4.2.3 Pressure buildups in the damper

In 4.2.1 it was shown that the shear associated with damping increases linearly with the inverse of damper gap size. We will show in this subsection that as the gap size is reduced, alignment errors associated with finite machine tolerances become more important, causing pressure to build up in the fluid. It is this pressure buildup that induces a radial load on the damper cup. Finally, we will show that increasing the gap size (while increasing the viscosity by the same proportion) greatly reduces the radial load on the damper cup while achieving the same amount of damping. Figure 24 shows a simplified cross section of the cup and vane of a damper. In this case, the vane is assumed to rotate around its geometric center. The cup, on the other hand, has been displaced by an amount,

, to the left. This eccentricity comes about due to finite tolerances in machining and assembly.

Figure 24. Cross section of the 1 DOF damper. The vane rotates at rate U around its geometric center, but the cup center is translated by

to the left. Both the gap size and eccentricity have been exaggerated for clarity.

To determine if eccentricity in the cup and/or the vane would lead to significant radial forces, we constructed a linearized model of the damper (Figure 25).

Figure 25. Linearized model (with exaggerated gap size) of the 1 DOF damper.

is the angle around the damper,

is the gap size around the damper (not yet specified), R is the radius of the damper, and U is the velocity of rotation. Note the boundary condition that the pressure at the two ends must be equal.

Recall that the volume flow can be calculated by integrating the velocity profile over the area through which it flows :

(29)

Substituting the velocity profile (17) into (29), and rearranging to solve for , we get

(30)

At this point, since we want to integrate with respect to x, we have to specify how the gap size varies with x. In this case, we want to see the effect of moving the cup an amount off of the center of rotation of the vane. Assuming the offset is small with respect to the radius, this offset leads to a gap size that varies with :

(31)

Unfortunately, the integration of (30) is extremely difficult if (31) represents the gap size. A good approximation, however, for (31) is (32), shown in Figure 26 along with the true gap size variation :

(32)

Figure 26. Actual and linear approximation of gap size in the 1 DOF damper.

and

are the radii of the vane and damper, respectively, and

is the eccentricity of the cup.

Substituting (32) into (30) and integrating with respect to x, we obtain the following pressure distribution around the damper :

(33)

where varies from to . Recall at this point that and are representing the same physical point in the damper, so their pressures must be equal. Enforcing this boundary condition allows us to solve for flow through the damper :

(34)

Substituting (34) into (33) leads to our final representation of the pressure in the damper :

(35)

The x and y components of the load on the cup are calculated by integrating pressure with respect to , along with a trigonometric coefficient to convert the forces back to the circular geometry. These two forces are added as vectors, yielding the following magnitude result :

(36)

Again, note that this force is actually force per unit length of the damper. Evaluating (36) numerically, we get a relationship between and . We non-dimensionalize this result with the following relationships :

(37)

where the eccentricity is normalized by the nominal gap size and the radial load is normalized by the shear force in the damper (20). In choosing these nondimensional parameters, we recognize two important concepts : 1) It is not the eccentricity of the damper that is important, but the percentage change in gap size around the damper, and 2) It is not the value of the radial force that is important, but the size of that force relative to the shear force associated with damping. Figure 27 shows the relationship between the nondimensional eccentricity ( ) and force ( ).

Figure 27. Non-dimensionalized radial load vs. non-dimensionalized cup eccentricity. Using this plot, we can see how changes in geometry affect the unwanted forces in the damper.

This relationship can be closely approximated with a second order function (38), so that increasing the gap size by a factor of three reduces the magnitude of the radial forces on the damper cup by a factor of nine (assuming the machining errors remain constant in value).

(38)

Of course, the viscosity must also be increased by a factor of three so that the damping torque remains the same.

In the next subsection, we will show how these radial forces can show up as noise in certain configurations of strain gauge circuitry. Finally, in 4.2.5, we will summarize how we redesigned the damper and strain gauge circuitry so that an accurate torque signal could be obtained.

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