surfaces were discretized into Ne × Me = 120 × 40 elements, where Ne and Me are the number of elements in the circumferential and axial direction, respectively.

$${\mathbf{K}}_{\mathbf{p}}\mathbf{p}={\mathbf{K}}_{{\mathbf{U}}_{\mathbf{x}}}{\mathbf{U}}_{\mathbf{x}}+{\mathbf{K}}_{\dot{\mathbf{h}}}\dot{\mathbf{h}}$$

(9)

where $\mathbf{p}$ is the pressure vector, ${\mathbf{K}}_{\mathbf{p}}$ is the pressure fluidity component ${\mathbf{K}}_{{\mathbf{U}}_{\mathbf{x}}}$ is the shear fluidity component, and ${\mathbf{K}}_{\dot{\mathbf{h}}}$ is the squeeze component.

3.3.2. Model of the TPJB

As displayed in Figure 2, the hydropower machine is supported by three bearings, and one of them is the TB. The TB is a 12-shoe TPJB, and the schematics and technical details are illustrated in Figure 4 and Table 2, respectively. The bearing consists of 12 pads, each pivot positioned at α = 0°, 30°, 60°… 330°. The X and Y axes represent the global coordinate with the origin at the center of the bearing. On the other hand, the local coordinate is represented by the ξ and η axes, where the ξ-axis is always parallel to the line joining the bearing and journal centers. The location of the journal center from the bearing center is given as a function of the eccentricity angle (α) and trajectory of the journal center ($e_j$).

The bearing was modeled as direct and cross-coupled stiffness and damping coefficients and included in the K and C matrices of Equation (1) for the rotordynamic simulations. The procedure that was used in this paper to determine the bearing coefficients closely resembles the method used in [24]. The bearing coefficients were precalculated by solving the fluid film lubrication model and represented using exponential functions as well as sine/cosine functions. Commercial software, RAPPID 3.22 [25], which employs a Navier–Stokes-based lubrication model, was used to calculate the bearing coefficients at eight predefined relative eccentricities, i.e., $ϵ$ = [0.01, 0.1, 0.2, …, 0.7]. For each relative eccentricity, the coefficients were calculated for 25 different eccentricity angles (α) in the interval α ∊ [−15°, +15°], and the maxima and minima coefficients were obtained. The bearing coefficients vary periodically between the maxima and minima coefficients as a function of α (Figure 5a), which can be expressed by sine/cosine functions (Equations (10) and (11)). The maxima and minima coefficients were approximated with exponential equations, i.e., $a{e}^{bϵ}+c{e}^{dϵ}$, where a, b, c, and d are constants obtained from a curve-fitting. The small circles in Figure 5b represent the maxima and minima coefficients obtained from RAPPID, whereas the solid lines show the exponential curve-fitting using a MATLAB inbuilt function exp2. The same procedure was applied for all stiffness and damping coefficients of the bearing.

$$K_{ij}\left(ϵ,\alpha ,\mathsf{\Omega }\right)=\frac{{\tilde{K}}_{ij}^{max}\left(ϵ,\mathsf{\Omega }\right)+{\tilde{K}}_{ij}^{min}\left(ϵ,\mathsf{\Omega }\right)}{2}sign\frac{{\tilde{K}}_{ij}^{max}\left(ϵ,\mathsf{\Omega }\right)-{\tilde{K}}_{ij}^{min}\left(ϵ,\mathsf{\Omega }\right)}{2}·\gamma$$

(10)

$$C_{ij}\left(ϵ,\alpha ,\mathsf{\Omega }\right)=\frac{C_{ij}^{max}\left(ϵ,\mathsf{\Omega }\right)+C_{ij}^{min}\left(ϵ,\mathsf{\Omega }\right)}{2}sign\frac{C_{ij}^{max}\left(ϵ,\mathsf{\Omega }\right)-C_{ij}^{min}\left(ϵ,\mathsf{\Omega }\right)}{2}·\gamma$$

(11)

where i and j are the local coordinates, and

$$sign=\left\{\begin{array}{l}+ij=\xi \xi \\ -otherwise\end{array}\right.$$

$$\gamma =\left\{\begin{array}{c}cos\left(n\alpha \right)i=j\\ \mathit{sin}\left(n\alpha \right)i\ne j\end{array}\right.,n:numberofpads,$$

$${\tilde{K}}_{ij}^k\left(ϵ,\mathsf{\Omega }\right)=\raisebox{1ex}{${\int }_0^{ϵ}{K}_{ij}^k\left(\lambda ,\mathsf{\Omega }\right)·d\lambda $}\!\left/ \!\raisebox{-1ex}{$ϵ$}\right.k:-maxormin$$

Furthermore, the bearing coefficients in Equations (10) and (11) are given in the local coordinates. To be able to use them in Equation (1), they must be transformed into the fixed coordinates using Equations (12) and (13):

$${\mathbf{K}}_{\mathbf{B}}={\mathbf{T}}^{\mathbf{T}}{\mathbf{K}}_{\mathsf{\beta }}\mathbf{T}$$

(12)

$${\mathbf{C}}_{\mathbf{B}}={\mathbf{T}}^{\mathbf{T}}{\mathbf{C}}_{\mathsf{\beta }}\mathbf{T}$$

(13)

where ${\mathbf{K}}_{\mathsf{\beta }}$ is the local bearing stiffness matrix, ${\mathbf{C}}_{\mathsf{\beta }}$ is the local bearing damping matrix, T is the transformation matrix, and ${\mathbf{K}}_{\mathbf{B}}$ and ${\mathbf{C}}_{\mathbf{B}}$ are the bearing stiffness and damping matrices in the fixed coordinate, respectively.

$$\mathbf{T}=\left[\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)& \mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)\\ -\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)& \mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha \right)\end{array}\right]$$

4. Results and Discussion

In this section, the dynamics of the hydropower machine are studied, and the contribution of the design modification on the LGB is investigated. Simulation and experimental results are presented in Section 4.1 and Section 4.2, respectively, and the results of the two models are compared. The unbalance responses were simulated, and modal parameters were identified for two cases: over-speeding condition and operation at nominal speed. In the former, the modal parameters were calculated for rotor speeds up to 3 × ${\mathsf{\Omega }}_0$, where the runaway speed is ${\mathsf{\Omega }}_{\mathrm{r}}$ = 335 rpm (2.23 × ${\mathsf{\Omega }}_0$). The magnetic stiffnesses produced by the generator and exciter were not included in the model, as the system was assumed to be disconnected from the electrical grid and to operate under no load. For the latter case, however, the magnetic stiffnesses were considered in the simulation. Unbalance response simulations were performed, and the results from the models were evaluated and compared. Furthermore, the modal parameters were identified at the nominal rotor speed.

4.1. Simulation Results

Rotor over-speeding condition without UMP: The Campbell diagram and stability map were plotted for rotor speeds up to 3 × ${\mathsf{\Omega }}_0$. An unbalance response simulation was first conducted at the nominal speed for both models (MOB and MMB), and the maximum amplitudes of the trajectory of the journal were obtained at each bearing location. Based on those values, the stiffness and damping coefficients of each bearing were computed at the prescribed relative eccentricity ratio for various rotor speeds, and subsequently, the modal parameters of the system were identified. Figure 6 and Figure 7 show the Campbell diagram and stability map of the models of the MOB and MMB, respectively. The first seven modes of the system are plotted as a function of the normalized rotor speed. Figure 8 displays the corresponding mode shapes plotted at 1 × ${\mathsf{\Omega }}_0$. For each plot in Figure 6 and Figure 7, the simulation results from the full and reduced models are overlaid and compared. The maximum deviation of the modal parameters due to model reduction (|1 − Reduced/Full| × 100%) was less than 1%, and thus, the reduced model can be considered as an accurate estimate of the full model.

The first mode of both models, which were the sub-synchronous modes, demonstrated high damping values, exceeding 70%. For the MOB, the synchronous line (1X) crossed the second mode at 2.05 × ${\mathsf{\Omega }}_0$, which is below the runaway speed. Furthermore, the second critical speed crossing the third mode at 2.25 × ${\mathsf{\Omega }}_0$ is considerably close to the runaway speed (${\mathsf{\Omega }}_{\mathrm{r}}$ = 2.23 × ${\mathsf{\Omega }}_0$). For the MMB, however, the damped natural frequencies of the second and third modes were high enough to surpass the 1X-synchronous line with a significant margin; subsequently, no critical speed was found within the frequency range. The synchronous line crossed the fourth mode at 2.29 × ${\mathsf{\Omega }}_0$, which exceeds the runaway speed.

Operation at nominal speed with UMP: An unbalance response was simulated at the nominal rotor speed with UMP. Figure 9 shows the results at the three bearing locations for MOB and MMB. The amplitudes of the orbits of the two models at the TB were relatively similar, and the maximum amplitude of the model of the MMB was ((1 − 0.0676/0.073) × 100%) 7% lower than those in the MOB. However, the amplitude differences were larger at the other two bearing locations. The maximum amplitudes of the model of MMB were ((1 − 0.127/0.178) × 100%) 29% and ((1 − 0.143/0.419) × 100%) 66% lower than those of the MOB at the UGB and LGB locations, respectively.

Furthermore, the damped natural frequency and damping ratio of the two systems at the nominal rotor speed were calculated and compared. For illustration purposes, the change in dynamics with the bearing clearance was investigated by gradually changing the values of C_b from 1.6 mm (original LGB) to 0.45 mm (modified LGB). The unbalance response simulations were performed, and Figure 10 displays the maximum trajectory of the journal center as a function of C_b. Note that the first points (C_b = 1.6 mm) refer to the simulation results for the MOB. Furthermore, the vertical axis of the right-hand side of each figure shows the percentile ratio of the response with respect to the reference (MOB). The magnitude of the trajectory of the journal decreases with the bearing clearance.

The stiffness and damping coefficients of each bearing were calculated based on prescribed eccentricities (Figure 10), and the modal parameters of the system were identified. In

Table 3. The first seven normalized damped natural frequencies ($r=\raisebox{1ex}{${\omega }_n$}\!\left/ \!\raisebox{-1ex}{${\omega }_0$}\right.$, where ${\omega }_0$ = 2.5 Hz) of the system for different diametric pad clearances of the LGB.

	Normalized Damped Natural Frequency, r (-)
Cb (mm)	1	2	3	4	5	6	7
1.6 (MOB)	0.65	1.53	1.6	2.75	3.73	3.87	4.65
1.4	0.65	1.59	1.65	2.75	3.71	3.87	4.63
1.2	0.65	1.64	1.92	2.75	3.67	3.87	4.59
1.1	0.63	1.63	1.86	2.75	3.68	3.87	4.6
1	0.63	1.76	5.03	2.75	3.59	3.87	4.43
0.8	0.61	1.96	7.89	2.75	3.61	3.87	4.33
0.6	0.57	2.25	8.3	2.76	3.62	3.86	4.24
0.45 (MMB)	0.54 (−16.9%) *	2.65 (+73%) *	8.38 (+423%) *	2.77 (+0.7%) *	3.56 (−4.6%) *	3.86 (−0.3%) *	4 (−14%) *

* ($r_{modified}/r_{original}$ − 1) × 100%.

and

Table 4. The damping ratio for the first seven modes of the system for different diametric pad clearances of the LGB.

	Damping Ratio, ζ (-)
Cb (mm)	1	2	3	4	5	6	7
1.6 (MOB)	0.303	0.369	0.252	0.032	0.142	0.061	0.08
1.4	0.323	0.387	0.486	0.033	0.15	0.062	0.096
1.2	0.377	0.385	0.643	0.034	0.158	0.064	0.115
1.1	0.348	0.387	0.583	0.035	0.156	0.064	0.109
1	0.454	0.384	0.696	0.036	0.149	0.067	0.132
0.8	0.524	0.420	0.290	0.039	0.133	0.071	0.129
0.6	0.568	0.495	0.179	0.045	0.139	0.074	0.147
0.45 (MMB)	0.589 (+94%) *	0.722 (+95%) *	0.151 (−40%) *	0.053 (+65%) *	0.160 (+13%) *	0.078 (+28%) *	0.151 (+89%) *

* (${\mathsf{\zeta }}_{modified}/{\mathsf{\zeta }}_{original}$ − 1) × 100%.

, the normalized damped natural frequencies and the damping ratios of the first seven modes are presented for models with different C_b values of the LGB, respectively. Unlike the first, fifth, sixth, and seventh modes, the natural frequencies of the second, third, and fourth modes increased when C_b decreased. The design modification employed in this paper significantly affected mainly the second and third modes. The numbers in the parentheses in the last row of Table 3 and Table 4 show the relative deviation of the modal parameters of the model with the modified LGB from those with the original LGB. Accordingly, due to the design modification, the damped natural frequencies of the second and third modes increased by 73% and 423%, respectively. Similarly, the corresponding damping ratios are shown in Table 4 of the model with different C_b values of the LGB. The damping ratio increased for all modes except for the third mode. Note that the damping ratio of the third mode decreased by 40% when its damped natural frequency increased by 423%.

4.2. Vibration Measurement

The vibration measurement results are discussed in this section, and Figure 11 displays the trajectory of the journal center at the three bearing locations. The results from the two measurements, i.e., before and after the design modification, are plotted together. Note that the static offsets of the orbits are removed for illustration purposes. The maximum peak-to-peak orbital amplitudes of the journal and at the bearing housing are presented in

Table 5. Summary of peak-to-peak amplitudes of MMB (C_b = 0.45 mm) at the three bearing locations. Values in parentheses show results from the MMO (C_b = 1.6 mm).

Cases	UGB		LGB		TB
	X (mm)	Y (mm)	X (mm)	Y (mm)	X (mm)	Y (mm)
0 MW 0 kV	0.11 (0.13)	0.12 (0.12)	0.1 (0.2)	0.1 (0.2)	0.04 (0.11)	0.04 (0.11)
0 MW 12.4 kV	0.23 (0.36)	0.23 (0.35)	0.3 (0.92)	0.3 (0.92)	0.15 (0.29)	0.15 (0.28)
13 MW	0.23 (0.36)	0.22 (0.36)	0.31 (0.91)	0.3 (0.91)	0.1 (0.29)	0.1 (0.29)
26 MW	0.21 (0.33)	0.21 (0.33)	0.3 (0.82)	0.29 (0.82)	0.04 (0.25)	0.04 (0.24)
38 MW	0.2 (0.32)	0.2 (0.31)	0.29 (0.82)	0.29 (0.81)	0.04 (0.19)	0.04 (0.18)
52 MW	0.19 (0.33)	0.19 (0.32)	0.28 (0.83)	0.26 (0.82)	0.04 (0.15)	0.04 (0.14)

and

Table 6. Summary of peak-to-peak amplitudes of MMB (C_b = 0.45 mm) at the three bearing housing locations. Values in parentheses show results from the MMO (C_b = 1.6 mm).

Cases	UGB		LGB		TB
	X (mm)	Y (mm)	X (mm)	Y (mm)	X (mm)	Y (mm)
0 MW 0 kV	0.02 (0.01)	0.02 (0.01)	<0.01 (0.01)	<0.01 (0.01)	<0.01 (0.01)	<0.01 (0.01)
0 MW 12.4 kV	0.025 (0.075)	0.02 (0.07)	0.03 (0.01)	0.03 (0.01)	<0.01 (0.01)	<0.01 (0.01)
13 MW	0.03 (0.07)	0.025 (0.07)	0.03 (0.02)	0.03 (0.02)	<0.01 (0.02)	<0.01 (0.02)
26 MW	0.025 (0.065)	0.02 (0.06)	0.03 (0.015)	0.03 (0.015)	<0.01 (0.015)	<0.01 (0.015)
38 MW	0.03 (0.06)	0.02 (0.055)	0.025 (0.015)	0.025 (0.015)	<0.01 (0.01)	<0.01 (0.01)
52 MW	0.02 (0.06)	0.02 (0.06)	0.025 (0.01)	0.025 (0.01)	<0.01 (0.01)	<0.01 (0.01)

for different cases. It was observed that the vibration amplitudes at the three bearing locations decreased due to the design modification. For operations at the nominal speed with UMP (0 MW 10.4 kV), the peak-to-peak amplitude of the MMB was ((1 − 0.230/0.355) × 100%) 35%, ((1 − 0.3/0.92) × 100%) 67%, and ((1 − 0.150/0.285) × 100%) 47% lower than those of the MOB at the UGB, LGB, and TB locations, respectively. Similarly, the bearing housing vibration showed a reduction at the UGB and TB. However, for almost all cases, the bearing housing vibration of the MMB (C_b = 0.45 mm) at the LGB was increased compared with those in the MMB (C_b = 1.6 mm), which is expected, as the modified bearing carries more load than the original bearing.

5. Conclusions

The design modification employed on the LGB of the hydropower machine improved the dynamics of the system. Both experimental and simulation results showed that the redesigned LGB, i.e., the four-lobe bearing with lower bearing clearance, provided a stability advantage and reduced the overall vibration amplitudes.

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The large bearing clearance of the original LGB allowed for greater shaft movement, resulted in the system’s elevated overall vibration amplitudes, and subjected the UGB and TB to carrying relatively larger loads.

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The design modification increased the second and third modes of the system above the 1X-synchronous line within the speed range, up to the runaway speed, and all the critical speeds were effectively eliminated.

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For operation at the nominal speed with UMP, the design modification significantly increased the second and third modes by 73% and 423%, respectively. Similarly, the damping ratios of the first seven modes increased, except for the third mode, which decreased by 40%. However, the damped natural frequency of the third mode increased from 1.6 × ${\mathsf{\Omega }}_0$ to 8.38 × ${\mathsf{\Omega }}_0$ (i.e., 423% increment), and the vibration amplitudes at this frequency were much smaller and were insignificant to producing mechanical failure.

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Furthermore, the simulated unbalance responses were reduced by 29%, 66%, and 7% at the UGB and LGB locations, respectively. In the field measurement, the peak-to-peak amplitudes at the three bearing locations were reduced by 35%, 67%, and 47%, respectively.