COMSOL 6.4 - Elastohydrodynamic Lubrication in a Cylindrical Journal Bearing

Elastohydrodynamic Lubrication in a Cylindrical Journal Bearing

It is commonly known that elastic deformations can be significant in heavily loaded bearings. Under these conditions, it is important to include the elastic effect of the lubricated surfaces. Disregarding this effect can potentially lead to incorrect estimations of the fluid pressure and underestimation of the film thickness. As a consequence, a suboptimal design might be the result of using the simpler hydrodynamic approach.

To show how important the effect of elastic deformations can be when simulating heavily loaded bearings, this model compares the results obtained considering the elastic behavior of the components with those obtained with a standard hydrodynamic analysis.

Model Definition

The model geometry is visualized in Figure 1. The geometry consists of a journal and the surrounding bearing. The bearing is equipped with two grooves intended for lubrication supply. Note that symmetry conditions will be used, which means that only half of the geometry will be used in the simulations.

Figure 1: Model geometry of the full journal bearing assembly.

The shaft as well as the bearing are assumed to be made of aluminum. The lubricant used for the simulations will be a generic mineral oil.

Figure 2 shows half of the journal and lubricated surface. A constant pressure will be imposed the groove regions. These regions are therefore not considered as a part of the lubricated surface. The journal has a radius of 10 cm, and a width of 6.7 cm. The cylindrical bearing has a uniform initial clearance of 200 μm. The speed of the journal will be simulated in the interval Ω ∈ [100, 10000] rpm, while being loaded with a total force of 20 kN.

Figure 2: Lubricated surface of the journal.

In many cases, a liquid lubricant can be assumed to possess pressure-independent fluid properties. However, in these heavily loaded conditions, the extreme pressure encountered will cause the lubricant to compress. As a consequence, the density and viscosity will increase. It is therefore of great importance that the density–pressure and viscosity–pressure relations are correctly described. In this model, the viscosity of the lubricant is assumed to follow the isothermal relation suggested by Barus (Ref. 1):

where μ0 is the viscosity at zero pressure, ξ is a lubricant dependent pressure-viscosity coefficient, and p is the pressure in the fluid.

In addition, the isothermal density–pressure relation proposed by Dowson and Higginson (Ref. 1) will be used:

where ρ0 is the density at zero pressure. The assumed parameters used in model is summarized in Table 1.

Table 1: FLUID PARAMETER
Parameter	Value	Unit
μ0	75	mPa⋅s
ξ	25	GPa-1
ρ0	850	kg/m3

In this analysis, the outermost bearing surface is assumed to be fixed, while the fluid loads are applied to the lubricated surfaces as shown in Figure 3, while a constant pressure of 2 bar is assigned to the groove regions.

Figure 3: Load distribution from the fluid film on journal and bearing surfaces.

Results and Discussion

Figure 4 shows the physical pressure in the journal bearing at 100 rpm. Under these heavily loaded condition, the journal will be located close to the bushing in the very bottom of the bearing. This causes the pressure to increase dramatically in this region. This is also indicated in the figure, which predicts the pressure to be well above 160 bar.

Figure 4: Physical fluid pressure at Ω = 100 rpm.

In Figure 5, you will find a visualization of the displacements normalized with respect to the initial clearance C. The plot depicts the conditions at a speed of 100 rpm. The deformations in the lower part of the bearing is more than 3 μm (approximately 0.15% of the initial clearance). What dominates the displacement field of the journal is naturally the translations toward the bottom of the bearing. However, it is clear that the displacements are smaller in the lower part the journal, which suggests that in that location the greatest deformations are expected.

Figure 5: Displacements of journal and bearing at Ω = 100 rpm, normalized with respect to clearance.

The von Mises stress is depicted in Figure 6. The maximum stress in the bearing appears beneath the surface chose to location with increased fluid pressure. This is commonly known to be the case in, for example, Hertzian contact theory, which shares many similarities with elastohydrodynamic problems. The stress field in the journal, on the other hand, is a combination of the same phenomenon and stress due to bending.

Figure 6: von Mises stress in journal and bearing at Ω = 100 rpm.

Figure 7 shows a comparison between the physical fluid pressure in the circumferential direction along the center of the bearing. The comparison is shown for five different rotational speeds in the investigated interval. At higher speeds, more oil is forced through the converging zone in the bearing, which leads to more distributed pressure profiles with a lower maximum pressure. In many cases, the more moderate pressure encountered at this high speed is replicated to an acceptable agreement by the simplified hydrodynamic simulation. However, it is evident that as the speed is reduced, the disagreement between the pressure estimates increases. This shows how important it is to include the effect of deformations when these become significant.

Figure 7: Comparison between physical pressure at bearing center along the circumferential direction.

The film thickness is visualized in Figure 8. It compares the results obtained with the two different modeling strategies for five different speeds in the simulated interval. The minimal clearance in the bearing reduces with decreasing speeds. Note that as the journal approaches the bushing, and the pressure increases dramatically, the film thickness flattens out due to the elastic deformations, while a small bump starts to form at the exit of the closure.

Figure 8: Comparison between film thickness at bearing center along the circumferential direction.

In Figure 9, an unwrapped plot of the physical fluid pressure is compared at 100 rpm. The maximum pressure computed by the hydrodynamic simulation is nearly twice as high as what is predicted in the elastohydrodynamic simulation. In addition, the shape of the pressure distribution is clearly different.

Figure 9: Comparison of unwrapped fluid pressure at Ω = 100 rpm.

Notes About the COMSOL Implementation

•

Use the interface along with the interface to set up the elastohydrodynamic model.

•

This model shows how to set up a manual coupling between the and interfaces. As an alternative approach, the built-in multiphysics coupling automatically couples the and interface. Note that the built-in coupling uses an approximated displacement field for the journal and bearing displacements, which makes it incapable to represent local deformations which are important for this model.

Reference

1. B.J. Hamrock, S.R. Schmid, and B.O. Jacobson, Fundamentals of Fluid Film Lubrications, Marcel Dekker, 2004.

Rotordynamics_Module/Tutorials/elastohydrodynamic_journal_bearing

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

The first step to build the model is to add the required physics interfaces and study.

In the window, click

In the tree, select > and > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Next, define the parameters that are needed for setting up the model.

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Omega	100[rpm]	1.6667 1/s	Rotational speed
C	200[um]	2E-4 m	Bearing clearance
W	20[kN]	20000 N	Bearing load
ps	2[bar]	2E5 Pa	Supply pressure
mu0	75[mPa*s]	0.075 Pa·s	Dynamic viscosity at zero pressure
xi	2.5e-8[m^2/N]	2.5E-8 1/Pa	Viscosity-pressure coefficient
rho0	850[kg/m^3]	850 kg/m³	Density at zero pressure

Root

Now, define a set of analytic functions for the density-pressure and viscosity-pressure dependencies.

Definitions

Fluid Density

In the toolbar, click

In the window for , type Fluid Density in the text field.

In the text field, type rho.

Locate the section. In the text field, type rho0*(1+0.6*x/(1.7*x+1e9)).

Locate the section. In the text field, type kg/m^3.

In the table, enter the following settings:


Argument	Unit
x	Pa

Fluid Viscosity

Right-click and choose .

In the window for , type Fluid Viscosity in the text field.

In the text field, type mu.

Locate the section. In the text field, type mu0*exp(xi*x).

Locate the section. In the text field, type Pa*s.

Import the geometry for this model. The file is prepared in the COMSOL Multiphysics geometry file format (.mphbin).

Geometry 1

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file elastohydrodynamic_journal_bearing.mphbin.

Click

Form Union (fin)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Select the checkbox.

In the toolbar, click

The geometry is now imported and appears in the window.

Next, define a set of selections. These will be useful when setting up the physics.

Definitions

Hydrodynamic Bearing (Journal)

In the toolbar, click

In the window for , type Hydrodynamic Bearing (Journal) in the text field.

Locate the section. From the list, choose .

Click the

button in the toolbar.

Click

In the dialog, type 14-19 in the text field.

Click .

Hydrodynamic Bearing (Bearing)

In the toolbar, click

In the window for , type Hydrodynamic Bearing (Bearing) in the text field.

Locate the section. From the list, choose .

Select the checkbox.

Click

In the dialog, type 41 in the text field.

Click .

Fixed Boundary

In the toolbar, click

In the window for , type Fixed Boundary in the text field.

Locate the section. From the list, choose .

Select the checkbox.

Click

In the dialog, type 30 in the text field.

Click .

Boundary (Supply Pressure)

In the toolbar, click

In the window for , type Boundary (Supply Pressure) in the text field.

Locate the section. From the list, choose .

Click

In the dialog, type 20-23, 47-58 in the text field.

Click .

Symmetry Plane

In the toolbar, click

In the window for , type Symmetry Plane in the text field.

In the window toolbar, click

next to

, then choose .

Click the

button in the toolbar.

Locate the section. From the list, choose .

Click

In the dialog, type 24-27, 59-62 in the text field.

Click .

Rigid Connector Edges

In the toolbar, click

In the window for , type Rigid Connector Edges in the text field.

Locate the section. From the list, choose .

Select the checkbox.

Click

In the dialog, type 5 in the text field.

Click .

Bearing Load

In the toolbar, click

In the window for , type Bearing Load in the text field.

Locate the section. From the list, choose .

Select the checkbox.

Click

In the dialog, type 1 in the text field.

Click .

Inlet Edges

In the toolbar, click

In the window for , type Inlet Edges in the text field.

Locate the section. From the list, choose .

Click

In the dialog, type 33 34 36 37 39-42 in the text field.

Click .

COMSOL Multiphysics is equipped with built-in material properties for a range of different materials. In this model, you will use the aluminum for the structural parts, whereas a user-defined material will be used for the lubricant.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Oil

In the window, under right-click and choose .

In the window for , type Oil in the text field.

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Dynamic viscosity	mu	mu(hdb.p)	Pa·s	Basic
Density	rho	rho(hdb.p)	kg/m³	Basic

The next step is to set up the interface. The outermost surface of the bearing is assumed to be fixed, a symmetry condition is used at the symmetry plane, and a is applied to prevent the journal from rotating around the x-axis. The supply pressure is applied to the groove regions and feeding channels, and the fluid forces will be applied to the lubricated surfaces. Lastly, mapping operators provided by the are used to apply the fluid forces from the to the bearing modeled using the interface.

Solid Mechanics (solid)

Continuity 1

In the window, under > click .

In the window for , locate the section.

Select the checkbox.

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Rigid Connector 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Select the checkbox.

Boundary Load (Supply Pressure)

In the toolbar, click

and choose .

In the window for , type Boundary Load (Supply Pressure) in the text field.

Locate the section. From the list, choose .

In the p text field, type ps.

Boundary Load (Fluid Load, Journal)

In the toolbar, click

and choose .

In the window for , type Boundary Load (Fluid Load, Journal) in the text field.

Locate the section. From the list, choose .

Locate the section. Specify the fA vector as


hdb.fjx	x
hdb.fjy	y
hdb.fjz	z

Boundary Load (Fluid Load, Bearing)

In the toolbar, click

and choose .

In the window for , type Boundary Load (Fluid Load, Bearing) in the text field.

Locate the section. From the list, choose .

Locate the section. Specify the fA vector as


src2dst_ap1(hdb.fbx)	x
src2dst_ap1(hdb.fby)	y
src2dst_ap1(hdb.fbz)	z

Boundary Load (Bearing Load)

In the toolbar, click

and choose .

In the window for , type Boundary Load (Bearing Load) in the text field.

Locate the section. From the list, choose .

Specify the Ftot vector as


-W/2	z

Solid Mechanics (solid)

In the window, collapse the > node.

Hydrodynamic Bearing (hdb)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Now, set up the interface. Here a symmetry condition is applied, and the supply pressure is assigned to the inlet edges. In addition, the displacements are used to describe the journal and foundation movements. The mapping operators are used to map the displacement field from the bearing.

Hydrodynamic Journal Bearing 1

In the window, under > click .

In the window for , locate the section.

In the C text field, type C.

Specify the vector as