Comparison of Different Hydrodynamic Bearings

This example compares the load bearing abilities of different hydrodynamic bearings. The simulation is performed using the Rotordynamics Module’s Hydrodynamic Bearing interface. This interface solves the Reynolds equation to compute the pressure developed in a thin fluid film for different bearing types. The bearings included in this example are of plain, elliptic, split-halves, and multilobe type (2, 3, and 4 lobes).

Model Definition

Eight bearings are compared: One each of plain, elliptic, and split-halves type, and five different multilobe bearings. The latter are one two-lobe bearing and two three-lobe and four-lobe bearings. The two three-lobe bearings differ from each other in their relative orientation with respect to the applied load direction, as do the two four-lobe bearings.

The journals rotate inside the bearing with an angular speed Ω (rad/s). The static position of the journal is obtained such that the net force due to the fluid film in the horizontal direction is zero whereas that in the vertical direction balances the journal weight, W.

The bearing configuration is shown in Figure 1 below.

Figure 1: Bearing configuration. Top row: Plain, elliptic, split halves, two-lobe. Bottom row: Three-lobe (LOP), three-lobe (LBP), four-lobe (LOP), four-lobe (LBP).

On the fluid side, the parameters needed for the computation are the dynamic viscosity, the density at cavitation pressure, and the compressibility. The fluid parameters, whose values are summarized in Table 1, are close to those of lubricating oils used in real bearings.

Table 1: Fluid properties.
Property	value
Density ρ	1000 kg/m3
Dynamic viscosityμ	0.072 Pa·s
Compressibility β	10-7 Pa-1

Bearing Data

The maximum and minimum clearance, Cmax and Cmin, respectively, of all the bearings are set to the same values in order to make them equivalent.

The initial clearance, hb, assuming that the journal is located at the center of the bearing, is listed in Table 2.

Table 2: Initial FILM thickness.
Bearing	Initial film thickness
Plain	hb = C
Elliptic
Split halves
Multilobe

The objective is to obtain various parameters in Table 2 for different bearings by setting maximum and minimum values to Cmax and Cmin respectively. Following sections provide these expressions.

Plain Bearing

Because the initial thickness is uniform, the best choice of C for the plain bearing is C = (Cmin + Cmin)/2.

Elliptic Bearing

The maximum and minimum clearance Cmax and Cmin are known.

Split-halves Bearing

For split-halves bearings, Cmin = C − d, Cmax = C + d, from which one finds C = (Cmax + Cmin)/2 and d = (Cmax − Cmin)/2.

Multilobe Bearings

For multilobe bearings, Cmax = C + d and Cmin = C + d cos(π/N).

Hence, one obtains C = (Cmin − cos(π/N)Cmax)/(1 − cos(π/N)), and d = (Cmax − Cmin)/(1 − cos(π/N)).

Results and Discussion

Figure 2 below shows the fluid pressure profile on the bearing.

Figure 2: Fluid film pressure profile.

Several results from the simulation provide valuable information about bearing performance. Figure 3 shows a plot of journal eccentricity versus load. The journals that exhibit lower eccentricity are the better ones. From the plot, it seems that the split-halves and 3-lobe LBP (load between pad) bearings have optimum eccentricity in the operating range. For loads higher than 5000 N, the 4-lobe bearing with load on pad (LOP) has the largest eccentricity, while the 4-lobe LBP has the smallest one. The performances of the other bearings lie somewhere in between.

Figure 3: Eccentricity vs. load.

The equilibrium position of the journals is shown in Figure 4 with increasing load W. The y-coordinate of the journal position is plotted on the x-axis, and the z-coordinate is plotted on the y-axis. When the load W is small, all journals tend to move more in the y direction while the movement in the z direction is small. However, as the load increases, they move significantly in the negative z direction and touch the bottom part of the bearing.

Figure 4: Journal position.

Figure 5 shows the plot of the fluid thickness profile when the journal is concentric with the bearing. The geometric parameters of bearings are set in such a way that the minimum and maximum clearances are the same for all bearings except for the plain bearing, which is kept at the mean position.

Figure 5: Initial thickness profile.

Figure 6 shows a plot of the steady-state (current) thickness profile of the fluid.

Figure 6: Current thickness profile.

Notes About the COMSOL Implementation

The position of the journals is computed using the interface in COMSOL Multiphysics. It computes the y and z positions of the bearing such that load balance is achieved in all directions.

In the computation, use an study extension on the load applied by the journal on the bearing to automatically run a loop over the parameter. The Auxiliary sweep functionality is activated in the study step settings.

Rotordynamics_Module/Tutorials/hydrodynamic_bearings_comparison

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

Global Definitions

Parameters

On the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Rj	0.1[m]	0.1 m	Radius of journal
H	0.1[m]	0.1 m	Height of journal
C	0.001[m]	0.001 m	Mean bearing clearance
d	0.1*C	1E-4 m	Pad center offset
Cmax	C+d	0.0011 m	Maximum bearing clearance
Cmin	C-d	9E-4 m	Minimum bearing clearance
Ow	200[rad/s]	200 rad/s	RPS
W	100[N]	100 N	Load on bearing, z component
mu	0.072[Pa*s]	0.072 Pa·s	Dynamic viscosity

Geometry 1

Cylinder 1 (cyl1)

On the toolbar, click .

In the window for , locate the section.

In the text field, type Rj.

In the text field, type H.

Locate the section. From the list, choose .

Click .

Array 1 (arr1)

Replicate 7 more cylinders along x direction by executing the following commands.

On the toolbar, click and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 8.

Locate the section. In the text field, type 2*H.

Form Union (fin)

On the toolbar, click .

Click the button on the toolbar.

Definitions

Explicit 1

On the toolbar, click .

In the window for , type Plain bearing in the text field.

Locate the section. From the list, choose .

Select Boundary 1 only.

In the window for , locate the section.

Select the check box.

Selecting this checkbox allows automatic selection of multiple surfaces across which tangent is continuous.

Plain bearing 1

Right-click and choose .

In the window for , type Elliptic bearing in the text field.

Locate the section. Click .

Select Boundary 5 only.

Explicit Selections

Repeat above sequence of commands to add more selections using the information given in the following table:


Name	Selection
Split halves bearing	9, 10, 11, 12
Two lobe bearing	13, 14, 15, 16
Three Lobe bearing (LOP)	17, 18, 19, 20
Three lobe bearing (LBP)	21, 22, 23, 24
Four lobe bearing (LOP)	25, 26, 27, 28
Four lobe bearing (LBP)	29, 30, 31, 32

In above table we display the entire selection for each bearing. But to create for example the selection, selecting surface 9 is enough. This is so because you duplicate the existing selection to create the new ones and the checkbox is already selected within the old.

In the window, collapse the node.

Hydrodynamic Bearing (hdb)

In the window’s toolbar, click the button and select in the menu.

In the window, under click .

In the window for , click to expand the section.

Select the check box.

You can change the compressibility β inside the bearing node.

Hydrodynamic Journal Bearing 1

In the window, under click .

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

Locate the section. In the C text field, type C.

Locate the section. In the Ω text field, type Ow.

Locate the section. From the μ list, choose . In the associated text field, type mu.

Hydrodynamic Journal Bearing (Plain) 1

Right-click and choose .

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

Locate the section. From the list, choose .

In the Cmin text field, type Cmin.

In the Cmax text field, type Cmax.

Hydrodynamic Journal Bearing (Elliptic) 1

Right-click and choose .

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

Locate the section. From the list, choose .

In the C text field, type C.

From the list, choose .

In the d text field, type d.

Hydrodynamic Journal Bearing (Split halves) 1

Right-click and choose .

In the window for , type Hydrodynamic Journal Bearing (2-lobe) in the text field.

Locate the section. From the list, choose .

In the C text field, type Cmax.

From the list, choose .

In the d text field, type 2*d.

Hydrodynamic Journal Bearing (2-lobe) 1

Right-click and choose .

In the window for , type Hydrodynamic Journal Bearing (3-lobe LOP) in the text field.

Locate the section. From the list, choose .

Locate the section. In the C text field, type 2*Cmax-Cmin.

In the d text field, type 4*d.

In the N text field, type 3.

Hydrodynamic Journal Bearing (3-lobe LOP) 1

Right-click and choose .

In the window for , type Hydrodynamic Journal Bearing (3-lobe LBP) in the text field.

Locate the section. From the list, choose .

Hydrodynamic Journal Bearing (3-lobe LBP) 1

Right-click and choose .

In the window for , type Hydrodynamic Journal Bearing (4-lobe LOP) in the text field.

Locate the section. From the list, choose .

Locate the section. In the C text field, type (sqrt(2)*Cmax-Cmin)/(sqrt(2)-1).

In the d text field, type sqrt(2)*(Cmax-Cmin)/(sqrt(2)-1).

In the N text field, type 4.

Hydrodynamic Journal Bearing (4-lobe LOP) 1

Right-click and choose .

In the window for , type Hydrodynamic Journal Bearing (4-lobe LBP) in the text field.

Locate the section. From the list, choose .

Next set the orientation of the bearings using following instructions.

Bearing Orientation 2

On the toolbar, click and choose .

In the window for , type Bearing Orientation Hydrodynamic Journal Bearing (3-lobe LOP) in the text field.

Locate the section. From the list, choose .

Locate the section. In the φ text field, type -pi/6.

Bearing Orientation Hydrodynamic Journal Bearing (3-lobe LOP) 1

Right-click and choose .

In the window for , type Bearing Orientation Hydrodynamic Journal Bearing (3-lobe LBP) in the text field.

Locate the section. From the list, choose .

Locate the section. In the φ text field, type pi/6.

Bearing Orientation Hydrodynamic Journal Bearing (3-lobe LBP) 1

Right-click and choose .

In the window for , type Bearing Orientation Hydrodynamic Journal Bearing (4-lobe LOP) in the text field.

Locate the section. From the list, choose .

Locate the section. In the φ text field, type pi/4.

Global Equations 1

Add a node.

On the toolbar, click and choose .

In the window for , locate the section.

Find the subsection. From the list, choose .

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


Name	f(u,ut,utt,t) (N)	Initial value (u_0) (m)	Initial value (u_t0) (m/s)	Description
u1y	-hdb.hjb1.Fjy	0	0	Plain bearing journal displacement, y component
u1z	W-hdb.hjb1.Fjz	-0.1*C	0	Plain bearing journal displacement, z component
u2y	-hdb.hjb2.Fjy	0	0	Elliptic bearing journal displacement, y component
u2z	W-hdb.hjb2.Fjz	-0.1*C	0	Elliptic bearing journal displacement, z component
u3y	-hdb.hjb3.Fjy	0	0	Split halves bearing journal displacement, y component
u3z	W-hdb.hjb3.Fjz	-0.1*C	0	Split halves bearing journal displacement, z component
u4y	-hdb.hjb4.Fjy	0	0	Two lobe bearing journal displacement, y component
u4z	W-hdb.hjb4.Fjz	-0.1*C	0	Two lobe bearing journal displacement, z component
u5y	-hdb.hjb5.Fjy	0	0	Three lobe (LOP) bearing journal displacement, y component
u5z	W-hdb.hjb5.Fjz	-0.1*C	0	Three lobe (LOP) bearing journal displacement, z component
u6y	-hdb.hjb6.Fjy	0	0	Three lobe (LBP) bearing journal displacement, y component
u6z	W-hdb.hjb6.Fjz	-0.1*C	0	Three lobe (LBP) bearing journal displacement, z component
u7y	-hdb.hjb7.Fjy	0	0	Four lobe (LOP) bearing journal displacement, y component
u7z	W-hdb.hjb7.Fjz	-0.1*C	0	Four lobe (LOP) bearing journal displacement, z component
u8y	-hdb.hjb8.Fjy	0	0	Four lobe (LBP) bearing journal displacement, y component
u8z	W-hdb.hjb8.Fjz	-0.1*C	0	Four lobe (LBP) bearing journal displacement, z component

Hydrodynamic Journal Bearing (Plain)

In the window, under click .

In the window for , locate the section.

Specify the uj vector as


0	x
u1y	y
u1z	z

Journal Displacements

Write similar expressions in the field of other bearings using the information given in the following table:


Name	Journal displacement
Hydrodynamic Journal Bearing (Elliptic)	(0,u2y,u2z)
Hydrodynamic Journal Bearing (Split halves)	(0,u3y,u3z)
Hydrodynamic Journal Bearing (2-lobe)	(0,u4y,u4z)
Hydrodynamic Journal Bearing (3-lobe LOP)	(0,u5y,u5z)
Hydrodynamic Journal Bearing (3-lobe LBP)	(0,u6y,u6z)
Hydrodynamic Journal Bearing (4-lobe LOP)	(0,u7y,u7z)
Hydrodynamic Journal Bearing (4-lobe LBP)	(0,u8y,u8z)

Mesh 1

Mapped 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Distribution 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type 15.

In the window, click .

In the window for , click .

Study 1

Step 1: Stationary

In the window for , click to expand the section.

Use following instructions to add an on load W.

Locate the section. Select the check box.

Click .

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
W	range(1000,200,10000)

On the toolbar, click .

Results

Fluid Pressure (hdb)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

In the text area, type Fluid Pressure (Pa), W = 10,000 N.

Contour 1

In the window, expand the node, then click .

In the window for , locate the section.

Clear the check box.

Click the button on the toolbar.

Fluid Pressure (hdb)

In the window, collapse the node.

Use the following instructions to plot eccentricity of the journals against load as shown in Figure 3.

1D Plot Group 2

On the toolbar, click and choose .

In the window for , type Eccentricity vs. Load in the text field.

Click to expand the section. From the list, choose .

In the text area, type Relative eccentricity vs Load.

Locate the section. Select the check box.

In the associated text field, type W (N).

Select the check box.

In the associated text field, type e/C (1).

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
sqrt(u1y^2+u1z^2)/C	1	Plain
sqrt(u2y^2+u2z^2)/C	1	Elliptic
sqrt(u3y^2+u3z^2)/C	1	Split halves
sqrt(u4y^2+u4z^2)/C	1	2-lobe
sqrt(u5y^2+u5z^2)/C	1	3-lobe (LOP)
sqrt(u6y^2+u6z^2)/C	1	3-lobe (LBP)
sqrt(u7y^2+u7z^2)/C	1	4-lobe (LOP)
sqrt(u8y^2+u8z^2)/C	1	4-lobe (LBP)

Click to expand the section. Locate the section. Find the subsection. From the list, choose .

In the text field, type 3.

Eccentricity vs. Load

In the window, under click .

In the window for , locate the section.

From the list, choose .

On the toolbar, click .

Click the button on the toolbar.

In the window, collapse the node.

Use the following instructions to plot attitude angle against load as shown in Figure 3.

1D Plot Group 3

On the toolbar, click and choose .

In the window for , type Attitude Angle vs Load in the text field.

Click to expand the section. From the list, choose .

In the text area, type Attitude angle vs Load.

Locate the section. Select the check box.

In the associated text field, type W (N).

Select the check box.

In the associated text field, type \phi (degree).

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
atan2(u1z,u1y)	deg	Plain
atan2(u2z,u2y)	deg	Elliptic
atan2(u3z,u3y)	deg	Split halves
atan2(u4z,u4y)	deg	2-lobe
atan2(u5z,u5y)	deg	3-lobe (LOP)
atan2(u6z,u6y)	deg	3-lobe (LBP)
atan2(u7z,u7y)	deg	4-lobe (LOP)
atan2(u8z,u8y)	deg	4-lobe (LBP)

Click to expand the section. Locate the section. Find the subsection. From the list, choose .

In the text field, type 3.

Attitude Angle vs Load

In the window, under click .

On the toolbar, click .

Click the button on the toolbar.

In the window, collapse the node.

Use the following instructions to plot journal position versus load as shown in Figure 4.

1D Plot Group 4

On the toolbar, click and choose .

In the window for , type Journal Position in the text field.

Click to expand the section. From the list, choose .

In the text area, type Journal position.

Locate the section. Select the check box.

In the associated text field, type u<sub>y</sub>/C (1).

Select the check box.

In the associated text field, type u<sub>z</sub>/C (1).

Global 1

Right-click and choose .

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

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


Expression	Unit	Description
u1z/C	1

Locate the section. From the list, choose .

In the text field, type u1y/C.

Click to expand the section. Locate the section. Find the subsection. From the list, choose .

In the text field, type 3.

Click to expand the section. From the list, choose .

In the table, enter the following settings:


Legends
Plain

Plain 1

Right-click and choose .

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

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


Expression	Unit	Description
u2z/C	1

Locate the section. In the text field, type u2y/C.

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


Legends
Elliptic

Global Display Nodes

Similarly add more display nodes using the information given in the following table:


Name	y axis Data	x axis Data	Legends
Split halves	u3z/C	u3y/C	Split halves
Two lobe	u4z/C	u4y/C	Two lobe
Three lobe LOP	u5z/C	u5y/C	Three lobe LOP
Three lobe LBP	u6z/C	u6y/C	Three lobe LBP
Four lobe LOP	u7z/C	u7y/C	Four lobe LOP
Four lobe LBP	u8z/C	u8y/C	Four lobe LBP

Journal Position

In the window, under click .

In the window for , locate the section.

From the list, choose .

On the toolbar, click .

Click the button on the toolbar.

In the window, collapse the node.

Use the following instructions to plot the initial thickness profile of the fluid film as shown in Figure 5.

Polar Plot Group 5

On the toolbar, click and choose .

In the window for , type Polar: Initial Thickness Profile in the text field.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

In the text area, type Initial thickness profile.

Locate the section. Select the check box.

In the text field, type 0.6.

In the text field, type 1.2.

Line Graph 1

Right-click and choose .

Select Edges 1, 2, 4, and 6 only.

In the window for , locate the section.

In the text field, type hdb.hB1/C.

Select the check box.

In the associated text field, type Plain.

Locate the θ section. From the list, choose .

In the text field, type hdb.Th+hdb.ang_bearing.

Click to expand the section. Locate the section. Find the subsection. From the list, choose .

In the text field, type 3.

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Plain

In the text field, type Plain.

Plain 1

Right-click and choose .

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

Locate the section. Click .

Select Edges 13, 14, 16, and 18 only.

Locate the section. In the text field, type Elliptic.

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


Legends
Elliptic

Line graph Nodes

Similarly add more nodes using the information given in the following table:


Name	Selection	r-Axis Data: Expression	Legends
Split halves	25, 26, 28, 30	Split halves	Split halves
Two lobe	37, 38, 40, 42	Two lobe	Two lobe
Three lobe LOP	49, 50, 52, 54	Three lobe LOP	Three lobe LOP
Three lobe LBP	61, 62, 64, 66	Three lobe LBP	(As is)
Four lobe LOP	73, 74, 76, 78	Four lobe LOP	Four lobe LOP
Four lobe LBP	85, 86, 88, 90	Four lobe LBP	Four lobe LBP

Polar: Initial Thickness Profile

In the window, under click .

On the toolbar, click .

In the window, collapse the node.

Finally, plot the initial thickness profile of the fluid film as shown in Figure 6 using the following instructions.

Polar: Initial Thickness Profile 1

In the window, right-click and choose .

In the window for , type Polar: Current Thickness Profile in the text field.

Locate the section. In the text area, type Thickness profile, W = 2800 N.

Line graph Nodes

Edit the existing nodes under using the information given in the following table:


Name	r-Axis Data: Expression	theta angle data: Expression
Plain	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Elliptic	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Split halves	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Two lobe	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Three lobe LOP	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Three lobe LBP	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Four lobe LOP	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)
Four lobe LBP	hdb.h/C	mod(hdb.Th+hdb.ang_bearing,2*pi)

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the text field, type 15.

Locate the section. In the text field, type 0.0.

In the text field, type 2.0.

On the toolbar, click .

In the window, collapse the node.