Summary
A spare inclinometer was fitted to the bottom of the left-hand A frame in order to verify the reliability of the inclinometers. The results were at first surprising - there was a difference between the LY and SY data, but it was consistent between runs. This paper looks at that difference and attempts to explain it in terms of flexure of the A frame. The conclusion is that there is a credible explanation in terms of flexure of the structure but that the details would need to be worked on in order to obtain a complete understanding.

Datasets used
I used the LY and SY data of 981120. Other work has shown that the difference between SY and LY is very consistent so I only looked at one dataset.

Background.
In an earlier note it was shown that we can reconstruct the data for LY or RY from the track profile given by radial-arm inclinometry. (See P/001/10 and P/001/11.) From the form of the LY-SY difference plot it seemed that it might be possible to construct SY in the same way, but using different weighting factors for the four wheels. The rationale for doing this would be that flexure of the structure was such that different points along the bottom of the A frame were tilted by different amounts depending upon which wheel moved.

Flexure according to our FE model
Ian Pain has run an FE model (JAC-FEA-6b) which gives results for deflection at various points along each beam for a lift of each wheel. The results are summarised in these tables and in figure 1. For each set of data a single wheel was lifted either at the front or the back by about 32 microns, and results are shown for nodes along the base of the relevant A frame and, for completeness, the A-frame on the other side.

(*) A simpleminded approach would lead us to expect a simple slope in the nearer A frame of 32/7.1 = 4.5E-6 Rad (0.93 Arcsec) and nothing on the far A frame, and in the tables above I have included numbers normalised to this amount.

As with all analyses, it is important not to over-interpret the results. This is a simple model, and the way the junctions of the beams are modeled is probably not realistic in detail. Nonetheless, the results for the middle portions of the beams should be reliable. Note also that these are NOT exaggerated displacement plots normally beloved of structural engineers- they are simply graphs of the magnitude of the node rotations.

Implications for LY and SY
From the results it should be possible to predict the influence of wheel movement on inclinometer slope. For the right-hand A-frame, for example, at a point near the middle (x=0), we should see:

Inc_right,X=0 = 0.94*(right front) - 1.0*(right rear) - 0.2*(left front)

(Allowing for the fact that in the figure the sign of the slope is reversed between the front and rear wheels for compactness, and all normalised to the simple-minded result)

 Similarly

 Inc_right,X=1 = 1.0*(right front) - 0.92*(right rear) + 0.04*(left front)

We use the radial arm inclinometry as our source data. As pointed out by IMC, the relationship between the lengths of the radial arm and the A frame base is root 2, and we should see the inverse relationship between inclinometers in those places. Thus the simple minded expectation is that a 1 arcsecond change in radial arm result gives a ~0.7 arcsec change in A-frame result. Allowing for this factor, assigning actual wheel numbers, and assuming that the LY and SY inclinometers are at X=0 and X=1 respectively, we should see

LY = -0.01 W1 -0.7 W3 +0.66 W4

SY = 0.03 W1 - 0.64 W3 + 0.7 W4

Hence LY-SY = -0.04 W1 -0.06 W3 -0.04 W4

The fact that wheel 1 (on the left of the telescope) features as strongly as wheel 4 in the LY-SY expression is directly related to the fact that the slopes of the two blue curves in figure 1 are similar in the region of interest.

Comparison of predictions and actual results
The predictions above for LY have already been tested in P/001/10 and P/001/11. There is fair agreement - see table below. The same analysis as was used in those papers has been applied to LY-SY, and the results are given here and in figure 2:

Commentary
There is some agreement between the predictions and the observed results. This suggests to me that the idea of relative flexure is along the right lines, but the details of the model and the positioning of the inclinometers would need more work in order to provide a more convincing correlation between theory and practice. The "actual" results are arrived at by fitting the parameters to the observed data, and it would be interesting to see how sensitive that fit was in order to see to what extent noise may be affecting the result.

Conclusion
In conclusion, the idea of flexure of the A-frame base is borne out as credible explanation of the observed LY-SY profile, although the details of the mechanism are not completely clear.

Figures follow: