COMSOL 6.4 - Validation of Dynamic Coefficients in Hydrodynamic Bearings

Validation of Dynamic Coefficients in Hydrodynamic Bearings

In this verification example, we will validate the built-in functionality for calculating dynamic coefficients, specifically, the stiffness and damping coefficients of hydrodynamic bearings. The verification is carried out on a generic cylindrical hydrodynamic journal bearing with a diameter-to-width ratio of 1.

The dynamic coefficients are first determined using the infinitesimal perturbation method available within the Hydrodynamic Bearing interface. Then, the dynamic coefficients are computed using the finite perturbation approach. Finally, the results from both methods are compared to evaluate consistency and accuracy.

Model Definition

The hydrodynamic bearing is modeled by a single cylindrical surface. No fluid domains are required because the thickness of the fluid film is modeled implicitly.

The journal has a width and diameter of 20 cm, with a constant bearing clearance of 100 μm. The runner spins at a constant speed of 1000 rad/s, and is subjected to an external load ranging from 100 N to 1 MN. The lubricant viscosity is assumed to follow the isothermal pressure–viscosity relation proposed by Barus (Ref. 1):

where μ0 is the viscosity at zero pressure, ξ is a lubricant-dependent pressure–viscosity coefficient, and p is the fluid pressure. Additionally, it is assumed that the density can be described by the isothermal pressure–density relation suggested by Dowson and Higgison (Ref. 1):

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

Table 1: FLUID PARAMETERs.
Parameter	Value	Unit
μ0	75	mPa⋅s
ξ	25	GPa-1
ρ0	860	kg/m3

Linearized dynamic coefficients of a fluid-film bearing can be determined using different approaches. This includes infinitesimal perturbation and finite perturbation, where the latter is sometime also referred to as numerical perturbation. In the Hydrodynamic Bearing interface, the computed dynamic coefficients are based on the infinitesimal perturbation approach.

The linearized stiffness and damping coefficients are defined as

where k and c are the stiffness and damping matrix, uj and vj are the displacement and velocity of the journal, and Fj is the total fluid force on the journal.

If the infinitesimal variations are replaced by finite differences, the stiffness and damping coefficients can be approximated by

where Fj0 is the total fluid force without the perturbation, δu and δv are the finite perturbations of the displacement and velocity, while Fδu and Fδv are the total fluid force including the perturbations.

This is the approach used for the finite perturbation.

Results and Discussion

Figure 1 compares the stiffness coefficients obtained using the built-in functionality with those calculated through finite perturbation. The results show excellent agreement, with the two methods matching to a very high degree of precision.

Similarly, Figure 2 shows the corresponding comparison of the associated damping coefficients. Once again, the results demonstrate a very close match between the two approaches.

Figure 3 illustrates a comparison between the pressure perturbation determined with the built-in functionality and the finite perturbation approach. The pressure perturbations obtained with the two distinct approaches are visually indistinguishable.

The small deviations observed can be attributed to the perturbation sizes selected for the finite perturbation method. In general, it is recommended that a convergence study is performed to determine proper perturbation sizes.

Figure 1: Comparison of stiffness coefficients.

Figure 2: Comparison of damping coefficients.

Figure 3: Comparison of perturbation pressure due to displacement perturbation in the z direction.

Reference

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

Rotordynamics_Module/Verification_Examples/dynamic_coefficients_finite_perturbation

Modeling Instructions

The first step in building the model is to add the required physics interface and study. This model uses a Hydrodynamic Bearing interface and a stationary study.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

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

Bearing Parameters

In the window, under click .

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

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


Name	Expression	Value	Description
C	100[um]	1E-4 m	Clearance
mu0	75[mPa*s]	0.075 Pa·s	Viscosity at zero pressure
xi	25e-9[m^2/N]	2.5E-8 1/Pa	Viscosity-pressure coefficient
rho0	860[kg/m^3]	860 kg/m³	Density at zero pressure
omega	1000[rad/s]	1000 rad/s	Rotational velocity
R	10[cm]	0.1 m	Radius of the journal
W	2*R	0.2 m	Width of the journal
Fz	1000[N]	1000 N	Static load

Perturbation Parameters

Add an additional parameter node for the parameters required to perform the finite perturbation.

In the toolbar, click

and choose > .

In the window for , type Perturbation Parameters in the text field.

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


Name	Expression	Value	Description
u_pert	0.1[nm]	1E-10 m	Displacement perturbation
v_pert	u_pert*omega	1E-7 m/s	Velocity perturbation
acuy	0	0	Activation of displacement perturbation in y direction
acuz	0	0	Activation of displacement perturbation in z direction
acvy	0	0	Activation of velocity perturbation in y direction
acvz	0	0	Activation of velocity perturbation in z direction

In the toolbar, click

In the window for , type Displacement Perturbation (Y) in the text field.

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


Name	Expression	Description
acuy	1	Activation of displacement perturbation in y direction

In the toolbar, click

In the window for , type Displacement Perturbation (Z) in the text field.

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


Name	Expression	Description
acuz	1	Activation of displacement perturbation in z direction

In the toolbar, click

In the window for , type Velocity Perturbation (Y) in the text field.

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


Name	Expression	Description
acvy	1	Activation of velocity perturbation in y direction

In the toolbar, click

In the window for , type Velocity Perturbation (Z) in the text field.

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


Name	Expression	Description
acvz	1	Activation of velocity perturbation in z direction

In the toolbar, click

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

Geometry 1

Create a geometry for the bearing.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

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

In the text field, type W.

Locate the section. From the list, choose .

Materials

Next, add a material node to specify the density and viscosity of the fluid.

Oil

In the window, under right-click and choose .

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

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


Property	Variable	Value	Unit	Property group
Dynamic viscosity	mu	mu0exp(xihdb.p)	Pa·s	Basic
Density	rho	rho0(1+0.6hdb.p/(1.7*hdb.p+1e9))	kg/m³	Basic

Hydrodynamic Bearing (hdb)

Set up the Hydrodynamic Bearing node, starting by activating the determination of dynamic coefficients.

In the window, under click .

In the window for , locate the section.

Select the checkbox.

Hydrodynamic Journal Bearing 1

Set up the Hydrodynamic Bearing node to perform a static analysis of the hydrodynamic bearing based on an external load.

In the window, under > click .

In the window for , locate the section.

In the C text field, type C.

From the Xc list, choose .

Locate the section. From the list, choose .

Specify the Wj vector as


-Fz	z

In the Ω text field, type omega.

Mesh 1

Next, create a structured mesh for the bearing.

Mapped 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Distribution 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 24.

Distribution 2

In the window, right-click and choose .

Select Edge 3 only.

In the window for , locate the section.

In the text field, type 30.

Click

The mesh should look similar to the mesh shown below.

Study: Infinitesimal Perturbation

Now, set up the stationary study. Use an auxiliary sweep to analyze the dynamic coefficients over a range of external loads.

In the window, click .

In the window for , type Study: Infinitesimal Perturbation in the text field.

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the checkbox.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
Fz (Static load)	10^range(2,0.2,6)	N

In the toolbar, click

Results

Follow the instruction below to create a plot showing how the stiffness coefficients evolve as function of the external load.

Stiffness Coefficients

In the toolbar, click

In the window for , type Stiffness Coefficients in the text field.

Locate the section.

Select the checkbox. In the associated text field, type Static load (N).

Select the checkbox. In the associated text field, type Stiffness Coefficient (N/m).

Locate the section. From the list, choose .

Built-in

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
hdb.hjb1.k22	N/m	k<sub>22</sub>, Built-in
abs(hdb.hjb1.k23)	N/m	\|k<sub>23</sub>\|, Built-in
-hdb.hjb1.k32	N/m	-k<sub>32</sub>, Built-in
hdb.hjb1.k33	N/m	k<sub>33</sub>, Built-in

In the text field, type Built-in.

Click the

button in the toolbar.

Click the

button in the toolbar.

In the toolbar, click

Damping Coefficients

Now, create a similar plot for the damping coefficients.

In the toolbar, click

In the window for , locate the section.

Select the checkbox. In the associated text field, type Static load (N).

Select the checkbox. In the associated text field, type Damping Coefficient (N*s/m).

In the text field, type Damping Coefficients.

Locate the section. From the list, choose .

Built-in

Right-click and choose .

In the window for , type Built-in in the text field.

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


Expression	Unit	Description
hdb.hjb1.c22	N*s/m	c<sub>22</sub>, Built-in
-hdb.hjb1.c23	N*s/m	-c<sub>23</sub>, Built-in
abs(hdb.hjb1.c32)	N*s/m	\|c<sub>32</sub>\|, Built-in
hdb.hjb1.c33	N*s/m	c<sub>33</sub>, Built-in

Click the

button in the toolbar.

Click the

button in the toolbar.

In the toolbar, click

Hydrodynamic Bearing (hdb)

To validate the dynamic coefficients determined by the built-in functionality, perform a finite perturbation of the fluid pressure. Start by deactivating the automatic determination of dynamic coefficients.

In the window, under click .

In the window for , locate the section.

Clear the checkbox.

Hydrodynamic Journal Bearing 1

Add a copy of the existing Hydrodynamic Journal Bearing node.

Hydrodynamic Journal Bearing 2

In the window, under > right-click and choose .

Make use of the withsol() operator to pick up the displacement and velocities from the first study. In addition, add a small perturbation multiplied by the activation condition.

In the window for , locate the section.

From the list, choose .

Specify the uj vector as