Ninth email to Philip Stewart

Date: Tue, 16 Apr 2002 01:21:32 -0400
From: Bob Burkhardt <bobwb@channel1.com>
Reply-To: Bob Burkhardt <bobnowbspam@lycos.com>
Subject: tensegrity structure analysis with member weights

Stu,

Here are my results for the analysis of your structure with the weights factored in. I had to make a few approximations, mostly to deal with software problems I ran into.

The approximations:

A strut's tendons now all meet at one point in the center of the strut instead of being offset a few inches from the center of the strut. A problem with my software forced me to do this. I'd never tried doing the external load computations with offsets factored in and it turned out it couldn't handle them.
I fixed the three base points (A), (C) and (D). The iterations I use to compute external loads will not converge otherwise. I've run into this problem with other people's models before. When I tested the software originally I worked with domes where the base points were fixed. In my previous calculation for this model, I just fixed six of the coordinate values to make the model determinate in space, but for some reason I haven't figured out, more needs to be done if the external load computations are to converge. Fixing three points means that now nine coordinate values must be fixed and that the forces for tendons (A)-(C), (C)-(D) and (A)-(D) can't be computed.
I gave the three base points common Z values. They were already pretty close. This allowed me to have simple force-of-gravity vectors going in the -Z direction. I made sure tendon lengths didn't change except the base tendons must have changed length a very slight amount.
Another approximation comes with the way the software is designed. As far as external load analysis is concerned, the software is made for *a priori* design rather than the *ex post* analysis we're trying to do here. It assumes the weight of the members distort the structure slightly. When I do an ex post analysis, I use your final configuration as the zero-gravity configuration in which the effect of member weights has not been figured in. When the effect of member weights is figured in, the model is distored away from the zero-gravity configuration which means it is distorted away from your final configuration. The inelasticity of the tendons (1% elongation at 10000 pounds is what I read off your table) makes this a reasonable approximation.

All told, I think these are all reasonable approximations and, as far as member force computations are concerned, didn't distort the model within the tolerances we're operating at.

Finally, I must say that I think in my previous calculations I mis-stated the strut member forces due to a software problem. I used a convex mapping so the ends of the struts extended past where the tendons connected. When I got rid of this extension (which should be purely cosmetic as far these calculations are concerned), the strut member forces changed. So, I don't think the force calculations are handled correctly when the extensions are present. The tendon member forces weren't affected.

So I had a little fudge factor that I could play with to get my computations in line with your results. The fudge factor is the scale factor I use to convert from model units to pounds. I used that so that my result for the (A)-(B) force got fairly close to your value of 2450 pounds. For an ex post analysis, there was really no other way to proceed since somehow one must decide how much is due to the zero-gravity pre-stressing and how much is due to the member weights. For your structure, the effect due to the pre-stressing predominates, so factoring in the weights doesn't make a whole lot of difference.

So my scale factor is .087 model units per pound, so you would divide all the values below by .087 to get the corresponding pound value.

The chart below summarizes the zero-gravity member forces with the new approximations. These are comparable to the figures I gave you last time and haven't changed significantly except for the strut values due to the problem I mentioned above. Except for the strut figures and scale changes, you can view any discrepancies between these figures and the ones I sent you before as an indicator of the effect of the various approximations I introduced above. Also, forces for the base tendons are missing due to another problem I mentioned above:

    (A)-A:     -400.254     (B)-B:     -285.459
    (C)-C:     -105.037     (D)-D:     -399.626
  (A)-(B):      204.161     (A)-D:      185.618
    A-(B):      330.975       A-B:      68.8091
    A-(C):      66.7029       A-C:      59.1437
    (B)-D:      118.398       B-C:      88.0917
      B-D:      148.015     C-(D):      229.898
      C-D:      167.092

Average tendon force (in model units): 151.537

With the member weights factored in, the values are:

    (A)-A:     -422.853     (B)-B:     -294.331
    (C)-C:     -125.965     (D)-D:     -409.799
  (A)-(B):      215.071     (A)-D:      182.758
    A-(B):      349.648       A-B:      63.5981
    A-(C):      49.4357       A-C:      74.1759
    (B)-D:      127.819       B-C:      77.4189
      B-D:      160.099     C-(D):      224.071
      C-D:       165.25

Average tendon force (in model units): 153.577

So the average tendon force increases by (153.577 - 151.537)/.087 = 23.4483 pounds which seems reasonable but is fairly insignificant in comparison with the zero-gravity forces present. Although on average there is an increase, for several tendons the member force decreases when the member weights are factored in, C-D for example.

Overall, I don't think I've told you much more. The effect due to the member weights is so insignificant for this structure, I don't think there's much in your measurements that could tell me whether I'm right or wrong on this score. I would call the last set of values my best guess though. It almost seems that the noise introduced by the approximations outweighs any changes due to factoring in the weights. The best scenario would be if your measurements were accurate enough that it could be shown that factoring in the member weights seemed to close any discrepancy observed with the first set of calculations I gave you.

Perhaps I could do better if I used the previous calculations as my base somehow and used these calculations to indicate the changes introduced by factoring in the member weights. I would calculate my deltas in pounds and choose a scale factor for the previous set of results so that the (A)-(B) value comes out right when the deltas are added in. Hopefully the other members would come closer to their actuals as well.

So I hope I haven't worn out your eyes. I've certainly done a job on mine. I'd be glad to give any clarifications for the above if you need them or address any reservations you might have.

Bob

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