The first half of 2009 is etched in my memory for two reasons. First, it was the ebb of my career as a researcher. Two years had passed since I joined the doctoral program, and I had not identified, let aside solved, a single problem which could measure itself to my standards. Second, I did find one problem and I did solve it, much to my satisfaction. This is the story of that problem, which concerned the prediction of spin susceptibility of aircraft. It is almost two years to this day…
Naira Hovakimyan was a good research supervisor by several standards – intelligent, determined, honest and somebody who genuinely cared for her students. However, we disagreed on major matters related to my research, and as a consequence, my two years with her had been stressful for both of us. I had not been able to focus myself on research, and as a consequence my progress towards a PhD was stalled to the point where it had become a source of additional pressure.
In the meantime, in the autumn of 2008, I revisited the problem of spin prediction for a course project. The problem of spin was one of those which had been tackled from the other direction: a recovery strategy was formally developed, followed by one to enter spin systematically. Numerous attempts to predict the spin susceptibility of aircraft had met with only limited success. Narayan Ananthkrishnan, AKN to his friends, was my research supervisor while I was still an undergrad at IIT Bombay. He had developed an effective and well-received spin recovery algorithm, and I had worked with him on spin entry using steep post-stall turns. We had casually discussed analytical spin prediction, but had never quite come to the point of launching a systematic effort. Over the years, AKN had become a mentor and a good friend. I frequently turned to him for advice on all sorts of matters – academic, philosophical, mundane – and continue to do so.
Back to April 2009, and I was in a state of despair. At that critical juncture, I decided to invest my energies on problems in my home turf: flight dynamics. The class project on spin gave me a few ideas on spin prediction, and they centred largely around Hopf bifurcations. If only to focus myself better and work with a like-minded researcher, I ran my idea past him, vague as it was. He wrote back saying something about saddle node bifurcations (SNBs). It was in the shower one morning, a few days after I received AKN’s mail about SNBS, that the implications of his suggestion dawned on me: we were looking for multiple attractors. It should have been obvious, but I had missed it altogether.
From a bifurcation analysis of an 8th order model of the F-18 HARV, we knew where to expect saddle nodes in the state-parameter space. However, as anyone who has worked with systems of differential equations knows very well, high order models (greater than order 3) are mostly analytically intractable. We rapidly agreed on a standard fifth order model called the pseudo-steady state (PSS) model. A bifurcation analysis of the PSS yielded the saddle nodes observed in the complete (8th order) model. We were clearly on the right track, but analytical tractability was not within reach of the PSS model.
At this stage, we made two critical observations. First, sideslip and pitch rate should not matter insofar as spin susceptibility is concerned, and second, the problem of spin prediction was primarily concerned with the topology of steady states instead of stability.
The reasons for the assumptions on sideslip and pitch rate are beyond the scope of this article. The first assumption helped the reduce the model order to 4. The simplified set of equations yielded the expected SNBs, as well as an analytical criterion which was, however, far too tedious to be meaningful. The second assumption yielded an even simpler model, and a much more tractable analytical criterion. All seemed well, until I tested the newly obtained analytical criterion against the data.
The criterion failed to predict the SNBs. It didn’t make any sense: the SNBs which were visible in a bifurcation diagram were not identified by the most elementary analytical test for SNBs. I spent two frustrating weeks trying to fine-tune the criterion. I had obviously ignored certain terms, but not without a good reason, and I accounted for them without success. Something fundamental was amiss in our derivation.
Finally, one evening, I noticed something peculiar: the expression which constituted my criterion had zero slope around the expected location of the SNBs in the state-parameter space. I examined my derivation and concluded that I had ignored only the second order derivatives following the standard practice in flight dynamics. The next morning, I derived the criterion from scratch, without ignoring the derivatives. I had to rewrite my code to evaluate the derivatives using a higher order finite difference scheme which I derived afresh. I asked Matlab to evaluate my criterion again, and I waited nervously as Matlab evaluated the criterion at nearly 10000 grid points. Then the plot came up: the criterion attained a value of zero within a 1% neighbourhood of the SNB locations. That was it! I had a lunch plan with friends, and I had to wait a few hours before I got the chance to review the morning’s work.
I went over my derivation, generalised it further, and verified that it was indeed correct. I called AKN that night, and dictated the five terms whose sum was zero at SNBs. I am sure he wasn’t any less excited than I was although he did a much better job of containing it. We agreed to meet the next morning in Van Vihar guest house’s foyer. In less than 5 minutes, I showed him the derivation and ran the code. “Excellent. Looks like we have it. Let’s celebrate.” It isn’t common for AKN to announce a celebration, and I knew then that we had solved the problem completely. It was August 2009.
We wrote a paper which detailed our work. It was accepted for publication, although not without critical comments which only helped improve our write-up. Our criterion is the first to illustrate the dependence of stability on second order aerodynamic derivatives. It would be most delightful to see this criterion make its way to aircraft design handbooks, but that matter does not rest with me.
My work on spin prediction restored my self-confidence and made for a fulfilling experience. I changed my adviser in August 2009, and have had a much more fruitful time since then working on a few problem in flight dynamics. It was a turning point for my research, and I can claim with some assurance that I shall graduate with a PhD by the end of this calendar year.
AKN is one of the most formidable intellectuals I have known and it is always a privilege to see him in action. I will always cherish our collaboration on spin prediction and it will remain a benchmark for my future endeavours.
