Thrust Bearing
The purpose of thrust bearings is to restrict the axial motion of the rotor. Due to the finite area of the thrust bearings, as a side effect, they also offer a resistance to the bending of the rotor about the two lateral directions. Just as for journal bearings, modeling of thrust bearings can be done in two ways — either by specifying the equivalent stiffness and damping constants or by specifying the equivalent resistive forces and moments. There is also a special category where the thrust bearing is assumed to offer infinite resistance to the axial and bending motion of the collar. This is called a no-clearance thrust bearing.
For the purpose of implementation, the motion of the collar in the Solid Rotor interface is represented by the displacement at the center of the collar and a rotation around it. To obtain these quantities from the displacement field of the collar, it is assumed that the relative displacement of the collar with respect to the foundation can be written in terms of the displacement and rotation at the center of the collar in the following way:
(3-18)
The displacement at the center of the collar ur,c can be obtained by:
and the rotation of the collar can be obtained by first taking the cross product by X-Xc on both sides of Equation 3-18 and then integrating over the area:
Here
In these expressions, Xc is the coordinate of the center of the journal given by
The displacement of the collar in the spatial (fixed) frame for the Solid Rotor interface is given by
For a Solid Rotor, Fixed Frame interface the transformation with the rotation matrix is not required.
Further, since the thrust bearing only supports the axial motion of the collar, the displacement component in the axial direction is given as
Here, the fact that axial direction is independent of the rotation is used. The relative axial displacement of the collar with respect to the foundation is then given by
The components of the collar relative to the foundation in local bearing directions are obtained differently for the Solid Rotor and Solid Rotor, Fixed Frame interfaces. In a Solid Rotor interface, an additional transformation of the collar displacement with respect to rotation matrix is needed to convert it into the space-fixed frame. For the Solid Rotor, Fixed Frame interface this transformation is not required. Details for the Solid Rotor interface are provided below.
The components of the collar rotation 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.
If the deformational displacement of the collar in the axial direction is important, then instead of the linearized displacement, the full displacement field of the collar is used:
Here, ufd is the displacement field of the foundation in the spatial frame.
For the Solid Rotor interface, the following methods are provided to model thrust bearings:
•
No Clearance
•
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 collar and the foundation, and therefore the relative axial motion between the collar and foundation is constrained to zero. The following constraint is applied:
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 for both the spring constant and the damping constant. In a general case, the following inputs are needed: ku (1x1), kθ (2x2), kuθ (1x2), and kθu (2x1) for stiffness; and cu (1x1), cθ (2x2), cuθ (1x2), and cθu (2x1) for damping. The contribution to the virtual work is:
where
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 collar varies linearly in the plane of the collar. The default values for these inputs are
and
where
Total Force and Moment
A bearing can also be modeled by directly specifying the total axial force and the bending moment it applies on the collar. These forces and moments are generally functions of the collar displacement and rotation and their time derivatives. Total force Fax (1x1) and total moment M (2x1) are the inputs. The contribution to the virtual work is
Force Per Unit Area
Instead of specifying the total force and moment on the journal, a distributed force on the journal surface can be specified. Force per unit area, Fax,A (1x1), is the input. In such a case, the contribution to the virtual work is written as