Turbulence Effects on the Dynamic Characteristics of Non-Circular Journal Bearings Under Large Sommerfeld Number Conditions

Abstract

Turbulence and instability problems are unavoidable challenges for fluid film bearings as rotation speed continues to rise. This paper investigates the effect of turbulence on the dynamic characteristics of two non-circular journal bearings, hybrid two-lobe bearing (TLB) and hybrid three-step recess bearing (TSRB), under large Sommerfeld number conditions. The linear perturbation method and nonlinear trajectory method are employed in this work. The bearing stiffness coefficients, damping coefficients, and threshold speed are determined by solving the perturbed Reynolds equation using the finite element method. Additionally, the bearing nonlinear trajectories are obtained by solving the motion equation and the dynamic Reynolds equation simultaneously. The threshold speed and trajectory spectrum are utilized to evaluate the bearing dynamic characteristics, and the results derived from laminar and turbulence models are compared. The research on those two different types of bearings has yielded a consistent conclusion: under large Sommerfeld number conditions, the onset of turbulence significantly enhances both the stiffness and damping coefficients of the bearings, as well as the onset speed for the appearance of half-frequency components in the trajectory. The findings suggest that turbulence lubrication is beneficial for improving the dynamic characteristics of these non-circular bearings.

1. Introduction

Fluid film bearings are well known for their advantages of low power consumption, extended service life, and low noise characteristics, which have led to their widespread application in various engineering fields, such as machine tool spindles [1,2], turbine machinery [3,4], ship propulsion systems [5,6], and precision machines [7]. However, as modern industrial equipment advances toward high performance, high speed, and high efficiency, the use of fluid film bearings presents several obstacles. On the one hand, as the rotation speed increases, the stability problem of the lubricating film caused by the inherent cross-coupled stiffness of the fluid film bearing gradually emerges, which restricts the further increase in the rotation speed. On the other hand, to achieve higher efficiency and reduce bearing power consumption, low-viscosity fluids such as water [8,9] and refrigerants [10,11] are employed as lubricants, resulting in the transformation of the fluid film state from laminar lubrication to turbulence lubrication at high rotation speeds. The alteration in the fluid film state not only increases power consumption but also leads to changes in dynamic characteristics, thereby affecting the performance and stability of the rotor bearing system.

The dynamic characteristics of the journal bearings are significantly influenced by their structural designs. Consequently, researchers have proposed various groove configurations to enhance the performance of these bearings. Song [12] established a stability model of a spiral-grooved hybrid journal bearing rotor system in high-speed conditions. It is found that the length of the spiral groove has a significant influence on the stability of the spindle system. Bättig [13] theoretically and experimentally investigated the potential of enhanced groove geometries to increase the stability threshold of herringbone grooved journal bearing. The analysis suggested a rotor with enhanced groove geometry to have a higher stability threshold. Lin [14] compared the static and dynamic performances of pumping-out herringbone grooved journal bearings and pump-in herringbone grooved journal bearings. The results show that the stability of the pumping-in type is better than the pumping-out one. In the dynamic analysis taken by Soni [15] on the floating ring bearing, the stability of the bearing has been significantly enhanced by the modified non-circular outer housing. In the high-speed spindle developed by Wang [16], a water-lubricated hybrid bearing with four stepped recesses was employed. The experimental results indicated that the application of stepped recesses effectively reduced the temperature rise and improved the spindle stability. Ren [17] proposed an analytical procedure to investigate the effect of multi-axial grooves on the bearing load capacity, stiffness, and damping. The study revealed that the number of grooves had a minimal impact on the coefficient k_yy and c_yy while it significantly influenced other dynamic coefficients. In the authors’ previous study [8], the lobe pocket with Archimedes helix profile was designed for the water-lubricated hybrid journal bearing. The proposed bearing can operate stably at no less than 100 krpm. In the aforementioned studies, they primarily focused on bearings with rigid constructions. Furthermore, numerous studies have demonstrated that bearings with elastic or adjustable structures—such as gas foil bearings [18,19], tilting pad bearings [20,21], and adjustable bearings [22,23]—exhibit remarkable stability. To sum up, the structure has a significant impact on the stability of fluid film bearings. Reasonable design can effectively enhance the overall dynamic characteristics of the bearings and their associated systems.

Turbulence lubrication is an important challenge that fluid film bearings encounter as the rotation speed continuously increases. Changes in the lubrication state will directly affect the performance of the bearings, including the load capacity, film pressure, stiffness, and damping [24,25]. In the theoretical and experimental studies on the circular journal bearing taken by Hashimoto [26,27], it is found that the turbulence effects on the dynamic behavior of rotor bearing systems become much more significant as the rotation speed increases. Therefore, under high rotation speed conditions, it may be dangerous to predict the behavior of the systems using the conventional laminar lubrication theory. It should be pointed out that this research was conducted under the condition of a relatively small Sommerfeld number, specifically involving heavy loads, large radius clearances, and low rotation speeds. Zhang [23] studied the dynamic behavior of adjustable journal bearings in turbulence and laminar flow regimes. It is found that the bearing exhibits better lubrication characteristics under laminar than under turbulence flow conditions, and in some operating conditions, the stable velocity range calculated according to laminar flow theory may become unstable in turbulence flow conditions. Feng [28] investigated the turbulence effect on the static and dynamic performances of the water-lubricated bearings. The results showed that the turbulence promotes the load capacity, stiffness, and damping coefficients. Koondilogpiboon [29] analyzed the nonlinear vibration of a symmetrical rigid rotor supported by two identical journal bearings. It is found that the turbulence models decreased the onset speed of instability of the circular and two-lobe bearing without pad preload when compared to the results yielded by the laminar model, while the onset speed of instability of turbulence models can be increased under high pad preload and high length to diameter ratio conditions. The above studies indicate that the influence of turbulence on the dynamic characteristics of bearings is rather complex and does not necessarily lead to a reduction in stability; under certain conditions, it may even enhance stability.

The above literature shows that the influence of turbulence on stability is closely related to the bearing structure and operation conditions. This paper studies the fluid film journal bearings utilized in small air compressors and machine tool spindles, which are required to maintain stable operation across a wide range of speeds, specifically under conditions characterized by both high rotation speed and light load, i.e., large Sommerfeld number conditions. This study analyzes the effects of turbulence on the dynamic characteristics of two non-circular journal bearings: hybrid two-lobe bearing (TLB) and hybrid three-step recess bearing (TSRB). The linear perturbation method and the nonlinear trajectory method are two approaches commonly used in the investigation of the dynamic behavior of fluid film bearings. The linear perturbation method treats the bearing lubrication film as a spring-damper system, assuming that both the bearing load and eccentricities remain constant. This approach neglects the nonlinear characteristics of the fluid film bearing’s load capacity, stiffness, and damping, which inevitably introduces a certain degree of error. While the nonlinear trajectory method involves fewer simplifications and yields more accurate and intuitive theoretical results [30,31], the calculation of the trajectory is very time-consuming. In this study, both methods are employed to achieve a more comprehensive understanding of the bearing dynamic characteristics. Furthermore, through the comparison of the results of the laminar model and the turbulence model, the influences of turbulence on the dynamic characteristics of the bearing are discussed.

2. Theoretical Model

2.1. Bearing Configuration

This paper investigates two types of non-circular journal bearings with distinct geometrical configurations: the hydrodynamic and hydrostatic hybrid two-lobe bearing (TLB) and the hybrid three-step recess bearing (TSRB). Figure 1 shows the geometries of these two journal bearings. The contour line of the TLB’s lobe is characterized as an Archimedean helix, with the lobe depth decreasing along the rotating direction of the bearing. The design can effectively enhance the dynamic effect of the fluid film in TLB. Cavitation in the lubricating film usually occurs in the deeper areas of the clearance. Therefore, the inlet hole of the lubricant is located at the boundary where the lobe depth is greater, which not only effectively ensures the supply of lubricant but also significantly eliminates the cavitation phenomenon. The TSRB adopted a similar design concept, and the inlet hole is located in the narrow, deep recess. The depths of the deep recess and the shallow recess of TSRB are fixed values and do not change with the circumferential angle. The two bearings have the same diameter, length, and clearance, while their geometrical parameters are different, as provided in

Table 1. Parameters of the journal bearings.

Item	TLB	TSRB
Bearing radius R/mm	7.5
Bearing length B/mm	14
Radius clearance h0/mm	0.02
Lobe/recess depth hp/mm	0.08	0.025
Deep recess depth hpd/mm	-	0.08
Lobe/recess axial width Bp/mm	10
Lobe/recess circumferential width θp/°	162	80
Deep recess circumferential width θpd/°	-	15
Diameter of supply orifice d0/mm	1.0
Water supply pressure Ps/bar	1.7
Water viscosity μ/Pa·s	1.0 × 10−3
Water density ρ/kg·m−3	1000

. In the study, to verify the influence of turbulence, low-viscosity water is adopted as the lubricant.

2.2. Governing Equation

The modified Reynolds equation of the water-lubricated journal bearing, which incorporates the effects of turbulence, is expressed as follows:

$${\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{h^3}{G_{\theta }\mu }}{\displaystyle \frac{\partial P}{R\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{h^3}{G_z\mu }}{\displaystyle \frac{\partial P}{\partial z}}\right)={\displaystyle \frac{\Omega R}{2}}{\displaystyle \frac{\partial h}{R\partial \theta }}+{\displaystyle \frac{\partial h}{\partial t}}$$

(1)

$$h=h_0+e_x\cos \theta +e_y\sin \theta +\Delta h_p$$

(2)

where θ and z represent the coordinates along the circumferential and axial directions of the journal bearing, respectively. R represents the radius of the journal bearing, P represents the pressure of the fluid film, Ω represents the rotation speed, and μ represents the viscosity of water. h represents the thickness of the fluid film. h₀ is the radius clearance of the journal bearing, and h_p is the depth of the local pocket/recess. e_x and e_y are the bearing eccentricities along the x and y directions, respectively. The turbulence coefficients, G_θ and G_z, are given by Ng and Pan [32]. In a laminar regime, G_θ = G_z = 12. In the turbulence regime, they can be calculated by

$$G_{\theta }=12+0.0136{\mathrm{Re}}^{0.9}$$

(3)

$$G_z=12+0.0043{\mathrm{Re}}^{0.98}$$

(4)

where Re represents the Reynolds number, which is defined by Re = ΩRρh/μ. The critical Reynolds number that signifies the transition from laminar flow to turbulence flow is a fundamental parameter in the analysis, which can be calculated by

$${\mathrm{Re}}_c=41.1\sqrt{R/h_0}$$

(5)

For hybrid bearing with an external supply of water, the flow rate of lubricating water introduced into the bearing through the orifice can be calculated as follows:

$$Q_{in}=\lambda A\sqrt{{\displaystyle \frac{2\left|P_s-P_d\right|}{\rho }}}$$

(6)

where λ represents the constant contraction coefficient, which is assigned a value of 0.7. A represents the throttle area, which can be calculated by A = πd₀²/4. P_S and P_d represent the pressure of the supply water and the lubricating film at the orifice, respectively. To ensure mass conservation within the flow, the flow into the bearing must equal the outflow Q_out, which can be determined through the integration of the flow at the boundary of the bearing:

$$Q_{in}=Q_{out}$$

(7)

$$Q_{out}={\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{h}{\int }}\frac{Rh}{G_{\theta }\mu }\frac{\partial P}{\partial z}dhd\theta }}$$

(8)

The film pressure at both ends of the bearing is equivalent to the ambient pressure, P (θ, z = ±B/2) = 0 bar. The load capacities of the bearing in the x and y directions are determined through the integration of the water film pressure:

$$\left\{\begin{array}{c}{F}_x={\displaystyle \underset{-B/2}{\overset{B/2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}P\cos \theta d\theta dz}}\\ F_y={\displaystyle \underset{-B/2}{\overset{B/2}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}P\sin \theta d\theta dz}}\\ F=\sqrt{F_x^2+F_y^2}\end{array}\right.$$

(9)

where B is the axial width of the journal bearing.

The bearing Sommerfeld number is defined by

$$S={\displaystyle \frac{2\mu NB{R}^3}{F{h}_0^2}}$$

(10)

where N is the rotation speed in Hz.

2.3. Linear Perturbation Method

In this paper, the linear perturbation method is employed to investigate the stability performance of the fluid film bearings. The load capacities of the journal bearings are linearized around the steady-state equilibrium position:

$$\left\{\begin{array}{l}{F}_x^{\prime }=F_x+k_{xx}x+k_{xy}y+c_{xx}\dot{x}+c_{xy}\dot{y}\\ F_y^{\prime }=F_y+k_{yx}x+k_{yy}y+c_{yx}\dot{x}+c_{yy}\dot{y}\end{array}\right.$$

(11)

The stiffness and damping coefficients, k_ij and c_ij (i, j = x, y), are determined by the perturbation method. The perturbation Reynolds equations of the journal bearings are expressed as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{h^3}{G_{\theta }\mu }}{\displaystyle \frac{\partial P_x}{R\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{h^3}{G_z\mu }}{\displaystyle \frac{\partial P_x}{\partial z}}\right)=-{\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{3h^2\cos \theta }{G_{\theta }\mu }}{\displaystyle \frac{\partial P}{R\partial \theta }}\right)-{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{3h^2\cos \theta }{G_z\mu }}{\displaystyle \frac{\partial P}{\partial z}}\right)-{\displaystyle \frac{\Omega R}{2}}\sin \theta \hfill \\ {\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{h^3}{G_{\theta }\mu }}{\displaystyle \frac{\partial P_y}{R\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{h^3}{G_z\mu }}{\displaystyle \frac{\partial P_y}{\partial z}}\right)=-{\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{3h^2\sin \theta }{G_{\theta }\mu }}{\displaystyle \frac{\partial P}{R\partial \theta }}\right)-{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{3h^2\sin \theta }{G_z\mu }}{\displaystyle \frac{\partial P}{\partial z}}\right)+{\displaystyle \frac{\Omega R}{2}}\cos \theta \hfill \\ {\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{h^3}{G_{\theta }\mu }}{\displaystyle \frac{\partial P_{\overline{x}}}{R\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{h^3}{G_z\mu }}{\displaystyle \frac{\partial P_{\overline{x}}}{\partial z}}\right)=\cos \theta \hfill \\ {\displaystyle \frac{\partial }{R\partial \theta }}\left({\displaystyle \frac{h^3}{G_{\theta }\mu }}{\displaystyle \frac{\partial P_{\overline{y}}}{R\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{h^3}{G_z\mu }}{\displaystyle \frac{\partial P_{\overline{y}}}{\partial z}}\right)=\sin \theta \hfill \end{array}\right.$$

(12)

The stiffness and damping coefficients are

$$\left\{\begin{array}{l}\left(\begin{array}{cc}{k}_{xx}& k_{yx}\\ k_{xy}& k_{yy}\end{array}\right)={\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\begin{array}{c}{P}_x\\ P_y\end{array}\right)\left(\begin{array}{cc}\cos \theta & \sin \theta \end{array}\right)d\theta dz}}\\ \left(\begin{array}{cc}{c}_{xx}& c_{yx}\\ c_{xy}& c_{yy}\end{array}\right)={\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\begin{array}{c}{P}_{\overline{x}}\\ P_{\overline{y}}\end{array}\right)\left(\begin{array}{cc}\cos \theta & \sin \theta \end{array}\right)d\theta dz}}\end{array}\right.$$

(13)

In the analysis, the dimensionless stiffness and damping coefficients are calculated by

$${\overline{k}}_{ij}=k_{ij}{\displaystyle \frac{h_0^3}{\mu B\Omega R^3}}, {\overline{c}}_{ij}=c_{ij}{\displaystyle \frac{h_0^3}{\mu B{R}^3}}, (i,j=x,y)$$

(14)

In the case of a rigid Jeffcott rotor, which is horizontally supported by two identical journal bearings, the threshold speed of the bearing depends on the stiffness and damping coefficients.

$${\overline{k}}_{eq}=\left({\overline{c}}_{xx}{\overline{k}}_{yy}+{\overline{c}}_{yy}{\overline{k}}_{xx}-{\overline{c}}_{xy}{\overline{k}}_{yx}-{\overline{c}}_{yx}{\overline{k}}_{xy}\right)/\left({\overline{c}}_{xx}+{\overline{c}}_{yy}\right)$$

(15)

$${\gamma }^2=\left[\left({\overline{k}}_{eq}-{\overline{k}}_{xx}\right)\left({\overline{k}}_{eq}-{\overline{k}}_{yy}\right)-{\overline{k}}_{xy}{\overline{k}}_{yx}\right]/\left({\overline{c}}_{xx}{\overline{c}}_{yy}-{\overline{c}}_{xy}{\overline{c}}_{yx}\right)$$

(16)

$${\omega }_s=\sqrt{{\displaystyle \raisebox{1ex}{${\overline{k}}_{eq}$}\!\left/ \!\raisebox{-1ex}{${\gamma }^2$}\right.}}$$

(17)

where ${\overline{k}}_{eq}$ and γ represent the equivalent stiffness coefficient and whirl-frequency ratio, respectively, ω_s denotes the dimensionless threshold speed.

2.4. Nonlinear Trajectory Method

In the nonlinear trajectory method, it is essential to simultaneously solve the nonlinear motion equations of the journal along with the dynamic Reynolds equation. In the Cartesian coordinate system o-xyz with its origin point at the bearing center, the nonlinear motion equations of the journal considering rotor unbalance can be expressed as

$$\left\{\begin{array}{c}m{\ddot{e}}_x=m{e}_u{\Omega }^2\cos \theta +F_x\\ m{\ddot{e}}_y=m{e}_u{\Omega }^2\sin \theta -mg+F_y\end{array}\right.$$

(18)

where m is the rotor mass, e_u is the mass eccentricity of the rotor at the centroid, g is the gravitational acceleration, and F_x and F_y are the bearing reactive forces calculated by Equation (9).

2.5. Numerical Algorithms

A program is developed for the calculation of the bearing static and dynamic properties. The flow chart is shown in Figure 2. Initially, the film thickness (Equation (2)) and Reynolds number (Equations (3)–(5)) are calculated based on specified geometric parameters and operational conditions. Subsequently, the modified Reynolds equation (Equation (1)) is solved using the finite element method. For hybrid journal bearings with an external supply of lubricant, the inflow and outflow rates of the lubricant are determined through Equations (6)–(8). The fluid film pressure is adjusted to ensure compliance with mass conservation principles.

After obtaining the static pressure distribution of the fluid film, the perturbation Reynolds equations (Equation (12)) are solved to obtain the partial derivatives of pressure P. Then, Equations (13)–(17) are solved to determine the dynamic coefficients, equivalent stiffness coefficient, whirl-frequency ratio, and threshold speed. Concurrently, the motion equations of the journal bearing (Equation (18)) are solved through the Runge–Kutta method utilizing the acquired force and motion parameters. This iterative process continues with updates on motion and forces until reaching a terminal time T_max, resulting in a nonlinear trajectory.

In the program, the iterative coefficient α is 0.8. The time step Δt used in the calculation of bearing trajectory is given as Δt = π/(1000Ω) so that there are 2000 steps in one revolution. The terminal time T_max is given as T_max = 16π/Ω.

3. Verification

The validity of the model and algorithm proposed in this paper is verified by comparing with the results in the literature [26,27], which conducted theoretical and experimental studies on the dynamic characteristics of a circular journal bearing without grooves. The radius and length of the bearing are both 35 mm, while the bearing radius clearance is 0.25 mm.

Figure 3 shows the variation of dimensionless threshold speed with Sommerfeld number, as calculated by the laminar flow model and turbulence model. As can be seen, the threshold speeds ω_s decrease with an increase in the Sommerfeld number. Under small Sommerfeld number conditions, ω_s decreases with an increase in the Reynolds number. In contrast, at large Sommerfeld number conditions, the turbulence does not affect the value of ω_s. The values of ω_s calculated in this paper are larger than those in the literature under small Sommerfeld number conditions, while the opposite relationship is observed under larger Sommerfeld number conditions. This may be because the results in the literature are analytical solutions based on short bearing theory, whereas the results of this paper are numerical solutions using the finite element method. Differences in the calculation method contribute to differences in the results.

The trajectories of the bearing are calculated at the rotation speeds of 3000 rpm, 5000 rpm, and 6000 rpm. The corresponding Reynolds numbers are 2749, 4581, and 5498, respectively. At these specified rotation speeds, the bearing operates in distinct lubrication states: laminar, transitional, and turbulence. The rotor mass and unbalanced eccentricity used in the calculation are 2.65 kg and 2.5 × 10⁻⁵ m, respectively. Figure 4 shows the numerical nonlinear trajectories obtained in this paper and the experimental results presented in the literature [27]. As can be seen in Figure 4a, convergent trajectories are observed regardless of whether the turbulence effect is taken into account at a rotation speed of 3000 rpm, while the experimental trajectory manifests as a small elliptical orbit. Despite discrepancies between theoretical and experimental equilibrium positions, the results clearly indicate that the journal can operate stably at this rotational speed. In Figure 4b, the trajectory calculated using the laminar model gradually approaches the equilibrium position at a rotation speed of 5000 rpm, indicating that the journal can operate stably. In contrast, the trajectory derived from the turbulence model exhibits a significantly larger elliptical orbit, which aligns more closely with experimental results. As the rotation speed increases to 6000 rpm, the measured journal trajectory attains a limit cycle, indicating that the journal becomes unstable, as shown in Figure 4c. In comparison to the trajectory predicted by the laminar model, the numerical results obtained from the turbulence model exhibit greater consistency with experimental data. Through a thorough examination of both theoretical and experimental outcomes, it can be concluded that utilizing the laminar model in the analysis may lead to an underestimation of bearing stability.

The comparative analysis presented above demonstrates that the calculation results obtained from the model proposed in this paper align with both theoretical and experimental data presented in the literature. This finding substantiates the validity of the proposed model and algorithm.

Furthermore, the analysis of the circular bearing without grooves, conducted using both the linear perturbation method and the nonlinear trajectory method, indicates that turbulence adversely affects the dynamic performance of the bearing under low Sommerfeld numbers, i.e., low rotation speeds and heavy load conditions.

4. Results and Discussion

4.1. TLB

4.1.1. Dynamic Coefficients and Threshold Speed

The dynamic characteristics of TLB are numerically analyzed using the laminar model at the rotation speed of 100 krpm. For comparison, the turbulence model is adopted in the analysis of the bearing performance at the rotation speeds of 60 krpm, 80 krpm, and 100 krpm. The corresponding Reynolds numbers for these conditions are 942, 1257, and 1571.

Figure 5 shows the variation of stiffness coefficients of TLB with Sommerfeld number under different lubrication models. As it can be seen, with the increment of the Sommerfeld number, the direct stiffness coefficient ${\overline{k}}_{xx}$ first decreases, then increases, and finally decreases gradually to a stable value. The direct stiffness coefficient ${\overline{k}}_{yy}$ first decreases and then increases to a stable value. While the cross-coupled stiffness ${\overline{k}}_{xy}$ and ${-\overline{k}}_{yx}$ exhibit an opposing trend, they first increase and then decrease with the Sommerfeld number. Furthermore, it is noteworthy that the stiffness coefficients ${\overline{k}}_{xx}$, ${\overline{k}}_{xy}$, and ${-\overline{k}}_{yx}$ calculated using the turbulence model yield higher values compared to those calculated using the laminar model. As for ${\overline{k}}_{xy}$, its value decreases with the Reynolds number when the Sommerfeld number is small (S < 0.4) and increases with the Reynolds number when S > 0.6.

Figure 6 shows the variation of damping coefficients of TLB with Sommerfeld number. The direct damping coefficient ${\overline{c}}_{xx}$, the cross-coupled damping coefficients ${\overline{c}}_{xy}$ and ${\overline{c}}_{yx}$ show the same variation trend. With the increment of the Sommerfeld number, ${\overline{c}}_{xx}$, ${\overline{c}}_{xy}$, and ${\overline{c}}_{yx}$ first increase and then decrease. While the direct damping coefficient ${\overline{c}}_{yy}$ shows complex variation with the Sommerfeld number, it first decreases, then increases, and finally decreases to a stable value. In addition, the ${\overline{c}}_{yy}$ calculated using the turbulence model have larger values than those calculated by the laminar model in various Sommerfeld numbers. The values of ${\overline{c}}_{xx}$, ${\overline{c}}_{xy}$, and ${\overline{c}}_{yx}$ increase with the Reynolds number when S < 0.7 and decrease with the Reynolds number when S > 2.

It can also be seen in Figure 5 and Figure 6 that all the dynamic coefficients, with the exception of the cross-coupled damping coefficients ${\overline{c}}_{xy}$ and ${\overline{c}}_{yx}$, increase with the increment of the Reynolds number. This leads to an increase in the equivalent stiffness coefficient, as shown in Figure 7a. With the increment of the Sommerfeld number, the whirl-frequency ratio γ² first increases and then decreases, as shown in Figure 7b. The turbulence has a different influence on γ² depending on the value of the Sommerfeld number. When 0.5 < S < 2, γ² decreases with the Reynolds number. In contrast, γ² increases with the Reynolds number in the ranges of S < 0.4 and S > 2. It should be pointed out that when S > 2, although ${\overline{k}}_{eq}$ and γ² both increase with the Reynolds number, the magnitude of the increase in ${\overline{k}}_{eq}$ is significantly greater than that of γ², leading to the increment of threshold speed ω_s with the Reynolds number, as shown in Figure 7c. When S < 0.2, the threshold speed ω_s decreases with the Reynolds number; these trends are the same as in the previous analysis on the circular bearing. In addition, with the increment of the Sommerfeld number, ω_s shows a trend of first decreasing and then increasing.

The investigation of the dynamic coefficients of TLB shows that turbulence plays a positive role in the bearing dynamic characteristics under large Sommerfeld number conditions, i.e., high rotation speed and light load operation conditions, and its stability performance will be underestimated without the consideration of turbulence.

4.1.2. Nonlinear Trajectory

Figure 8 shows the nonlinear trajectories of TLB at various rotation speeds ranging from 20 krpm to 100 krpm. In the calculation, the rotor mass and unbalanced eccentricity in the calculation are 1.0 kg and 1.0 × 10⁻⁶ m, respectively. The corresponding Sommerfeld numbers vary from 0.98 at 20 krpm to 4.92 at 100 krpm. As can be seen, the trajectories calculated using the turbulence model consistently exhibit elliptical shapes across the entire speed range. Conversely, the trajectory calculated using the laminar model varies with changes in rotation speed. When the rotation speed is below 50 krpm, elliptical trajectories are observed, as shown in Figure 8a–c. However, once the rotation speed exceeds 50 krpm, sub-synchronous whirling motion occurs, resulting in heart-shaped patterns, as shown in Figure 8d–i.

Figure 9 shows the waterfall plots of the bearing trajectories. As can be seen, only synchronous components are observed in the waterfalls obtained by the turbulence model. In contrast, the sub-synchronous component is observed in the results calculated using the laminar model when the rotation speed reaches 50 krpm, with its amplitude increasing with the increment of rotation speed. For validation purposes, the experimental setup and results from a previous study [8] are presented in Figure 10 and Figure 11, where no sub-synchronous component is found in the waterfall plots of the bearing vibration. The results calculated using the turbulence model are in better agreement with the experimental data.

Both the linear and nonlinear analysis presented above indicated that, under large Sommerfeld number conditions, the stability performance of TLB will be significantly underestimated if turbulence effects are not taken into account. This finding is quite different from that obtained from the study on the hydrodynamic circular bearing. In order to investigate whether this finding is specific to bearings with two lobes, the performance of the hybrid bearing with three-step recesses is further studied.

4.2. TSRB

4.2.1. Dynamic Coefficients and Threshold Speed

Figure 12 shows the variation of stiffness coefficients of TSRB with Sommerfeld number. With the increment of the Sommerfeld number, the direct stiffness coefficient ${\overline{k}}_{xx}$ first decreases and then increases, as shown in Figure 12a. While the cross-coupled stiffness ${-\overline{k}}_{yx}$ exhibits an opposite trend, it first increases and then decreases with an increasing Sommerfeld number, as shown in Figure 12c. As for the direct stiffness coefficient ${\overline{k}}_{yy}$ and the cross-coupled stiffness ${\overline{k}}_{xy}$, a consistent decrease is observed with an increasing Sommerfeld number, as shown in Figure 12b,d. In addition, all four stiffness coefficients increase with the Reynolds number.

Figure 13 shows the variation of damping coefficients of TSRB with the Sommerfeld number. As can be seen, the direct damping coefficient ${\overline{c}}_{yy}$ and the cross-coupled damping coefficients ${\overline{c}}_{xy}$ and ${\overline{c}}_{yx}$ show the same variation trend. Their values decrease with an increasing Sommerfeld number. The direct damping coefficient ${\overline{c}}_{xx}$ shows a more complex variation trend; it first decreases, then increases, and finally decreases to a stable value. In addition, ${\overline{c}}_{xx}$ and ${\overline{c}}_{yy}$ calculated using the turbulence model have larger values than those calculated using the laminar model. In contrast, the values of ${\overline{c}}_{xy}$ and ${\overline{c}}_{yx}$ decreases with the Reynolds number at small Sommerfeld number conditions. When S > 2, the turbulence effect on ${\overline{c}}_{xy}$ and ${\overline{c}}_{yx}$ can be considered negligible.

The analysis presented above indicates that when S > 2, the stiffness and damping coefficients change little with the Sommerfeld number. As for the influence of turbulence, the stiffness coefficients and direct damping coefficients increase with the Reynolds number. Consequently, the equivalent stiffness coefficient ${\overline{k}}_{eq}$ and whirl-frequency ratio γ² both increase with the Reynolds number, as shown in Figure 14a,b. Generally, an increase in ${\overline{k}}_{eq}$ signifies enhanced stability, whereas a rise in γ² suggests diminished stability. Under the combined influence of these two mechanisms, it can be noted that for S < 0.1, the threshold speed ω_s decreases with increasing the Reynolds number; conversely, for S > 0.1, ω_s increases with increasing the Reynolds number, as shown in Figure 14c.

4.2.2. Nonlinear Trajectory

Figure 15 and Figure 16 show the nonlinear trajectories and the waterfall plots of TSRB at different rotation speeds ranging from 20 krpm to 100 krpm. The rotor mass and unbalanced eccentricity used in the calculation are the same as those employed for the TLB, which are set at 1.0 kg and 1.0 × 10⁻⁶ m, respectively. The corresponding Sommerfeld numbers vary from 0.98 at 20 krpm to 4.92 at 100 krpm. When the rotation speed does not exceed 30 krpm, the trajectories calculated using turbulence and laminar models both exhibit convergence and are approximately circular in shape. Only synchronous components are found in the frequency spectra, as shown in Figure 15a,b and Figure 16.

As the rotation speed increases to 40 krpm, the trajectory calculated using the turbulence model remains nearly circular; however, the result obtained from the laminar model transitions into a heart-shaped pattern, as shown in Figure 15c. A half-frequency component is observed in the frequency spectrum obtained from the results calculated using the laminar model. With further increment in rotation speed, this half-frequency component becomes predominant in bearing movement, leading to a dispersive tendency of the trajectory calculated by the laminar model, as shown in Figure 15d,i. Conversely, while maintaining a heart-shaped pattern for trajectories predicted by the turbulence model at an increased rotation speed of 80 krpm, half-frequency components are also identified within their corresponding spectra. The results show that the theoretical threshold speeds predicted by the turbulence model and laminar model when the sub-synchronous whirling motion appears are 40 krpm and 80 krpm, respectively.

To verify the numerical results of the analysis of TRSB, experiments were conducted using the test rig presented in the previous study [8]. The bearing vibrations are tested within a speed range from 20 krpm to 96 krpm, and the obtained waterfall plot of the bearing vibration is shown in Figure 17. As can be seen, half-frequency vibrations appear at the frequency of 1333 Hz, corresponding to a rotation speed of 80 krpm. The experimental threshold speed is in good agreement with the theoretical prediction by the turbulence model, suggesting that the dynamic performance of TRSB will be underestimated if the turbulence effect is not considered under the large Sommerfeld number condition. The finding corroborates conclusions drawn from previous research on TLB.

5. Conclusions

In this work, the effect of turbulence on the dynamic characteristics of two non-circular journal bearings under large Sommerfeld number conditions is investigated. The linear perturbation method and nonlinear trajectory method are both used in the study. The dynamic characteristics of the two bearings are analyzed. The following conclusions are drawn:

(1)

As the Sommerfeld number increases, the equivalent stiffnesses of the two bearings exhibit a general decreasing trend. The whirl ratios demonstrate an increasing tendency, and the threshold speeds decrease. However, when the Sommerfeld number exceeds 2, its influence diminishes;

(2)

The influence of turbulence on the dynamic characteristics of two non-circular bearings is closely related to the Sommerfeld number. Under small Sommerfeld number conditions, turbulence has a negative effect on the bearing stability performance; however, under large Sommerfeld number conditions, turbulence is beneficial for improving the bearing dynamic characteristic;

(3)

The effect of turbulence on the two different bearings is the same. However, the analysis of the threshold speed and the nonlinear trajectory both show that TLB has better stability performance than TRSB. Geometry design is an important aspect in the development of high-speed fluid film journal bearings.