Dutch Roll

The video link shows a textbook example of Dutch roll instability in a Tupolev 154 aircraft which was supposedly flown last month (April 2011) after 10 years.

Let’s sum up very quickly what we observe in the video: (1) the aircraft rolls and yaws periodically, (2) roll and yaw are clearly out of phase (3) the aircraft has a vertical tail and the rudder does not appear to be moving significantly, and (4) the aircraft seems to be at a moderate angle of attack.

Background: typically, the dynamic behaviour of an aircraft (or any other system for that matter) can be decomposed into several modes, i.e., the the dynamic behaviour is a weighted sum of the different modes. The Dutch roll (DR) is one such mode.

Although the name suggests that it is a roll-based mode, DR primarily involves aircraft yaw. The rolland yaw oscillations that characerise a DR instability have a moderate frequency (typically 0.5 – 1Hz) and the rolling and yawing motions are out of phase. It is primarily the side-to-side (yawing) motion that is a source of piloting difficulties.

DR instability is encountered at moderate to high angles of attack, and under some circumstances, is a precursor to another instability called wing rock (where the dominant yaw motion gives way to rapid and disorienting oscillations in roll).

Forensics: For an aircraft with a tail like the Tu-154′s, it is hard to imagine that the DR was destabilised at all. As I said, it is usually stable at low angles of attack and even when it is destabilised, it can be stabilised by the aircraft’s control system. This leads us to three possible causes for the observed behaviour:

(a) The control system failed. This would explain the onset of instability, but not why it persisted.

(b) The steering mechanism was locked. This may explain why the pilots struggled to check the instability once it was triggered.

(c) The elevator was rendered partially ineffective. This prevented them from pitching down to low angles of attack where the DR would have stabilised itself.

Incidentally, one way to attenuate unstable DR oscillations is to deploy the flaps so that the increased drag on the wings damps out the oscillations.  This may not always damp out the oscillations altogether, but it certainly makes them a lot more benign. In fact, flap deployment also creates a pitch-down moment which would be favourable under the circumstances.

References:

Text: Etkin and Reid “Dynamics of Flight: Stability and Control”

Paper: Ananthkrishnan, Unnikrishnan and Shah, “Approximate Analytical Criterion for Aircraft Wing Rock Onset” Journal of Guidance, Control and Dynamics, 2004.

“Good” Research

What is a good research project?

This is a question that gets asked often enough. The broad concept of a good research project, although universally understood and accepted per se, differs in the minutae across the disciplines and depends strongly on time.

One striking example is the following observation in Gribbin’s “In Search of Schrodinger’s Cat” regarding the quality of research in quantum mechanics: “Back then, (the 1920′s and 30′s) it was easy for a second rate physicist to do first rate work; nowadays, it’s hard even for first rate physicists to get second rate work.” The implicit notion of “first rate research” seems to be the following: it is a project which makes a major advance in the concerned field.

Hence, the poor “first rate” physicists today would get half as much credit as they would ideally deserve, at least in the 30′s. Why call them first-rate physicists  then? Their claim to the epithet, and this may be apparent to the reader, rests on their grasp of the subject in general. Their contributions represent, for the most part, an important incremental advance in the field. I do mean incremental, but not major. Their work may improve the state-of-the-art, but the field could nevertheless do without the improvement. This brings us to a more contemporary notion of good research: it helps make important incremental advances in the state of the art.

I am aware of two other notions of good research:

1. It creates a revolution or a paradigm shift. This notion does not fit into either of the two above, and is the most widely perceived and imagined notion outside the technical community and occasionally even within.
2. It can elicit interest in the community at large and find application in the real world. For the first time, we have talked about the application of an idea rather than a knowledge of the idea itself.

There are probably other notions of good research, not far removed from the four stated here, but let me leave them aside and bring two points to the reader’s attention. First, all four notions concern the end product. Second, they all relate to an entire field or a sub-field, but never to an individual or his aspirations. In other words, the concept of good research has a very consumerist dimension. I have lost count of the number of times I have heard people dismiss projects as “reinventing the wheel.”

This brings me to a more personal part of this post: my own notions of good research.

1. I prefer to take a more individual-centric perspective. For me, a good research problem is one which grows an individual’s knowledge base and strengthens its foundations. As an example,consider the “first rate physicists struggling to conduct second rate research” in the above anecdote.
2. The research problem should be grounded in experiments or in rigorous observations of nature whenever possible.
3. The research should be in keeping with the highest standards of morality and integrity. Ethics are insufficient in my opinion because more often than not, they do not evaluate specific long-term effects of the research such as its potential to harm the nature.
4. Lastly, should the research be judged by an end product which is a deliverable, the product should satisfy all assumptions and meet all specifications laid at the outset. Novelty, in my opinion, does not constitute the specifications. It is merely the culmination of the effort, the driving assumptions and the circumstances.

The spin story

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 8^th 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 (8^th 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.

And another maiden post

I have learnt a few important lessons in my hitherto unsuccessful attempts at maintaining a blog:
1. Write only about matters close to you.
2. It makes no sense whatsoever to restate what is being said in a million posts elsewhere.
3. If you feel tempted to publish, sleep over it and measure the inclination is next morning.
4. Do not try to be a Jack of all trades.

With these lessons in mind, here’s a fresh start. Science and engineering are two broad areas wherein I can claim a considerable grasp of the matters. I don’t claim to be in Einstein or Feynman’s pedigree, but I am not among the run-of-the-mill practitioners either. This blog is dedicated then to experiences in science and engineering.

If all goes to the plan, the contents of this blog will be technical, but comprehensible to an interested reader without a formal training in the area addressed by the blog.

Finally, I need to justify my motivation to maintain a blog in the first place.  There are two main reasons: (1) There are times when I get an insight into matters which I deem worth sharing, and (2) I enjoy sharing ideas with people.