Theoretical and Experimental Study of Flexible Structure Tilting Pad Bearings Considering Deformation

Abstract

In high-speed and heavy-load conditions, ordinary rigid tilting pad journal bearings experience significant contact stress at the pad pivot points, leading to severe pad deformation and increased wear. A flexible structure tilting pad bearing (FSTPB) is presented in this paper, using spring supports to replace the traditional pivot supports and flexible hinge supports. A theoretical calculation model for tilting pad radial journal bearings considering flexible structure deformation is established, and the impact of elastic deformation on the performance of the flexible structure tilting pad bearings is discussed. Based on theoretical research, vibration experiments on flexible tilting pad bearings under different loading conditions were conducted. The influence of various structural parameters on the vibration characteristics of the flexible tilting pad radial bearings was studied. The results indicate that, compared to ordinary tilting pad bearings, flexible structure tilting pad bearings exhibit excellent vibration reduction characteristics at high speeds. Reducing the bearing clearance, lowering the stiffness of the flexible structure, and increasing the offset angle of the flexible structure contribute to enhancing the operational stability of the bearing–rotor system.

1. Introduction

Sliding bearing–rotor systems are crucial components of rotating machinery, widely used in various precision rotating equipment. In recent years, with rapid technological advancements, rotating machinery is developing towards larger sizes and higher speeds. Factors such as internal friction and mass imbalance have led to increasing system instability [1]. Therefore, it is of significant engineering importance to reduce the vibration levels and amplitudes of rotor systems by altering the bearing structure, thereby enhancing the system’s operational stability.

Due to their high stability, tilting pad (TP) bearings have been researched and designed by many scholars [2,3]. However, they also have undesirable characteristics, such as stress concentration at the pivot points and difficulties in manufacturing and installation [4]. The flexible pivot journal bearing (FPJB) shown in Figure 1a uses an electric discharge machining (EDM) structure. This design replaces the pivot of the journal pad with a flexible pivot support, inheriting the excellent performance of conventional tilting pad bearings, while simplifying the bearing structure, reducing the manufacturing difficulty, and improving the bearing manufacturing and installation precision [5,6]. Plantegenet et al. [7] conducted experimental analysis on the thermal imbalance effects of rotors supported by flexible pivot tilting pad bearings, highlighting the Morton effect characteristics induced by these bearings. This refers to the increase in the rotor’s synchronous vibration amplitude caused by heat. The results indicated that large amplitude synchronous vibrations can lead to instability in the rotor–bearing system and that reducing the bearing’s radial clearance is beneficial for system stability. Wilkes and Childs [8] discussed the importance of pad and pivot flexibility in predicting the impedance coefficients of the tilting pad journal bearings, presenting measurements of how the bearing clearance changes with operating temperature, and summarizing the differences between the measured and predicted frequency dependencies of the dynamic impedance coefficients. The measured thermal bearing clearance was approximately 30% smaller than the measured cold bearing clearance and was inversely proportional to the pad surface temperature. Zhang et al. [9] experimentally studied the variations in pad temperature and oil film thickness in elastic pivot-supported tilting pad bearings. Koondilogpiboon et al. [10] analyzed the nonlinear rotor dynamics in different directions of a flexible rotor supported by a four-pad flexible pivot tilting pad bearing and compared it with a fixed geometry journal bearing of the same shape. The experimental results showed that, apart from the flexible tilting pad bearing’s onset of instability speed being higher than the maximum test bench speed, sub-critical speeds occurred near the calculated onset of the instability speed in all cases. The frequency of the rotating orbit was the same as the first critical speed and proceeded in the direction of the shaft’s rotation. Vannini et al. [11] designed, manufactured, and tested a large four-pad flexible pivot bearing with a diameter of 280 mm and an L/D ratio of 0.7. The experiments were conducted at a speed of 7000 rpm and a load of 0.75 MPa, measuring all the relevant test boundary conditions, as well as the static and dynamic characteristics of the bearing. The experimental results were discussed and compared with predictions from existing numerical programs. Varela and Santos [12] described the process of establishing a mathematical model for the equivalent stiffness and damping coefficients of flexible pivot tilting pad bearings based on active lubrication control and set up an experimental platform to test the active lubrication performance of the bearings. The servo valve could change the flow rate of the pressurized oil entering the bearing clearance to achieve active control. The research results indicated that using an active lubrication system to control the flexible pivot tilting pad bearings could improve the stiffness and damping characteristics of the bearings. Peng et al. [13] used a flexible pivot tilting pad bearing as the basis and conducted experiments with piezoelectric actuators at different circumferential angles and radial displacements of the tilting pads. The results showed that introducing active control in the flexible pivot tilting pad bearings by manipulating the control variables related to the circumferential angle and radial displacement of the bearing pads could significantly reduce the amplitude of the rotor journal. Zemella et al. [14] experimentally studied the effects of the pad preload and pivot offset angle on the frequency-dependent characteristics of the dynamic stiffness and damping coefficients of the tilting pad journal bearings.

In flexible pad tilting pad bearings, the elastic deformation of the pad is equally critical. Khelifa et al. [15] introduced theoretical research results for a five-pad tilting pad bearing, calculating the total oil film thickness, including the elastic deformation and thermal effects. The Reynolds equation for dynamic loads was solved using the finite element method. The resulting pressure, oil film forces, and torque components were expressed as functions of the journal coordinates. By differentiating the oil film forces, the dynamic coefficients of the oil film were determined, leading to the final determination of the pad elastic deformation. Boubendir et al. [16] studied the effect of the pad elastic deformation on the static characteristics of finite porous journal bearings. Suh and Palazzolo [3] in 2015 considered the influence of temperature on the lubricating oil viscosity, obtaining the thermal and elastic deformation of the pads and revising the traditional formula for calculating the oil film thickness at the nodes. Suh et al. [17] established a model for flexible pivot tilting pad bearings based on three-dimensional elastohydrodynamic lubrication methods and calculated the fluid forces acting on the bearing and rotating journal using the finite element method to analyze the static and dynamic behavior caused by pad deformation. Mehdi et al. [18] predicted the static and dynamic characteristics of tilting pad journal bearings (TPJBs) using two different models, elucidating the dominant role of contact point flexibility on TPJB performance. They calculated the pivot stiffness using the Hertz contact theory and coupled it with fluid film hydrodynamics. Zhang et al. [19] used the Matlab PDE toolbox and ANSYS to perform static characteristic calculations and simulations for fixed tilting pad journal bearings considering the deformation of the pads and their pivot structures. Arihara et al. [20] introduced a thermoelastohydrodynamic analysis model for tilting pad bearings, predicting their steady-state performance and comparing it with experimental results to validate the model’s effectiveness. Khalifa et al. [21] studied the impact of multilayer surfaces on the hydrodynamic performance of journal bearings. Radial loads applied to the journal bearing affected the surface area of the bearing, leading to the formation of radial stress. Lee et al. [22] used the isoviscous Reynolds equation, which ignored the temperature rise caused by the viscous shear in the thin film and the resulting thermal deformation of the bearing structure. They predicted the elastic deformation of the bearing housing using a three-dimensional finite element model and conducted transient analysis, allowing the bearing housing and journal to converge to a state of static equilibrium. Wang et al. [23] studied the lubrication characteristics of diesel engine main bearings under elastic deformation conditions, establishing an elastohydrodynamic simulation model for diesel engine main bearings. The results indicated that, as the main bearing clearance increased, the minimum oil film thickness gradually decreased and the maximum oil film pressure gradually increased. Chun et al. [24] studied the quantitative impact of thermoelastic deformation on bearing preload variation. By considering the relationship between viscosity and temperature, they coupled the variable viscosity Reynolds equation with the energy equation and solved it using the finite element method. They took into account heat transfer between the rotating journal, oil film, and bearing pads and used a three-dimensional finite element model to calculate the deformation of the bearing structure.

In summary, many scholars have conducted extensive theoretical and experimental research on conventional tilting pad bearings and flexible hinge tilting pad bearings, emphasizing the necessity of considering elastic deformation in theoretical models when evaluating the performance of flexible hinge tilting pad bearings. This paper proposes a tilting pad bearing structure that replaces flexible hinges with springs. It establishes a theoretical model of flexible structure tilting pad bearings incorporating elastic deformation using the coefficient method. Vibration performance experiments on flexible structure tilting pad bearings with various structural parameters were conducted, studying the influence of multiple factors on the lubrication characteristics of the flexible structure tilting pad radial sliding bearings. This research provides design methods and foundations to enhance the stability of tilting pad bearings.

2. Flexible Structure Tilting Pad Bearing

The photograph of the flexible structure tilting pad bearing designed in this study is shown in Figure 1b. This design adopts a pair of asymmetrically arranged spring supports to replace the traditional pivot support and flexible hinge support methods. This arrangement provides a certain amount of radial damping while allowing circumferential deflection. The bearing is manufactured using wire-cutting technology, which achieves structural integration, greatly reduces installation errors, and improves installation precision.

3. Theoretical Calculation Model for the Flexible Structure Tilting Pad Bearing

3.1. The Reynolds Equation for Bearing Lubrication

Relative motion is generated between the rotating journal and the inner surface of the bearing shell, causing pressure in the oil film within the wedge-shaped gap. This pressure supports the external load and prevents direct contact between the journal and the bearing shell. The dynamic pressure of the oil film is described by the Reynolds equation. The simplified form of this equation is given as follows:

$${\displaystyle \frac{\partial }{\partial x}}\left(h^3{\displaystyle \frac{\partial p}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(h^3{\displaystyle \frac{\partial p}{\partial z}}\right)=6\mu \omega r{\displaystyle \frac{\partial h}{\partial x}}$$

(1)

In the above equation x and z represent the bearing coordinates, h represents the oil film thickness, p represents the oil film pressure, μ represents the dynamic viscosity of the lubricating oil, ω represents the journal speed, and r represents the journal radius.

3.2. The Expression for Oil Film Thickness

The flexible tilting pad bearing employs flexible hinges and spring supports, similar in working principle to conventional tilting pad bearings. When solving for the oil film thickness, it is necessary to consider the bearing’s eccentricity, the preload coefficient, and the tilting angles of each pad. Additionally, the pads in the flexible structure do not operate with pure tilting motion. The bearing calculation model is shown in Figure 2.

The origin of the coordinate system xO_by is fixed at the center O_b of the bearing. O_j is the center of the journal, O_p is the center of the pad support, e is the eccentricity between the center of the journal and the center of the bearing, ω is the rotational speed of the shaft, δ_i is the tilt angle of the ith pad, Φ_pi is the pivot position angle of the ith pad, θ is the tilt angle of the bearing, Φ is the rotation angle starting from the y-axis, φ is the rotation angle starting from the tilt angle, and η is the swing angle of the pad around the pivot point. According to the calculation model, the expression for the oil film thickness of the tilting pad bearing is as follows:

$$h=c+e\mathit{cos}(\Phi -\theta )+(\xi -mc)\mathit{cos}(\Phi -{\Phi }_{pi})+(\eta -r{\delta }_i)\mathit{sin}(\Phi -{\Phi }_{pi})+\mathsf{\Delta }h$$

(2)

In the equation m represents the preload coefficient, $m={\displaystyle \frac{c-c^{\prime }}{c}},c^{\prime }$ represents the bearing assembly clearance, c represents the bearing clearance, and Δh represents the deformation of the pad caused by flexible support. When $H={\displaystyle \frac{h}{c}},\epsilon ={\displaystyle \frac{e}{c}},\overline{\xi }={\displaystyle \frac{\xi }{c}},\overline{\eta }={\displaystyle \frac{\eta }{c}}$,$\delta ={\displaystyle \frac{\mathsf{\Delta }h}{c}}$, nondimensionalization Equation (2) yields the nondimensional oil film thickness of the bearing as follows:

$$H=1+\epsilon \mathit{cos}(\Phi -\theta )+(\overline{\xi }-m)\mathit{cos}(\Phi -{\Phi }_{pi})+(\overline{\eta }-{\displaystyle \frac{{\delta }_i}{\psi }})\mathit{sin}(\Phi -{\Phi }_{pi})+\delta$$

(3)

3.3. The Coefficient Method Calculates the Elastic Deformation of a Flexible Structure Tilting Pad Bearing

The elastic deformation of the pad blocks significantly influences the bearing characteristics. Therefore, this paper proposes using the coefficient method to calculate the elastic deformation of the pads. Compared to the conventional theoretical formulae for the deformation calculation, the coefficient method excels in accurately accounting for the three-dimensional elastic deformations of complex elastic support structures. The finite element model of the flexible tilting pad bearing, discretized using eight-node hexahedral elements and divided into a mesh, is shown in Figure 3.

To verify the correctness of the stiffness matrix extraction process, random positions in the mesh are selected, where eight nodes are each subjected to forces of 1 N in the x, y, and z directions. After applying the boundary conditions, the deformation of these eight points is solved and exported, as shown in Figure 4.

Another method involves extracting the stiffness matrix for these eight nodes and then establishing a force matrix where each entry is 1 N. Multiplying the stiffness matrix by this force matrix gives the displacements. The displacement results obtained from both methods are summarized in

Table 1. Node deformation values.

Node	ANSYS	The Coefficient Method	Error
3542	2.9527 × 10−4	2.95267 × 10−4	0.00105%
3543	3.0164 × 10−4	3.01635 × 10−4	0.00164%
3544	3.1128 × 10−4	3.11282 × 10−4	0.000619%
3545	3.2024 × 10−4	3.20242 × 10−4	0.000692%
3546	3.3429 × 10−4	3.34288 × 10−4	0.0006%
3547	3.4553 × 10−4	3.45533 × 10−4	0.000915%
3548	3.5777 × 10−4	3.57765 × 10−4	0.00133%
3549	3.7226 × 10−4	3.72259 × 10−4	0.00103%
3544	2.9527 × 10−4	2.95267 × 10−4	0.00105%
3545	3.0164 × 10−4	3.01635 × 10−4	0.00164%

.

Comparing the results in Table 1, the deformations at each node are equal. This demonstrates that the coefficient method for calculation is accurate, reliable, and trustworthy.

3.4. The Process of Solving Static Characteristics Considering the Elastic Deformation of the Bearing Pad

The solution process for the deformation of the pad blocks in a flexible tilting pad bearing is shown in Figure 5. The process of solving the static characteristics considering the pad elastic deformation starts with calculating the initial oil film thickness using Equation (3) and solving the lubrication equation using finite differences. The elastic deformations on the bearing surface are then computed using the stiffness matrix. These deformations are iteratively incorporated to adjust the initial film thickness until convergence of the elastic deformations is achieved. At static equilibrium, the oil film thickness stabilizes and stops changing. In the elastohydrodynamic coupling algorithm, as the number of iterations increases, the initial film thickness continuously improves.

3.5. The Impact of Elastic Deformation on the Static Characteristics of Flexible Structure Tilting Pad Bearings

The elastic deformation of the pad blocks is incorporated into the influence on the oil film pressure. As shown in Figure 1, the overall spring structure connecting the pad blocks is identical. When the shaft is at static equilibrium, the tilt angle is 0°. This study considers the pad pressure over a circumferential angle range of 0–360°. The required bearing parameters for the calculation are shown in

Table 2. Parameter table of the flexible tilting pad bearing.

Parameters	Value
Bearing width (mm)	25
Shaft diameter (mm)	30
Bearing radial clearance (mm)	0.05
Number of pads	4
Pad arc angle (°)	72
Thickness of pads (mm)	5
Preload factor	0.375
Type of lubricant	ISO-VG32
Supply oil pressure (MPa)	0.1
Oil inlet temperature (°C)	10

. The overall spring stiffness of the flexible tilting pad bearings calculated using ANSYS are 2.97 × 10⁶ N/m and 2.42 × 10⁶ N/m, respectively. These two types of flexible bearings are compared with conventional tilting pad bearings under the same operating conditions.

The results are shown in Figure 6, Figure 7, Figure 8 and Figure 9. From these figures, it can be seen that under the same load and speed conditions, the oil film thickness is thinnest on rigid surfaces (without considering the pad block elastic deformation). Due to the designed offset spring, which adjusts the swing angle adaptively based on external loads, speed, and other actual operating conditions, both types of flexible structure tilting pad bearings with different overall spring stiffnesses can form converging oil wedges. The graphs illustrate that the elastic deformation of the pad blocks noticeably reduces the peak oil film pressure. As the overall spring stiffness decreases, the stiffness of the bearing’s flexible structure increases, leading to gradually larger deformations. This causes the oil film thickness on both the upper and lower pads to gradually move away from the rigid surface scenario. However, the bearing’s load capacity does not decrease, meaning the difference between the load-bearing and non-load-bearing pads does not diminish. Figure 6 and Figure 7 depict the oil film thickness and pressure distribution of the flexible tilting pad bearings under the same 100 N load but different speeds and overall spring stiffnesses. From these graphs, it is evident that the peak pressure decreases with increasing speed, while the minimum oil film thickness increases with increasing speed. Figure 7, Figure 8 and Figure 9 illustrate the oil film thickness and pressure distribution of the flexible tilting pad bearings at the same speed of 3000 rpm but different loads and overall spring stiffnesses. These graphs show that the peak pressure increases with increasing load, while the minimum oil film thickness decreases with increasing load.

3.6. Dynamic Characteristic Calculation of Flexible Tilting Pad Radial Sliding Bearings

The small perturbation method is used to solve the stiffness and damping coefficients of the inner oil film. When the journal is perturbed by a small displacement (±∆x, ±∆y) and small velocity (±$\mathsf{\Delta }\dot{x}$, ±$\mathsf{\Delta }\dot{y}$), new oil forces can be obtained. The stiffness and damping coefficients are given by the following equations:

$$\left\{\begin{array}{c}{K}_{xx}=-{\displaystyle \frac{F_{x1}-F_{x2}}{2\left|\mathsf{\Delta }x\right|}},K_{yy}=-{\displaystyle \frac{F_{y1}-F_{y2}}{2\left|\mathsf{\Delta }y\right|}}\\ C_{xx}=-{\displaystyle \frac{F_{x3}-F_{x4}}{2\left|\mathsf{\Delta }\dot{x}\right|}},C_{yy}=-{\displaystyle \frac{F_{y3}-F_{y4}}{2\left|\mathsf{\Delta }\dot{y}\right|}}\end{array}\right.$$

(4)

The equivalent stiffness and damping coefficients of the FSTPB can be obtained as follows:

$$\left\{\begin{array}{c}{K}_{eq}={\displaystyle \frac{K_1{K}_2{K}_3}{K_1{K}_2+K_1{K}_3+K_2{K}_3}}\\ C_{eq}={\displaystyle \frac{C_1{C}_2{C}_3}{C_1{C}_2+C_1{C}_3+C_2{C}_3}}\end{array}\right.$$

(5)

The dynamic characteristics play a crucial role in the calculation of the rotor dynamics and the stability of rotor systems. The stiffness and damping characteristics of the FSTPB are calculated using the small perturbation method, and the critical speed of the experimental bearing–rotor system is determined using the MADYN 2000 rotor dynamics calculation software. This critical speed is then compared with the experimental results to verify the accuracy of the theoretical research.

4. Experimental Procedure

4.1. Test Bearings

The experimental bearings designed in this study consist of six types. One type is an ordinary tilting pad bearing supported by flexible hinges, as shown in Figure 1a. To investigate the impact of flexible support on the bearing lubrication characteristics, five types of spring-supported bearings were fabricated, as shown in Figure 1b. These are all supported by a layer of asymmetrically arranged springs, with variations in the spring position and stiffness. The basic parameters of the bearings are shown in

Table 3. Basic bearing parameters.

Parameters	Value
Bearing length (mm)	25
Preload factor	0.375
Pad arc angle (°)	72
Number of pads	4
Thickness of pads (mm)	5
Oil inlet diameter (mm)	3
Thickness of Babbitt alloy	1
Bearing material	40CrMo

. The six types of bearings are labeled as shown in

Table 4. Bearing types.

Types	Bearing Radial Clearance	Spring Stiffness	Spring Position
	/mm	/(N/m)	a1/°	a2/°
FPJB	0.05	/	/	/
FSTPB 1	0.05	2.97 × 106	25	30
FSTPB 2	0.00375	2.97 × 106	25	30
FSTPB 3	0.05	2.42 × 106	25	30
FSTPB 4	0.05	2.97 × 106	20	30
FSTPB 5	0.05	2.97 × 106	25	25

. The bearings were made of 40CrMo, which has high yield strength, tensile strength, and toughness. The bearings were processed using wire cutting, enabling high precision for the springs and hinge structures. The bearing pads were centrifugally cast with a 1 mm thick layer of Babbitt alloy, which provides good tribological properties and prevents shaft wear.

4.2. Experiment Setup

The flexible tilting pad bearing test rig, as shown in Figure 10, primarily consists of a high-speed electric spindle, flexible coupling, dual-disk rotor, sliding bearing housing, loading device, and oil supply system. The high-speed electric spindle is fixed on an adjustable position base, cooled by a water circulation cooling system, and capable of providing speeds ranging from 0 to 12,000 rpm. The oil supply system includes a 5 L oil tank, an oil pump, and oil supply and return lines and is capable of providing a stable oil pressure of 0.1 MPa. The loading device uses weights for loading, applying standard mass weights during the performance testing of the bearing. The loading system includes a support frame, fixed pulley, and steel tension rope. The specific loading scheme is as follows: the steel rope is wound around the test bearing and connected to the fixed pulley, with the other end of the steel tension rope connected to standard mass weights, keeping the steel rope in a freely tightened state. In this way, the fixed pulley converts the weight of the mass into the tensile force of the rope acting on the test bearing, achieving static force loading on the flexible tilting pad radial sliding bearing. The advantage of weight loading over other loading schemes lies in the standard mass of the weights, ensuring high accuracy and controllability, with stable and nearly fluctuation-free tensile force, providing strong reliability.

Two eddy current displacement sensors (TR81 model: 810503; sensitivity: (5%) 5 V/mm) are fixed onto the sensor bracket with nuts, positioned vertically and horizontally to the rotor, monitoring the trajectory changes of the rotor center in real time. The installation of the acceleration sensors (PCB model: 352C33; sensitivity: (10%) 100 mV/g) are glued to the base of the bearing housing, mounted in both horizontal and vertical positions, to detect vibration signals in two directions. The experiment uses ISO-VG32-grade lubricating oil, supplied to the bearing from the top of the bearing housing through an oil supply line. Some key operating parameters of the system are listed in

Table 5. Operating parameters.

Parameters	Value
Density (kg/m3)	7850
Poisson’s ratio	0.3
Young’s modulus (GPa)	210
Shaft diameter (mm)	30
Disc diameter (mm)	140
Shaft length (mm)	750
Disc width (mm)	25
Mass of shaft (kg)	4.17
Mass of disc (kg)	5.76
Supply oil pressure (MPa)	0.1
Type of lubricant	ISO-VG32
Oil inlet temperature (°C)	10

.

5. Experimental Results and Discussion

As indicated in Table 5, the experiments are conducted with the combined weight of the rotor and disk being 99.3 N. The rotor–bearing system is not subjected to additional loading. The motor speed is adjusted from 0 rpm to 6000 rpm using the motor speed control device, with the vibration data collected and analyzed at every 1000 rpm interval. The vibration tests are conducted and compared for six different bearing structures. The experimental results are presented as follows.

5.1. Comparison of Experimental Results for Different Types of Bearings

To select the appropriate bearing type, both conventional tilting pad bearings and flexible support tilting pad bearings were designed and fabricated. Vibration tests were conducted to compare their performance, with the specific data presented in

Table 6. Experimental control bearings with different bearing type parameters.

Types	Bearing Radial Clearance	Spring Stiffness	Spring Position
	/mm	/(N/m)	a1/°	a2/°
FPJB	0.05	/	/	/
FSTPB1	0.05	2.97 × 106	25	30

and the structures illustrated in Figure 1.

In a study of low-noise machinery, the vibration level of the bearing housing is crucial in evaluating the quality of the machinery [25]. Due to the current limitations that prevent the direct measurement of the external forces transmitted to the bearing housing by the rotor system, it is common to evaluate the reduction in the external forces and vibrations exerted by the bearings on the system through the method of testing the acceleration of the machine foot. Therefore, in this experiment, the vibration acceleration signals were collected using the acceleration sensors mounted on the bearing housing. The collected acceleration signals were processed into vibration acceleration levels (VAL) in units of decibels (dB). This transformation allows the level values, obtained through logarithmic and division operations, to effectively reflect the signal values over a wide dynamic range. The calculation formula is given by Equation (6).

$$VAL=20\lg {\displaystyle \frac{\alpha }{{\alpha }_0}}\left(\mathrm{dB}\right)$$

(6)

where α is the effective value of the acceleration and ${\alpha }_0$ is the reference acceleration, with ${\alpha }_0=10^{-6}\mathrm{m}/{\mathrm{s}}^2$.

Figure 11 shows the fundamental frequency vibration acceleration values at the machine foot of the bearing housing for both the FPJB and FSTPB 1. In Figure 11a, the vertical direction vibration acceleration values are shown. It can be observed that the vibration levels initially increase with the rotational speed until reaching the vicinity of the critical speed. Beyond the critical speed, the vibration levels exhibit a decreasing trend, followed by a subsequent increase. Figure 11b illustrates the vibration acceleration values in the horizontal direction, which initially increase and then decrease. Considering both directions of the vibration values, it can be noted that the first-order critical speed of the FPJB is around 5000 rpm, while that of the FSTPB 1 is approximately 4500 rpm. Additionally, comparing the vibration levels of the two bearing types, it can be observed that the FPJB and FSTPB 1 exhibit similar vibration levels at low speeds. However, in the medium- to high-speed range, the vibration levels of the FSTPB 1 notably decrease, with the maximum vibration levels in the vertical and horizontal directions decreasing by 7.1 dB and 6 dB, respectively.

Although Figure 11 depicts the trend of the vibration levels with the rotational speed, it does not provide spectral information. Therefore, the acceleration time domain signals were subjected to fast Fourier transform (FFT), and waterfall plots were generated. These plots vividly describe the trend of the acceleration vibration amplitude with rotational speed changes. They are also effective in illustrating the rotor synchronous vibration and nonsynchronous vibration caused by the oil film effects [26].

Figure 12 displays the waterfall plots of the horizontal vibration acceleration at the machine foot for the two bearing types. From both plots, it can be observed that, as the rotational speed increases, the amplitude of the fundamental frequency (1×) vibration gradually increases, reaching a peak before declining. Due to differences in the bearing stiffness, the positions of the peak amplitudes differ between the two bearing types, with that of the FPJB occurring near 5000 rpm and the FSTPB 1 occurring near 4500 rpm. Comparing the fundamental frequency vibration amplitudes of the two bearings, it can be noted that at low speeds (below 3000 rpm), the vibration amplitudes are relatively low. Near the medium-speed range, the vibration reduction effect of the FSTPB 1 is not ideal due to the single-layer spring structure reducing the bearing stiffness, leading to an earlier critical speed. However, at high speeds, the FSTPB 1 exhibits good vibration reduction characteristics, significantly reducing the vibration levels compared to the conventional bearing.

Figure 13 compares the shaft center trajectories of the two types of bearings at different rotational speeds. The elliptical shape indicates stable rotor operation. By comparing the shaft center trajectories of the two bearings, it can be observed that at 5000 r/min and 6000 r/min the area enclosed by the shaft center trajectory of the conventional tilting pad bearing is larger, indicating the largest shaft displacement amplitude and the most unstable operation. The FSTPB 1, compared to the conventional bearing, can reduce the displacement amplitude and improve the stability of the system operation. However, at 4000 r/min, the area of the shaft center trajectory of the FSTPB 1 is larger than that of the conventional bearing. This is due to the lower stiffness of the single-layer spring structure, which causes the critical speed to decrease and the vibration peak of the displacement to occur earlier.

Similarly, plotting the displacement signal spectra as waterfall plots facilitates the analysis of the bearing response at all rotational speeds, as shown in Figure 14. By comparing the fundamental frequency displacement amplitudes, it can be found that the critical speed of the conventional bearing is 5000 r/min, while the critical speed of the FSTPB 1 is around 4500 r/min, with the conventional bearing having the largest peak amplitude. The experimental results indicate that the flexible bearing 1, with its spring structure, can significantly reduce the vibration response and improve the stability of the rotor system.

5.2. Different Bearing Clearance Experimental Results Comparison

To design and select the appropriate bearing clearance, two types of flexible bearings with different bearing radial clearances were designed and fabricated. Vibration tests were conducted to compare them, and the specific parameters are shown in

Table 7. Experimental comparison of different bearing clearance parameters.

Types	Bearing Radial Clearance	Spring Stiffness	Spring Position
	/mm	/(N/m)	a1/°	a2/°
FSTPB1	0.05	2.97 × 106	25	30
FSTPB2	0.0375	2.97 × 106	25	30

.

Figure 15 compares the center orbits of the two bearings at different speeds. The elliptical shape indicates stable rotor operation. A comparison of the center orbits of the two bearings reveals that, at 4000 rpm and 6000 rpm, the area of the center orbit for the FSTPB 2 is smaller than that of the FSTPB 1, with little difference between them at 5000 rpm. The experimental results indicate that reducing the bearing clearance appropriately helps to reduce the critical amplitude and improve the system stability. This is because reducing the bearing radial clearance changes the oil film thickness, affecting the oil film pressure and load-carrying capacity. This results in improvements in the bearing stiffness and damping characteristics, causing the orbit of the rotor center to approach the bearing center, thereby enhancing the stability of the system.

5.3. Different Spring Stiffnesses of the Bearing Experimental Results Comparison

To design and select the appropriate spring stiffness for flexible bearings, two types of flexible bearings with different spring stiffnesses were designed and fabricated. Vibration tests were conducted to compare them, with the specific structures detailed in

Table 8. Experimental comparison of bearings with different spring thicknesses.

Types	Bearing Radial Clearance	Spring Stiffness	Spring Position
	/mm	/(N/m)	a1/°	a2/°
FSTPB1	0.05	2.97 × 106	25	30
FSTPB3	0.05	2.42 × 106	25	30

.

Plotting the spectrum of displacement signals as waterfall plots facilitates the analysis of the bearing responses across the entire range of rotational speeds, as shown in Figure 16. Similarly, comparing the fundamental frequency displacement amplitudes reveals that at speeds of 3000 rpm and below, both the FSTPB 1 and FSTPB 3 exhibit relatively small displacement amplitudes. Additionally, between the critical speeds of 4000 and 5000 rpm, the FSTPB 1 exhibits the largest peak amplitude. The experimental results indicate that appropriately reducing the spring stiffness of the flexible supports affects the clearance of the inner oil film, thereby improving the bearing stiffness and damping characteristics. This can reduce the displacement amplitude of the system, enhancing the stability of the bearing–rotor system operation.

5.4. Different Spring Positions Experimental Results Comparison

To design and select the appropriate support position for flexible bearings, three different spring positions of the flexible bearings were designed and fabricated. Vibration tests were conducted to compare them, with the specific parameters detailed in

Table 9. Experimental control of bearings with different spring positions.

Types	Bearing Radial Clearance	Spring Stiffness	Spring Position
	/mm	/(N/m)	a1/°	a2/°
FSTPB1	0.05	2.97 × 106	25	30
FSTPB4	0.05	2.97 × 106	20	30
FSTPB5	0.05	2.97 × 106	25	35

.

Wavelet analysis based on multi-resolution analysis has the advantage of more accurately and effectively reflecting the local characteristics of the signals in both the time and frequency domains. Figure 17 shows the wavelet time–frequency diagram of the acceleration signal. The rotational speeds of 4000 r/min, 5000 r/min, and 6000 r/min were selected for comparison. In the figure, the vibration energy is represented by the colors, with red indicating the strongest vibration, green representing about half of the maximum vibration energy, and blue as the background value representing the weakest energy. The energy at the rotational frequency is relatively large and will have a distinct red peak. By comparing the color intensity representing the energy magnitude at the rotational frequency, the vibration level can be determined.

From Figure 17, it can be observed that at 4000 rpm, FSTPBs 1, 4, and 5 exhibit higher energy levels, indicated by noticeable red energy values. This is attributed to the introduction of the spring structures in the bearings, leading to an earlier onset of critical speed, resulting in higher vibration levels at 4000 rpm. At 5000 rpm, the FSTPB 1 shows relatively higher vibration levels, while at 6000 rpm the FSTPB 5 exhibits lower vibration levels. The experimental results indicate that altering the spring position of the bearings changes the distribution of the oil film pressure, thereby influencing the stiffness and damping of the bearings. This is beneficial for reducing the vibration levels during system operation.

5.5. Rotor Experimental Results Under Different Loading Conditions Comparison

To further investigate the impact of the support positions on the lubrication characteristics of flexible bearings, additional loads were applied to the aforementioned three different spring positions of the flexible bearings using the loading apparatus illustrated in Figure 10. The spindle speed was first increased to 3000 rpm, and, after the system stabilized, the loading was conducted. Each type of bearing was subjected to five loadings ranging from 50 N to 250 N. The displacement signals and vibration acceleration signals were collected and analyzed every 1000 rpm for comparison. The specific structural data are shown in Table 9.

Figure 18 illustrates the waterfall plots of the vibration acceleration at the foot of three types of bearings under different loading conditions. From the waterfall plots, it can be observed that with increasing rotational speed the amplitude of the fundamental frequency (1×) vibration gradually increases, reaching a peak before declining. The peak amplitudes of the three bearings all occur at 4000 rpm. Under loading conditions of 150N, 200N, and 250N, the FSTPB 4 exhibits the smallest peak vibration amplitude, while the FSTPB 5 shows the smallest peak vibration amplitude under loading conditions of 50N and 100N. Moreover, at speeds of 3000 rpm, 5000 rpm, and 6000 rpm, the fundamental frequency (1×) amplitudes of the three types of flexible bearings are relatively small. The experimental results indicate that, under most loading conditions, the FSTPB 4 exhibits the smallest peak vibration amplitude. This is attributed to the largest spring offset angle in the FSTPB 4 under loading conditions. The offset spring structure steepens the wedge shape between the bearing pad and the rotor shaft, improving the pressure distribution and load-bearing capacity, thereby reducing the system’s vibration amplitude.

6. Conclusions

An experimental study of the vibration performance of flexible structure tilting pad bearings with different structural parameters was conducted, and the collected experimental data were analyzed, leading to the following conclusions:

(1)

When the bearings are only subjected to the weight of the rotor and the components on the shaft, the critical speeds of the two types of bearings are different. The critical speed of the conventional bearing is 5000 rpm, while that of the FSTPB 1 is 4500 rpm. In the mid- to high-speed range, the vibration levels of the flexible tilting pad bearings are significantly reduced, with the maximum vibration values decreasing by 7.1 dB and 6 dB in the vertical and horizontal directions, respectively. However, due to the reduction in the bearing stiffness, the critical speed of the flexible tilting pad bearings is advanced. Vibration acceleration signals indicate that the flexible tilting pad bearings can significantly reduce the critical speed and vibration levels at high speeds. For flexible bearings, increasing the bearing clearance appropriately and reducing the spring stiffness can improve the oil film thickness and pressure distribution of the bearings, thereby affecting the stiffness and damping characteristics of the bearings, reducing the vibration levels of the system and improving the stability of the system operation.

(2)

Loads of 50 N, 100 N, 150 N, 200 N, and 250 N were applied to the center of the rotor. The vibration acceleration signals indicate that the spring-supported flexible structure tilting pad bearings significantly reduce the vibration levels and exhibit good vibration characteristics at all speeds. Furthermore, as the load gradually increases, the bearings with larger spring offset angles show better vibration reduction effects. This is because appropriately increasing the spring offset angle can steepen the wedge shape between the bearing and the shaft, thereby improving the pressure distribution and load-carrying capacity, as well as enhancing the stiffness and damping characteristics to some extent.

(3)

The results from multiple experiments indicate that, compared to flexible pivot journal bearings, flexible structure tilting pad bearings exhibit excellent vibration damping characteristics at high speeds, resulting in reduced vibration levels. For the flexible structure tilting pad bearings, reducing the bearing clearance appropriately helps in the oil film convergence, enhancing the oil film pressure and load capacity, reducing the critical amplitudes and improving the system stability during operation. Additionally, decreasing the stiffness of the flexible structure alters the oil film thickness and pressure inside the bearing, influencing its stiffness and damping characteristics, thereby bringing the axis center trajectory closer to the bearing center and enhancing the system stability. Lastly, increasing the offset angle of the flexible structure steepens the wedge shape between the bearing and the shaft, improving the pressure distribution and load capacity of the oil film, thus reducing the system’s vibration amplitudes.