Journal Bearing
The purpose of journal bearings is to support the rotor against its lateral movement. Due to the finite length of the bearings, as a side effect, they also offer a resistance to the bending of the rotor about the two lateral directions. If the bearings are very short when compared to the overall length of the rotor, the bending resistance of the bearings can be neglected. Modeling of the journal bearings in COMSOL Multiphysics can be done in two ways: get the equivalent stiffness and damping constants or consider the equivalent resistive forces and moments offered by the bearing. There is also a special category where the journal bearing is assumed to offer infinite resistance to the lateral and bending motion of the journal. This is called a no-clearance bearing.
For the purpose of implementation, the motion of the journal in the Solid Rotor interface is represented by a displacement at the center of the journal and a rotation around it. To obtain these quantities from the displacement field of the journal, it is assumed that the displacement of the journal in the rotating frame can be approximated by a rigid map:
(3-16)
The displacement at the center of the journal ur,c can be obtained as
and the rotation of the journal can be obtained by first taking the cross product by X − Xc on both sides of Equation 3-16and integrating over the area:
with
In the expressions above, Xc is the coordinate of the center of the journal given by
The displacement of the journal, which affects and in turn is affected by the bearing operation, is the displacement excluding the effect of the axial rotation of the rotor. In a spatial (fixed) frame this is, for the Solid Rotor interface, given by
In the Solid Rotor, Fixed Frame interface the above transformation using a rotation matrix is not required.
The relative displacement of the journal axis with respect to the foundation in the spatial frame is given by
Here, ufd is the displacement field of the foundation in the spatial frame.
In the cases where deformation of the journal is significant, the displacement field on the journal surface should be considered instead of a linearized displacement of the journal axis:
for a Solid Rotor interface, and
for a Solid Rotor, Fixed Frame interface.
The lateral displacement components at the center of the journal, with respect to the foundation in the bearing lateral directions, are
and
The rotation components of the journal relative to the foundation about the bearing lateral axes are
and
in which uf and θf are the displacement and rotation vectors, respectively, of the bearing foundation at the center of the bearing. The above definition of local displacement and rotation components are valid for the Solid Rotor interface. In a Solid Rotor, Fixed Frame interface the additional transformation with a rotation matrix to convert the rotating frame displacement to a fixed frame displacement is not required.
For the Solid Rotor and Solid Rotor, Fixed Frame interfaces, the following methods are provided to model journal bearings:
•
No Clearance
•
Cylindrical Hydrodynamic
•
Total Spring and Damping Constant
•
Total Force and Moment
•
Force Per Unit Area
No Clearance
In this method, it is assumed that there is no clearance between the journal and the bushing of the bearing. Therefore, the relative motion between the journal and foundation is constrained to zero in the lateral direction. The following constraints are applied:
Cylindrical Hydrodynamic
The cylindrical hydrodynamic approximates the fluid-film forces in a cylindrical hydrodynamic journal bearing based on the Reynolds equation with a short-width approximation and Gümbel boundary condition. Under these assumptions, the total forces acting on the journal in the two local radial directions are given by:
where Fr1 and Fr2 are the total force in the radial and tangential direction respectively, and ϕ is the attitude angle. The total radial and tangential force can be expressed as:
Here μ is the dynamic viscosity of the fluid, R and L are the journal radius and length respectively, C is the clearance, and ε is the relative eccentricity.
The contribution to the virtual work is given by:
Total Spring and Damping Constant
In this case, spring and damping constants are needed explicitly to model the bearing. There is also an option to include the translational-rotational coupling both for the spring constant and the damping constant. In the general case, the following inputs are needed: ku, kθ, kuθ, and kθu for stiffness and cu, cθ, cuθ, and cθu for damping. These inputs are all 2-by-2 matrices. The contribution to the virtual work is
where
and
In a frequency-domain analysis, the contribution to the virtual work is modified to
In a stationary analysis, the terms corresponding to damping are dropped.
Default values for the inputs kθ and cθ are provided assuming that ku and cu are constants and that the displacement of the journal varies linearly along the axis. The default values for these inputs are
and
where
Total Force and Moment
A bearing can also be modeled by directly specifying the total force and moment it applies on the journal. These forces and moments are generally functions of the journal displacement and rotation and their time derivatives. Total force F (2-by-1) and total moment M (2-by-1) are the inputs. The contribution to the virtual work in this case is
Force Per Unit Area
Instead of specifying the total force and moment on the journal, a distributed force on the journal surface can also be specified. The force per unit area, FA (2-by-1), is the input. In such a case, the contribution to the virtual work is written as