Journal Bearing: An Integrated CFD-Analytical Approach for the Estimation of the Trajectory and Equilibrium Position

Abstract

For decades, journal bearings have been designed based on the half-Sommerfeld equations. The semi-analytical solution of the conservation equations for mass and momentum leads to the pressure distribution along the journal. However, this approach admits negative values for the pressure, phenomenon without experimental evidence. To overcome this, negative values of the pressure are artificially substituted with the vaporization pressure. This hypothesis leads to reasonable results, even if for a deeper understanding of the physics behind the lubrication and the supporting effects, cavitation should be considered and included in the mathematical model. In a previous paper, the author has already shown the capability of computational fluid dynamics to accurately reproduce the experimental evidences including the Kunz cavitation model in the calculations. The computational fluid dynamics (CFD) results were compared in terms of pressure distribution with experimental data coming from different configurations. The CFD model was coupled with an analytical approach in order to calculate the equilibrium position and the trajectory of the journal. Specifically, the approach was used to study a bearing that was designed to operate within tight tolerances and speeds up to almost 30,000 rpm for operation in a gearbox.

1. Introduction

Journal bearings are among the most widespread mechanical components. Their function is to support shafts carrying radial loads and to reduce friction. The load-carrying capacity is strongly related to the pressure distribution in the convergent/divergent clearance between journal and ring. During operation, in fact, the axes of the journal and the ring are not coaxial. Therefore, the convergent clearance is significantly pressurized while after the smallest clear opening size the pressure decreases instantaneously. The minimum pressure achievable ($p_v$) is physically limited by the vaporization. Once this value is reached, the liquid phase evaporates impeding a further pressure decrement. The vaporization pressure is a physical property of the lubricant. The cavitation leads to an unbalance in the pressure distribution that is, in turn, responsible for the supporting effect.

One of the the first works on journal bearings was published by Sommerfeld [1] more than 100 years ago. The Reynold’s equations were solved for a bearing having an infinite length. Subsequent studies by Swift [2] improved the approach. The Sommerfeld solution can be applied to bearings for which the ratio between the axial length and the diameter L/D is significantly greater than one [3]. With this assumption, the pressure gradient in the axial direction can be neglected. Moreover, the equations can be solved only assuming that the convergent/divergent is fully lubricated. Cavitation effects and the consequent limitation of the pressure due to vaporization are also neglected. The full-Sommerfeld solution therefore predicts a symmetric distribution of the pressures with negative values in the divergent. Consequently, the equilibrium position of the journal is estimated to be just below the ring center, without any lateral shift. This result has been proven to be wrong [4] and never observed experimentally.

The full-Sommerfeld theory was extended by Gumbel [5] in 1914. He proposed to substitute the negative values with the vaporization pressure. Figure 1 shows an example of results achievable with the full- and the half-Sommerfeld models in terms of pressure distribution along the journal. A similar approach was presented by Dowson [6].

Another important work was presented by Raimondi and Boyd [7,8,9].

In the last years, thanks to the increasing computational power, more and more numerical methods were presented by several authors among which Mane et al. [11] and Chauhan et al. [12] who used computational fluid dynamics (CFD) to overcome the infinite length approximation. These models do still not include vaporization, and the artificial substitution of negative values with the vaporization pressure is required. A similar model was developed by Gao et al. [3]. Gandjalikhan Nassab et al. [13] validated their CFD approach using the experimental measurements by Pan et al. [14]. The work of Heshmat [15] is based on the hypothesis of keeping the pressure constant in the caviting region. Other numerical analyses where presented by several authors. Among them Sawicki et al. [16] and Riedel et al. [17] compared the numerical results with the measured data by Jakobsson et al. [4] and Vijayaraghavan et al. [18], respectively. Aitken et al. [19] developed a Newton–Raphson based approach including a simple vapor cavitation model for the study of big-end bearings.

The author has already experience with bearings [20]. He presented a full 3D numerical approach that includes the vaporization effects. The model was developed in the OpenFOAM^® [21] environment. Different cavitation models—e.g., Kunz [22], Merkle [23], and Sauer [24,25,26]—were compared. For sake of completeness, in the recent years new cavitation models capable to ensure the mass conservation are available but were not included in the present study. The CFD results in terms of pressure distributions were validated by comparison with the data by Gao et al. and by Jakobsson [27]. Additionally, some authors have considered the thermal effects [28,29,30], which were further studied by Wang et al. [31], Li et al. [32], and Yang et al. [29]. Other authors such as Liu et al. and Dhande et al. [33] investigated the fluid–structure interaction [34]. Other authors used standard CFD approaches to evaluate the effect of nano-fluids on the performances of bearings [35,36], magnetorheological fluids [37], simple water [38], or non-Newtonian lubricants [39]. Moreover, the effect of the roughness was included in some models [40]. Among them, the CFD model shows a good capability to estimate the pressure distribution along the journal, showing a difference with respect to the measurements of less than 5%. As drawback, the inclusion of a vaporization model and the need of performing time-dependent simulations leads to an increase in the computational effort by 350 times with respect to the full-lubricated 3D simulation (whose results have to be artificially corrected replacing the negative values).

2. Materials

In order to better explain the developed approach, a bearing (Figure 2) with journal radius equal to R = 2 mm (D = 4 mm) and a radial clearance equal to 10 µm was selected (

Table 1. Bearing parameters.

Journal Diameter D [mm]	Bearing Length L [mm]	Clearance e [µm]	Speed ω [rad/s]
4	13	10	328–3000
Oil Density (15 °C) [kg/m3]	Oil Viscosity (40 °C) [m2/s]	Oil Viscosity (100 °C) [m2/s]
1060	220 × 10−6	40 × 10−6

). The axial dimension is L = 13 mm. Rotational velocities between 328 and 3000 rad/s (3132–28650 RPM) were considered. The mass of the journal itself is $1.307·{10}^{-3}$ kg. An additional mass ranging from 0.25 to 10 kg was considered.

A commercial lubricant (Kluebersynth GH 6-22) having a density of ${\rho }_{15°\mathrm{C}}=1060 \mathrm{kg}/{\mathrm{m}}^3$ and a viscosity equal to ${\eta }_{40°\mathrm{C}}=220\times {10}^{-6} {\mathrm{m}}^2/\mathrm{s}$ at 40 °C and ${\eta }_{100°\mathrm{C}}=40\times {10}^{-6} {\mathrm{m}}^2/\mathrm{s}$ at 100 °C was considered.

3. Methods

A coupled numerical–analytical approach [41,42,43] was used. Coupled approaches applied to journal bearings were presented also by Li et al. [44].

3.1. Computational Fluid Dynamics

The system was modeled within the OpenFOAM^® environment [45,46,47].

The method relied on finite volumes (FV) [48]. The computational domain, namely the clearance between ring and journal, was discretized into small cells by means of the mesh generator blockMesh with about 500k hexahedrons (15 cells in the radial direction).

Equilibrium equations are applied to each cell of the grid in order to ensure the mass and the momentum conservations [49].

$$\frac{\partial \rho }{\partial t}+\nabla \left(\rho v\right)=0$$

(1)

$$\frac{\partial \rho }{\partial t}+\nabla \left(\rho \mathit{v}\right)=-\nabla p+\nabla \left[\mu \left(\nabla \mathit{v}+\nabla {\mathit{v}}^T\right)\right]+\rho \mathit{g}+\mathit{F}$$

(2)

where $\rho$ is the lubricant density, $v$ is the velocity vector, $p$ is the pressure, $g$ is the gravitational acceleration and $F$ represents the external forces. The mean temperature was imposed as a variation of the fluid properties according, for the density, to the following formula [50].

$${\rho }_{\theta }={\rho }_{15°C}-\left(\theta -15\right)·0.0007$$

(3)

and by using a logarithmic relation between the known viscosity values at 40 °C and 100 °C [51].

$${\nu }_{\theta }={\nu }_{40°\mathrm{C}}+\left(\frac{\theta -313}{60}\right)·\left({\nu }_{100°\mathrm{C}}-{\nu }_{40°\mathrm{C}}\right)$$

(4)

The fluid is regarded as a continuum and its behavior is described by means of its macroscopic properties. The Boundary Conditions (BC) are reported in

Table 2. BC.

	Volume Fraction	Pressure	Velocity
Journal	∇α = 0	∇p = 0	ω = 0
Bearing	∇α = 0	∇p = 0	ω = U
Sides	αv = 0	p = 105 kg/ms2	∇u = 0

.

The partial differential equations are solved using a PIMPLE (merged PISO-SIMPLE) scheme. The PIMPLE scheme combines the structure of the SIMPLE algorithm allowing the under-relaxation and a fast convergence of the solution with the velocity correction loops that are typically of the PISO mode.

The solver uses a phase-fraction Eulerian fixed-grid technique to describe the interface between the two phases. The tracking of the interface is accomplished by the solution of a continuity equation for the volume fraction of one of the phases (in this case the vapor–sub-index _v).

$$\frac{\partial \alpha }{\partial t}+\nabla ·\left({\alpha }_{v}v\right)+\nabla ·\left(v_C{\alpha }_v\left(1-{\alpha }_v\right)\right)=\dot{m}$$

(5)

Several different approaches to define the compressive velocity field are reported in literature, e.g., CICSAM [14] and the currently used multi-dimensional universal limiter with explicit solution [15], which limits the flux of the variables to guarantee a bounded solution. $\dot{m}$ is a source term that will mimic the phase-change rate and is defined as the difference between vaporization $\dot{m^{+}}$ and condensation $\dot{m^{-}}$ defined according to Kunz [22].

$$\dot{m^{+}}=C^{+}{\rho }_v\left(1-\alpha \right)\frac{\min \left[0,\overline{p}-p_v\right]}{\frac{1}{2}{\rho }_l{U}_{\infty }^2{t}_{\infty }}$$

(6)

$$\dot{m^{-}}=\frac{C^{-}{\rho }_v{\alpha }_v{\alpha }_l^2}{t_{\infty }}$$

(7)

The constants C⁺ and C⁻ are set to 100. The Kunz model was proved to be accurate and the results were validated with experimental data (Figure 3). However, the pressure diagrams show small discrepancies in the cavitation region that are, in the real bearing, wider than the one predicted by the simulations. This evidence can be probably explained by the calibration of the Kunz model whose constants were taken from literature. For the sake of completeness, it should be mentioned that in literature more advanced numerical approaches are available [48]. These include the effect of surface texturing, slip boundary, realistic deformations of the bearing with fluid structure interaction, thermal effects and viscosity variation, shaft misalignment and other effects that on complex systems. However, for the purposes of this paper, the present approach was considered sufficiently representative of the selected bearing.

Simulations (Figure 4) were performed for different levels of eccentricity and the results reported in terms of Drag and Lift forces.

Figure 5, Figure 6 and Figure 7 show the forces and the attitude angle for different eccentricities and rotational speeds. Both the CFD and the half-Sommerfeld results are reported. These data are the basis for the calculation of the trajectory and equilibrium position of the journal.

Figure 7 reports the resultant forces and attitude angle obtained by integrating the pressure distributions calculated with the different models. The Half-Sommerfeld solution underestimates the forces, especially for what concerns the drag effects.

3.2. Dynamics

The Force-eccentricity relations obtained from the numerical models were used into a force balance (Newton equation) to estimate the equilibrium position and the trajectory of the journal.

The involved forces are the fluid dynamic ones (${\mathit{F}}_{CFD}$), the gravitational- ($\mathit{g}·m$) and the inertial-effects ($\mathit{a}·m$).

$$\sum \mathit{F} ={\mathit{F}}_{CFD}+\mathit{a}·m+ \mathit{g}·m=0$$

(8)

${\mathit{F}}_{CFD}$ and $\mathit{g}$ are known both in terms of direction and magnitude (in the nominal position x = 0 and y = 0, no hydraulic forces act on the journal and its first motion is only due to gravitational effects). The acceleration $\mathit{a}$ of the journal (+ additional mass) can be therefore calculated according to Equation (8). The equilibrium position ($\mathit{s}$) after a small-time interval $dt$ is calculated as

$$\mathit{s}={\mathit{s}}_0+\frac{\mathit{F}·d{t}^2}{m}$$

(9)

The solution was simulated for 1500 timesteps of 0.0001 s.

4. Discussion

Scilab [52] was used for solving the Newton equations.

Figure 8, Figure 9 and Figure 10 show the trajectories of the journal from the geometrical center of the bearing (x = 0, y = 0) to the equilibrium position for different external load and operating conditions. For the smallest rotational speed (i.e., 328 rad/s), additional masses of 0.25, 0.5, and 1 kg were considered. For the intermediate velocity (i.e., 1500 rad/s), masses of 1, 2.5, and 5 kg were considered. Finally, for the higher speed (i.e., 3000 rad/s), masses of 2.5, 5, and 10 kg were considered.

The right diagrams show the trajectory of the journal from the nominal point (0,0) to the equilibrium position. It can be clearly observed that the equilibrium position does not lie on the vertical axis but is slightly shifted also tangentially. This effect is well known and occurs due to the cavitation that produces an unbalance of the pressures between the convergent and the divergent gaps.

The diagrams on the left show the amplitude of the oscillations (variation of the eccentricity).

The equilibrium position and the trajectory calculated with the half-Sommerfeld and the CFD approaches differ. The differences can be significant depending on the rotational speed and the load of the bearing. Figure 11 shows the percentage error in terms of predicted positions. While for the intermediate rotational speed the differences are negligible, for the low and the high velocities the differences can rise to about 30%. This result is aligned with the differences in terms of pressure distributions observed with the two approaches (Figure 3).

The eccentricity at equilibrium depends both on the load and the rotational speed. An increase of the rotational speed promotes an increase of the drag and lift forces. Consequently, for the same load, the higher the speed, the lower the eccentricity. On the contrary, for a given speed the eccentricity increases together with the mass. This phenomenon is predicted by both approaches (CFD vs. half-Sommerfeld), even if the latter overestimate the eccentricity for a given force.

Figure 12 shows the damping times and the related excited frequencies during the transient startup. It can be appreciated that the damping time reduces for higher values of the mass. A higher mass implies a higher acceleration and an offset from the nominal position (0,0). This is in turn related to higher viscous forces that promote the damping. Even for what concerns the excited frequencies the CFD and the Half-Sommerfeld predictions differ. In particular, the latter predicts lower frequencies. This is mainly related to the lower drag forces.

While the damping time is useful to estimate the wear of the journal related to each stop, the knowledge of the frequencies has the same importance and is fundamental for a preliminary evaluation of the NVH (noise vibration harshness) behavior of the system. The acceleration from the steady state to the operating conditions introduces energy in the system and excites different frequencies f that, in turn, interact with the eigenfrequencies of the system causing noise and undesired vibrations. For the studied configurations, the excited frequencies results in the range 4–24 Hz.

5. Performance Enhancements

The CFD simulations were implemented in the open-source OpenFOAM^® environment. Each simulation (for a specific eccentricity and rotational speed) took approximately 1 h on a 100 GFLOPS workstation. The solution of the Newton equation was performed in the open-source software Scilab. The computational effort was of few seconds. Considering that with nine simulations the whole operating range of the bearing was characterized, the method is sufficiently efficient and could be potentially very interesting for engineers in the design stage.

6. Conclusions

In this work a coupled analytical-CFD approach is shown. It combines the numerical solution of the Reynolds equations of fluid dynamics and the Newton equation for equilibrium. Both the equilibrium position and the trajectory of the journal, the damping time and the excited frequencies can be predicted. This latter information can support the design of new concepts, providing preliminary indication about the Noise Vibration Harshness (NVH) behavior.

In order to validate the approach, the results of the CFD simulations were compared with experimental data available in literature. Moreover, the numerical predictions were also compared with those predicted from the Half-Sommerfeld theory which, in turn, seems to underestimate the experimental measurements.

Specifically, the presented method was developed coupling OpenFOAM^® and Scilab [34], two open-source software. The choice of an open-source software was made due to the flexibility to customize the code with the implementation of specific models for the analysis of the physical problem of interest. Moreover, the calculations are not subjected to restrictions on the number of CPUs.