Unraveling Uncertainty


Uncertainty is present all around us, from the change in climate around the world to the oscillations of nerves in the nerve-muscle junction. Observing the uncertainties in wing oscillations could help with devising early warning systems to make safer aircrafts.

The term uncertainty strikes us as something that is not predictable or definite. When mechanical systems are designed, a lot of idealisations are made – mathematical models used to represent these systems are not exact representatives of their behavior in real life. This is due to the presence of uncertainties or states of randomness. Dr Sayan Gupta’s lab – “The Uncertainty Lab”, in the Department of Applied Mechanics, is dedicated to exploring uncertainties in modelling various systems inspired from real life. Dr Gupta and his students study the behaviour of systems when under the influence of uncertain forces. He remarks about the name of his lab, saying,“It’s not an uncertain lab. It’s an uncertainty lab!”

Dr Gupta’s work is highly collaborative and interdisciplinary, as he often works alongside Dr Sunetra Sarkar of the Aerospace Department and Dr Shaikh Faruque Ali of the Applied Mechanics Department, among others. Uncertainty arises due to the ignorance of the random or stochastic nature of a system in real life. Dr Gupta points out that “everything is uncertain”. During the modelling of a system, idealisations bring in a lot of errors. It is therefore, not clear whether the end result obtained is accurate or not. In order to study such behaviour, there needs to be some kind of mechanism to know how a stochastically or randomly varying parameter affects a system’s response.

S Krishna Kumar, one of Dr Gupta’s PhD students, studies and models how randomness in wind flow affects the movement of the wings of an airplane, modelled using an airfoil. The kind of study that he does can also be used to understand the complications in modelling other systems, like a bridge or a skyscraper. He explains to us that an aircraft wing system has two states – static and oscillatory. As the name suggests, in the static state, the aircraft’s wing is considered to be at rest, and the in the oscillatory state the wing is observed to move vigorously. When the aircraft speed or its angle of inclination are relatively low, the the state of the wing is found to be static. As the speed of the aircraft or the incline angle increases beyond a threshold, the “static state” of the wing transitions into the “oscillatory state”.

In a realistic model, the static state of the wing is not truly still or stationary. It is modelled as having random fluctuations – this is where uncertainty is an input into the model. To consider the static state as one with fluctuations is, in fact, a very realistic assumption. So, the static state is considered to have irregular behaviour, on account of random fluctuations, whereas the oscillatory state can be modelled as having a definite frequency and amplitude. A transition from the static to a completely oscillatory state is considered to be a transition from irregular to regular behavior. This transition is observed to happen via a series of intermediate states known as intermittencies.

A time series illustrating intermittency: Coexistence of periodic behaviour and random behaviour

An intermittent state is a mix of seemingly random and seemingly oscillatory states, as illustrated in the figure above. Intermittency has usually been reported as curious discoveries, and in-depth studies of this phenomena has been ignored for a long time.

In the system of interest for the present study, the vigorous oscillations observed in the airfoil are is called “flutter”. Aircraft wings are thin, slender, and light structures, and thus cannot withstand these vigorous oscillations, which can cause them to break away from the fuselage of the aircraft.

One method to handle this problem is to use controllers to minimize the flutter once it occurs. Controlling the flutter involves applying mechanical techniques to reduce the oscillations once they begin to occur. Krishna Kumar tell us that controllers are present in most present day aircrafts.

The alternative, which is what Dr Gupta has worked on, is to predict the onset of flutter in order to remain in a safe flying regime. Krishna Kumar tells us that they want to identify precursors before the onset of flutter. At low speed of the aircraft, the wing is almost static, or has small or weak oscillations that eventually decay. Once the flow speed past the wing increases past a critical value, the airfoil oscillations cross a threshold, and cannot be ignore. With a further increase in these oscillations due to high speed of the aircraft, these oscillations become more dominant and vigorous. The natural question to ask is – at what speed should one fly an aircraft so that flutter does not occur?

“Theoretically, this sounds very simple”, Krishna Kumar says. He explains that an aircraft model can be tested to find the critical value of this transition, and extrapolating to real life situations, speeds past this critical value can be avoided. However, the simulations of wind flow and aircraft speed in a controlled laboratory setting are very idealistic. In reality, these factors continuously fluctuate. For example, the speed of an aircraft varies based on external environmental conditions like wind speed and extent of cloud cover.

So, as part of their work, Dr Gupta and his students find precursors before the wing begins to flutter. These precursors are the intermittent states that occur during the transition from the randomly fluctuating and the oscillatory state. The idea is that if one can measure the presence of intermittency (which occurs well before flutter), it can serve as an early warning signal.

Dr Sayan and his students have carried out wind tunnel experiments, as well as numerical studies, to study how an airfoil behaves in different wind flow regimes. They observed that as the flow speed past the airfoil is increased, there were intermittent bursts of periodic oscillations amid the background of random fluctuations. As the flow speed in increased further, the intermittent bursts of oscillations last longer, and soon the behaviour of the airfoil transitions to fully developed periodic oscillations (or flutter).

Airfoil plunge response from experiments done by Dr Gupta’s group, with increasing flow speeds from (a) to (d), showing a transition from random fluctuations to fully developed periodic oscillations, via intermittent stages.
Plunge response of airfoil in the presence of fluctuations, from numerical calculations carried out by Dr Gupta’s group. Flow speeds increase in the order of (a), (c), and (d), showing corresponding increase in frequency of periodic oscillations, until fully developed oscillations are observed. (b) is a Zoomed-in view of (a), highlighting the on-off intermittency.

In order to study the intermittent states, a mathematical tool called the recurrence plot is used. This tool picks up whether a system exhibits regular, irregular, or intermittent behavior. As the name suggests, this tool looks for recurrences, or repetitions, in the system’s dynamics.

Different time series with corresponding recurrence plots

A recurrence plot of a given time series of length N is a matrix of dimension NxN (see image above). The ijth element in this recurrence matrix indicates whether the state of the system at time i was similar to the state of the system at time j – if it was, then the corresponding element is shown with a black dot in the plot.

An irregular system appears grainy in the recurrence plot – without any clear pattern or message about the original data used to draw the plot. Whereas, for a regular system, its recurrence plot shows repeated structure. For an intermittent system, the recurrence plot has a mix of both repeated structure and grainy structure. This structure, or lack thereof, can be captured using some mathematical measure calculated on the recurrence matrix, and the irregularities or regularities can be captured and quantified. The changing values of irregularities to regularities can thus be used as a precursor to flutter in the aircraft wing.

Dr Gupta and his students want to go beyond devising effective warning methods using intermittency.They also want to understand why the phenomenon of intermittency occurs. Dr Gupta tells us that it is interesting to note that intermittencies are also seen in other physical systems – like sunspot data in solar flares giving information as to when sunspots occur, nerve oscillations in muscles, and many more.

Krishna Kumar adds that once they understood that the randomness, or uncertainty, was what drives intermittency in many systems, they started exploring the possible mechanisms involved. Among many things, they found that the nature of the intermittency depends on the time scales governing the input. In other words, for their airfoil system, the type or pattern of intermittency depends on the nature of the flow velocity. If the flow velocity varies rapidly, it causes ‘burst type intermittency’. However, when it is slowly varying, it causes ‘on-off intermittency’. Both these types of intermittencies have same origin, but they show up differently in the output behaviour. This kind of understanding of intermittency can help prepare more robust and practically useful precursors, among other applications.

To conclude this journey into understanding the world of uncertainties, we are certain of a few things. One route to chaos is through states of intermittency, and understanding this uncharted region helps us pick precursors in order to devise early warning systems.

Nikitha Damaraju

Nikhita is an third year undergrad from Biological Sciences. She aims to probe answers to questions in the field of Cancer Biology using Data Sciences. Aside from this, she loves playing the violin, running and listening to Carnatic Music.


Dr. Sayan Gupta

Dr. Sayan Gupta is an Associate Professor in the Department of Applied Mechanics. His research interests lie in the field of stochastic dynamics and structural risk assessment. He broadly focuses on nonlinear dynamics, probabilistic mechanics, stochastic load modeling, structural reliability and life prediction, structural system identification and health monitoring from vibration measurements, and energy harvesting from wind.

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