The aim of this paper is to look at the FE results of Ian Pain (p/001/18) and see how they can be integrated with recent discoveries about the wheel/track interactions to generate a model of the way the so-called "spikes" are being formed. The conclusions include a plausible theory for the origin of the spikes.

Data and tools

I used the results of FEA run carried out by IP, described in p/001/18. I also used track profile data from the radial arm inclinometry of May 1998, as described in p/001/11 together with the conversion factor 0.015mm/arcsec given in that paper.

Key FEA results

From the point of view of this paper, the key results are:
 

And for a 1 degree temperature differential between the members in the base frame (essentially the ones below the observing floor) and the rest, with the back two wheels held on the track, the front two wheels lifted by 0.163 mm.

 A simple antenna model

In order to consider the possibilities of the wheels lifting off under some combination of track undulations and temperature fluctuations, I made a very simple model of the telescope. It does not model the way the whole structure tilts but it should be sufficient to capture the changing forces on the wheels. The model (figure 1) considers the telescope to be a rigid board pivoted at the central bearing (like a child's balancing toy) and stabilised by a spring at each corner leading down to the track. What happens if we lift one corner, as shown in the figure? Spring 1 compresses by an amount (T1-d1) and spring 3 by (d3). If the structure were completely square , it would simply tilt about the axis joining the other two conrners. This is not quite true, but for the purposes of this argument I will assume it to be so. Since the structure is very symmetrical we don't need to introduce distances when balancing moments about the axis joining wheels 2 and four, so we can say that:
 

F1 = (T1-d1)k1 = d1k3
 And hence the effective stiffness 

 
This expression is symmetrical as regards k1 and k3, so that the effective stiffness at wheel 3 will be the same as that at wheel 1. This means that we have a choice to make about k1 and k3 (I will make them the same, although it would be equally valid to remove one or the other). Further, by left-right symmetry, the effective stiffness at all four corners is the same. This in turn means that the stiffness of each spring is the same, and from the expression above it is equal to twice the effective stiffness. The FEA has given the effective stiffness at about 3 tonnes/mm (results above), so the stiffness of each spring in my model is 6 tonnes/mm. Note that this model makes no distinction in terms of forces between an event in which wheel 1 goes over a bump, and one in which wheel 3 goes over a bump - in fact the forces at wheels 1 and 3 are the same at all times in order to provide the right balance.

In the real telescope, however, there is a moment provided by the eccentricity of the telescope weight on the central bearing. This means that all of the results above should be considered as offsets from some initial set of F1, F2, F3 and F4 which we can derive by measurement.

 After some manipulation it will be seen that

ΔF1 = ΔF3 = k (ΔT1 + ΔT3)/2

And similarly

ΔF2 = ΔF4 = k (ΔT2 + ΔT4)/2

So the wheels behave as diagonal pairs with no interaction between them. (This is the rsult of the assumption made about about the telescope being square.) Encouragingly, the results for the reaction forces on the FE model show just such decoupled behaviour to a large extent.

Forces on wheels and moments on the telescope

From the radial-arm inclinometry and Leica data we know the form of the track profile, and so we can plot the (ΔT1 + ΔT3) terms against azimuth. From the strain gauge data (TR 001/51.1/IP) we know roughly what the forces are at particular azimuths for each wheel, including the correcting moment for the weight eccentricity. We can thus produce a plot of force in each wheel against azimuth in the absence of disturbing factors such as temperature differentials.

The strain gauge azimuth location has been checked by IMC, and so we can say that the following loads and azimuths are known:

Figure 2 shows the result, with the reference points above chown by red crosses. Note that this is the result we would have got if we had tried to predict wheel loads in the absence of any history of spikes or anything else, and that it shows wheel 1 lifting off (just) at two azimuth locations.

Effect of temperature

A temperature differential of 1 degree lifts each front wheel by 168 microns. From the same arguments as above, ΔF = k(ΔT)/2 so all forces decrease by .168*3 = .5 tonnes. To think about it another way, lifting the front wheels by 160 microns is the same in terms of forces as lifting all wheels by 80 microns. Now to lift one wheel by 30 microns requires 100 kgf on that wheel AND THE DIAGONALLY OPPOSITE WHEEL, so to lift one wheel by 80 microns would require ~250kgf, so to lift a wheel and its diagonal opposite would require 500 kgf on BOTH of them. This has the effect of moving the "zero load" axis up by 0.5t for every 1 degree of such temperature differential between the time the loads were measured and the time the inclinometry was done. As can be seen from the graph, that would leave the front wheels carrying no load for a significant fraction of the time. In practice the wheels would I think stay on the track but a gap would open up somewhere in the bogie mechanism. And, significantly, the change in the path taken by the corner of the telescope (which is now flying over the track joint features rather than following them) is just the sort of thing we see in the difference plots.

Conclusions

From the best data we currently have on the shape of the track, the measured wheel loads, and the stiffness of the JCMT, this paper predicts that the loads on the front wheels will be very small if not zero at some places around the track. If the members of the structure above the base square are cooled to one degree below the members in the base (essentially, the members above the observing floor are 1 deg cooler than those below the observing floor) then the load on each wheel is decreased by about 500kgf. Such an effect would reduce the load on the front two wheels to zero over a significant portion of the track. The path taken by the corners of the telescope would no longer follow the track profile, and the changes in A-frame tilt that would result are very similar in form to those we see in the "spike"-laden difference plots.
 

Appendix - IMC's email:
 

Date: Thu, 7 Jan 1999 21:12:29 -1000 (HST)
From: Iain Coulson <i.coulson@jach.hawaii.edu>
To: Justin Greenhalgh <justin@jach.hawaii.edu>
Cc: Ian Pain <i.pain@jach.hawaii.edu>, Dwight Chan <d.chan@jach.hawaii.edu>
Subject: strain gauge location
Hi Justin -
As judged from inside the plinth area, the strain gauge
location is near the middle of sector 6, more precisely
54% of the way from joint 5/6 to joint 6/7 - i.e.
at absolute azimuth 54 degrees.

It should be encountered by wheel 1 at antenna azimuth 5.6
                                  2                   88.8
                                  3                  185.6
                                  4                  268.8
                                  1                  365.6

 
Figures follow: (For figure 1 see Word version)