Design and Evaluation of Integral Damping Bearing for Vibration Suppression in Multi-Disk Rotor Systems

Abstract

In response to vibration problems caused by unbalanced faults in multi-disk rotor systems of compressors, this research study proposes a new type of integral damping bearing (IDB) with an integral structure and better damping performance. The IDB’s stiffness and damping coefficients were acquired through computational fluid dynamics (CFD) analysis. When the IDB was applied to a multi-disk rotor system, the analysis results show that it effectively lowered the bending stress and strain energy of the shaft and enhanced the stability of the system. The bending stress was reduced from 45 MPa (with traditional ball bearings) to 8.63 MPa (with the IDB). By using the IDB, the strain energy borne by the shaft and bearing was only 10.44%. Subsequently, experimental research was conducted on the suppression of unbalanced vibration by using the IDB under various working conditions. The peak amplitude of the rotor system with the IDB was reduced by 19.94%, and the amplitude at the 1X frequency was reduced by 32.73% compared with ball bearings. The experimental results indicate that the IDB can efficiently dampen the vibration of the rotor system. And it was found that the IDB has stable vibration reduction performance under various working conditions.

1. Introduction

Rotating machinery such as compressors, aircraft engines, and steam turbines often use rotor systems with a multi-disk structure. As a result of residual unbalance inherent in rotor systems from manufacturing, installation, and other processes, vibration is generated during rotor operation. When the speed nears the critical speed, rotor systems can enter resonance, with even minor residual unbalance leading to severe rotor vibration. In severe cases, it can cause shaft cracks, bearing failures, and even serious accidents, such as machine damage and fatalities. Hence, measures need to be implemented to dampen the rotor system’s vibration and enhance its stability.

In order to reveal the mechanism of vibration generation in multi-disk rotor systems, Merels et al. proposed a CSM method to analyze the response of natural frequencies and modes, unbalance, and nonlinear forces in multi-disk rotor systems [1]. Zhang et al. presented a novel dynamic modeling approach for a composite rod fastening rotor system, demonstrating the impact of fault severity and its parameters on the nonlinear dynamics of rotor systems [2]. Subsequently, Phadatare et al. conducted frequency response analysis to identify the necessary conditions for instability and chaotic motion in rotor systems [3]. Zhao et al. conducted a bifurcation state analysis of a pull rod structure rotor system with a multi-disk [4]. Han et al. employed the theory of parameter instability and the Taylor expansion technique to examine the distinctive instability features of asymmetric rotor systems with two disks in contrast to a single-disk rotor system [5]. Tarkashvand et al. developed a numerical analysis technique for an intricate multi-disk rotor system, investigating the impact of varying bearing stiffness and damping parameters on system vibration response and stability [6]. Saeed et al. proposed a vibration reduction technique based on active magnetic bearings for the nonlinear vibration problem in rotor systems. Through simulation analysis and experimental research, they found significant vibration reduction effects [7]. In summary, these studies indicate that the vibration response of multi-disk rotor systems is closely related to the stiffness and damping parameters of systems when there are unbalanced faults.

To increase system damping and reduce rotor vibration, a common measure taken is to install a damper in rotor systems. The squeeze oil film damper (SFD) is commonly utilized in aerospace engines due to its uncomplicated design and superior shock absorption capabilities [8,9,10]. However, Ma et al. found that SFDs with improper parameters accelerated system vibration, and too little oil film clearance led to excessive oil film stiffness in the SFD, thus increasing resonance amplitude [11]. In the presence of eccentricities, SFDs may cause increases in critical speed, amplitude jumps, and displacement amplitude oscillations during start/deceleration [12]. This indicates that SFDs with unreasonable parameters easily cause nonlinear vibration in rotor systems [13].

To solve the nonlinear vibration problem in rotor systems caused by SFDs, Ri et al. optimized the design of an SFD rotor system based on a genetic algorithm with amplitude as the objective function and determined the optimal parameters of SFD clearance and oil viscosity [14]. CholUk et al. demonstrated in their research study that an optimized SFD can successfully decrease a rotor system’s amplitude and enhance the rotor’s stability [15,16]. Ahn et al. created an optimal design for vibration response in SFD rotor systems based on a hybrid GA-SA algorithm. The numerical results showed that an optimized SFD can effectively reduce the transmission load of linear and nonlinear responses of rotor systems [17].

With the successive proposals of elastic ring squeeze film dampers (ESFDs), hybrid squeeze film dampers (HSFDs), ISFDs, and other dampers, research has found that the nonlinear problem of traditional SFDs can be effectively improved by changing the structure of SFDs [18,19,20,21,22,23]. Among these studies, researchers have carried out much research on the damping characteristics and vibration reduction applications of integral squeeze film dampers (ISFDs) [24,25]. Lu et al. found that ISFDs have better linear dynamic characteristics through analyses and experiments, revealing the influence of end seal clearance on ISFDs’ damping characteristics [26]. Ertas et al. utilized an ISFD on a 46 MW steam turbine to address sub-synchronous vibrations in rotors triggered by flow excitation. This intervention enhanced the stability of rotor systems, allowing units to operate securely and steadily at maximum capacity [27]. Yan et al. applied an ISFD to various rotor system structures and observed its effective vibration reduction capabilities. And a G-type ISFD structure with a better vibration reduction effect was developed [28,29,30,31]. However, ISFDs have not been applied to multi-disk rotor systems.

To solve the vibration problem of multi-disk rotor systems in compressors, an integral damping bearing (IDB) is proposed in this research study. First, simplified mechanical models of a ball bearing and the IDB are established, and the mechanical characteristics of the two bearings are analyzed. Next, a simulation model of a rotor system utilizing traditional ball bearings and IDBs is developed to assess the rotor dynamics. The model is used to analyze the impact of IDBs on bending stress, strain energy, and system stability. Following this, an experimental investigation is conducted to study the effectiveness of IDBs in suppressing unbalanced vibration in the rotor. This research study explores how the vibration reduction performance of IDBs varies at different levels of unbalance and rotational speeds.

2. Materials and Methods

2.1. Integral Damping Bearing

The ball bearing is a frequently utilized type of bearing in rotor systems. During operation, the bearing provides support to the rotor and has high stiffness dynamic characteristics. Under loading, the balls of the bearing deform. So, the ball bearing can be simplified into a spring element with a certain stiffness coefficient k_B. Figure 1a depicts the mechanical model equivalent of the ball bearing.

The integral damping bearing (IDB) is an integral structure that integrates an integral damper (ID) and a ball bearing. This structure not only provides support for the rotor system but also plays a role in vibration reduction. The physical image is shown in Figure 1b. The ID is made up of four parts: squeeze film area, S-shaped elastomer, inner ring, and outer ring. The eight S-shaped elastomers with uniform and symmetric circumferential distribution connect the inner and outer rings while providing elastic support. The space between the inner and outer rings creates the squeeze film area. Under the action of compression, the oil film in the squeeze film area produces a damping effect, which is used to suppress the rotor’s vibration. Based on the ID’s dynamic characteristics, the ID is simplified into the spring damping element with stiffness coefficient k_ID and damping coefficient c_ID. By incorporating the dynamic attributes of a ball bearing and an ID, the mechanical model of an IDB is depicted in Figure 1b.

When comparing the mechanical models of the ball bearing and IDB, it can be seen that the IDB adds a spring and damping unit on the basis of the traditional ball bearing. Therefore, the IDB is an elastic damping support, which can provide additional damping to dissipate the rotor’s vibration energy, to achieve the purpose of vibration reduction. Before designing the IDB, it is necessary to determine the IDB’s stiffness and damping coefficients based on the dynamic parameters of a rotor system and complete the design of the IDB’s structural parameters accordingly.

2.2. Analysis of Dynamic Characteristics of Integral Damping Bearing

To determine the stiffness and damping coefficients, dynamic characteristic analysis of the ball bearing and IDB was carried out. The traditional ball bearing produced by OKO in Sweden, model 6210-2Z/P4 was considered. The ball bearing’s finite element model with specified boundary conditions as set up (Figure 2a). The ball bearing’s outer ring was set as the fixed constraint, while a load force of 2000 N was applied on the inner ring. We generated the displacement cloud map of the ball bearing (Figure 2b). According to the displacement cloud map, the ball bearing’s stiffness coefficient can be calculated (k_B = 2.78 × 10⁸ N/m).

Based on the dynamic parameters of the rotor system, the IDB was designed. The IDB’s key structural parameters are shown in

Table 1. Key structural parameters of the IDB.

Structural Parameters	Value	Structural Parameters	Value
Outer diameter of ID	174 mm	Thickness of S-shaped elastomer	1.0 mm
Inner diameter of ID	90 mm	Height of S-shaped elastomer	14.6 mm
Axial thickness of ID	40 mm	Radial position of S-shaped elastomer	57.2 mm
Oil film clearance	0.5 mm	Distribution angle of S-shaped elastomer	48°

. The IDB’s finite element model was created, as depicted in Figure 3a. The IDB’s boundary condition can be established based on its operational parameters. The ID’s outer ring in the IDB was defined as a fixed constraint, and the bearing’s inner ring applied a load force of 2000 N. A mechanical performance analysis was conducted on the IDB, and the displacement cloud map of the IDB was obtained as shown in Figure 3b. According to the displacement obtained from the analysis, it can be used to calculate the IDB’s stiffness coefficient (k_IDB = 2.84 × 10⁶ N/m) [32].

Due to the close correlation between the damping characteristics of the IDB and the squeeze film region, before establishing the fluid dynamics equation of the IDB’s squeeze film region, the following assumptions must be made: (1) the lubricating oil is a Newtonian liquid; (2) during the flow process, the lubricating oil is incompressible and constant in temperature and flows in a laminar state. According to the research by Andrés and Ertas, the mechanical response of an ID during vortex motion is similar to that of an SFD [25,26]. Based on the structural characteristics of the ID, the IDB’s Reynolds-like equation in the squeeze film area can be listed as follows:

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(h^3{\displaystyle \frac{\mathsf{\partial }P}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}}\left(h^3{\displaystyle \frac{\mathsf{\partial }P}{\mathsf{\partial }y}}\right)=12\mu {\displaystyle \frac{\mathsf{\partial }h}{\mathsf{\partial }x}}+\left(\rho h^2\right){\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{t}^2}}h.$$

(1)

From the above equation, it can be seen that the IDB generates a dynamic pressure field in the squeeze film region during vortex motion. By integrating, the oil film force components can be obtained as follows:

$$\left\{\begin{array}{l}{F}_x\\ F_z\end{array}\right\}=2{\displaystyle {\sum }_{i=1}^4{\displaystyle {\int }_0^L{\displaystyle {\int }_{{\theta }_{0i}}^{{\theta }_{1i}}\left[P\left\{\begin{array}{l}\cos \left(\theta \right)\\ \sin \left(\theta \right)\end{array}\right\}dx\right]}}}dz.$$

(2)

where (x, y, z) are the Cartesian coordinates, P is the oil film pressure, and h is the oil film gap. ρ and μ are the density and dynamic viscosity of the lubricating oil, respectively. F_x and F_z represent the oil film force components in the x- and z-directions, where L is the axial length of the IDB and i denotes the four-stage squeeze oil film region of the IDB (i = 1, 2, 3, 4). (θ₀, θ₁) is the integral limit for a certain section of film area. According to Equation (1), we can calculate the oil film pressure in the IDB’s squeeze film region. By integrating the pressure according to Equation (2), we can calculate the force shown in Figure 4. To reveal the IDB’s damping characteristics, the oil film force variation curve in the squeezed oil film region is obtained by CFD analysis, as shown in Figure 4. For the CFD analysis of the IDB, we modeled and performed it with Fluent software 2020 R2.

During the squeeze process, the position of the inner ring relative to the outer ring constantly changes, so the value and direction of the normal and tangential oil film forces in the IDB also change accordingly. The normal oil film force F_r is directed towards the center from the inner ring extrusion position, whereas the tangential oil film force F_τ acts in the opposite direction to the inner ring extrusion position. Therefore, F_r and F_τ can be calculated from the equation below.

$$\left\{\begin{array}{l}{F}_r=F_x\cos \left(\mathsf{\Omega }t\right)+F_z\sin \left(\mathsf{\Omega }t\right)\\ F_{\tau }=F_x\sin \left(\mathsf{\Omega }t\right)-F_z\cos \left(\mathsf{\Omega }t\right)\end{array}\right..$$

(3)

The damping properties of the IDB primarily depend on the oil film force within the internal flow structure. The damping coefficient can be determined by the oil film force components. The calculation formula for the equivalent stiffness coefficient K and the equivalent damping coefficient C is as follows:

$$\left\{\begin{array}{l}K={\displaystyle \frac{F_r}{e}}\\ C=-{\displaystyle \frac{F_{\tau }}{e\mathsf{\Omega }}}\end{array}\right..$$

(4)

where F_r is the normal oil film force, F_τ is the tangential oil film force, e is the eccentricity of the inner edge vortex of the IDB, and Ω is the eddy angular velocity.

By substituting the force components in Figure 4 into Equations (3) and (4), the IDB’s damping coefficient values can be calculated based on the calculated oil film force. The IDB’s damping coefficient were calculated as c_IDB = 9.70 × 10⁴ N·s/m by using numerical analysis [32].

Based on the analysis results, the dynamic coefficients of the ball bearing and IDB can be obtained, as shown in

Table 2. Table of dynamic coefficients of ball bearing and IDB.

Parameter	Ball Bearing	IDB
Stiffness coefficient (N/m)	2.78 × 108	2.84 × 106
Damping coefficient (N·s/m)	-	9.70 × 104

. Compared with the traditional ball bearing, it can be seen that the IDB can provide additional damping, improving the damping ratio of the rotor system.

2.3. Simulation Model of Rotor System

Based on the structural characteristics of a compressor, it was simplified into the multi-disk rotor system through equivalent simplification, as shown in Figure 5.

By simulating and analyzing the dynamic characteristics of the rotor system changes before and after implementing the IDB, the impact of the IDB on the dynamic characteristics can be investigated. The structural parameters of the rotor are presented in Figure 5a. The rotor system was modeled and simulated with commercial software, DyRoBes V18.3. Based on the structural parameters, we established a simulation model as shown in Figure 5b. The multi-disk rotor system mainly included components such as coupling, shaft, disks, ball bearings, or IDBs. All four disks were equipped with counterweight holes, which were used to install many counterweight blocks with different mass values at the different phases to simulate different unbalanced faults.

2.4. Test Rig of Multi-Disk Rotor System

The test rig of the multi-disk rotor system was constructed with a focus on vibration reduction. It included two main components, i.e., the multi-disk rotor system and the data acquisition system, as illustrated in Figure 6. By applying the IDB to the test rig, vibration reduction experiments of the rotor system were conducted under various unbalance and speed conditions. The IDB’s impact on the unbalanced vibration of rotor was investigated.

As shown in Figure 6, the multi-disk rotor system was composed of an AC motor, a coupling, two supports equipped with IDBs, four disks, and a shaft. The AC motor powered the rotation of the shaft and disk via a flexible coupling. Mass blocks with eccentricity led to the unbalanced loads. In the experiment, a specific mass was installed as a counter-weight block at the counterweight hole in disk No.2. The unbalanced load was created by installing mass blocks in the counterweight holes of disk No.2, with the radius distance between the counterweight holes and the rotor axis being 120 mm. An unbalanced load results in an unbalanced fault within a rotor system. By replacing counterweights with different mass values, different unbalanced faults can be simulated. Before the experiment, the rotor system was dynamically balanced to minimize residual unbalance.

The data acquisition system mainly consisted of a computer, a data collector, two displacement sensors, a speed sensor, and so on. The speed sensor was mounted within the coupling to gauge the rotational speed. The two displacement sensors measured disk No.2’s displacement in operation. The displacement sensor was an eddy current displacement sensor. The sensor had a measurement range of 2 mm, a linear error of ≤±0.25%FS, a resolution of 0.1 μm, and a frequency response range of 0~10 KHz. The data collector uploaded the speed signal and vibration displacement signal measured by the speed sensor and displacement sensor to the computer, completing signal acquisition, analysis, and storage.

In the experiment, the vibration reduction tests were carried out on the rotor system with two different support structures by swapping out the two bearings. Figure 7 shows physical images of the supports with the ball bearing and the IDB.

Furthermore, the support was equipped with oil inlet and outlet apertures. The oil pump delivered lubricating oil to the oil film area of the IDB via the oil inlet. Subsequently, the lubricating oil exited the end face of the IDB and returned to the oil pump’s oil tank through the oil outlet. Therefore, it could form a stable and circulating oil supply condition for the IDB to ensure that the IDB provides stable damping for the rotor system.

3. Results

3.1. Influence of IDB on Bending Stress Distribution and Strain Energy of Rotor System

As the speed nears the critical speed, the shaft experiences substantial flexible deformation. The bending stress and strain energy of the shaft increase. Excessive bending stress and strain energy may reduce the service life of the shaft. To better observe the critical speeds of the rotor system, we derived the rotor system’s Campbell diagram. As shown in Figure 8, under different support conditions, the first critical speed values of the rotor system were 4657 rpm (with the ball bearing) and 1919 rpm (with the IDB). Rotor deformation and strain energy were analyzed by using commercial software, DyRoBeS V18.3. Figure 9 displays the bending stress and strain energy of the shaft at first-order critical speed.

The first-order vibration mode and bending stress can be observed in Figure 9, which shows that the deformation of the shaft was also significantly improved by using the IDB. When the traditional ball bearings were employed in the rotor system, the shaft experienced significant bending deformation, with the bending stress peaking at 45 MPa in the central position. However, when using the IDB, the bending deformation of shaft was significantly improved, and the maximum bending stress was reduced to 8.63 MPa. For comparison, it can be seen that the maximum reduction in the shaft bending stress after adopting the IDB reached 80.82%.

As for the strain energy distribution chart shown in Figure 10, the strain energy was distributed among the shaft and the two bearings. When the ball bearings were used, 86.53% of the strain energy was distributed on the shaft. Proportions of 6.09% and 7.38% of the strain energy were distributed on ball bearings A and B. However, after adopting the IDB in the rotor system, the strain energy of the shaft decreased from 86.53% to 8.50%. The strain energy proportions of 50.78% and 38.78% were distributed in dampers A and B of the IDBs.

In summary, this study indicates that the deformation of first-order vibration mode was significantly improved after adopting the IDB in the rotor system. The peak bending stress of the shaft was greatly reduced. The reduction in strain energy in the shaft has been successful, resulting in improved durability of the shaft.

3.2. Influence of IDB on Rotor System Stability

Logarithmic decrement is an important indicator for measuring the stability of rotor systems. To investigate the influence of the IDB on the logarithmic decrement, a stability analysis of the multi-disk rotor systems was conducted. The speed range was from 0 to 6000 rpm. According to the analysis, the logarithmic decrement of the rotor system under the two bearings varied with the speed, as shown in Figure 11.

Figure 11 indicates that as the speed increased, the logarithmic decrement of the rotor system in first-order forward precession increased, while the logarithmic decrement in first-order reverse precession decreased. Compared with the ball bearing, the logarithmic decrement of the rotor system in the first-order forward and reverse precession was increased after adopting the IDB. At a speed of 3000 rpm, the logarithmic decrement in the first-order forward precession increased from 0.8 × 10⁻⁴ to 0.15. It can be observed that the IDB enhances the logarithmic decrement, thereby aiding in the enhancement of the rotor system stability across various operating conditions.

3.3. Experimental Study on Suppressing Vibration of Rotor System

To explore the IDB’s influence on unbalanced vibration, vibration reduction experiments were conducted on the test rig of the multi-disk rotor system with unbalanced faults. Before the experiment, a mass block with an unbalance of 0.48 kg·cm was installed at the 0° phase of disk No.2. The experiment was able to gather data on vibration displacement for disk No.2. The time-domain comparison diagram of the rotor system’s vibration displacement is shown in Figure 12a. To reveal the vibration mechanism, the vibration displacement data at disk No.2 were processed by using the Fourier transform. The amplitude of the rotor system under the two bearings was compared in the frequency domain, as illustrated in Figure 12b.

As shown in Figure 12a, when using ball bearings in the rotor system, the peak vibration displacement was 29.69 μm. By adopting the IDB, the peak vibration displacement was reduced to 23.77 μm. Compared with the ball bearing, the reduction in vibration was 19.94% when using the IDB. It is evident that the IDB effectively dampened the rotor system’s vibration.

Figure 12b indicates that the amplitude was the highest at the 1X frequency (i.e., 20 Hz) in the spectrum. There was a significant unbalanced fault in the rotor system, which was caused by the unbalanced mass set on the disk. It can be inferred that the vibration was mainly caused by the unbalanced faults. In addition, there were high harmonic components, such as the 2X frequency and 3X frequency, in the spectrum. Although adjustments were made to the test rig before the experiment, there were still some faults, such as misalignment and rubbing in the rotor system.

When using the ball bearing, the amplitude at the 1X frequency was 25.74 μm. By implementing the IDB, the amplitude at the 1X frequency was 17.31 μm. In comparison, the amplitude at the 1X frequency decreased by 32.73% when implementing the IDB. From this, it is evident that the IDB efficiently mitigated unbalanced vibration in the multi-disk rotor system. Simultaneously, it was observed that the amplitudes at high harmonics, like the 2X frequency and 3X frequency, in the spectrum were diminished to different extents, suggesting that the IDB can also ameliorate issues such as misalignment and rubbing in the rotor system.

3.4. Performance of Vibration Reduction of IDB Under Different Working Conditions

Unbalanced vibration in a rotor system is strongly influenced by both the level of unbalance and the rotational speed. We explored the impact of unbalance and speed on the vibration reduction performance of the IDB by conducting vibration reduction experiments on the multi-disk rotor system. During the experiments, vibration displacement data were collected at the measurement point for different unbalance conditions (speed of 1200 rpm and unbalance of 0.16 kg·cm–0.73 kg·cm) and different rotational speed conditions (speeds of 600 rpm–1200 rpm and unbalance of 0.73 kg·cm). The comparison of peak vibration displacement on the rotor system with two bearings under different working conditions is obtained as shown in Figure 13.

As shown in Figure 13a, the peak amplitude at disk No.2 almost linearly increased with the increase in unbalance. It increased relatively slowly when using the IDB. Compared with the ball bearing, the vibration was significantly reduced when using the IDB. At the same time, it was found that as the unbalance increased, the IDB’s damping continued to increase. Among them, the damping was better when the unbalance was 0.72 kg∙cm. As shown in Figure 13b, similar to different unbalance conditions, the vibration displacement of the rotor system with the IDB under different speed conditions was well suppressed. The IDB’s damping improved as the speed increased.

According to the peak amplitude values under different unbalance conditions in Figure 13, it was found that the peak value of the rotor system with ball bearings reached 25.39 μm when the unbalance was 0.24 kg∙cm. However, when the IDB was applied to the rotor system with the unbalance of 0.73 kg∙cm, the peak value was only 24.42 μm. From this, it can be seen that even if a large, unbalanced fault were configured for the rotor system with the IDB, the amplitude would still remain low during operation. The IDB has the capability to effectively mitigate the rotor system’s sensitivity to mass unbalance.

To reveal the IDB’s damping mechanism and explore the influence mechanism of the different unbalance values and rotational speeds on the rotor system’s vibration, the measured vibration data were processed to obtain the spectral waterfall diagrams of the rotor system under different working conditions, as shown in Figure 14.

From Figure 14, it was found that the amplitude at the 1X frequency was the largest in the rotor system’s spectral waterfall diagram. This was due to the large, unbalanced mass on disk No.2, causing the rotor system to exhibit an unbalanced fault. As the unbalance and rotational speed increased, the amplitude also increased at the 1X frequency. In addition, the amplitudes at the 2X, 3X, and other high multiplying frequencies only changed with the change in rotational speed and had nothing to do with the unbalance value.

To compare and analyze the vibration reduction performance of the IDB under different working conditions, the amplitude at the 1X frequency in Figure 14 was extracted.

Table 3. Amplitudes at 1X frequency under different unbalance conditions.

Unbalance (kg∙cm)	Amplitudes at 1X Frequency (μm)		Reduction (%)
	Ball Bearing	IDB
0.16	19.40	13.68	29.48
0.24	21.42	14.84	30.72
0.35	22.85	15.67	31.42
0.48	25.74	17.32	32.71
0.60	27.30	18.12	33.63
0.73	29.71	19.23	35.27

and

Table 4. Amplitudes at 1X frequency under different rotational speed conditions.

Rotational Speed (rpm)	Amplitudes at 1X Frequency (μm)		Reduction (%)
	Ball Bearing	IDB
600	18.27	13.09	28.35
720	19.71	13.75	30.24
840	21.79	14.86	31.80
960	23.51	15.92	32.28
1200	29.71	19.23	35.27

display the amplitudes at the 1X frequency under varying working conditions. By analyzing the data in Table 3 and Table 4, it is evident that the IDB was able to significantly decrease the rotor system’s amplitude at the 1X frequency across various unbalance and rotational speed conditions. When the unbalance was 0.73 kg·cm and the rotational speed was 1200 rpm, the vibration reduction effect of the IDB was better. The amplitude at the 1X frequency decreased by 35.27%, going from 29.71 μm to 19.23 μm.

To sum up, it can be seen from the above analysis that the IDB effectively suppressed the unbalanced vibration of a rotor system. The unbalanced faults caused by a series of unbalanced mass values and rotational speeds were improved to varying degrees. The IDB exhibited consistent damping performance across various operating conditions.

4. Conclusions

To solve the vibration problem of the multi-disk rotor system, a simplified analysis model was established based on the structural characteristics. Simulation analysis was carried out to explore the influence on a rotor system by a IDB. Then, the test rig of a rotor system was built. To explore the influence of the IDB on unbalanced vibration, experimental research on suppressing vibration of the rotor system under various working conditions was carried out. The following conclusions are drawn:

(1)

The simulation analysis showed that the first-order mode deformation of the rotor can be significantly improved by the IDB. It can obviously reduce the maximum bending stress and strain energy of the shaft. The bending stress was reduced from 45 MPa (with traditional ball bearings) to 8.63 MPa (with IDBs). By using the IDB, the strain energy borne by the shaft and bearing was only 10.44%. Simultaneously, the IDB markedly boosted the logarithmic decrement of the rotor system, thereby enhancing system stability to accommodate various working conditions.

(2)

In experimental studies on vibration reduction in the multi-disk rotor system, it was discovered that the introduction of an IDB significantly reduced unbalanced vibration. The peak amplitude of the rotor system with the IDB was reduced by 19.94%, and the amplitude at the 1X frequency was reduced by 32.73% compared with ball bearings.

(3)

The experimental results indicate that unbalanced vibration can be successfully mitigated by IDBs. IDBs can improve the unbalanced failure of the rotor system to various degrees and effectively reduce the sensitivity of unbalanced mass. Across a series of unbalance values and a wide range of rotational speeds, the IDB has stable vibration reduction performance.