46
47
(a) (b)
Figure 4.2-1: (a) Rubber element, (b) Spring setup
The force acting on the coupling was a function of time. This meant that the compression of the springs had to be calculated at each time step for each spring. The number of springs was denoted by N. The position of the dots and crosses had to be calculated as the coupling rotates. The amount of misalignment was denoted by Δy and Δx these were the vertical and horizontal displacements of the shaft and coupling on the one side. Gravity was not considered in this model so the misalignment in the horizontal or vertical direction or a combination of the two will give the same solution. For this research the horizontal displacement was equal to zero, and only vertical displacement was considered. Figure 4.2-2 shows the layout used to setup the misalignment fault. This also only shows one spring in the system where the complete system had six springs.
Figure 4.2-2: Representation of one spring
48 Position of the dot j was given by:
( ) where ( ) 4.2-3 ( )
And the position of the cross j was given by:
( ) where ( ) 4.2-4 ( )
The change in length of the spring was given by:
√( ) ( ) 4.2-5
The force due to the springs in the coupling was given by:
∑ 4.2-6
The force acting in the coupling was in local coordinates and had to be transformed to global coordinates. Looking at the direction that the force acted in it was seen that the angle is also a function of time. The angle was given by:
(
) 4.2-7
Then the force in the x and y direction was given by:
∑ ( ) ∑ ( ) 4.2-8
This force was due to the compression and tension in the springs. To accurately model the coupling, the springs in the coupling had to be constrained so that it can never be extended past its original length. This was done due to the fact that the rubber insert of the coupling was not attached to the jaws and was a loose fit. This constraint was programmed into the Matlab code for the simulation.
4.2.2.2 Angular misalignment
Angular misalignment is when a rotor centre axis has an angular offset to the motor shaft.
Misalignment is dampened to a certain degree by using a flexible coupling but will not eliminate the problem completely. Misalignment will cause forces to be present in the coupling and rotor bearings. These forces will cause the vibration in the rotor bearing system.
49
the coupling had such a configuration that the force acting on the coupling varied as a function of time. For consistency the same model setup was used for the angular misalignment that was used for the parallel misalignment in the previous section. The only difference was that parallel misalignment had an offset between the two centres and angular had an angle. This meant that on the node of the coupling there was no offset. This node acted as a pivot point.
Looking at Figure 4.2-3 it was seen that angular misalignment created a compression of the spider element to the left and to the right of the pivot point. The forces present in the coupling due to the angular misalignment are shown in Figure 4.2-3 by FL for the force on the left hand side and FR for the force on the right hand side. This figure only shows an example of the left hand force and right hand force that is present at two of the spider legs. These forces are present throughout the rubber element and vary as the rotor rotates.
Figure 4.2-3: Flexible coupling showing angular misalignment forces
There was also shear in the coupling rubber element but will was not considered in this research. Shear forces will only show up in the axial direction and no axial measurement were considered. Angular misalignment was analysed similar to the parallel misalignment in the sense that only one rubber leg of the spider was considered and that the others are exactly the same but at a different position on the coupling. The only difference with angular misalignment is that there were two forces considered now which created a moment around the coupling node. The forces were derived using the initial angular misalignment and then noting that the misalignment seen by a spider leg changes as the coupling rotates. Figure 4.2.4 shows how the degree of angular misalignment changes as the coupling rotates.
50
Figure 4.2-4: Tracking coupling misalignment
For the left hand side of the coupling the position is given by:
( ) where ( ) 4.2-9
then the change in the y axis is given by:
4.2-10
For the right hand side of the coupling the position is given by:
( ) where ( ) 4.2-11
then the change in the y axis is given by:
4.2-12
Figure 4.2-5 shows the tip of the spider leg and where the angular misalignment forces act.
These forces are dependent on the angle which was created by the misalignment which varies as the rotor rotates.
Figure 4.2-5: Spider tip deflection
51 Where the left hand side angle was given by:
. / 4.2-13
Then the change in the rubber element was given by:
4.2-14
For the right hand side the angle was given by:
. / 4.2-15
Then the change in the rubber element was given by:
4.2-16
The force due to the springs in the coupling was given by:
∑ 4.2-17
The force acting in the coupling was in local coordinates and had to be transformed to global coordinates. Looking at the direction that the force acted in it was seen that the angle is also a function of time. The angle was given by:
(
) 4.2-18
The force in the x and y direction for the left hand side was given by:
∑ ( ) ∑ ( ) 4.2-19
Then the force in the x and y direction for the right hand side was given by:
∑ ( ) ∑ ( ) 4.2-20